Problem 46031. Construct dimensionless parameters

Created by ChrisR

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Summary</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function to construct dimensionless parameters from a list of variables, a matrix indicating their dimensions, and a vector identifying the variables to use to make the other variables dimensionless. More details are given in \"Problem statement\" below.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Introduction</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>Dimensional analysis and the Buckingham </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"pi\"/></w:customXmlPr><w:r><w:t>\\pi</w:t></w:r></w:customXml><w:r><w:rPr><w:i/></w:rPr><w:t> theorem</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Buckingham_π_theorem\"><w:r><w:t>Buckingham π theorem</w:t></w:r></w:hyperlink><w:r><w:t> is a key result exploited in</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Dimensional_analysis\"><w:r><w:t>dimensional analysis</w:t></w:r></w:hyperlink><w:r><w:t> in engineering, physics, and applied mathematics. It states that if a problem involves</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"m\"/></w:customXmlPr><w:r><w:t>m</w:t></w:r></w:customXml><w:r><w:t> variables and</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> physical dimensions (e.g., length, time, mass), then the problem can be described with</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"m-n\"/></w:customXmlPr><w:r><w:t>m-n</w:t></w:r></w:customXml><w:r><w:t> dimensionless groups, called </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"pi\"/></w:customXmlPr><w:r><w:t>\\pi</w:t></w:r></w:customXml><w:r><w:t> groups. In</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Buckingham_π_theorem\"><w:r><w:t>other words</w:t></w:r></w:hyperlink><w:r><w:t>, the \"physics does not depend on a specific unit system\".</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>To find the dimensionless groups, we follow these steps:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Choose</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"n\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:t> variables that cannot form a dimensionless group by themselves</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Form the product of one of the remaining</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"m-n\"/></w:customXmlPr><w:r><w:t>m-n</w:t></w:r></w:customXml><w:r><w:t> variables with the variables in step 1, each raised to different exponents</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Determine the exponents that would make the product (or group) dimensionless</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"2\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Repeat steps 2 and 3 for the remaining variables</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:i/></w:rPr><w:t>Example</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Suppose we are interested in the drag</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"F\"/></w:customXmlPr><w:r><w:t>F</w:t></w:r></w:customXml><w:r><w:t> (i.e., a force) on a cylinder of length</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"L\"/></w:customXmlPr><w:r><w:t>L</w:t></w:r></w:customXml><w:r><w:t> and diameter</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t> immersed in a fluid of density</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"rho\"/></w:customXmlPr><w:r><w:t>\\rho</w:t></w:r></w:customXml><w:r><w:t> and dynamic viscosity</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"mu\"/></w:customXmlPr><w:r><w:t>\\mu</w:t></w:r></w:customXml><w:r><w:t> flowing at velocity</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"V\"/></w:customXmlPr><w:r><w:t>V</w:t></w:r></w:customXml><w:r><w:t>. Because the problem involves 6 variables and 3 dimensions (length, time, and mass), it can be described with 3 dimensionless groups. For step 1 above, choose</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"V\"/></w:customXmlPr><w:r><w:t>V</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t>, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"rho\"/></w:customXmlPr><w:r><w:t>\\rho</w:t></w:r></w:customXml><w:r><w:t>. We could choose other lists of 3, but notice that we could not choose</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"V\"/></w:customXmlPr><w:r><w:t>V</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t>, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"L\"/></w:customXmlPr><w:r><w:t>L</w:t></w:r></w:customXml><w:r><w:t> because</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"L\"/></w:customXmlPr><w:r><w:t>L</w:t></w:r></w:customXml><w:r><w:t> form a dimensionless group themselves (and we have no way of removing mass dimensions.)