Perform Dimensional Analysis
Perform Dimensional Analysis
setupBPT[terms_List] := With[{ud = Union[Flatten[UnitDimensions /@ terms, 1][[All, 1]]]}, If[MemberQ[ud, _UnitDimensions], Throw[None]];{Length@terms - Length[ud], ud}]
UDToExp[dimensions_List] := Times@@(dimensions /. List[d_String, n_] :> Power[d, n])
Nondimensionalize[expr_, vars_List] := With[{exp = expr /. {Power[q_Quantity, n_] :> UDToExp[UnitDimensions[q]] ^ n, qu_Quantity :> UDToExp[UnitDimensions[qu]]}}, With[{res = Solve[Thread[((List@@PowerExpand[exp]) /. Power[unit_String, n_] :> n) == 0], vars]}, If[Flatten[res] === {}, Throw[None], With[{nondim = expr /. q_Quantity :> QuantityMagnitude[q]}, First[nondim /. res]]]]]
Buckinghamπ[terms_List, const : {__Integer}] := Catch[With[{res = setupBPT[terms]}, If[Length[Union[const]] =!= First[res], Throw[None]];If[MemberQ[const, x_ /; x > Length[terms]], Throw[None]];With[{c = #}, Module[{temp, vars = {}}, Check[temp = If[Not[MemberQ[const, #]], With[{u = Unique[]}, AppendTo[vars, u];terms[[#]] ^ u], If[c =!= #, 1, terms[[#]]]]& /@ Range[Length[terms]], Throw[None], {Part::partw}];Nondimensionalize[Times@@temp, vars]]]]& /@ const]Find the various non-dimensional expressions associated with the following five physical quantities:
BPTMatrixLabels = {"pressure", "dynamic viscosity", "velocity", "characteristic length", "mass density"};
P = Quantity[p, "Newtons" / "Meters" ^ 2];
mu = Quantity[μ, ("Pascals" * "Seconds")];
V = Quantity[v, "Meters" / "Seconds"];
L = Quantity[d, "Meters"];
Rho = Quantity[ρ, "Kilograms" / "Meters" ^ 3];
terms = {P, mu, V, L, Rho};MatrixForm[Outer[Buckinghamπ[terms, {#1, #2}]&, Range[5], Range[5]], TableHeadings -> {BPTMatrixLabels, BPTMatrixLabels}]