I use mathematica to confirm a result from a paper ("Unsteady unidirectional flow of Bingham fluid between parallel plates with different given volume flow rate conditions
") which should be simplified to a real expression. However, I have tried many many times but all failed. The following code showed my efforts:
ClearAll["Global`*"]
Γ[s_] := Module[{m, delta, ee},
m = Sqrt[s]/Sqrt[ν];
delta = Sinh[m h] Sinh[m h0] - Cosh[m h] Cosh[m h0];
ee = Cosh[m h0] (Sinh[m h] - Sinh[m h0]) -
Sinh[m h0] (Cosh[m h] - Cosh[m h0]);
(m h delta)/(
m h delta + m h0 (Cosh[m h0]^2 - Sinh[m h0]^2) + ee) ];
Simplify[Im[
E^( I ω t) Γ[I ω] +
E^(- I ω t) Γ[-I ω] // ComplexExpand],
TimeConstraint -> Infinity]
Actually, the complex expression in the above code is the first term of dp/dx as shown in the following picture, and dp/dx is a pressure gradient which should be a expression of real values. So the image part of this expression should vanish, but the above code gives no result after a hour.
Thanks.
Edit 1: -----------------------------------
As suggested by @Bob Hanlon, I tried to include all the variable constraints in Simplify
, but still failed to get a real-valued expression.
expr = E^(I \[Omega] t) \[CapitalGamma][I \[Omega]] + E^(- I \[Omega] t) \[CapitalGamma][-I \[Omega]]
Simplify[ComplexExpand[expr], {h > h0 > 0,t > 0, \[Omega] > 0, \[Nu] > 0}]
I
-- being real (implied by your use ofComplexExpand
), are there any additional known constraints? For example, are any of the variables/constants positive or at least nonnegative? Any known constraints should be included in theSimplify
or wrap theSimplify
withAssuming
. $\endgroup$