</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For step 2, start with the drag</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>F</w:t></w:r><w:r><w:t>. Following the step, we get</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"pi_1 = F V^{a1} d^{a2} rho^{a3}\"/></w:customXmlPr><w:r><w:t>\\pi_1 = F V^{a_1} d^{a_2} \\rho^{a3}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>where</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>a1</w:t></w:r><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>a2</w:t></w:r><w:r><w:t>, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:i/></w:rPr><w:t>a3</w:t></w:r><w:r><w:t> are exponents. Because </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"pi1\"/></w:customXmlPr><w:r><w:t>\\pi_1</w:t></w:r></w:customXml><w:r><w:t> must be dimensionless and because the dimensionless of</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"F\"/></w:customXmlPr><w:r><w:t>F</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"V\"/></w:customXmlPr><w:r><w:t>V</w:t></w:r></w:customXml><w:r><w:t>,</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"d\"/></w:customXmlPr><w:r><w:t>d</w:t></w:r></w:customXml><w:r><w:t>, and</w:t></w:r><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"rho\"/></w:customXmlPr><w:r><w:t>\\rho</w:t></w:r></w:customXml><w:r><w:t> are </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"ML/T^2\"/></w:customXmlPr><w:r><w:t>\\rm ML/T^2</w:t></w:r></w:customXml><w:r><w:t>, L/T, L, and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/><w:attr w:name=\"altTextString\" w:val=\"M/L^3\"/></w:customXmlPr><w:r><w:t>\\rm M/L^3</w:t></w:r></w:customXml><w:r><w:t>, respectively, we can solve a system of three equations to find the exponents and write</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"pi1 = F V^{-2} d^{-2} rho^{-1}\"/></w:customXmlPr><w:r><w:t>\\pi_1 = F V^{-2}d^{-2}\\rho^{-1}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>which is related to a quantity in fluid mechanics called the drag coefficient. We then repeat steps 2 and 3 with the two remaining variables and find</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"pi2 = L/d\"/></w:customXmlPr><w:r><w:t>\\pi_2 = L d^{-1}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>and</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t> </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"true\"/><w:attr w:name=\"altTextString\" w:val=\"pi3 = mu/(V d rho)\"/></w:customXmlPr><w:r><w:t>\\pi_3 = \\mu V^{-1} d^{-1} \\rho^{-1}</w:t></w:r></w:customXml></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>or a geometric ratio and the inverse of the</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Reynolds_number\"><w:r><w:t>Reynolds number</w:t></w:r></w:hyperlink><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Problem statement</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Write a function that constructs dimensionless parameters from a cell array</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>vars</w:t></w:r><w:r><w:t> with the variable names, a matrix</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>dims</w:t></w:r><w:r><w:t> with the exponents for the dimensions in rows (length, time, mass) for each variable in the columns, and a vector</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>indx</w:t></w:r><w:r><w:t> indicating the variables chosen in step 1 above. The function should return a cell array</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>Pi</w:t></w:r><w:r><w:t> with the dimensionless groups in strings and a vector</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a</w:t></w:r><w:r><w:t> of size</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>m x m-n</w:t></w:r><w:r><w:t> with the exponents of the normalizing variables. For the example above, the input could be specified as</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ vars = {'F', 'V', 'd', 'L', 'rho', 'mu'};\n indx = [2 3 5];\n dims = [ 1 1 1 1 -3 -1;\n -2 -1 0 0 0 -1;\n 1 0 0 0 1 1];]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>and the output would be</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[ Pi = {'F V^{-2} d^{-2} rho^{-1}', 'L d^{-1}', 'mu V^{-1} d^{-1} rho^{-1}'} \n a = [-2 0 -1;\n -2 -1 -1;\n -1 0 -1]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>Further specifications:</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the variables chosen in</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>indx</w:t></w:r><w:r><w:t> will not work (i.e., the number is incorrect or some form a dimensionless group themselves), return</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>a = NaN</w:t></w:r><w:r><w:t> and</w:t></w:r><w:r><w:t> </w:t></w:r><w:r><w:rPr><w:rFonts w:cs=\"monospace\"/></w:rPr><w:t>Pi = {'Error in choosing normalizing variables'}</w:t></w:r><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the exponent of a variable is zero, omit the variable from the string.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the exponent of a variable is one, keep the variable but do not show the exponent.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the exponent of a variable is a positive integer, indicate it with a caret (^) and no braces.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the exponent is anything else, indicate it with a caret and braces.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"ListParagraph\"/><w:numPr><w:numId w:val=\"1\"/></w:numPr><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>If the exponent is not an integer, express it as a fraction. Hint: MATLAB has a function that will help.