# Fluid Dynamics

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Paper 1, Section II, A

commentA two-dimensional flow is given by a velocity potential

$\phi(x, y, t)=\epsilon y \sin (x-t)$

where $\epsilon$ is a constant.

(a) Find the corresponding velocity field $\mathbf{u}(x, y, t)$. Determine $\boldsymbol{\nabla} \cdot \mathbf{u}$.

(b) The time-average $\langle\psi\rangle(x, y)$ of a quantity $\psi(x, y, t)$ is defined as

$\langle\psi\rangle(x, y)=\frac{1}{2 \pi} \int_{0}^{2 \pi} \psi(x, y, t) d t .$

Show that the time-average of this velocity field is zero everywhere. Write down an expression for the acceleration of fluid particles, and find the time-average of this expression at a fixed point $(x, y)$.

(c) Now assume that $|\epsilon| \ll 1$. The material particle at $(0,0)$ at $t=0$ is marked with dye. Write down equations for its subsequent motion. Verify that its position $(x, y)$ for $t>0$ is given (correct to terms of order $\epsilon^{2}$ ) by

$\begin{aligned} x &=\epsilon^{2}\left(\frac{1}{4} \sin 2 t+\frac{t}{2}-\sin t\right) \\ y &=\epsilon(\cos t-1) \end{aligned}$

Deduce the time-average velocity of the dyed particle correct to this order.

Paper 2, Section I, A

commentConsider an axisymmetric container, initially filled with water to a depth $h_{I}$. A small circular hole of radius $r_{0}$ is opened in the base of the container at $z=0$.

(a) Determine how the radius $r$ of the container should vary with $z<h_{I}$ so that the depth of the water will decrease at a constant rate.

(b) For such a container, determine how the cross-sectional area $A$ of the free surface should decrease with time.

[You may assume that the flow rate through the opening is sufficiently small that Bernoulli's theorem for steady flows can be applied.]

Paper 3, Section I, A

commentA two-dimensional flow $\mathbf{u}=(u, v)$ has a velocity field given by

$u=\frac{x^{2}-y^{2}}{\left(x^{2}+y^{2}\right)^{2}} \quad \text { and } \quad v=\frac{2 x y}{\left(x^{2}+y^{2}\right)^{2}}$

(a) Show explicitly that this flow is incompressible and irrotational away from the origin.

(b) Find the stream function for this flow.

(c) Find the velocity potential for this flow.

Paper 3, Section II, A

commentA two-dimensional layer of viscous fluid lies between two rigid boundaries at $y=\pm L_{0}$. The boundary at $y=L_{0}$ oscillates in its own plane with velocity $\left(U_{0} \cos \omega t, 0\right)$, while the boundary at $y=-L_{0}$ oscillates in its own plane with velocity $\left(-U_{0} \cos \omega t, 0\right)$. Assume that there is no pressure gradient and that the fluid flows parallel to the boundary with velocity $(u(y, t), 0)$, where $u(y, t)$ can be written as $u(y, t)=\operatorname{Re}\left[U_{0} f(y) \exp (i \omega t)\right]$.

(a) By exploiting the symmetry of the system or otherwise, show that

$f(y)=\frac{\sinh [(1+i) \Delta \hat{y}]}{\sinh [(1+i) \Delta]}, \text { where } \hat{y}=\frac{y}{L_{0}} \text { and } \Delta=\left(\frac{\omega L_{0}^{2}}{2 \nu}\right)^{1 / 2}$

(b) Hence or otherwise, show that

where $\Delta_{\pm}=\Delta(1 \pm \hat{y})$.

(c) Show that, for $\Delta \ll 1$,

$u(y, t) \simeq \frac{U_{0} y}{L_{0}} \cos \omega t$

and briefly interpret this result physically.

$\begin{aligned} & \frac{u(y, t)}{U_{0}}=\frac{\cos \omega t\left[\cosh \Delta_{+} \cos \Delta_{-}-\cosh \Delta_{-} \cos \Delta_{+}\right]}{(\cosh 2 \Delta-\cos 2 \Delta)} \\ & +\frac{\sin \omega t\left[\sinh \Delta_{+} \sin \Delta_{-}-\sinh \Delta_{-} \sin \Delta_{+}\right]}{(\cosh 2 \Delta-\cos 2 \Delta)}, \end{aligned}$

Paper 4, Section II, A

commentConsider the spherically symmetric motion induced by the collapse of a spherical cavity of radius $a(t)$, centred on the origin. For $r<a$, there is a vacuum, while for $r>a$, there is an inviscid incompressible fluid with constant density $\rho$. At time $t=0, a=a_{0}$, and the fluid is at rest and at constant pressure $p_{0}$.

