# Fluid Dynamics Ii

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Paper 1, Section II, 39A

comment(a) Write down the Stokes equations for the motion of an incompressible viscous fluid with negligible inertia (in the absence of body forces). What does it mean that Stokes flow is linear and reversible?

(b) The region $a<r<b$ between two concentric rigid spheres of radii $a$ and $b$ is filled with fluid of large viscosity $\mu$. The outer sphere is held stationary, while the inner sphere is made to rotate with angular velocity $\boldsymbol{\Omega}$.

(i) Use symmetry and the properties of Stokes flow to deduce that $p=0$, where $p$ is the pressure due to the flow.

(ii) Verify that both solid-body rotation and $\mathbf{u}(\mathbf{x})=\boldsymbol{\Omega} \wedge \boldsymbol{\nabla}(1 / r)$ satisfy the Stokes equations with $p=0$. Hence determine the fluid velocity between the spheres.

(iii) Calculate the stress tensor $\sigma_{i j}$ in the flow.

(iv) Deduce that the couple $\mathbf{G}$ exerted by the fluid in $r<c$ on the fluid in $r>c$, where $a<c<b$, is given by

$\mathbf{G}=\frac{8 \pi \mu a^{3} b^{3} \mathbf{\Omega}}{b^{3}-a^{3}}$

independent of the value of $c$. [Hint: Do not substitute the form of $A$ and $B$ in $A+B r^{-3}$ until the end of the calculation.]

Comment on the form of this result for $a \ll b$ and for $b-a \ll a$.

$\left[Y o u\right.$ may use $\int_{r=R} n_{i} n_{j} d S=\frac{4}{3} \pi R^{2} \delta_{i j}$, where $\mathbf{n}$ is the normal to $\left.r=R .\right]$

Paper 2, Section II, 39A

comment(a) Incompressible fluid of viscosity $\mu$ fills the thin, slowly varying gap between rigid boundaries at $z=0$ and $z=h(x, y)>0$. The boundary at $z=0$ translates in its own plane with a constant velocity $\mathbf{U}=(U, 0,0)$, while the other boundary is stationary. If $h$ has typical magnitude $H$ and varies on a lengthscale $L$, state conditions for the lubrication approximation to be appropriate.

Write down the lubrication equations for this problem and show that the horizontal volume flux $\mathbf{q}=\left(q_{x}, q_{y}, 0\right)$ is given by

$\mathbf{q}=\frac{\mathbf{U} h}{2}-\frac{h^{3}}{12 \mu} \nabla p$

where $p(x, y)$ is the pressure.

Explain why $\mathbf{q}=\nabla \wedge(0,0, \psi)$ for some function $\psi(x, y)$. Deduce that $\psi$ satisfies the equation

$\nabla \cdot\left(\frac{1}{h^{3}} \nabla \psi\right)=-\frac{U}{h^{3}} \frac{\partial h}{\partial y}$

(b) Now consider the case $\mathbf{U}=\mathbf{0}, h=h_{0}$ for $r>a$ and $h=h_{1}$ for $r<a$, where $h_{0}, h_{1}$ and $a$ are constants, and $(r, \theta)$ are polar coordinates. A uniform pressure gradient $\nabla p=-G \mathbf{e}_{x}$ is applied at infinity. Show that $\psi \sim A r \sin \theta$ as $r \rightarrow \infty$, where the constant $A$ is to be determined.

Given that $a \gg h_{0}, h_{1}$, you may assume that the equations of part (a) apply for $r<a$ and $r>a$, and are subject to conditions that the radial component $q_{r}$ of the volume flux and the pressure $p$ are both continuous across $r=a$. Show that these continuity conditions imply that

$\left[\frac{\partial \psi}{\partial \theta}\right]_{-}^{+}=0 \quad \text { and }\left[\frac{1}{h^{3}} \frac{\partial \psi}{\partial r}\right]_{-}^{+}=0$

respectively, where []$_{-}^{+}$denotes the jump across $r=a$.

Hence determine $\psi(r, \theta)$ and deduce that the total flux through $r=a$ is given by

$\frac{4 A a h_{1}^{3}}{h_{0}^{3}+h_{1}^{3}}$

Paper 3, Section II, 38A

commentViscous fluid occupying $z>0$ is bounded by a rigid plane at $z=0$ and is extracted through a small hole at the origin at a constant flow rate $Q=2 \pi A$. Assume that for sufficiently small values of $R=|\mathbf{x}|$ the velocity $\mathbf{u}(\mathbf{x})$ is well-approximated by

$\mathbf{u}=-\frac{A \mathbf{x}}{R^{3}}$

except within a thin axisymmetric boundary layer near $z=0$.

(a) Estimate the Reynolds number of the flow as a function of $R$, and thus give an estimate for how small $R$ needs to be for such a solution to be applicable. Show that the radial pressure gradient is proportional to $R^{-5}$.

(b) In cylindrical polar coordinates $(r, \theta, z)$, the steady axisymmetric boundary-layer equations for the velocity components $(u, 0, w)$ can be written as

$u \frac{\partial u}{\partial r}+w \frac{\partial u}{\partial z}=-\frac{1}{\rho} \frac{d P}{d r}+\nu \frac{\partial^{2} u}{\partial z^{2}}, \quad \text { where } \quad u=-\frac{1}{r} \frac{\partial \Psi}{\partial z}, \quad w=\frac{1}{r} \frac{\partial \Psi}{\partial r}$

and $\Psi(r, z)$ is the Stokes streamfunction. Verify that the condition of incompressibility is satisfied by the use of $\Psi$.

Use scaling arguments to estimate the thickness $\delta(r)$ of the boundary layer near $z=0$ and then to motivate seeking a similarity solution of the form

$\Psi=(A \nu r)^{1 / 2} F(\eta), \quad \text { where } \quad \eta=z / \delta(r)$

(c) Obtain the differential equation satisfied by $F$, and state the conditions that would determine its solution. [You are not required to find this solution.]

By considering the flux in the boundary layer, explain why there should be a correction to the approximation $(*)$ of relative magnitude $(\nu R / A)^{1 / 2} \ll 1$.

Paper 4, Section II, A

commentConsider a steady axisymmetric flow with components $(-\alpha r, v(r), 2 \alpha z)$ in cylindrical polar coordinates $(r, \theta, z)$, where $\alpha$ is a positive constant. The fluid has density $\rho$ and kinematic viscosity $\nu$.

(a) Briefly describe the flow and confirm that it is incompressible.

(b) Show that the vorticity has one component $\omega(r)$, in the $z$ direction. Write down the corresponding vorticity equation and derive the solution

$\omega=\omega_{0} e^{-\alpha r^{2} /(2 \nu)}$

Hence find $v(r)$ and show that it has a maximum at some finite radius $r^{*}$, indicating how $r^{*}$ scales with $\nu$ and $\alpha$.

(c) Find an expression for the net advection of angular momentum, prv, into the finite cylinder defined by $r \leqslant r_{0}$ and $-z_{0} \leqslant z \leqslant z_{0}$. Show that this is always positive and asymptotes to the value

$\frac{8 \pi \rho z_{0} \omega_{0} \nu^{2}}{\alpha}$

as $r_{0} \rightarrow \infty$

(d) Show that the torque exerted on the cylinder of part (c) by the exterior flow is always negative and demonstrate that it exactly balances the net advection of angular momentum. Comment on why this has to be so.

