Couette flow

Couette flow is viscous shear flow driven by relative tangential motion of parallel boundaries. In the steady no-pressure-gradient case its velocity profile is linear.

A rotating inner cylinder of radius $R$ and height $H$ has a uniform gap $h$ from a fixed housing at its side and both ends. For $h\ll R,H$, no-slip boundary condition and laminar Couette flow give side torque $2\pi \eta \omega R^{3}H/h$. Each end contributes ${\int }_0^R(\eta \omega s/h)(2\pi sds)s=\pi \eta \omega R^4/(2h)$. Summing both end faces proves the displayed expression. Shaft area, corner regions, gas slip and viscous heating are neglected.

In the thin-gap, laminar approximation, a rotating disc of radius $a$ separated by $d$ from a stationary disc shears a Newtonian fluid. The resisting torque is $\pi \eta \omega a^4/(2d)$, neglecting edge effects and secondary flows. This gives a way of inferring dynamic viscosity from a measured torque.

Taylor-Couette flow is the flow between coaxial rotating cylinders. Its simplest steady homogeneous purely azimuthal viscous state has steady circular Couette profile $v_{\varphi }=Ar+B/r$. Stability of that base state depends on rotation rates, geometry and viscosity; the inviscid Rayleigh centrifugal stability criterion addresses axisymmetric centrifugal disturbances, not every viscous instability.

For a homogeneous Newtonian fluid with constant kinematic viscosity, a steady axisymmetric purely azimuthal flow obeys $v_{\varphi }^{\prime \prime }+v_{\varphi }^{\prime }/r-v_{\varphi }/r^2=0$. Writing $v_{\varphi }=r\Omega$ gives $(r^3{\Omega }^{\prime }{)}^{\prime }=0$, so $\Omega =A+B/r^2$. A domain containing the rotation axis requires $B=0$ for regularity; rotating walls in an annulus can sustain $B\ne 0$. An arbitrary time-dependent profile instead evolves by ${\partial }_t\Omega =\nu ({\Omega }^{\prime \prime }+3{\Omega }^{\prime }/r)$ within this ansatz.

Couette-Poiseuille flow superposes wall-driven shear and pressure-driven parabolic flow between no-slip parallel plates. With one plate moving at $U$, separation $a$, and $dp/dx=-\rho \nu G$, the displayed profile has volume flux $Ua/2+G{a}^3/12$ per unit width. Requiring zero flux gives $G=-6U/a^2$ and $u=U[3(y/a{)}^2-2y/a]$, with a pressure gradient induced by return-flow constraints.

Between a resting wall at $y=0$ and a wall moving at speed $U$ at $y=h$, an adverse pressure gradient $p_x=G$ givesZero volume flux requires $G=6\mu U/h^2$, yielding $u/U=3(y/h{)}^2-2y/h$. Reverse flow below $2h/3$ balances the positive flux near the moving wall. The minimum is $-U/3$ at $h/3$.

In two-layer Couette flow, each Newtonian-fluid layer has a linear velocity profile. Continuity of velocity and shear stress at the interface makes the velocity drop across each layer proportional to its thickness divided by its dynamic viscosity.