(a) Consider the radial volume transport in the fluid $Q(R, t)$, defined as

$Q(R, t)=\int_{r=R} u d S$

where $u$ is the radial velocity, and $d S$ is an infinitesimal element of the surface of a sphere of radius $R \geqslant a$. Use the incompressibility condition to establish that $Q$ is a function of time alone.

(b) Using the expression for pressure in potential flow or otherwise, establish that

$\frac{1}{4 \pi a} \frac{d Q}{d t}-\frac{(\dot{a})^{2}}{2}=-\frac{p_{0}}{\rho}$

where $\dot{a}(t)$ is the radial velocity of the cavity boundary.

(c) By expressing $Q(t)$ in terms of $a$ and $\dot{a}$, show that

$\dot{a}=-\sqrt{\frac{2 p_{0}}{3 \rho}\left(\frac{a_{0}^{3}}{a^{3}}-1\right)}$

[Hint: You may find it useful to assume $\dot{a}(t)$ is an explicit function of a from the outset.]

(d) Hence write down an integral expression for the implosion time $\tau$, i.e. the time for the radius of the cavity $a \rightarrow 0$. [Do not attempt to evaluate the integral.]

Paper 1, Section II, C

commentSteady two-dimensional potential flow of an incompressible fluid is confined to the wedge $0<\theta<\alpha$, where $(r, \theta)$ are polar coordinates centred on the vertex of the wedge and $0<\alpha<\pi$.

(a) Show that a velocity potential $\phi$ of the form

$\phi(r, \theta)=A r^{\gamma} \cos (\lambda \theta),$

where $A, \gamma$ and $\lambda$ are positive constants, satisfies the condition of incompressible flow, provided that $\gamma$ and $\lambda$ satisfy a certain relation to be determined.

Assuming that $u_{\theta}$, the $\theta$-component of velocity, does not change sign within the wedge, determine the values of $\gamma$ and $\lambda$ by using the boundary conditions.

(b) Calculate the shape of the streamlines of this flow, labelling them by the distance $r_{\min }$ of closest approach to the vertex. Sketch the streamlines.

(c) Show that the speed $|\mathbf{u}|$ and pressure $p$ are independent of $\theta$. Assuming that at some radius $r=r_{0}$ the speed and pressure are $u_{0}$ and $p_{0}$, respectively, find the pressure difference in the flow between the vertex of the wedge and $r_{0}$.

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

$\nabla f=\left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}\right) \quad \text { and } \quad \nabla \cdot \mathbf{F}=\frac{1}{r} \frac{\partial}{\partial r}\left(r F_{r}\right)+\frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}$

for a scalar $f$ and a vector $\mathbf{F}=\left(F_{r}, F_{\theta}\right)$.]

Paper 2, Section I, C

commentIncompressible fluid of constant viscosity $\mu$ is confined to the region $0<y<h$ between two parallel rigid plates. Consider two parallel viscous flows: flow A is driven by the motion of one plate in the $x$-direction with the other plate at rest; flow B is driven by a constant pressure gradient in the $x$-direction with both plates at rest. The velocity mid-way between the plates is the same for both flows.

The viscous friction in these flows is known to produce heat locally at a rate

$Q=\mu\left(\frac{\partial u}{\partial y}\right)^{2}$

per unit volume, where $u$ is the $x$-component of the velocity. Determine the ratio of the total rate of heat production in flow A to that in flow B.

Paper 2, Section II, C

commentA vertical cylindrical container of radius $R$ is partly filled with fluid of constant density to depth $h$. The free surface is perturbed so that the fluid occupies the region

$0<r<R, \quad-h<z<\zeta(r, \theta, t)$

where $(r, \theta, z)$ are cylindrical coordinates and $\zeta$ is the perturbed height of the free surface. For small perturbations, a linearised description of surface waves in the cylinder yields the following system of equations for $\zeta$ and the velocity potential $\phi$ :

$\begin{aligned} \nabla^{2} \phi &=0, \quad 0<r<R, \quad-h<z<0 \\ \frac{\partial \phi}{\partial t}+g \zeta &=0 \quad \text { on } \quad z=0 \\ \frac{\partial \zeta}{\partial t}-\frac{\partial \phi}{\partial z} &=0 \quad \text { on } \quad z=0 \\ \frac{\partial \phi}{\partial z} &=0 \quad \text { on } \quad z=-h \\ \frac{\partial \phi}{\partial r} &=0 \quad \text { on } \quad r=R \end{aligned}$

(a) Describe briefly the physical meaning of each equation.