[You may assume that for a flow $(u, v, w)$ in cylindrical polar coordinates

$\begin{aligned} & e_{r \theta}=\frac{r}{2} \frac{\partial}{\partial r}\left(\frac{v}{r}\right)+\frac{1}{2 r} \frac{\partial u}{\partial \theta}, \quad e_{\theta z}=\frac{1}{2 r} \frac{\partial w}{\partial \theta}+\frac{1}{2} \frac{\partial v}{\partial z}, \quad e_{r z}=\frac{1}{2} \frac{\partial u}{\partial z}+\frac{1}{2} \frac{\partial w}{\partial r} \\ & \text { and } \left.\boldsymbol{\omega}=\frac{1}{r}\left|\begin{array}{ccc}\mathbf{e}_{r} & r \mathbf{e}_{\theta} & \mathbf{e}_{z} \\\partial / \partial r & \partial / \partial \theta & \partial / \partial z \\u & r v & w\end{array}\right| .\right] \end{aligned}$

Paper 1, Section II, 39B

commentA viscous fluid is confined between an inner, impermeable cylinder of radius $a$ with centre at $(x, y)=(0, a)$ and another outer, impermeable cylinder of radius $2 a$ with centre at $(0,2 a)$ (so they touch at the origin and both have their axes in the $z$ direction). The inner cylinder rotates about its axis with angular velocity $\Omega$ and the outer cylinder rotates about its axis with angular velocity $-\Omega / 4$. The fluid motion is two-dimensional and slow enough that the Stokes approximation is appropriate.

(i) Show that the boundary of the inner cylinder is described by the relationship

$r=2 a \sin \theta,$

where $(r, \theta)$ are the usual polar coordinates centred on $(x, y)=(0,0)$. Show also that on this cylinder the boundary condition on the tangential velocity can be written as

$u_{r} \cos \theta+u_{\theta} \sin \theta=a \Omega,$

where $u_{r}$ and $u_{\theta}$ are the components of the velocity in the $r$ and $\theta$ directions respectively. Explain why the boundary condition $\psi=0$ (where $\psi$ is the streamfunction such that $u_{r}=\frac{1}{r} \frac{\partial \psi}{\partial \theta}$ and $\left.u_{\theta}=-\frac{\partial \psi}{\partial r}\right)$ can be imposed.

(ii) Write down the boundary conditions to be satisfied on the outer cylinder $r=4 a \sin \theta$, explaining carefully why $\psi=0$ can also be imposed on this cylinder as well.

(iii) It is given that the streamfunction is of the form

$\psi=A \sin ^{2} \theta+B r^{2}+C r \sin \theta+D \sin ^{3} \theta / r$

where $A, B, C$ and $D$ are constants, which satisfies $\nabla^{4} \psi=0$. Using the fact that $B=0$ due to the symmetry of the problem, show that the streamfunction is

$\psi=\frac{\alpha \sin \theta}{r}(r-2 a \sin \theta)(r-4 a \sin \theta)$

where the constant $\alpha$ is to be found.

(iv) Sketch the streamline pattern between the cylinders and determine the $(x, y)$ coordinates of the stagnation point in the flow.

Paper 2, Section II, 38B

commentConsider a two-dimensional flow of a viscous fluid down a plane inclined at an angle $\alpha$ to the horizontal. Initially, the fluid, which has a volume $V$, occupies a region $0 \leqslant x \leqslant x^{*}$ with $x$ increasing down the slope. At large times the flow becomes thin-layer flow.

(i) Write down the two-dimensional Navier-Stokes equations and simplify them using the lubrication approximation. Show that the governing equation for the height of the film, $h=h(x, t)$, is

$\tag{†} \frac{\partial h}{\partial t}+\frac{\partial}{\partial x}\left(\frac{g h^{3} \sin \alpha}{3 \nu}\right)=0$

where $\nu$ is the kinematic viscosity of the fluid and $g$ is the acceleration due to gravity, being careful to justify why the streamwise pressure gradient has been ignored compared to the gravitational body force.

(ii) Develop a similarity solution to $(†)$ and, using the fact that the volume of fluid is conserved over time, derive an expression for the position and height of the head of the current downstream.

(iii) Fluid is now continuously supplied at $x=0$. By using scaling analysis, estimate the rate at which fluid would have to be supplied for the head height to asymptote to a constant value at large times.

Paper 3, Section II, 38B

comment(a) Briefly outline the derivation of the boundary layer equation

$u u_{x}+v u_{y}=U d U / d x+\nu u_{y y}$

explaining the significance of the symbols used and what sets the $x$-direction.

(b) Viscous fluid occupies the sector $0<\theta<\alpha$ in cylindrical coordinates which is bounded by rigid walls and there is a line sink at the origin of strength $\alpha Q$ with $Q / \nu \gg 1$. Assume that vorticity is confined to boundary layers along the rigid walls $\theta=0$ $(x>0, y=0)$ and $\theta=\alpha$.

(i) Find the flow outside the boundary layers and clarify why boundary layers exist at all.

(ii) Show that the boundary layer thickness along the wall $y=0$ is proportional to

$\delta:=\left(\frac{\nu}{Q}\right)^{1 / 2} x$

(iii) Show that the boundary layer equation admits a similarity solution for the streamfunction $\psi(x, y)$ of the form

$\psi=(\nu Q)^{1 / 2} f(\eta)$

where $\eta=y / \delta$. You should find the equation and boundary conditions satisfied by $f$.

(iv) Verify that

$\frac{d f}{d \eta}=\frac{5-\cosh (\sqrt{2} \eta+c)}{1+\cosh (\sqrt{2} \eta+c)}$

yields a solution provided the constant $c$ has one of two possible values. Which is the likely physical choice?

Paper 4 , Section II, 38B

commentConsider a two-dimensional horizontal vortex sheet of strength $U$ in a homogeneous inviscid fluid at height $h$ above a horizontal rigid boundary at $y=0$ so that the fluid velocity is

$\boldsymbol{u}=\left\{\begin{array}{cr} U \hat{\boldsymbol{x}}, & 0<y<h \\ \mathbf{0}, & h<y \end{array}\right.$

(i) Investigate the linear instability of the sheet by determining the relevant dispersion relation for small, inviscid, irrotational perturbations. For what wavelengths is the sheet unstable?

(ii) Evaluate the temporal growth rate and the wave propagation speed in the limits of both short and long waves. Using these results, sketch how the growth rate varies with the wavenumber.

(iii) Comment briefly on how the introduction of a stable density difference (fluid in $y>h$ is less dense than that in $0<y<h$ ) and surface tension at the interface would affect the growth rates.

Paper 1, Section II, A

commentA disc of radius $R$ and weight $W$ hovers at a height $h$ on a cushion of air above a horizontal air table - a fine porous plate through which air of density $\rho$ and dynamic viscosity $\mu$ is pumped upward at constant speed $V$. You may assume that the air flow is axisymmetric with no flow in the azimuthal direction, and that the effect of gravity on the air may be ignored.

