Past exam of the topics in fluid mechanics course of the University of Oxford / 2008 / 3

Define ${\chi }_1$ to be one in phase 1 and zero in phase 2, including its dependence on both transverse coordinates even though the averaged fields depend only on $x,t$. With no interphase mass transfer, its distributional phase-mass equation is ${\partial }_t(\rho {\chi }_1)+\nabla \cdot (\rho \mathbf{v}{\chi }_1)=0$. At a pipe cross section $\Sigma$ of area $A$, setThus ${\rho }_1$ is an intrinsic phase density and $u_1$ is the density-weighted mean axial velocity. This weighting is necessary when density varies within the phase; it becomes the ordinary intrinsic mean for an incompressible phase of constant density.

Integrate over the cross section. The transverse divergence contributes no boundary mass flux at the impermeable pipe wall, and the material interface has no mass leakage. The section area is constant, soFor phase 2 use ${\chi }_2=1-{\chi }_1$ and fraction $1-\alpha$, with the analogous intrinsic density and density-weighted velocity:The intrinsic quantities are defined where their phase fraction is nonzero. These are the intrinsic phase averages in a two-fluid pipe model.

Freeze a uniform state with $0<\alpha <1$, ${\rho }_i>0$, and seek a finite characteristic speed $c$. Denote its perturbation amplitudes by $a,v_1,v_2,\pi$. The two phase-mass equations and two momentum equations giveEliminating the amplitudes givesWith $s^2={\rho }_1(1-\alpha )/({\rho }_2\alpha )>0$, factor the quadratic over the complex numbers to obtainFor nonzero relative velocity the speeds are a nonreal conjugate pair. A Fourier perturbation $e^{ik(x-ct)}$ on the growing branch has magnitude $e^{|k||\mathrm{Im}c|t}$. Arbitrarily short waves therefore grow arbitrarily quickly, preventing a continuous dependence estimate in the usual finite-order Sobolev norms. This generic slip state is ill-posed as an initial value problem.

The pressure has no independent time-evolution equation; it enforces the incompressible volume constraint, so the request concerns the finite characteristic speeds rather than a freely propagating pressure mode. At $u_1=u_2$ the two speeds coincide and this particular nonreal-speed argument disappears; further degeneracy analysis is needed there, rather than calling coincident real speeds complex. The issue is complex characteristic speeds of a pressure-coupled two-fluid model.

For constant phase density, let $u_1$ be the intrinsic mean axial speed and defineThe profile coefficient measures the momentum flux relative to a uniform velocity profile with the same mean. Cauchy-Schwarz gives $D_1\ge 1$ where the mean is nonzero, with equality for a uniform phase profile. At zero mean it is safer to retain the second moment itself rather than divide by $u_1^2$.

In the stated common-pressure, no-extra-force closure, the averaged conservative momentum equation isFor constant $D_1$, expand its derivatives and use ${\alpha }_t+u_1{\alpha }_x+\alpha (u_1{)}_x=0$. The terms combine asThereforeThis is the phase momentum profile coefficient correction. It accounts for the second moment of velocity, not merely a rescaling of the convective speed. A variable coefficient would also produce ${\rho }_1\alpha u_1^2(D_1{)}_x$.