(b) Consider axisymmetric normal modes of the form

$\phi=\operatorname{Re}\left(\hat{\phi}(r, z) e^{-i \sigma t}\right), \quad \zeta=\operatorname{Re}\left(\hat{\zeta}(r) e^{-i \sigma t}\right)$

Show that the system of equations $(1)-(5)$ admits a solution for $\hat{\phi}$ of the form

$\hat{\phi}(r, z)=A J_{0}\left(k_{n} r\right) Z(z)$

where $A$ is an arbitrary amplitude, $J_{0}(x)$ satisfies the equation

$\frac{d^{2} J_{0}}{d x^{2}}+\frac{1}{x} \frac{d J_{0}}{d x}+J_{0}=0$

the wavenumber $k_{n}, n=1,2, \ldots$ is such that $x_{n}=k_{n} R$ is one of the zeros of the function $d J_{0} / d x$, and the function $Z(z)$ should be determined explicitly.

(c) Show that the frequency $\sigma_{n}$ of the $n$-th mode is given by

$\sigma_{n}^{2}=\frac{g}{h} \Psi\left(k_{n} h\right)$

where the function $\Psi(x)$ is to be determined.

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

$\left.\nabla^{2}=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2}}{\partial \theta^{2}}+\frac{\partial^{2}}{\partial z^{2}} \cdot\right]$

Paper 1, Section I, C

commentA viscous fluid flows steadily down a plane that is inclined at an angle $\alpha$ to the horizontal. The fluid layer is of uniform thickness and has a free upper surface. Determine the velocity profile in the direction perpendicular to the plane and also the volume flux (per unit width), in terms of the gravitational acceleration $g$, the angle $\alpha$, the kinematic viscosity $\nu$ and the thickness $h$ of the fluid layer.

Show that the volume flux is reduced if the free upper surface is replaced by a stationary plane boundary, and give a physical explanation for this.

Paper 1, Section II, C

commentExplain why the irrotational flow of an incompressible fluid can be expressed in terms of a velocity potential $\phi$ that satisfies Laplace's equation.

The axis of a stationary cylinder of radius $a$ coincides with the $z$-axis of a Cartesian coordinate system $(x, y, z)$ with unit vectors $\left(\mathbf{e}_{x}, \mathbf{e}_{y}, \mathbf{e}_{z}\right)$. A fluid of density $\rho$ flows steadily past the cylinder such that the velocity field $\mathbf{u}$ is independent of $z$ and has no component in the $z$-direction. The flow is irrotational but there is a constant non-zero circulation

$\oint \mathbf{u} \cdot d \mathbf{r}=\kappa$

around every closed curve that encloses the cylinder once in a positive sense. Far from the cylinder, the velocity field tends towards the uniform flow $\mathbf{u}=U \mathbf{e}_{x}$, where $U$ is a constant.

State the boundary conditions on the velocity potential, in terms of polar coordinates $(r, \theta)$ in the $(x, y)$-plane. Explain why the velocity potential is not required to be a single-valued function of position. Hence obtain the appropriate solution $\phi(r, \theta)$, in terms of $a, U$ and $\kappa$.

Neglecting gravity, show that the net force on the cylinder, per unit length in the $z$-direction, is

$-\rho \kappa U \mathbf{e}_{y}$

Determine the number and location of stagnation points in the flow as a function of the dimensionless parameter

$\lambda=\frac{\kappa}{4 \pi U a}$

Paper 2, Section I, C

commentConsider the steady flow

$u_{x}=\sin x \cos y, \quad u_{y}=-\cos x \sin y, \quad u_{z}=0$

where $(x, y, z)$ are Cartesian coordinates. Show that $\boldsymbol{\nabla} \cdot \mathbf{u}=0$ and determine the streamfunction. Calculate the vorticity and verify that the vorticity equation is satisfied in the absence of viscosity. Sketch the streamlines in the region $0<x<2 \pi, 0<y<2 \pi$.

Paper 3, Section II, C

commentA cubic box of side $2 h$, enclosing the region $0<x<2 h, 0<y<2 h,-h<z<h$, contains equal volumes of two incompressible fluids that remain distinct. The system is initially at rest, with the fluid of density $\rho_{1}$ occupying the region $0<z<h$ and the fluid of density $\rho_{2}$ occupying the region $-h<z<0$, and with gravity $(0,0,-g)$. The interface between the fluids is then slightly perturbed. Derive the linearized equations and boundary conditions governing small disturbances to the initial state.

