A deep pool of oil (density

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Viscous flow and dynamical similarity

Fluid Dynamics, Ch Scaling, similarity and similarity solutions

Problem

A deep pool of oil (density ρ\rho, dynamic viscosity μ\mu, kinematic viscosity ν=μ/ρ\nu=\mu/\rho) is at rest for t<0t<0 and occupies y>0y>0, with a flat free surface at y=0y=0 (yy measured downward into the oil). At t=0t=0 a wind begins to blow over the surface and, for t>0t>0, maintains a constant uniform tangential surface stress of magnitude τ0\tau_0 on the oil, consistent with the tangential-stress continuity condition at an interface. The resulting unidirectional flow u(y,t)u(y,t) obeys the one-dimensional diffusion equation used for the unsteady exact solutions, ∂u∂t=ν∂2u∂y2,\frac{\partial u}{\partial t}=\nu\frac{\partial^2 u}{\partial y^2}, with u(y,0)=0u(y,0)=0, u→0u\to0 as y→∞y\to\infty, and the surface condition −μ∂u∂y∣y=0=τ0,t>0,-\mu\left.\frac{\partial u}{\partial y}\right|_{y=0}=\tau_0,\qquad t>0, with uu taken positive in the direction of the wind's stress; the minus sign appears because the oil's outward normal at the surface is −y^-\hat{\mathbf{y}}. ((a)) List the parameters that fix uu at given (y,t)(y,t). Show that τ0\tau_0, μ\mu and ν\nu together do form a time, μ/τ0\mu/\tau_0, and a length, νμ/τ0\sqrt{\nu\mu/\tau_0}, so that units alone do not force a similarity form. ((b)) The equation and surface condition are linear in τ0\tau_0. Use this to argue that uu is proportional to τ0\tau_0, and hence find a similarity form u(y,t)=τ0μνt  F(η)u(y,t)=\frac{\tau_0}{\mu}\sqrt{\nu t}\;F(\eta) with a similarity variable η\eta you should identify, explaining why, once linearity has divided out τ0\tau_0, no other length or time scale can appear. ((c)) Substitute this form into the diffusion equation to show that FF satisfies F′′(η)+η2F′(η)−12F(η)=0,F''(\eta)+\frac{\eta}{2}F'(\eta)-\frac12F(\eta)=0, and show that the surface condition reduces to the tt-independent statement F′(0)=−1F'(0)=-1.

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