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 1429.07px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 714.533px; transform-origin: 407px 714.533px; vertical-align: baseline; "><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.9px 7.8px; transform-origin: 31.9px 7.8px; unicode-bidi: normal; ">Summary</div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 376.783px 7.8px; transform-origin: 376.783px 7.8px; unicode-bidi: normal; ">Write a function to construct dimensionless parameters from a list of variables, a matrix indicating their dimensions, and a vector identifying the variables to use to make the other variables dimensionless. More details are given in "Problem statement" below.</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 40.8167px 7.8px; transform-origin: 40.8167px 7.8px; unicode-bidi: normal; ">Introduction</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 133.45px 7.8px; transform-origin: 133.45px 7.8px; unicode-bidi: normal; ">Dimensional analysis and the Buckingham π<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.6167px 7.8px; transform-origin: 27.6167px 7.8px; unicode-bidi: normal; "> theorem</div><div style="block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.0667px 7.8px; transform-origin: 12.0667px 7.8px; unicode-bidi: normal; ">The<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem">Buckingham π theorem</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.0833px 7.8px; transform-origin: 82.0833px 7.8px; unicode-bidi: normal; "> is a key result exploited in<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Dimensional_analysis">dimensional analysis</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 115.167px 7.8px; transform-origin: 115.167px 7.8px; unicode-bidi: normal; "> in engineering, physics, and applied mathematics. It states that if a problem involves<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> m<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 43.5833px 7.8px; transform-origin: 43.5833px 7.8px; unicode-bidi: normal; "> variables and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.633px 7.8px; transform-origin: 174.633px 7.8px; unicode-bidi: normal; "> physical dimensions (e.g., length, time, mass), then the problem can be described with<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="m-n" style="width: 38.5px; height: 18px;" width="38.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 93.7667px 7.8px; transform-origin: 93.7667px 7.8px; unicode-bidi: normal; "> dimensionless groups, called π<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 33.0667px 7.8px; transform-origin: 33.0667px 7.8px; unicode-bidi: normal; "> groups. In<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Buckingham_%CF%80_theorem">other words</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 96.65px 7.8px; transform-origin: 96.65px 7.8px; unicode-bidi: normal; ">, the "physics does not depend on a specific unit system".</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 174.267px 7.8px; transform-origin: 174.267px 7.8px; unicode-bidi: normal; ">To find the dimensionless groups, we follow these steps:</div><ol style="block-size: 102.167px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 51.0833px; transform-origin: 391px 51.0833px; margin-top: 10px; margin-bottom: 20px; "><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">Choose n variables that cannot form a dimensionless group by themselves</li><li style="block-size: 40.8667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.4333px; text-align: left; transform-origin: 363px 20.4333px; white-space: pre-wrap; margin-left: 56px; ">Form the product of one of the remaining <img src="data:image/png;base64,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" alt="m-n" style="width: 38.5px; height: 18px;" width="38.5" height="18"> variables with the variables in step 1, each raised to different exponents</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">Determine the exponents that would make the product (or group) dimensionless</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">Repeat steps 2 and 3 for the remaining variables</li></ol><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 27.2333px 7.8px; transform-origin: 27.2333px 7.8px; unicode-bidi: normal; ">Example</div><div style="block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.5px; text-align: left; transform-origin: 384px 52.