(a) Write down the relevant components of the Navier-Stokes equations. By estimating the size of the individual terms, simplify these equations when $\varepsilon:=h / R \ll 1$ and $R e:=\rho V h / \mu \ll 1$.

(b) Explain briefly why it is reasonable to expect that the vertical velocity of the air below the disc is a function of distance above the air table alone, and thus find the steady pressure distribution below the disc. Hence show that

$W=\frac{3 \pi \mu V R}{2 \varepsilon^{3}}$

Paper 2, Section II, A

commentA viscous fluid is contained in a channel between rigid planes $y=-h$ and $y=h$. The fluid in the upper region $\sigma<y<h$ (with $-h<\sigma<h$ ) has dynamic viscosity $\mu_{-}$ while the fluid in the lower region $-h<y<\sigma$ has dynamic viscosity $\mu_{+}>\mu_{-}$. The plane at $y=h$ moves with velocity $U_{-}$and the plane at $y=-h$ moves with velocity $U_{+}$, both in the $x$ direction. You may ignore the effect of gravity.

(a) Find the steady, unidirectional solution of the Navier-Stokes equations in which the interface between the two fluids remains at $y=\sigma$.

(b) Using the solution from part (a):

(i) calculate the stress exerted by the fluids on the two boundaries;

(ii) calculate the total viscous dissipation rate in the fluids;

(iii) demonstrate that the rate of working by boundaries balances the viscous dissipation rate in the fluids.

(c) Consider the situation where $U_{+}+U_{-}=0$. Defining the volume flux in the upper region as $Q_{-}$and the volume flux in the lower region as $Q_{+}$, show that their ratio is independent of $\sigma$ and satisfies

$\frac{Q_{-}}{Q_{+}}=-\frac{\mu_{-}}{\mu_{+}}$

Paper 3, Section II, A

commentFor a fluid with kinematic viscosity $\nu$, the steady axisymmetric boundary-layer equations for flow primarily in the $z$-direction are

$\begin{aligned} u \frac{\partial w}{\partial r}+w \frac{\partial w}{\partial z} &=\frac{\nu}{r} \frac{\partial}{\partial r}\left(r \frac{\partial w}{\partial r}\right) \\ \frac{1}{r} \frac{\partial(r u)}{\partial r}+\frac{\partial w}{\partial z} &=0 \end{aligned}$

where $u$ is the fluid velocity in the $r$-direction and $w$ is the fluid velocity in the $z$-direction. A thin, steady, axisymmetric jet emerges from a point at the origin and flows along the $z$-axis in a fluid which is at rest far from the $z$-axis.

(a) Show that the momentum flux

$M:=\int_{0}^{\infty} r w^{2} d r$

is independent of the position $z$ along the jet. Deduce that the thickness $\delta(z)$ of the jet increases linearly with $z$. Determine the scaling dependence on $z$ of the centre-line velocity $W(z)$. Hence show that the jet entrains fluid.

(b) A similarity solution for the streamfunction,

$\psi(x, y, z)=\nu z g(\eta) \quad \text { with } \quad \eta:=r / z$

exists if $g$ satisfies the second order differential equation

$\eta g^{\prime \prime}-g^{\prime}+g g^{\prime}=0$

Using appropriate boundary and normalisation conditions (which you should state clearly) to solve this equation, show that

$g(\eta)=\frac{12 M \eta^{2}}{32 \nu^{2}+3 M \eta^{2}}$

Paper 4, Section II, A

comment(a) Show that the Stokes flow around a rigid moving sphere has the minimum viscous dissipation rate of all incompressible flows which satisfy the no-slip boundary conditions on the sphere.

(b) Let $\boldsymbol{u}=\boldsymbol{\nabla}(\boldsymbol{x} \cdot \boldsymbol{\Phi}+\chi)-2 \boldsymbol{\Phi}$, where $\boldsymbol{\Phi}$ and $\chi$ are solutions of Laplace's equation, i.e. $\nabla^{2} \boldsymbol{\Phi}=\mathbf{0}$ and $\nabla^{2} \chi=0$.

(i) Show that $\boldsymbol{u}$ is incompressible.

(ii) Show that $\boldsymbol{u}$ satisfies Stokes equation if the pressure $p=2 \mu \boldsymbol{\nabla} \cdot \boldsymbol{\Phi}$.

(c) Consider a rigid sphere moving with velocity $\boldsymbol{U}$. The Stokes flow around the sphere is given by

$\boldsymbol{\Phi}=\alpha \frac{\boldsymbol{U}}{r} \quad \text { and } \quad \chi=\beta \boldsymbol{U} \cdot \boldsymbol{\nabla}\left(\frac{1}{r}\right)$

where the origin is chosen to be at the centre of the sphere. Find the values for $\alpha$ and $\beta$ which ensure no-slip conditions are satisfied on the sphere.

Paper 1, Section II, C

commentA two-dimensional layer of very viscous fluid of uniform thickness $h(t)$ sits on a stationary, rigid surface $y=0$. It is impacted by a stream of air (which can be assumed inviscid) such that the air pressure at $y=h$ is $p_{0}-\frac{1}{2} \rho_{a} E^{2} x^{2}$, where $p_{0}$ and $E$ are constants, $\rho_{a}$ is the density of the air, and $x$ is the coordinate parallel to the surface.

What boundary conditions apply to the velocity $\mathbf{u}=(u, v)$ and stress tensor $\sigma$ of the viscous fluid at $y=0$ and $y=h$ ?

By assuming the form $\psi=x f(y)$ for the stream function of the flow, or otherwise, solve the Stokes equations for the velocity and pressure fields. Show that the layer thins at a rate

$V=-\frac{\mathrm{d} h}{\mathrm{~d} t}=\frac{1}{3} \frac{\rho_{a}}{\mu} E^{2} h^{3}$

Paper 2, Section II, C

commentAn initially unperturbed two-dimensional inviscid jet in $-h<y<h$ has uniform speed $U$ in the $x$ direction, while the surrounding fluid is stationary. The unperturbed velocity field $\mathbf{u}=(u, v)$ is therefore given by

$\begin{array}{ll} u=0 & \text { in } \quad y>h \\ u=U & \text { in } \quad-h<y<h \\ u=0 & \text { in } \quad y<-h \end{array}$

Consider separately disturbances in which the layer occupies $-h-\eta<y<h+\eta($ varicose disturbances) and disturbances in which the layer occupies $-h+\eta<y<h+\eta($ sinuous disturbances $)$, where $\eta(x, t)=\hat{\eta} e^{i k x+\sigma t}$, and determine the dispersion relation $\sigma(k)$ in each case.

Find asymptotic expressions for the real part $\sigma_{R}$ of $\sigma$ in the limits $k \rightarrow 0$ and $k \rightarrow \infty$ and draw sketches of $\sigma_{R}(k)$ in each case.

Compare the rates of growth of the two types of disturbance.

