# 3774 - take into account the boundary conditions F-1 = 0...

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The equation of motion with non-degenerated terms would be ( 29 2 2 cos y u t a t u + = μ ϖ ρ (1) Introduce non-dimensional variables ; a u u ρϖ = t t = ; h y y = (further, primes will be omitted for simplicity). Then, the non-dimensional equation one will reduce to 2 2 2 cos y u h t t u + = (2) where R h = 2 is the Reynolds number. Thus, the boundary value problem becomes 2 2 1 cos y u R t t u + = (3) at y = -1 u = 0 at y = 1 u = 0 Solve this problem in complex form by introducing ( 29 { } ) ( Re , y F e t y u it = , where Re denotes the real part. Substituting into (3), we have

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2 2 dy F d R iRF + = (4) Look for the solution as ( 29 ( 29 i y iR C y iR C F - - + = exp exp 2 1 , which satisfies (4), and
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Unformatted text preview: take into account the boundary conditions F (-1) = 0, F (1) = 0. from where ( 29 ( 29 i iR C iR C =-+ exp exp 2 1 ( 29 ( 29 i iR C iR C = +-exp exp 2 1 ( 29 iR i C C cosh 2 2 1 = = ( 29 ( 29 -× = 1 cosh cosh iR y iR i F , and ( 29 ( 29 -= 1 cosh cosh Re iR y iR ie u it , from where the answer can be derived by simplification (using Matlab, for example, or manually)...
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3774 - take into account the boundary conditions F-1 = 0...

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