In the case $\rho_{2}>\rho_{1}$, show that the angular frequencies $\omega$ of the normal modes are given by

$\omega^{2}=\left(\frac{\rho_{2}-\rho_{1}}{\rho_{1}+\rho_{2}}\right) g k \tanh (k h)$

and express the allowable values of the wavenumber $k$ in terms of $h$. Identify the lowestfrequency non-trivial mode $(\mathrm{s})$. Comment on the limit $\rho_{1} \ll \rho_{2}$. What physical behaviour is expected in the case $\rho_{1}>\rho_{2}$ ?

Paper 4, Section II, C

commentThe linear shallow-water equations governing the motion of a fluid layer in the neighbourhood of a point on the Earth's surface in the northern hemisphere are

$\begin{aligned} \frac{\partial u}{\partial t}-f v &=-g \frac{\partial \eta}{\partial x} \\ \frac{\partial v}{\partial t}+f u &=-g \frac{\partial \eta}{\partial y} \\ \frac{\partial \eta}{\partial t} &=-h\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right) \end{aligned}$

where $u(x, y, t)$ and $v(x, y, t)$ are the horizontal velocity components and $\eta(x, y, t)$ is the perturbation of the height of the free surface.

(a) Explain the meaning of the three positive constants $f, g$ and $h$ appearing in the equations above and outline the assumptions made in deriving these equations.

(b) Show that $\zeta$, the $z$-component of vorticity, satisfies

$\frac{\partial \zeta}{\partial t}=-f\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)$

and deduce that the potential vorticity

$q=\zeta-\frac{f}{h} \eta$

satisfies

$\frac{\partial q}{\partial t}=0$

(c) Consider a steady geostrophic flow that is uniform in the latitudinal $(y)$ direction. Show that

$\frac{d^{2} \eta}{d x^{2}}-\frac{f^{2}}{g h} \eta=\frac{f}{g} q .$

Given that the potential vorticity has the piecewise constant profile

$q= \begin{cases}q_{1}, & x<0 \\ q_{2}, & x>0\end{cases}$

where $q_{1}$ and $q_{2}$ are constants, and that $v \rightarrow 0$ as $x \rightarrow \pm \infty$, solve for $\eta(x)$ and $v(x)$ in terms of the Rossby radius $R=\sqrt{g h} / f$. Sketch the functions $\eta(x)$ and $v(x)$ in the case $q_{1}>q_{2}$.

Paper 1, Section I, D

commentShow that the flow with velocity potential

$\phi=\frac{q}{2 \pi} \ln r$

in two-dimensional, plane-polar coordinates $(r, \theta)$ is incompressible in $r>0$. Determine the flux of fluid across a closed contour $C$ that encloses the origin. What does this flow represent?

Show that the flow with velocity potential

$\phi=\frac{q}{4 \pi} \ln \left(x^{2}+(y-a)^{2}\right)+\frac{q}{4 \pi} \ln \left(x^{2}+(y+a)^{2}\right)$

has no normal flow across the line $y=0$. What fluid flow does this represent in the unbounded plane? What flow does it represent for fluid occupying the domain $y>0$ ?

Paper 1, Section II, D

commentA layer of fluid of dynamic viscosity $\mu$, density $\rho$ and uniform thickness $h$ flows down a rigid vertical plane. The adjacent air has uniform pressure $p_{0}$ and exerts a tangential stress on the fluid that is proportional to the surface velocity and opposes the flow, with constant of proportionality $k$. The acceleration due to gravity is $g$.

(a) Draw a diagram of this situation, including indications of the applied stresses and body forces, a suitable coordinate system and a representation of the expected velocity profile.

(b) Write down the equations and boundary conditions governing the flow, with a brief description of each, paying careful attention to signs. Solve these equations to determine the pressure and velocity fields in terms of the parameters given above.

(c) Show that the surface velocity of the fluid layer is $\frac{\rho g h^{2}}{2 \mu}\left(1+\frac{k h}{\mu}\right)^{-1}$.

(d) Determine the volume flux per unit width of the plane for general values of $k$ and its limiting values when $k \rightarrow 0$ and $k \rightarrow \infty$.

Paper 2, Section I, D

commentThe Euler equations for steady fluid flow $\mathbf{u}$ in a rapidly rotating system can be written

$\rho \mathbf{f} \times \mathbf{u}=-\nabla p+\rho \mathbf{g},$

where $\rho$ is the density of the fluid, $p$ is its pressure, $\mathbf{g}$ is the acceleration due to gravity and $\mathbf{f}=(0,0, f)$ is the constant Coriolis parameter in a Cartesian frame of reference $(x, y, z)$, with $z$ pointing vertically upwards.