Paper 3, Section II, C

commentFor two Stokes flows $\mathbf{u}^{(1)}(\mathbf{x})$ and $\mathbf{u}^{(2)}(\mathbf{x})$ inside the same volume $V$ with different boundary conditions on its boundary $S$, prove the reciprocal theorem

$\int_{S} u_{i}^{(1)} \sigma_{i j}^{(2)} n_{j} d S=\int_{S} u_{i}^{(2)} \sigma_{i j}^{(1)} n_{j} d S$

where $\sigma^{(1)}$ and $\sigma^{(2)}$ are the stress tensors associated with the flows.

Stating clearly any properties of Stokes flow that you require, use the reciprocal theorem to prove that the drag $\mathbf{F}$ on a body translating with uniform velocity $\mathbf{U}$ is given by

$F_{i}=A_{i j} U_{j},$

where $\mathbf{A}$ is a symmetric second-rank tensor that depends only on the geometry of the body.

A slender rod falls slowly through very viscous fluid with its axis inclined to the vertical. Explain why the rod does not rotate, stating any properties of Stokes flow that you use.

When the axis of the rod is inclined at an angle $\theta$ to the vertical, the centre of mass of the rod travels at an angle $\phi$ to the vertical. Given that the rod falls twice as quickly when its axis is vertical as when its axis is horizontal, show that

$\tan \phi=\frac{\sin \theta \cos \theta}{1+\cos ^{2} \theta}$

Paper 4, Section II, C

commentA cylinder of radius $a$ rotates about its axis with angular velocity $\Omega$ while its axis is fixed parallel to and at a distance $a+h_{0}$ from a rigid plane, where $h_{0} \ll a$. Fluid of kinematic viscosity $\nu$ fills the space between the cylinder and the plane. Determine the gap width $h$ between the cylinder and the plane as a function of a coordinate $x$ parallel to the surface of the wall and orthogonal to the axis of the cylinder. What is the characteristic length scale, in the $x$ direction, for changes in the gap width? Taking an appropriate approximation for $h(x)$, valid in the region where the gap width $h$ is small, use lubrication theory to determine that the volume flux between the wall and the cylinder (per unit length along the axis) has magnitude $\frac{2}{3} a \Omega h_{0}$, and state its direction.

Evaluate the tangential shear stress $\tau$ on the surface of the cylinder. Approximating the torque on the cylinder (per unit length along the axis) in the form of an integral $T=a \int_{-\infty}^{\infty} \tau d x$, find the torque $T$ to leading order in $h_{0} / a \ll 1$.

Explain the restriction $a^{1 / 2} \Omega h_{0}^{3 / 2} / \nu \ll 1$ for the theory to be valid.

[You may use the facts that $\int_{-\infty}^{\infty} \frac{d x}{\left(1+x^{2}\right)^{2}}=\frac{\pi}{2}$ and $\left.\int_{-\infty}^{\infty} \frac{d x}{\left(1+x^{2}\right)^{3}}=\frac{3 \pi}{8} .\right]$

Paper 1, Section II, B

commentFluid of density $\rho$ and dynamic viscosity $\mu$ occupies the region $y>0$ in Cartesian coordinates $(x, y, z)$. A semi-infinite, dense array of cilia occupy the half plane $y=0$, $x>0$ and apply a stress in the $x$-direction on the adjacent fluid, working at a constant and uniform rate $\rho P$ per unit area, which causes the fluid to move with steady velocity $\mathbf{u}=(u(x, y), v(x, y), 0)$. Give a careful physical explanation of the boundary condition

$\left.u \frac{\partial u}{\partial y}\right|_{y=0}=-\frac{P}{\nu} \quad \text { for } \quad x>0$

paying particular attention to signs, where $\nu$ is the kinematic viscosity of the fluid. Why would you expect the fluid motion to be confined to a thin region near $y=0$ for sufficiently large values of $x$ ?

Write down the viscous-boundary-layer equations governing the thin region of fluid motion. Show that the flow can be approximated by a stream function

$\psi(x, y)=U(x) \delta(x) f(\eta), \quad \text { where } \quad \eta=\frac{y}{\delta(x)}$

Determine the functions $U(x)$ and $\delta(x)$. Show that the dimensionless function $f(\eta)$ satisfies

$f^{\prime \prime \prime}=\frac{1}{5} f^{\prime 2}-\frac{3}{5} f f^{\prime \prime}$

What boundary conditions must be satisfied by $f(\eta)$ ? By considering how the volume flux varies with downstream location $x$, or otherwise, determine (with justification) the sign of the transverse flow $v$.

Paper 2, Section II, B

commentA cylinder of radius $a$ falls at speed $U$ without rotating through viscous fluid adjacent to a vertical plane wall, with its axis horizontal and parallel to the wall. The distance between the cylinder and the wall is $h_{0} \ll a$. Use lubrication theory in a frame of reference moving with the cylinder to determine that the two-dimensional volume flux between the cylinder and the wall is

$q=\frac{2 h_{0} U}{3}$

upwards, relative to the cylinder.

Determine an expression for the viscous shear stress on the cylinder. Use this to calculate the viscous force and hence the torque on the cylinder. If the cylinder is free to rotate, what does your result say about the sense of rotation of the cylinder?

[Hint: You may quote the following integrals:

$\left.\int_{-\infty}^{\infty} \frac{d t}{1+t^{2}}=\pi, \quad \int_{-\infty}^{\infty} \frac{d t}{\left(1+t^{2}\right)^{2}}=\frac{\pi}{2}, \quad \int_{-\infty}^{\infty} \frac{d t}{\left(1+t^{2}\right)^{3}}=\frac{3 \pi}{8}\right]$

Paper 3, Section II, B

commentA spherical bubble of radius a moves with velocity $\mathbf{U}$ through a viscous fluid that is at rest far from the bubble. The pressure and velocity fields outside the bubble are given by

$p=\mu \frac{a}{r^{3}} \mathbf{U} \cdot \mathbf{x} \quad \text { and } \quad \mathbf{u}=\frac{a}{2 r} \mathbf{U}+\frac{a}{2 r^{3}}(\mathbf{U} \cdot \mathbf{x}) \mathbf{x}$

respectively, where $\mu$ is the dynamic viscosity of the fluid, $\mathbf{x}$ is the position vector from the centre of the bubble and $r=|\mathbf{x}|$. Using suffix notation, or otherwise, show that these fields satisfy the Stokes equations.

Obtain an expression for the stress tensor for the fluid outside the bubble and show that the velocity field above also satisfies all the appropriate boundary conditions.

Compute the drag force on the bubble.

[Hint: You may use

$\int_{S} n_{i} n_{j} d S=\frac{4}{3} \pi a^{2} \delta_{i j}$

where the integral is taken over the surface of a sphere of radius a and $\mathbf{n}$ is the outward unit normal to the surface.]

Paper 4, Section II, B

commentA horizontal layer of inviscid fluid of density $\rho_{1}$ occupying $0<y<h$ flows with velocity $(U, 0)$ above a horizontal layer of inviscid fluid of density $\rho_{2}>\rho_{1}$ occupying $-h<y<0$ and flowing with velocity $(-U, 0)$, in Cartesian coordinates $(x, y)$. There are rigid boundaries at $y=\pm h$. The interface between the two layers is perturbed to position $y=\operatorname{Re}\left(A e^{i k x+\sigma t}\right)$.

