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convective_boundary_value_problems

# convective_boundary_value_problems - CONVECTIVE BOUNDARY...

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New Chapter Page 1 K. Haghighi CONVECTIVE BOUNDARY VALUE PROBLEMS u and v are the velocity components in the x and y directions, ρ φ is a coefficient depending on φ . If φ is temperature ρ φ is the product of the density and heat capacity. u and v are not variable. The boundary conditions are the same. 0 2 2 2 2 = + φ φ ρ φ ρ φ + φ φ φ Q G y v x u y D x D y x

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New Chapter Page 2 K. Haghighi Integral Equations for the Element Matrices ( ) { } [ ] dA Q G y v x u y D x D N R y x T A e + φ φ ρ φ ρ φ + φ = φ φ 2 2 2 2 [ ] [ ] [ ] x x N x N x N x T T T φ + φ = φ 2 2
New Chapter Page 3 K. Haghighi [ ] [ ] [ ] dA x x N D dA x N x D dA x D N T A x T A x x T A φ + φ = φ 2 2 [ ] [ ] [ ] dA x x N D d cos x N D dA x N D T A x T x T x A φ + Γ θ φ = φ Γ 2 2 The Galerkin formulation of the first order derivatives of the convective terms is: [ ] [ ] dA y v N x u N A T T φ ρ + φ ρ φ φ

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New Chapter Page 4 K. Haghighi Substituting ( ) { } [ ] [ ] [ ] [ ] [ ] [ ] [ ] φ + φ ρ + φ ρ +
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convective_boundary_value_problems - CONVECTIVE BOUNDARY...

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