Water ( ) flows steadily and fully developed between two fixed

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

Fluid Dynamics, Ch The Navier-Stokes equations and unsteady exact solutions

Problem

Water (μ=1.00×10−3 Pa s\mu=1.00\times10^{-3}\ \mathrm{Pa\,s}) flows steadily and fully developed between two fixed, horizontal, parallel plates a distance h=2.0 mmh=2.0\ \mathrm{mm} apart, driven by a uniform pressure gradient −dp/dx=G=300 Pa m−1-\mathrm dp/\mathrm dx=G=300\ \mathrm{Pa\,m^{-1}} along the channel. Both plates satisfy no slip, so u=0u=0 at y=0y=0 and y=hy=h, where yy is measured from one plate. The velocity profile is u(y)=G2μy(h−y)u(y)=\dfrac{G}{2\mu}y(h-y). ((a)) Starting from the identity 2eijeij=∣ω∣2+2 ∂iuj ∂jui2e_{ij}e_{ij}=|\boldsymbol\omega|^2+2\,\partial_iu_j\,\partial_ju_i together with ∇⋅u=0\nabla\cdot\mathbf u=0, derive the dissipation theorem ∫VΦ dV=μ∫V∣ω∣2 dV+2μ∮S(u⋅∇)u⋅n^ dS,\int_V\Phi\,\mathrm dV=\mu\int_V|\boldsymbol\omega|^2\,\mathrm dV+2\mu\oint_S(\mathbf u\cdot\nabla)\mathbf u\cdot\hat{\mathbf n}\,\mathrm dS, where Φ=2μeijeij\Phi=2\mu e_{ij}e_{ij} is the viscous dissipation function per unit volume, and state the condition on u\mathbf u at a boundary under which its contribution to the surface integral vanishes. ((b)) Confirm that condition holds at both plates of the channel described. ((c)) Compute, per unit plan area of the channel (that is, per unit length in xx and per unit width in zz), both μ∫0hωz2 dy\mu\int_0^h\omega_z^2\,\mathrm dy from the given profile and the rate of working of the pressure gradient G×QG\times Q, where Q=∫0hu dyQ=\int_0^hu\,\mathrm dy is the volume flow rate per unit width, and show that the two agree.

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