Fluid occupies a layer of slowly-varying height $h(x, y)$. Given that the pressure $p=p_{0}$ is constant at $z=h$ and that the flow is approximately horizontal with components $\mathbf{u}=(u, v, 0)$, show that the contours of $h$ are streamlines of the horizontal flow. What is the leading-order horizontal volume flux of fluid between two locations at which $h=h_{0}$ and $h=h_{0}+\Delta h$, where $\Delta h \ll h_{0}$ ?

Identify the dimensions of all the quantities involved in your expression for the volume flux and show that your expression is dimensionally consistent.

Paper 3, Section II, D

commentA soap bubble of radius $a(t)$ is attached to the end of a long, narrow straw of internal radius $\epsilon$ and length $L$, the other end of which is open to the atmosphere. The pressure difference between the inside and outside of the bubble is $2 \gamma / a$, where $\gamma$ is the surface tension of the soap bubble. At time $t=0, a=a_{0}$ and the air in the straw is at rest. Assume that the flow of air through the straw is irrotational and consider the pressure drop along the straw to show that subsequently

$a^{3} \ddot{a}+2 a^{2} \dot{a}^{2}=-\frac{\gamma \epsilon^{2}}{2 \rho L},$

where $\rho$ is the density of air.

By multiplying the equation by $2 a \dot{a}$ and integrating, or otherwise, determine an implicit equation for $a(t)$ and show that the bubble disappears in a time

$t=\frac{\pi}{2} \frac{a_{0}^{2}}{\epsilon}\left(\frac{\rho L}{2 \gamma}\right)^{1 / 2}$

[Hint: The substitution $a=a_{0} \sin \theta$ can be used.]

Paper 4, Section II, D

commentA deep layer of inviscid fluid is initially confined to the region $0<x<a, 0<y<a$, $z<0$ in Cartesian coordinates, with $z$ directed vertically upwards. An irrotational disturbance is caused to the fluid so that its upper surface takes position $z=\eta(x, y, t)$. Determine the linear normal modes of the system and the dispersion relation between the frequencies of the normal modes and their wavenumbers.

If the interface is initially displaced to position $z=\epsilon \cos \frac{3 \pi x}{a} \cos \frac{4 \pi y}{a}$ and released from rest, where $\epsilon$ is a small constant, determine its position for subsequent times. How far below the surface will the velocity have decayed to $1 / e$ times its surface value?

Paper 1, Section I, D

commentFor each of the flows

(i) $\mathbf{u}=\left(2 x y, x^{2}+y^{2}\right)$

(ii) $\mathbf{u}=(-2 y,-2 x)$

determine whether or not the flow is incompressible and/or irrotational. Find the associated velocity potential and/or stream function when appropriate. For either one of the flows, sketch the streamlines of the flow, indicating the direction of the flow.

Paper 1, Section II, D

commentA layer of thickness $h$ of fluid of density $\rho$ and dynamic viscosity $\mu$ flows steadily down and parallel to a rigid plane inclined at angle $\alpha$ to the horizontal. Wind blows over the surface of the fluid and exerts a stress $S$ on the surface of the fluid in the upslope direction.

(a) Draw a diagram of this situation, including indications of the applied stresses and body forces, a suitable coordinate system and a representation of the expected velocity profile.

(b) Write down the equations and boundary conditions governing the flow, with a brief description of each, paying careful attention to signs. Solve these equations to determine the pressure and velocity fields.

(c) Determine the volume flux and show that there is no net flux if

$S=\frac{2}{3} \rho g h \sin \alpha$

Draw a sketch of the corresponding velocity profile.

(d) Determine the value of $S$ for which the shear stress on the rigid plane is zero and draw a sketch of the corresponding velocity profile.

Paper 2, Section I, $7 \mathrm{D}$

commentFrom Euler's equations describing steady inviscid fluid flow under the action of a conservative force, derive Bernoulli's equation for the pressure along a streamline of the flow, defining all variables that you introduce.

Water fills an inverted, open, circular cone (radius increasing upwards) of half angle $\pi / 4$ to a height $h_{0}$ above its apex. At time $t=0$, the tip of the cone is removed to leave a small hole of radius $\epsilon \ll h_{0}$. Assuming that the flow is approximately steady while the depth of water $h(t)$ is much larger than $\epsilon$, show that the time taken for the water to drain is approximately

$\left(\frac{2}{25} \frac{h_{0}^{5}}{\epsilon^{4} g}\right)^{1 / 2}$

Paper 3, Section II, D

commentUse Euler's equations to derive the vorticity equation

$\frac{D \boldsymbol{\omega}}{D t}=\boldsymbol{\omega} \cdot \nabla \mathbf{u}$

where $\mathbf{u}$ is the fluid velocity and $\boldsymbol{\omega}$ is the vorticity.