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 118.267px 7.8px; transform-origin: 118.267px 7.8px; unicode-bidi: normal; ">Suppose we are interested in the drag<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> F<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 111.633px 7.8px; transform-origin: 111.633px 7.8px; unicode-bidi: normal; "> (i.e., a force) on a cylinder of length<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> L<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 42.8px 7.8px; transform-origin: 42.8px 7.8px; unicode-bidi: normal; "> and diameter<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> d<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70.4px 7.8px; transform-origin: 70.4px 7.8px; unicode-bidi: normal; "> immersed in a fluid of density<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> ρ<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70.0167px 7.8px; transform-origin: 70.0167px 7.8px; unicode-bidi: normal; "> and dynamic viscosity<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> μ<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 56.8px 7.8px; transform-origin: 56.8px 7.8px; unicode-bidi: normal; "> flowing at velocity<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> V<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 191.783px 7.8px; transform-origin: 191.783px 7.8px; unicode-bidi: normal; ">. Because the problem involves 6 variables and 3 dimensions (length, time, and mass), it can be described with 3 dimensionless groups. For step 1 above, choose<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> V<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> d<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; ">, and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> ρ<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.2px 7.8px; transform-origin: 16.2px 7.8px; unicode-bidi: normal; ">. We could choose other lists of 3, but notice that we could not choose<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> V<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> d<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; ">, and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> L<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28.4px 7.8px; transform-origin: 28.4px 7.8px; unicode-bidi: normal; "> because<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> d<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6167px 7.8px; transform-origin: 13.6167px 7.8px; unicode-bidi: normal; "> and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> L<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 70.0167px 7.8px; transform-origin: 70.0167px 7.8px; unicode-bidi: normal; "> form a dimensionless group themselves (and we have no way of removing mass dimensions.)</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 91px 7.8px; transform-origin: 91px 7.8px; unicode-bidi: normal; ">For step 2, start with the drag<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4.28333px 7.8px; transform-origin: 4.28333px 7.8px; unicode-bidi: normal; ">F<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 84.8px 7.8px; transform-origin: 84.8px 7.8px; unicode-bidi: normal; ">. Following the step, we get</div><div style="block-size: 26px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 13px; text-align: left; transform-origin: 384px 13px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="pi_1 = F V^{a1} d^{a2} rho^{a3}" style="width: 102px; height: 26px;" width="102" height="26"></div><div style="block-size: 42.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.4167px; text-align: left; transform-origin: 384px 21.4167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 19.0667px 7.8px; transform-origin: 19.0667px 7.8px; unicode-bidi: normal; ">where<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 7.8px; transform-origin: 7.78333px 7.8px; unicode-bidi: normal; ">a1<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 7.8px; transform-origin: 7.78333px 7.8px; unicode-bidi: normal; ">a2<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; ">, and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.78333px 7.8px; transform-origin: 7.78333px 7.8px; unicode-bidi: normal; ">a3<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 79.35px 7.8px; transform-origin: 79.35px 7.8px; unicode-bidi: normal; "> are exponents. Because <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB8AAAAoCAYAAAAG0SEsAAABCUlEQVRYhe2WUQ2EMAyGPw84wAAGUHAKcDAHOMACGpBwHrCABixwD7ShtwMCt+Verl/SBDq2f/SnC+A4juM4P6YEGqC9EFUu0QLogeVGlLmER2ACnmbxp8Qo95PJ9TmEkcWCXFdGWAmSa3MJKgXv3mnpg8lNksvm8dFGYj+1EtOF+TXw+FZcy2tL3kmuO5lXAsOF507R8lpvZ8nVB3Nq1k0niddsJVdvH3I/s1pyRpsiri02mtywk+vY7++vxK1fC+vppqgN+rGFaCNJ4rbU8RsSjWkceX9bvDGLznz2sa1I3PvJ4rB+VIF9HwsZaw/Gk8Vz4eL/J66H1PBL0ZKtIzQaMv3lOI5zmxdP4XNErSZ5SAAAAABJRU5ErkJggg==" alt="pi1" style="width: 15.5px; height: 20px;" width="15.5" height="20"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 180.9px 7.8px; transform-origin: 180.9px 7.8px; unicode-bidi: normal; "> must be dimensionless and because the dimensionless of<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> F<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> V<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">,<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> d<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5667px 7.8px; transform-origin: 15.5667px 7.8px; unicode-bidi: normal; ">, and<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> ρ<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14px 7.8px; transform-origin: 14px 7.8px; unicode-bidi: normal; "> are <img src="data:image/png;base64,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" alt="ML/T^2" style="width: 47px; height: 19.5px;" width="47" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 38.5px 7.8px; transform-origin: 38.5px 7.8px; unicode-bidi: normal; ">, L/T, L, and <img src="data:image/png;base64,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" alt="M/L^3" style="width: 38px; height: 19.5px;" width="38" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 269.8px 7.8px; transform-origin: 269.8px 7.8px; unicode-bidi: normal; ">, respectively, we can solve a system of three equations to find the exponents and write</div><div style="block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="pi1 = F V^{-2} d^{-2} rho^{-1}" style="width: 107.5px; height: 21px;" width="107.5" height="21"></div><div style="block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 363.433px 7.8px; transform-origin: 363.433px 7.8px; unicode-bidi: normal; ">which is related to a quantity in fluid mechanics called the drag coefficient. We then repeat steps 2 and 3 with the two remaining variables and find</div><div style="block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="pi2 = L/d" style="width: 64px; height: 21px;" width="64" height="21"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 11.6833px 7.8px; transform-origin: 11.6833px 7.8px; unicode-bidi: normal; ">and</div><div style="block-size: 21.8167px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.9167px; text-align: left; transform-origin: 384px 10.9167px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 13.6px 7.8px; transform-origin: 13.6px 7.8px; unicode-bidi: normal; "> <img src="data:image/png;base64,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" alt="pi3 = mu/(V d rho)" style="width: 105.5px; height: 21px;" width="105.5" height="21"></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 129.517px 7.8px; transform-origin: 129.517px 7.8px; unicode-bidi: normal; ">or a geometric ratio and the inverse of the<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <a target='_blank' href = "https://en.wikipedia.org/wiki/Reynolds_number">Reynolds number</a><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; ">.</div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63.0167px 7.8px; transform-origin: 63.0167px 7.8px; unicode-bidi: normal; ">Problem statement</div><div style="block-size: 106.8px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 53.4px; text-align: left; transform-origin: 384px 53.4px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 231.7px 7.8px; transform-origin: 231.7px 7.8px; unicode-bidi: normal; ">Write a function that constructs dimensionless parameters from a cell array<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.8px; transform-origin: 15.4px 7.8px; unicode-bidi: normal; ">vars<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 104.25px 7.8px; transform-origin: 104.25px 7.8px; unicode-bidi: normal; "> with the variable names, a matrix<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.8px; transform-origin: 15.4px 7.8px; unicode-bidi: normal; ">dims<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> with the exponents for the dimensions in rows (length, time, mass) for each variable in the columns, and a vector<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.4px 7.8px; transform-origin: 15.4px 7.8px; unicode-bidi: normal; ">indx<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> indicating the variables chosen in step 1 above. The function should return a cell array<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 7.7px 7.8px; transform-origin: 7.7px 7.8px; unicode-bidi: normal; ">Pi<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 105.033px 7.8px; transform-origin: 105.033px 7.8px; unicode-bidi: normal; "> with the dimensionless groups in strings and a vector<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 3.85px 7.8px; transform-origin: 3.85px 7.8px; unicode-bidi: normal; ">a<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 22.1667px 7.8px; transform-origin: 22.1667px 7.8px; unicode-bidi: normal; "> of size<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 1.95px 7.8px; transform-origin: 1.95px 7.8px; unicode-bidi: normal; "> <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.95px 7.8px; transform-origin: 26.95px 7.8px; unicode-bidi: normal; ">m x m-n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 255.567px 7.8px; transform-origin: 255.567px 7.8px; unicode-bidi: normal; "> with the exponents of the normalizing variables. For the example above, the input could be specified as</div><div style="background-color: rgb(247, 247, 247); block-size: 102.167px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 51.0833px; transform-origin: 404px 51.0833px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 161.7px 8.25px; transform-origin: 161.7px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> vars = {<span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 11.55px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 11.55px 8.25px; ">'F', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 11.55px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 11.55px 8.25px; ">'V', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 11.55px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 11.55px 8.25px; ">'d', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 11.55px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 11.55px 