Write down the full set of equations and boundary conditions governing this flow. Derive the linearised boundary conditions appropriate in the limit $A \rightarrow 0$. Solve the linearised equations to show that the perturbation to the interface grows exponentially in time if

$U^{2}>\frac{\rho_{2}^{2}-\rho_{1}^{2}}{\rho_{1} \rho_{2}} \frac{g}{4 k} \tanh k h .$

Sketch the right-hand side of this inequality as a function of $k$. Thereby deduce the minimum value of $U$ that makes the system unstable for all wavelengths.

Paper 1, Section II, B

commentState the vorticity equation and interpret the meaning of each term.

A planar vortex sheet is diffusing in the presence of a perpendicular straining flow. The flow is everywhere of the form $\mathbf{u}=(U(y, t),-E y, E z)$, where $U \rightarrow \pm U_{0}$ as $y \rightarrow \pm \infty$, and $U_{0}$ and $E>0$ are constants. Show that the vorticity has the form $\boldsymbol{\omega}=\omega(y, t) \mathbf{e}_{z}$, and obtain a scalar equation describing the evolution of $\omega$.

Explain physically why the solution approaches a steady state in which the vorticity is concentrated near $y=0$. Use scaling to estimate the thickness $\delta$ of the steady vorticity layer as a function of $E$ and the kinematic viscosity $\nu$.

Determine the steady vorticity profile, $\omega(y)$, and the steady velocity profile, $U(y)$.

$\left[\right.$ Hint: $\left.\quad \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-u^{2}} \mathrm{~d} u .\right]$

State, with a brief physical justification, why you might expect this steady flow to be unstable to long-wavelength perturbations, defining what you mean by long.

Paper 2, Section II, B

commentFor a two-dimensional flow in plane polar coordinates $(r, \theta)$, state the relationship between the streamfunction $\psi(r, \theta)$ and the flow components $u_{r}$ and $u_{\theta}$. Show that the vorticity $\omega$ is given by $\omega=-\nabla^{2} \psi$, and deduce that the streamfunction for a steady two-dimensional Stokes flow satisfies the biharmonic equation

$\nabla^{4} \psi=0$

A rigid stationary circular disk of radius $a$ occupies the region $r \leqslant a$. The flow far from the disk tends to a steady straining flow $\mathbf{u}_{\infty}=(-E x, E y)$, where $E$ is a constant. Inertial forces may be neglected. Calculate the streamfunction, $\psi_{\infty}(r, \theta)$, for the far-field flow.

By making an appropriate assumption about its dependence on $\theta$, find the streamfunction $\psi$ for the flow around the disk, and deduce the flow components, $u_{r}(r, \theta)$ and $u_{\theta}(r, \theta)$.

Calculate the tangential surface stress, $\sigma_{r \theta}$, acting on the boundary of the disk.

$[$ Hints: In plane polar coordinates $(r, \theta)$,

$\begin{gathered} \boldsymbol{\nabla} \cdot \mathbf{u}=\frac{1}{r} \frac{\partial\left(r u_{r}\right)}{\partial r}+\frac{1}{r} \frac{\partial u_{\theta}}{\partial \theta}, \quad \omega=\frac{1}{r} \frac{\partial\left(r u_{\theta}\right)}{\partial r}-\frac{1}{r} \frac{\partial u_{r}}{\partial \theta} \\ \nabla^{2} V=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial V}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} V}{\partial \theta^{2}}, \quad e_{r \theta}=\frac{1}{2}\left(r \frac{\partial}{\partial r}\left(\frac{u_{\theta}}{r}\right)+\frac{1}{r} \frac{\partial u_{r}}{\partial \theta}\right) \end{gathered}$

Paper 3, Section II, B

commentA cylindrical pipe of radius $a$ and length $L \gg a$ contains two viscous fluids arranged axisymmetrically with fluid 1 of viscosity $\mu_{1}$ occupying the central region $r<\beta a$, where $0<\beta<1$, and fluid 2 of viscosity $\mu_{2}$ occupying the surrounding annular region $\beta a<r<a$. The flow in each fluid is assumed to be steady and unidirectional, with velocities $u_{1}(r) \mathbf{e}_{z}$ and $u_{2}(r) \mathbf{e}_{z}$ respectively, with respect to cylindrical coordinates $(r, \theta, z)$ aligned with the pipe. A fixed pressure drop $\Delta p$ is applied between the ends of the pipe.

Starting from the Navier-Stokes equations, derive the equations satisfied by $u_{1}(r)$ and $u_{2}(r)$, and state all the boundary conditions. Show that the pressure gradient is constant.

Solve for the velocity profile in each fluid and calculate the corresponding flow rates, $Q_{1}$ and $Q_{2}$.

Derive the relationship between $\beta$ and $\mu_{2} / \mu_{1}$ that yields the same flow rate in each fluid. Comment on the behaviour of $\beta$ in the limits $\mu_{2} / \mu_{1} \gg 1$ and $\mu_{2} / \mu_{1} \ll 1$, illustrating your comment by sketching the flow profiles.

$[$ Hint: In cylindrical coordinates $(r, \theta, z)$,

$\nabla^{2} u=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \theta^{2}}+\frac{\partial^{2} u}{\partial z^{2}}, \quad e_{r z}=\frac{1}{2}\left(\frac{\partial u_{r}}{\partial z}+\frac{\partial u_{z}}{\partial r}\right)$

Paper 4 , Section II, B

commentA thin layer of fluid of viscosity $\mu$ occupies the gap between a rigid flat plate at $y=0$ and a flexible no-slip boundary at $y=h(x, t)$. The flat plate moves with constant velocity $U \mathbf{e}_{x}$ and the flexible boundary moves with no component of velocity in the $x$-direction.

State the two-dimensional lubrication equations governing the dynamics of the thin layer of fluid. Given a pressure gradient $\mathrm{d} p / \mathrm{d} x$, solve for the velocity profile $u(x, y, t)$ in the fluid and calculate the flux $q(x, t)$. Deduce that the pressure gradient satisfies

$\frac{\partial}{\partial x}\left(\frac{h^{3}}{12 \mu} \frac{\mathrm{d} p}{\mathrm{~d} x}\right)=\frac{\partial h}{\partial t}+\frac{U}{2} \frac{\partial h}{\partial x}$

The shape of the flexible boundary is a periodic travelling wave, i.e. $h(x, t)=$ $h(x-c t)$ and $h(\xi+L)=h(\xi)$, where $c$ and $L$ are constants. There is no applied average pressure gradient, so the pressure is also periodic with $p(\xi+L)=p(\xi)$. Show that

$\frac{\mathrm{d} p}{\mathrm{~d} x}=6 \mu(U-2 c)\left(\frac{1}{h^{2}}-\frac{\left\langle h^{-2}\right\rangle}{\left\langle h^{-3}\right\rangle} \frac{1}{h^{3}}\right)$

where $\langle\ldots\rangle=\frac{1}{L} \int_{0}^{L} \ldots \mathrm{d} x$ denotes the average over a period. Calculate the shear stress $\sigma_{x y}$ on the plate.

