𝜕v𝜕y=0⟹v(y)=constant=−v0partial v over partial y end-fraction equals 0 ⟹ v open paren y close paren equals constant equals negative v sub 0 Substitute into the momentum equation:
δ∼νLU∞delta tilde the square root of the fraction with numerator nu cap L and denominator cap U sub infinity end-sub end-fraction end-root
ρ(𝜕u𝜕t+u⋅∇u)=−∇p+μ∇2u+(ζ+μ3)∇(∇⋅u)+frho open paren the fraction with numerator partial bold u and denominator partial t end-fraction plus bold u center dot nabla bold u close paren equals negative nabla p plus mu nabla squared bold u plus open paren zeta plus the fraction with numerator mu and denominator 3 end-fraction close paren nabla open paren nabla center dot bold u close paren plus bold f is fluid density is the velocity vector is thermodynamic pressure is dynamic viscosity is bulk viscosity represents body forces (like gravity) advanced fluid mechanics problems and solutions
𝜕η𝜕y=U∞νx,𝜕η𝜕x=−y2xU∞νx=−η2xpartial eta over partial y end-fraction equals the square root of the fraction with numerator cap U sub infinity end-sub and denominator nu x end-fraction end-root comma space partial eta over partial x end-fraction equals negative y over 2 x end-fraction the square root of the fraction with numerator cap U sub infinity end-sub and denominator nu x end-fraction end-root equals negative the fraction with numerator eta and denominator 2 x end-fraction Step 2: Differentiate to find Find the horizontal velocity component
u=𝜕ψ𝜕y,v=−𝜕ψ𝜕xu equals partial psi over partial y end-fraction comma space v equals negative partial psi over partial x end-fraction Using the chain rule, evaluate Can’t copy the link right now
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Combine the terms back into the time-dependent expression: their policies apply.
u(y)=Uyh−h22μ(dpdx)[yh−(yh)2]u open paren y close paren equals the fraction with numerator cap U y and denominator h end-fraction minus the fraction with numerator h squared and denominator 2 mu end-fraction open paren d p over d x end-fraction close paren open bracket y over h end-fraction minus open paren y over h end-fraction close paren squared close bracket