Consider axisymmetric, incompressible, inviscid flow between two rigid plates at $z=h(t)$ and $z=-h(t)$ in cylindrical polar coordinates $(r, \theta, z)$, where $t$ is time. Using mass conservation, or otherwise, find the complete flow field whose radial component is independent of $z$.

Now suppose that the flow has angular velocity $\boldsymbol{\Omega}=\Omega(t) \mathbf{e}_{z}$ and that $\Omega=\Omega_{0}$ when $h=h_{0}$. Use the vorticity equation to determine the angular velocity for subsequent times as a function of $h$. What physical principle does your result illustrate?

Paper 4, Section II, D

commentThe linearised equations governing the horizontal components of flow $\mathbf{u}(x, y, t)$ in a rapidly rotating shallow layer of depth $h=h_{0}+\eta(x, y, t)$, where $\eta \ll h_{0}$, are

$\begin{gathered} \frac{\partial \mathbf{u}}{\partial t}+\mathbf{f} \times \mathbf{u}=-g \nabla \eta \\ \frac{\partial \eta}{\partial t}+h_{0} \nabla \cdot \mathbf{u}=0 \end{gathered}$

where $\mathbf{f}=f \mathbf{e}_{z}$ is the constant Coriolis parameter, and $\mathbf{e}_{z}$ is the unit vector in the vertical direction.

Use these equations, either in vector form or using Cartesian components, to show that the potential vorticity

$\mathbf{Q}=\zeta-\frac{\eta}{h_{0}} \mathbf{f}$

is independent of time, where $\zeta=\nabla \times \mathbf{u}$ is the relative vorticity.

Derive the equation

$\frac{\partial^{2} \eta}{\partial t^{2}}-g h_{0} \nabla^{2} \eta+f^{2} \eta=-h_{0} \mathbf{f} \cdot \mathbf{Q}$

In the case that $\mathbf{Q} \equiv 0$, determine and sketch the dispersion relation $\omega(k)$ for plane waves with $\eta=A e^{i(k x+\omega t)}$, where $A$ is constant. Discuss the nature of the waves qualitatively: do long waves propagate faster or slower than short waves; how does the phase speed depend on wavelength; does rotation have more effect on long waves or short waves; how dispersive are the waves?

Paper 1, Section I, C

commentConsider the flow field in cartesian coordinates $(x, y, z)$ given by

$\mathbf{u}=\left(-\frac{A y}{x^{2}+y^{2}}, \frac{A x}{x^{2}+y^{2}}, U(z)\right)$

where $A$ is a constant. Let $\mathcal{D}$ denote the whole of $\mathbb{R}^{3}$ excluding the $z$ axis.

(a) Determine the conditions on $A$ and $U(z)$ for the flow to be both incompressible and irrotational in $\mathcal{D}$.

(b) Calculate the circulation along any closed curve enclosing the $z$ axis.

Paper 1, Section II, $17 \mathrm{C}$

comment(a) For a velocity field $\mathbf{u}$, show that $\mathbf{u} \cdot \boldsymbol{\nabla} \mathbf{u}=\boldsymbol{\nabla}\left(\frac{1}{2} \mathbf{u}^{2}\right)-\mathbf{u} \times \boldsymbol{\omega}$, where $\boldsymbol{\omega}$ is the flow vorticity.

(b) For a scalar field $H(\mathbf{r})$, show that if $\mathbf{u} \cdot \nabla H=0$, then $H$ is constant along the flow streamlines.

(c) State the Euler equations satisfied by an inviscid fluid of constant density subject to conservative body forces.

(i) If the flow is irrotational, show that an exact first integral of the Euler equations may be obtained.

(ii) If the flow is not irrotational, show that although an exact first integral of the Euler equations may not be obtained, a similar quantity is constant along the flow streamlines provided the flow is steady.

(iii) If the flow is now in a frame rotating with steady angular velocity $\Omega \mathbf{e}_{z}$, establish that a similar quantity is constant along the flow streamlines with an extra term due to the centrifugal force when the flow is steady.

Paper 2, Section I, C

commentA steady, two-dimensional unidirectional flow of a fluid with dynamic viscosity $\mu$ is set up between two plates at $y=0$ and $y=h$. The plate at $y=0$ is stationary while the plate at $y=h$ moves with constant speed $U \mathbf{e}_{x}$. The fluid is also subject to a constant pressure gradient $-G \mathbf{e}_{x}$. You may assume that the fluid velocity $\mathbf{u}$ has the form $\mathbf{u}=u(y) \mathbf{e}_{x}$.