The speed $U$ is such that there is no need to apply an external tangential force to the plate in order to maintain its motion. Show that

$U=6 c \frac{\left\langle h^{-2}\right\rangle\left\langle h^{-2}\right\rangle-\left\langle h^{-1}\right\rangle\left\langle h^{-3}\right\rangle}{3\left\langle h^{-2}\right\rangle\left\langle h^{-2}\right\rangle-4\left\langle h^{-1}\right\rangle\left\langle h^{-3}\right\rangle} .$

Paper 1, Section II, E

comment(i) In a Newtonian fluid, the deviatoric stress tensor is linearly related to the velocity gradient so that the total stress tensor is

$\sigma_{i j}=-p \delta_{i j}+A_{i j k l} \frac{\partial u_{k}}{\partial x_{l}}$

Show that for an incompressible isotropic fluid with a symmetric stress tensor we necessarily have

$A_{i j k l} \frac{\partial u_{k}}{\partial x_{l}}=2 \mu e_{i j} \text {, }$

where $\mu$ is a constant which we call the dynamic viscosity and $e_{i j}$ is the symmetric part of $\partial u_{i} / \partial x_{j}$.

(ii) Consider Stokes flow due to the translation of a rigid sphere $S_{a}$ of radius $a$ so that the sphere exerts a force $\mathbf{F}$ on the fluid. At distances much larger than the radius of the sphere, the instantaneous velocity and pressure fields are

$u_{i}(\mathbf{x})=\frac{1}{8 \mu \pi}\left(\frac{F_{i}}{r}+\frac{F_{m} x_{m} x_{i}}{r^{3}}\right), \quad p(\mathbf{x})=\frac{1}{4 \pi} \frac{F_{m} x_{m}}{r^{3}},$

where $\mathbf{x}$ is measured with respect to an origin located at the centre of the sphere, and $r=|\mathbf{x}|$.

Consider a sphere $S_{R}$ of radius $R \gg a$ instantaneously concentric with $S_{a}$. By explicitly computing the tractions and integrating them, show that the force $G$ exerted by the fluid located in $r>R$ on $S_{R}$ is constant and independent of $R$, and evaluate it.

(iii) Explain why the Stokes equations in the absence of body forces can be written

$\frac{\partial \sigma_{i j}}{\partial x_{j}}=0$

Show that by integrating this equation in the fluid volume located instantaneously between $S_{a}$ and $S_{R}$, you can recover the result in (ii) directly.

Paper 2, Section II, E

commentConsider an infinite rigid cylinder of radius a parallel to a horizontal rigid stationary surface. Let $\mathbf{e}_{x}$ be the direction along the surface perpendicular to the cylinder axis, $\mathbf{e}_{y}$ the direction normal to the surface (the surface is at $y=0$ ) and $\mathbf{e}_{z}$ the direction along the axis of the cylinder. The cylinder moves with constant velocity $U \mathbf{e}_{x}$. The minimum separation between the cylinder and the surface is denoted by $h_{0} \ll a$.

(i) What are the conditions for the flow in the thin gap between the cylinder and the surface to be described by the lubrication equations? State carefully the relevant length scale in the $\mathbf{e}_{x}$ direction.

(ii) Without doing any calculation, explain carefully why, in the lubrication limit, the net fluid force $\mathbf{F}$ acting on the stationary surface at $y=0$ has no component in the $\mathbf{e}_{y}$ direction.

(iii) Using the lubrication approximation, calculate the $\mathbf{e}_{x}$ component of the velocity field in the gap between the cylinder and the surface, and determine the pressure gradient as a function of the gap thickness $h(x)$.

(iv) Compute the tangential component of the force, $\mathbf{e}_{x} \cdot \mathbf{F}$, acting on the bottom surface per unit length in the $\mathbf{e}_{z}$ direction.

[You may quote the following integrals:

$\int_{-\infty}^{\infty} \frac{d u}{\left(1+u^{2}\right)}=\pi, \quad \int_{-\infty}^{\infty} \frac{d u}{\left(1+u^{2}\right)^{2}}=\frac{\pi}{2}, \quad \int_{-\infty}^{\infty} \frac{d u}{\left(1+u^{2}\right)^{3}}=\frac{3 \pi}{8}$

Paper 3, Section II, E

commentConsider a three-dimensional high-Reynolds number jet without swirl induced by a force $\mathbf{F}=F \mathbf{e}_{z}$ imposed at the origin in a fluid at rest. The velocity in the jet, described using cylindrical coordinates $(r, \theta, z)$, is assumed to remain steady and axisymmetric, and described by a boundary layer analysis.

(i) Explain briefly why the flow in the jet can be described by the boundary layer equations

$u_{r} \frac{\partial u_{z}}{\partial r}+u_{z} \frac{\partial u_{z}}{\partial z}=\nu \frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u_{z}}{\partial r}\right)$

(ii) Show that the momentum flux in the jet, $F=\int_{S} \rho u_{z}^{2} d S$, where $S$ is an infinite surface perpendicular to $\mathbf{e}_{z}$, is not a function of $z$. Combining this result with scalings from the boundary layer equations, derive the scalings for the unknown width $\delta(z)$ and typical velocity $U(z)$ of the jet as functions of $z$ and the other parameters of the problem $(\rho, \nu, F)$.

(iii) Solving for the flow using a self-similar Stokes streamfunction

$\psi(r, z)=U(z) \delta^{2}(z) f(\eta), \quad \eta=r / \delta(z)$

show that $f(\eta)$ satisfies the differential equation

$f f^{\prime}-\eta\left(f^{\prime 2}+f f^{\prime \prime}\right)=f^{\prime}-\eta f^{\prime \prime}+\eta^{2} f^{\prime \prime \prime} .$

What boundary conditions should be applied to this equation? Give physical reasons for them.

[Hint: In cylindrical coordinates for axisymmetric incompressible flow $\left(u_{r}(r, z), 0, u_{z}(r, z)\right)$ you are given the incompressibility condition as

$\frac{1}{r} \frac{\partial}{\partial r}\left(r u_{r}\right)+\frac{\partial u_{z}}{\partial z}=0$

the $z$-component of the Navier-Stokes equation as

$\rho\left(\frac{\partial u_{z}}{\partial t}+u_{r} \frac{\partial u_{z}}{\partial r}+u_{z} \frac{\partial u_{z}}{\partial z}\right)=-\frac{\partial p}{\partial z}+\mu\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u_{z}}{\partial r}\right)+\frac{\partial^{2} u_{z}}{\partial z^{2}}\right]$

and the relationship between the Stokes streamfunction, $\psi(r, z)$, and the velocity components as

$\left.u_{r}=-\frac{1}{r} \frac{\partial \psi}{\partial z}, \quad u_{z}=\frac{1}{r} \frac{\partial \psi}{\partial r}\right]$

Paper 4, Section II, E

commentA stationary inviscid fluid of thickness $h$ and density $\rho$ is located below a free surface at $y=h$ and above a deep layer of inviscid fluid of the same density in $y<0$ flowing with uniform velocity $U>0$ in the $\mathbf{e}_{x}$ direction. The base velocity profile is thus

$u=U, y<0 ; \quad u=0,0<y<h$

while the free surface at $y=h$ is maintained flat by gravity.

