A velocity distribution function in cylindrical coordinates for an irrorational vortex confined at a boundary layer is:
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Expression 13: "u" left parenthesis, "x" , right parenthesis equals StartFraction, "w" Subscript, 0 , Baseline "R" squared Over "x" , EndFraction left parenthesis, 1 minus "e" Superscript, negative StartFraction, "x" squared Over "R" squared , EndFraction left parenthesis, 1 minus StartFraction, "x" squared Over "R" squared , EndFraction , right parenthesis , Baseline , right parenthesisux=w0R20x1−e−x2R21−x2R2f
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Inverse error function:
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Expression 15: "e" Subscript, "r" "f" "i" "n" "v" "s" "y" "m" "m" "e" "t" "r" "i" "c" , Baseline left parenthesis, "x" , right parenthesis equals left parenthesis, StartFraction, "R" Subscript, "f" , Baseline Over StartRoot, 3 , EndRoot , EndFraction , right parenthesis StartFraction, 1 Over StartRoot, 2 , EndRoot , EndFraction normal distribution left parenthesis, 0 , 1 , right parenthesis .inversecdf left parenthesis, StartFraction, "x" left parenthesis, StartNestedFraction, StartRoot, 3 , EndRoot NestedOver "R" Subscript, "f" , Baseline , EndNestedFraction , right parenthesis plus 1 Over 2 , EndFraction , right parenthesiserfinvsymmetricx=Rf312normaldist0,1.inversecdfx3Rf+12
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Expression 16: "u" Subscript, "D" "r" , Baseline left parenthesis, "x" , right parenthesis equals StartFraction, "w" Subscript, 0 , Baseline "R" squared Over "x" , EndFraction left parenthesis, 1 minus "e" Superscript, negative StartFraction, "x" squared Over left parenthesis, "e" Subscript, "r" "f" "i" "n" "v" "s" "y" "m" "m" "e" "t" "r" "i" "c" , Baseline left parenthesis, "x" , right parenthesis , right parenthesis squared , EndFraction left parenthesis, 1 minus StartFraction, "x" squared Over "R" squared , EndFraction , right parenthesis , Baseline , right parenthesisuDrx=w0R20x1−e−x2erfinvsymmetricx21−x2R2f
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Expression 17: "q" left parenthesis, "t" , right parenthesis equals StartFraction, "R" Subscript, "f" , Baseline Over StartRoot, 3 , EndRoot , EndFraction error function left parenthesis, StartRoot, 4 nu "t" plus "R" squared , EndRoot StartFraction, StartRoot, 3 , EndRoot Over "R" Subscript, "f" , Baseline , EndFraction , right parenthesisqt=Rf3erf4νt+R203Rf
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Hidden Label: left parenthesis, "q" left parenthesis, "T" , right parenthesis , "u" Subscript, "D" "r" , Baseline left parenthesis, "q" left parenthesis, "T" , right parenthesis , right parenthesis , right parenthesisqT,uDrqT