A viscous fluid of density occupies the half-space above a flat plate at

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

Mathematical Methods for Physics, Ch The classical partial differential equations of physics and their boundary conditions

Problem

A viscous fluid of density ρ\rho occupies the half-space y>0y > 0 above a flat plate at y=0y=0. Consider only motion in the xx direction, with fluid velocity u(y,t)u(y,t) depending on height yy above the plate and on time, but not on xx or zz. Newton's second law applied to a thin layer of fluid between yy and y+Δyy+\Delta y (per unit area of the plate) gives, in the limit Δy→0\Delta y\to0, ρ ∂u∂t=−∂τ∂y,\rho\,\frac{\partial u}{\partial t} = -\frac{\partial \tau}{\partial y}, where τ(y,t)\tau(y,t) is the shear stress transmitted across a horizontal plane at height yy. Newton's law of viscosity, an experimental fit, states τ=−μ ∂u∂y,\tau = -\mu\,\frac{\partial u}{\partial y}, with μ\mu the (constant) viscosity of the fluid. ((a)) Combine the two stated relations to derive the partial differential equation obeyed by u(y,t)u(y,t), and identify the quantity ν≡μ/ρ\nu \equiv \mu/\rho that plays the role of the diffusivity DD of heat conduction. ((b)) Confirm that ν\nu has units m2 s−1\mathrm{m^2\,s^{-1}}. ((c)) For glycerine, μ=1.5 Pa s\mu = 1.5\,\mathrm{Pa\,s} and ρ=1260 kg m−3\rho = 1260\,\mathrm{kg\,m^{-3}}. A layer of glycerine L=0.10 mL = 0.10\,\mathrm{m} thick sits above a plate that is suddenly set in motion at t=0t=0. Estimate the time scale L2/νL^2/\nu on which the motion diffuses across this layer.

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