By considering small perturbations of the vortex sheet at $y=0$ of the form $\eta=\eta_{0} e^{i k x+\sigma t}, k>0$, calculate the dispersion relationship between $k$ and $\sigma$ in the irrotational limit. By explicitly deriving that

$\operatorname{Re}(\sigma)=\pm \frac{\sqrt{\tanh (h k)}}{1+\tanh (h k)} U k$

deduce that the vortex sheet is unstable at all wavelengths. Show that the growth rates of the unstable modes are approximately $U k / 2$ when $h k \gg 1$ and $U k \sqrt{h k}$ when $h k \ll 1$.

Paper 1, Section II, B

commentA particle of arbitrary shape and volume $4 \pi a^{3} / 3$ moves at velocity $\mathbf{U}(t)$ through an unbounded incompressible fluid of density $\rho$ and viscosity $\mu$. The Reynolds number of the flow is very small so that the inertia of the fluid can be neglected. Show that the particle experiences a force $\mathbf{F}(t)$ due to the surface stresses given by

$F_{i}(t)=-\mu a A_{i j} U_{j}(t)$

where $A_{i j}$ is a dimensionless second-rank tensor determined solely by the shape and orientation of the particle. State the reason why $A_{i j}$ must be positive definite.

Show further that, if the particle has the same reflectional symmetries as a cube, then

$A_{i j}=\lambda \delta_{i j}$

Let $b$ be the radius of the smallest sphere that contains the particle (still assuming cubic symmetry). By considering the Stokes flow associated with this sphere, suitably extended, and using the minimum dissipation theorem (which should be stated carefully), show that

$\lambda \leqslant 6 \pi b / a .$

[You may assume the expression for the Stokes drag on a sphere.]

Paper 2, Section II, B

commentAir is blown over the surface of a large, deep reservoir of water in such a way as to exert a tangential stress in the $x$-direction of magnitude $K x^{2}$ for $x>0$, with $K>0$. The water is otherwise at rest and occupies the region $y>0$. The surface $y=0$ remains flat.

Find order-of-magnitude estimates for the boundary-layer thickness $\delta(x)$ and tangential surface velocity $U(x)$ in terms of the relevant physical parameters.

Using the boundary-layer equations, find the ordinary differential equation governing the dimensionless function $f$ defined in the streamfunction

$\psi(x, y)=U(x) \delta(x) f(\eta), \quad \text { where } \eta=y / \delta(x)$

What are the boundary conditions on $f$ ?

Does $f \rightarrow 0$ as $\eta \rightarrow \infty$ ? Why, or why not?

The total horizontal momentum flux $P(X)$ across the vertical line $x=X$ is proportional to $X^{a}$ for $X>0$. Find the exponent $a$. By considering the steadiness of the momentum balance in the region $0<x<X$, explain why the value of $a$ is consistent with the form of the stress exerted on the boundary.

Paper 3, Section II, B

commentA rigid sphere of radius $a$ falls under gravity through an incompressible fluid of density $\rho$ and viscosity $\mu$ towards a rigid horizontal plane. The minimum gap $h_{0}(t)$ between the sphere and the plane satisfies $h_{0} \ll a$. Find an approximation for the gap thickness $h(r, t)$ between the sphere and the plane in the region $r \ll a$, where $r$ is the distance from the axis of symmetry.

For a prescribed value of $\dot{h}_{0}=d h_{0} / d t$, use lubrication theory to find the radial velocity and the fluid pressure in the region $r \ll a$. Explain why the approximations of lubrication theory require $h_{0} \ll a$ and $\rho h_{0} \dot{h}_{0} \ll \mu$.

Calculate the total vertical force due to the motion that is exerted by the fluid on the sphere. Deduce that if the sphere is settling under its own weight (corrected for buoyancy) then $h_{0}(t)$ decreases exponentially. What is the exponential decay rate for a solid sphere of density $\rho_{s}$ in a fluid of density $\rho_{f}$ ?

Paper 4, Section II, B

commentAn incompressible fluid of density $\rho$ and kinematic viscosity $\nu$ is confined in a channel with rigid stationary walls at $y=\pm h$. A spatially uniform pressure gradient $-G \cos \omega t$ is applied in the $x$-direction. What is the physical significance of the dimensionless number $S=\omega h^{2} / \nu ?$

Assuming that the flow is unidirectional and time-harmonic, obtain expressions for the velocity profile and the total flux. [You may leave your answers as the real parts of complex functions.]

In each of the limits $S \rightarrow 0$ and $S \rightarrow \infty$, find and sketch the flow profiles, find leading-order asymptotic expressions for the total flux, and give a physical interpretation.

Suppose now that $G=0$ and that the channel walls oscillate in their own plane with velocity $U \cos \omega t$ in the $x$-direction. Without explicit calculation of the solution, sketch the flow profile in each of the limits $S \rightarrow 0$ and $S \rightarrow \infty$.

Paper 1, Section II, A

commentThe velocity field $\mathbf{u}$ and stress tensor $\sigma$ satisfy the Stokes equations in a volume $V$ bounded by a surface $S$. Let $\hat{\mathbf{u}}$ be another solenoidal velocity field. Show that

$\int_{S} \sigma_{i j} n_{j} \hat{u}_{i} d S=\int_{V} 2 \mu e_{i j} \hat{e}_{i j} d V$

where $\mathbf{e}$ and $\hat{\mathbf{e}}$ are the strain-rates corresponding to the velocity fields $\mathbf{u}$ and $\hat{\mathbf{u}}$ respectively, and $\mathbf{n}$ is the unit normal vector out of $V$. Hence, or otherwise, prove the minimum dissipation theorem for Stokes flow.

A particle moves at velocity $\mathbf{U}$ through a highly viscous fluid of viscosity $\mu$ contained in a stationary vessel. As the particle moves, the fluid exerts a drag force $\mathbf{F}$ on it. Show that

$-\mathbf{F} \cdot \mathbf{U}=\int_{V} 2 \mu e_{i j} e_{i j} d V .$

Consider now the case when the particle is a small cube, with sides of length $\ell$, moving in a very large vessel. You may assume that

$\mathbf{F}=-k \mu \ell \mathbf{U}$

for some constant $k$. Use the minimum dissipation theorem, being careful to declare the domain(s) involved, to show that

$3 \pi \leqslant k \leqslant 3 \sqrt{3} \pi .$

[You may assume Stokes' result for the drag on a sphere of radius $a, \mathbf{F}=-6 \pi \mu a \mathbf{U}$.]

Paper 2, Section II, A

commentWrite down the boundary-layer equations for steady two-dimensional flow of a viscous incompressible fluid with velocity $U(x)$ outside the boundary layer. Find the boundary layer thickness $\delta(x)$ when $U(x)=U_{0}$, a constant. Show that the boundarylayer equations can be satisfied in this case by a streamfunction $\psi(x, y)=g(x) f(\eta)$ with suitable scaling function $g(x)$ and similarity variable $\eta$. Find the equation satisfied by $f$ and the associated boundary conditions.

