Paper 3, Section II, 38B

Fluid Dynamics II | Part II, 2020

(a) Briefly outline the derivation of the boundary layer equation

uux+vuy=UdU/dx+νuyyu u_{x}+v u_{y}=U d U / d x+\nu u_{y y}

explaining the significance of the symbols used and what sets the xx-direction.

(b) Viscous fluid occupies the sector 0<θ<α0<\theta<\alpha in cylindrical coordinates which is bounded by rigid walls and there is a line sink at the origin of strength αQ\alpha Q with Q/ν1Q / \nu \gg 1. Assume that vorticity is confined to boundary layers along the rigid walls θ=0\theta=0 (x>0,y=0)(x>0, y=0) and θ=α\theta=\alpha.

(i) Find the flow outside the boundary layers and clarify why boundary layers exist at all.

(ii) Show that the boundary layer thickness along the wall y=0y=0 is proportional to

δ:=(νQ)1/2x\delta:=\left(\frac{\nu}{Q}\right)^{1 / 2} x

(iii) Show that the boundary layer equation admits a similarity solution for the streamfunction ψ(x,y)\psi(x, y) of the form

ψ=(νQ)1/2f(η)\psi=(\nu Q)^{1 / 2} f(\eta)

where η=y/δ\eta=y / \delta. You should find the equation and boundary conditions satisfied by ff.

(iv) Verify that

dfdη=5cosh(2η+c)1+cosh(2η+c)\frac{d f}{d \eta}=\frac{5-\cosh (\sqrt{2} \eta+c)}{1+\cosh (\sqrt{2} \eta+c)}

yields a solution provided the constant cc has one of two possible values. Which is the likely physical choice?

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