u=𝜕ψ𝜕y=U∞f′(η)u equals partial psi over partial y end-fraction equals cap U sub infinity end-sub f prime of open paren eta close paren
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Integrate (assuming $\delta=0$ at $x=0$): $$ \frac\delta^22 = \frac15 \nu xU_\infty $$ $$ \delta(x) = \sqrt\frac30 \nu xU_\infty = \frac5.48 x\sqrtRe_x $$ advanced fluid mechanics problems and solutions
1 over r end-fraction d over d r end-fraction open paren r d u over d r end-fraction close paren equals the fraction with numerator 1 and denominator mu end-fraction the fraction with numerator d cap P and denominator d x end-fraction Since the pressure gradient is constant, we write:
ψ(r,θ)=f(r)sin2θpsi open paren r comma theta close paren equals f of r sine squared theta Substituting this into we write: ψ(r
tanθ=2cotβ[M12sin2β−1M12(γ+cos2β)+2]tangent theta equals 2 cotangent beta open bracket the fraction with numerator cap M sub 1 squared sine squared beta minus 1 and denominator cap M sub 1 squared open paren gamma plus cosine 2 beta close paren plus 2 end-fraction close bracket Since evaluating this implicitly for
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(low frequency), the velocity profile matches the parabolic shape of steady Poiseuille flow. When
are Bessel functions of the first and second kind, respectively. Regularity at the centerline: At . Therefore, No-slip condition at the wall: At
u(y)=UyB+12μ(dPdx)(y2−By)u open paren y close paren equals the fraction with numerator cap U y and denominator cap B end-fraction plus the fraction with numerator 1 and denominator 2 mu end-fraction open paren the fraction with numerator d cap P and denominator d x end-fraction close paren open paren y squared minus cap B y close paren
ϕ(r,θ)=m2πlnr+Γ2πθphi open paren r comma theta close paren equals the fraction with numerator m and denominator 2 pi end-fraction l n r plus the fraction with numerator cap gamma and denominator 2 pi end-fraction theta