(a) State the equation satisfied by $u(y)$ and its boundary conditions.

(b) Calculate $u(y)$.

(c) Show that the value of $U$ may be chosen to lead to zero viscous shear stress acting on the bottom plate and calculate the resulting flow rate.

Paper 3, Section II, C

commentA layer of thickness $h_{1}$ of a fluid of density $\rho_{1}$ is located above a layer of thickness $h_{2}$ of a fluid of density $\rho_{2}>\rho_{1}$. The two-fluid system is bounded by two impenetrable surfaces at $y=h_{1}$ and $y=-h_{2}$ and is assumed to be two-dimensional (i.e. independent of $z$ ). The fluid is subsequently perturbed, and the interface between the two fluids is denoted $y=\eta(x, t)$.

(a) Assuming irrotational motion in each fluid, state the equations and boundary conditions satisfied by the flow potentials, $\varphi_{1}$ and $\varphi_{2}$.

(b) The interface is perturbed by small-amplitude waves of the form $\eta=\eta_{0} e^{i(k x-\omega t)}$, with $\eta_{0} k \ll 1$. State the equations and boundary conditions satisfied by the linearised system.

(c) Calculate the dispersion relation of the waves relating the frequency $\omega$ to the wavenumber $k$.

Paper 4, Section II, C

comment(a) Show that for an incompressible fluid, $\nabla \times \boldsymbol{\omega}=-\nabla^{2} \mathbf{u}$, where $\boldsymbol{\omega}$ is the flow vorticity,

(b) State the equation of motion for an inviscid flow of constant density in a rotating frame subject to gravity. Show that, on Earth, the local vertical component of the centrifugal force is small compared to gravity. Present a scaling argument to justify the linearisation of the Euler equations for sufficiently large rotation rates, and hence deduce the linearised version of the Euler equations in a rapidly rotating frame.

(c) Denoting the rotation rate of the frame as $\boldsymbol{\Omega}=\Omega \mathbf{e}_{z}$, show that the linearised Euler equations may be manipulated to obtain an equation for the velocity field $\mathbf{u}$ in the form

$\frac{\partial^{2} \nabla^{2} \mathbf{u}}{\partial t^{2}}+4 \Omega^{2} \frac{\partial^{2} \mathbf{u}}{\partial z^{2}}=\mathbf{0}$

(d) Assume that there exist solutions of the form $\mathbf{u}=\mathbf{U}_{0} \exp [i(\mathbf{k} \cdot \mathbf{x}-\omega t)]$. Show that $\omega=\pm 2 \Omega \cos \theta$ where the angle $\theta$ is to be determined.

Paper 1, Section I, B

commentConsider a spherical bubble of radius $a$ in an inviscid fluid in the absence of gravity. The flow at infinity is at rest and the bubble undergoes translation with velocity $\mathbf{U}=U(t) \hat{\mathbf{x}}$. We assume that the flow is irrotational and derives from a potential given in spherical coordinates by

$\phi(r, \theta)=U(t) \frac{a^{3}}{2 r^{2}} \cos \theta,$

where $\theta$ is measured with respect to $\hat{\mathbf{x}}$. Compute the force, $\mathbf{F}$, acting on the bubble. Show that the formula for $\mathbf{F}$ can be interpreted as the acceleration force of a fraction $\alpha<1$ of the fluid displaced by the bubble, and determine the value of $\alpha$.

Paper 1, Section II, B

commentA fluid layer of depth $h_{1}$ and dynamic viscosity $\mu_{1}$ is located underneath a fluid layer of depth $h_{2}$ and dynamic viscosity $\mu_{2}$. The total fluid system of depth $h=h_{1}+h_{2}$ is positioned between a stationary rigid plate at $y=0$ and a rigid plate at $y=h$ moving with speed $\mathbf{U}=U \hat{\mathbf{x}}$, where $U$ is constant. Ignore the effects of gravity.

(i) Using dimensional analysis only, and the fact that the stress should be linear in $U$, derive the expected form of the shear stress acted by the fluid on the plate at $y=0$ as a function of $U, h_{1}, h_{2}, \mu_{1}$ and $\mu_{2}$.

(ii) Solve for the unidirectional velocity profile between the two plates. State clearly all boundary conditions you are using to solve this problem.

(iii) Compute the exact value of the shear stress acted by the fluid on the plate at $y=0$. Compare with the results in (i).

(iv) What is the condition on the viscosity of the bottom layer, $\mu_{1}$, for the stress in (iii) to be smaller than it would be if the fluid had constant viscosity $\mu_{2}$ in both layers?

(v) Show that the stress acting on the plate at $y=h$ is equal and opposite to the stress on the plate at $y=0$ and justify this result physically.

