Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 36B (Fluid Dynamics II)

The steady axial Navier-Stokes equation for the straight-pipe approximation is $0=G/\rho +\nu w^{\prime \prime }(y)$. No-slip boundary conditions are $w(\pm a)=0$. Integrating twice givesIgnoring the radial variation is justified away from the side walls by the given ordering $a\ll b$.

Independence of $x$ and incompressibility give $v_y=0$ for the secondary velocity $(u,v,0)$. No penetration at the upper and lower walls then forces $v=0$. The radial momentum equation, retaining the given centrifugal acceleration, is $-w^2/R=-p_x/\rho +\nu u^{\prime \prime }$. The transverse pressure gradient is constant in $y$ at this order. Integrating twice and absorbing the resulting quadratic coefficient into $C$ givesThe odd integration term vanishes because both no-slip conditions are equal. The remaining no-slip and zero-flux conditions areThe second equation is half of ${\int }_{-a}^{a}u(y)dy=0$. Optionally solving gives $C=-66K{a}^4/7$ and $D=5K{a}^6/7$. With $s=y/a$ the velocity is $K{a}^6(1-s^2)(s^4-4s^2+5/7)$, so the ratio to the primary velocity is bounded, including its limiting value at the walls, by a constant times $G{a}^4/(\rho {\nu }^{2}R)$. A sufficient parameter ordering isEquivalently the last condition is $\mathrm{Re}a/R\ll 1$ for $\mathrm{Re}=G{a}^3/(\rho {\nu }^2)$. It keeps the secondary circulation weak; the neglected radial return flow and side-wall structure supply closure outside this central wide-channel approximation.