8.25px; ">'L', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 19.25px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 19.25px 8.25px; ">'rho', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 15.4px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 15.4px 8.25px; ">'mu'};</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 61.6px 8.25px; transform-origin: 61.6px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> indx = [2 3 5];</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 111.65px 8.25px; transform-origin: 111.65px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> dims = [ 1 1 1 1 -3 -1;</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 111.65px 8.25px; transform-origin: 111.65px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> -2 -1 0 0 0 -1;</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 115.5px 8.25px; transform-origin: 115.5px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> 1 0 0 0 1 1];</div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 74.7px 7.8px; transform-origin: 74.7px 7.8px; unicode-bidi: normal; ">and the output would be</div><div style="background-color: rgb(247, 247, 247); block-size: 81.7333px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 40.8667px; transform-origin: 404px 40.8667px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; "><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 292.6px 8.25px; transform-origin: 292.6px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> Pi = {<span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 100.1px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 100.1px 8.25px; ">'F V^{-2} d^{-2} rho^{-1}', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 38.5px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 38.5px 8.25px; ">'L d^{-1}', <span style="border-block-end-color: rgb(170, 4, 249); border-block-start-color: rgb(170, 4, 249); border-bottom-color: rgb(170, 4, 249); border-inline-end-color: rgb(170, 4, 249); border-inline-start-color: rgb(170, 4, 249); border-left-color: rgb(170, 4, 249); border-right-color: rgb(170, 4, 249); border-top-color: rgb(170, 4, 249); caret-color: rgb(170, 4, 249); color: rgb(170, 4, 249); column-rule-color: rgb(170, 4, 249); margin-inline-end: 0px; margin-right: 0px; outline-color: rgb(170, 4, 249); perspective-origin: 103.95px 8.25px; text-decoration: none; text-decoration-color: rgb(170, 4, 249); text-emphasis-color: rgb(170, 4, 249); transform-origin: 103.95px 8.25px; ">'mu V^{-1} d^{-1} rho^{-1}'} </div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 69.3px 8.25px; transform-origin: 69.3px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> a = [-2 0 -1;</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 69.3px 8.25px; transform-origin: 69.3px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> -2 -1 -1;</div><div style="background-color: rgba(0, 0, 0, 0); block-size: 20.4333px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.916667px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.916667px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.916667px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.916667px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2167px; transform-origin: 404px 10.2167px; white-space: nowrap; "><span style="block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 69.3px 8.25px; transform-origin: 69.3px 8.25px; unicode-bidi: normal; white-space: pre; margin-right: 45px; "> -1 0 -1]</div></div><div style="block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68.4667px 7.8px; transform-origin: 68.4667px 7.8px; unicode-bidi: normal; ">Further specifications:</div><ul style="block-size: 143.933px; counter-reset: list-item 0; font-family: Helvetica, Arial, sans-serif; list-style-type: square; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 71.9667px; transform-origin: 391px 71.9667px; margin-top: 10px; margin-bottom: 20px; "><li style="block-size: 41.7667px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 20.8833px; text-align: left; transform-origin: 363px 20.8833px; white-space: pre-wrap; margin-left: 56px; ">If the variables chosen in indx will not work (i.e., the number is incorrect or some form a dimensionless group themselves), return a = NaN and Pi = {'Error in choosing normalizing variables'}.</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">If the exponent of a variable is zero, omit the variable from the string.</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">If the exponent of a variable is one, keep the variable but do not show the exponent.</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">If the exponent of a variable is a positive integer, indicate it with a caret (^) and no braces.</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">If the exponent is anything else, indicate it with a caret and braces.</li><li style="block-size: 20.4333px; counter-reset: none; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2167px; text-align: left; transform-origin: 363px 10.2167px; white-space: pre-wrap; margin-left: 56px; ">If the exponent is not an integer, express it as a fraction. Hint: MATLAB has a function that will help.</li></ul></div></div>