Find the drag on a thin two-dimensional flat plate of finite length $L$ placed parallel to a uniform flow. Why does the drag not increase in proportion to the length of the plate? [You may assume that the boundary-layer solution is applicable except in negligibly small regions near the leading and trailing edges. You may also assume that $f^{\prime \prime}(0)=0.33$.]

Paper 3, Section II, A

commentA disk hovers on a cushion of air above an air-table - a fine porous plate through which a constant flux of air is pumped. Let the disk have a radius $R$ and a weight $M g$ and hover at a low height $h \ll R$ above the air-table. Let the volume flux of air, which has density $\rho$ and viscosity $\mu$, be $w$ per unit surface area. The conditions are such that $\rho w h^{2} / \mu R \ll 1$. Explain the significance of this restriction.

Find the pressure distribution in the air under the disk. Show that this pressure balances the weight of the disk if

$h=R\left(\frac{3 \pi \mu R w}{2 M g}\right)^{1 / 3}$

Paper 4, Section II, A

commentConsider the flow of an incompressible fluid of uniform density $\rho$ and dynamic viscosity $\mu$. Show that the rate of viscous dissipation per unit volume is given by

$\Phi=2 \mu e_{i j} e_{i j}$

where $e_{i j}$ is the strain rate.

Determine expressions for $e_{i j}$ and $\Phi$ when the flow is irrotational with velocity potential $\phi$.

In deep water a linearised wave with a surface displacement $\eta=a \cos (k x-\omega t)$ has a velocity potential $\phi=-(\omega a / k) e^{-k z} \sin (k x-\omega t)$. Hence determine the rate of the viscous dissipation, averaged over a wave period $2 \pi / \omega$, for an irrotational surface wave of wavenumber $k$ and small amplitude $a \ll 1 / k$ in a fluid with very small viscosity $\mu \ll \rho \omega / k^{2}$ and great depth $H \gg 1 / k$.

Calculate the depth-integrated kinetic energy per unit wavelength. Assuming that the average potential energy is equal to the average kinetic energy, show that the total wave energy decreases to leading order as $e^{-\gamma t}$, where $\gamma$ should be found.

Paper 1, Section II, C

commentDefine the strain-rate tensor $e_{i j}$ in terms of the velocity components $u_{i}$. Write down the relation between $e_{i j}$, the pressure $p$ and the stress $\sigma_{i j}$ in an incompressible Newtonian fluid of viscosity $\mu$. Show that the local rate of stress-working $\sigma_{i j} \partial u_{i} / \partial x_{j}$ is equal to the local rate of dissipation $2 \mu e_{i j} e_{i j}$.

An incompressible fluid of density $\rho$ and viscosity $\mu$ occupies the semi-infinite region $y>0$ above a rigid plane boundary $y=0$ which oscillates with velocity $(V \cos \omega t, 0,0)$. The fluid is at rest at infinity. Determine the velocity field produced by the boundary motion after any transients have decayed.

Show that the time-averaged rate of dissipation is

$\frac{1}{4} \sqrt{2} V^{2}(\mu \rho \omega)^{1 / 2}$

per unit area of the boundary. Verify that this is equal to the time average of the rate of working by the boundary on the fluid per unit area.

Paper 2, Section II, C

commentAn incompressible viscous liquid occupies the long thin region $0 \leqslant y \leqslant h(x)$ for $0 \leqslant x \leqslant \ell$, where $h(x)=d_{1}+\alpha x$ with $h(0)=d_{1}, h(\ell)=d_{2}<d_{1}$ and $d_{1} \ll \ell$. The top boundary at $y=h(x)$ is rigid and stationary. The bottom boundary at $y=0$ is rigid and moving at velocity $(U, 0,0)$. Fluid can move in and out of the ends $x=0$ and $x=\ell$, where the pressure is the same, namely $p_{0}$.

Explaining the approximations of lubrication theory as you use them, find the velocity profile in the long thin region, and show that the volume flux $Q$ (per unit width in the $z$-direction) is

$Q=\frac{U d_{1} d_{2}}{d_{1}+d_{2}}$

Find also the value of $h(x)$ (i) where the pressure is maximum, (ii) where the tangential viscous stress on the bottom $y=0$ vanishes, and (iii) where the tangential viscous stress on the top $y=h(x)$ vanishes.

Paper 3, Section II, C

commentFor two Stokes flows $\mathbf{u}^{(1)}(\mathbf{x})$ and $\mathbf{u}^{(2)}(\mathbf{x})$ inside the same volume $V$ with different boundary conditions on its boundary $S$, prove the reciprocal theorem

$\int_{S} \sigma_{i j}^{(1)} n_{j} u_{i}^{(2)} d S=\int_{S} \sigma_{i j}^{(2)} n_{j} u_{i}^{(1)} d S$

where $\sigma^{(1)}$ and $\sigma^{(2)}$ are the stress fields associated with the flows.

When a rigid sphere of radius $a$ translates with velocity $\mathbf{U}$ through unbounded fluid at rest at infinity, it may be shown that the traction per unit area, $\boldsymbol{\sigma} \cdot \mathbf{n}$, exerted by the sphere on the fluid has the uniform value $3 \mu \mathbf{U} / 2 a$ over the sphere surface. Find the drag on the sphere.

Suppose that the same sphere is now free of external forces and is placed with its centre at the origin in an unbounded Stokes flow given in the absence of the sphere as $\mathbf{u}^{*}(\mathbf{x})$. By applying the reciprocal theorem to the perturbation to the flow generated by the presence of the sphere, and assuming this tends to zero sufficiently rapidly at infinity, show that the instantaneous velocity of the centre of the sphere is

$\frac{1}{4 \pi a^{2}} \int \mathbf{u}^{*}(\mathbf{x}) d S$

where the integral is taken over the sphere of radius $a$.

Paper 4, Section II, C

commentA steady, two-dimensional flow in the region $y>0$ takes the form $(u, v)=$ $(E x,-E y)$ at large $y$, where $E$ is a positive constant. The boundary at $y=0$ is rigid and no-slip. Consider the velocity field $u=\partial \psi / \partial y, v=-\partial \psi / \partial x$ with stream function $\psi=\operatorname{Ex} \delta f(\eta)$, where $\eta=y / \delta$ and $\delta=(\nu / E)^{1 / 2}$ and $\nu$ is the kinematic viscosity. Show that this velocity field satisfies the Navier-Stokes equations provided that $f(\eta)$ satisfies

$f^{\prime \prime \prime}+f f^{\prime \prime}-\left(f^{\prime}\right)^{2}=-1$

What are the conditions on $f$ at $\eta=0$ and as $\eta \rightarrow \infty$ ?

Paper 1, Section II, B

The steady two-dimensional boundary-layer equations for flow primarily in the $x$ direction are

$\begin{gathered} \rho\left(u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}\right)=-\frac{d P}{d x}+\mu \frac{\partial^{2} u}{\partial y^{2}} \\ \frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \end{gathered}$