Past exam of the theoretical mechanics course of the University of Oxford / 2009 / A / 1

For a material volume $V(t)$ moving with velocity $\mathbf{u}$, the Reynolds transport theorem isIt applies componentwise to vector quantities. Cauchy's stress theorem represents traction on a surface with outward unit normal $\mathbf{n}$ as $t_i={\sigma }_{ij}{n}_j$. Conservation of mass gives $d{\int }_{V(t)}\rho dV/dt=0$ and hence ${\partial }_t\rho +\nabla \cdot (\rho \mathbf{u})=0$. Constant density reduces this to $\boxed{\nabla \cdot \mathbf{u}=0}$.

With no body force, conservation of momentum and the divergence theorem giveApplying transport and then mass conservation cancels the terms proportional to $u_i$, leaving $\rho ({\partial }_t{u}_i+u_j{\partial }_j{u}_i)={\partial }_j{\sigma }_{ij}$. For the stated Newtonian stress, constant viscosity gives ${\partial }_j{\sigma }_{ij}=-{\partial }_{i}p+\mu \Delta u_i+\mu {\partial }_i(\nabla \cdot \mathbf{u})$. The last term vanishes. Since the material volume was arbitrary,A body force, if present, would have to be added to the momentum balance; it is absent from the specified model.

For $\mathbf{u}=u(y,t){\mathbf{e}}_x$, incompressibility is automatic and the nonlinear advection term is zero. With no pressure gradient the streamwise momentum equation is $\boxed{\rho u_t=\mu u_{yy}}$. Seek its periodic response as $u=\mathrm{Re}[Uf(y)e^{i\Omega t}]$. The amplitude satisfies $f^{\prime \prime }=i\rho \Omega f/\mu$ with $f(\pm d)=1$. Since $[k(1+i){]}^2=i\rho \Omega /\mu$ for $k=\sqrt{\rho \Omega /(2\mu )}$, even symmetry givesOdd components vanish because the two boundary amplitudes are equal. This is the unique periodic solution. Without a specified initial profile, decaying zero-wall diffusion transients may be added, so the formula should not be read as a claim that every transient is absent.

As $kd\to \infty$, the wall motions penetrate only a distance $k^{-1}=\sqrt{2\mu /(\rho \Omega )}$. Near either wall, with inward distance $s=d-|y|$, the leading velocity isThe two oscillatory viscous layers are separated by an almost motionless core; the phase lag increases inward. This is in-phase oscillating parallel-plate flow.