*Summary*

Write a function to construct dimensionless parameters from a list of variables, a matrix indicating their dimensions, and a vector identifying the variables to use to make the other variables dimensionless. More details are given in "Problem statement" below.

*Introduction*

_Dimensional analysis and the Buckingham π theorem_

The <https://en.wikipedia.org/wiki/Buckingham_π_theorem Buckingham π theorem> is a key result exploited in <https://en.wikipedia.org/wiki/Dimensional_analysis dimensional analysis> in engineering, physics, and applied mathematics. It states that if a problem involves _m_ variables and _n_ physical dimensions (e.g., length, time, mass), then the problem can be described with _m-n_ dimensionless groups, called π groups. In <https://en.wikipedia.org/wiki/Buckingham_π_theorem other words>, the "physics does not depend on a specific unit system".

To find the dimensionless groups, we follow these steps:

# Choose _n_ variables that cannot form a dimensionless group by themselves
# Form the product of one of the remaining _m-n_ variables with the variables in step 1, each raised to different exponents
# Determine the exponents that would make the product (or group) dimensionless
# Repeat steps 2 and 3 for the remaining variables

_Example_

Suppose we are interested in the drag _F_ (i.e., a force) on a cylinder of length _L_ and diameter _d_ immersed in a fluid of density _ρ_ and dynamic viscosity _μ_ flowing at velocity _V_. Because the problem involves 6 variables and 3 dimensions (length, time, and mass), it can be described with 3 dimensionless groups. For step 1 above, choose _V_, _d_, and _ρ_. We could choose other lists of 3, but notice that we could not choose _V_, _d_, and _L_ because _d_ and _L_ form a dimensionless group themselves (and we have no way of removing mass dimensions.)

For step 2, start with the drag _F_. Following the step, we get

π_1 = F V^a1 d^a2 ρ^a3

where _a1_, _a2_, and _a3_ are exponents. Because π_1 must be dimensionless and because the dimensionless of _F_, _V_, _d_, and _ρ_ are ML/T^2, L/T, L, and M/L^3, respectively, we can solve a system of three equations to find the exponents and write

π_1 = F V^{-2} d^{-2} ρ^{-1}

which is related to a quantity in fluid mechanics called the drag coefficient. We then repeat steps 2 and 3 with the two remaining variables and find

π_2 = L d^{-1}   and   π_3 = μ V^{-1} d^{-1} ρ^{-1}

or a geometric ratio and the inverse of the <https://en.wikipedia.org/wiki/Reynolds_number Reynolds number>.

*Problem statement*

vars = {'F', 'V', 'd', 'L', 'rho', 'mu'};
 indx = [2 3 5];
 dims = [ 1  1  1  1  -3  -1;
         -2 -1  0  0   0  -1;
          1  0  0  0   1   1];
  
and the output would be
  
 Pi = {'F V^{-2} d^{-2} rho^{-1}', 'L d^{-1}', 'mu V^{-1} d^{-1} rho^{-1}'} 
 a  = [-2   0  -1;
       -2  -1  -1;
       -1   0  -1]

Further specifications:

* If the variables chosen in |indx| will not work (i.e., the number is incorrect or some form a dimensionless group themselves), return |a = NaN| and |Pi = {'Error in choosing normalizing variables'}|.
* If the exponent of a variable is zero, omit the variable from the string.
* If the exponent of a variable is one, keep the variable but do not show the exponent.
* If the exponent of a variable is a positive integer, indicate it with a caret (^) and no braces.
* If the exponent is anything else, indicate it with a caret and braces. 
* If the exponent is not an integer, express it as a fraction. Hint: MATLAB has a function that will help.

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Problem 46031. Construct dimensionless parameters

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