Paper 2, Section I, B

commentConsider the two-dimensional velocity field $\mathbf{u}=(u, v)$ with

$u(x, y)=x^{2}-y^{2}, \quad v(x, y)=-2 x y$

(i) Show that the flow is incompressible and irrotational.

(ii) Derive the velocity potential, $\phi$, and the streamfunction, $\psi$.

(iii) Plot all streamlines passing through the origin.

(iv) Show that the complex function $w=\phi+i \psi$ (where $i^{2}=-1$ ) can be written solely as a function of the complex coordinate $z=x+i y$ and determine that function.

Paper 3, Section II, B

commentA source of sound induces a travelling wave of pressure above the free surface of a fluid located in the $z<0$ domain as

$p=p_{a t m}+p_{0} \cos (k x-\omega t),$

with $p_{0} \ll p_{a t m}$. Here $k$ and $\omega$ are fixed real numbers. We assume that the flow induced in the fluid is irrotational.

(i) State the linearized equation of motion for the fluid and the free surface, $z=h(x, t)$, with all boundary conditions.

(ii) Solve for the velocity potential, $\phi(x, z, t)$, and the height of the free surface, $h(x, t)$. Verify that your solutions are dimensionally correct.

(iii) Interpret physically the behaviour of the solution when $\omega^{2}=g k$.

Paper 4, Section II, B

commentConsider a steady inviscid, incompressible fluid of constant density $\rho$ in the absence of external body forces. A cylindrical jet of area $A$ and speed $U$ impinges fully on a stationary sphere of radius $R$ with $A<\pi R^{2}$. The flow is assumed to remain axisymmetric and be deflected into a conical sheet of vertex angle $\alpha>0$.

(i) Show that the speed of the fluid in the conical sheet is constant.

(ii) Use conservation of mass to show that the width $d(r)$ of the fluid sheet at a distance $r \gg R$ from point of impact is given by

$d(r)=\frac{A}{2 \pi r \sin \alpha}$

(iii) Use Euler's equation to derive the momentum integral equation

$\iint_{S}\left(p n_{i}+\rho n_{j} u_{j} u_{i}\right) d S=0$

for a closed surface $S$ with normal $\mathbf{n}$ where $u_{m}$ is the $m$ th component of the velocity field in cartesian coordinates and $p$ is the pressure.

(iv) Use the result from (iii) to calculate the net force on the sphere.

Paper 1, Section I, B

commentConstant density viscous fluid with dynamic viscosity $\mu$ flows in a two-dimensional horizontal channel of depth $h$. There is a constant pressure gradient $G>0$ in the horizontal $x$-direction. The upper horizontal boundary at $y=h$ is driven at constant horizontal speed $U>0$, with the lower boundary being held at rest. Show that the steady fluid velocity $u$ in the $x$-direction is

$u=\frac{-G}{2 \mu} y(h-y)+\frac{U y}{h}$

Show that it is possible to have $d u / d y<0$ at some point in the flow for sufficiently large pressure gradient. Derive a relationship between $G$ and $U$ so that there is no net volume flux along the channel. For the flow with no net volume flux, sketch the velocity profile.

Paper 1, Section II, B

commentConsider the purely two-dimensional steady flow of an inviscid incompressible constant density fluid in the absence of body forces. For velocity $\mathbf{u}$, the vorticity is $\boldsymbol{\nabla} \times \mathbf{u}=\boldsymbol{\omega}=(0,0, \omega)$. Show that

$\mathbf{u} \times \boldsymbol{\omega}=\boldsymbol{\nabla}\left[\frac{p}{\rho}+\frac{1}{2}|\mathbf{u}|^{2}\right]$

where $p$ is the pressure and $\rho$ is the fluid density. Hence show that, if $\omega$ is a constant in both space and time,

$\frac{1}{2}|\mathbf{u}|^{2}+\omega \psi+\frac{p}{\rho}=C,$

where $C$ is a constant and $\psi$ is the streamfunction. Here, $\psi$ is defined by $\mathbf{u}=\boldsymbol{\nabla} \times \boldsymbol{\Psi}$, where $\boldsymbol{\Psi}=(0,0, \psi)$.

Fluid in the annular region $a<r<2 a$ has constant (in both space and time) vorticity $\omega$. The streamlines are concentric circles, with the fluid speed zero on $r=2 a$ and $V>0$ on $r=a$. Calculate the velocity field, and hence show that

$\omega=\frac{-2 V}{3 a}$

Deduce that the pressure difference between the outer and inner edges of the annular region is

$\Delta p=\left(\frac{15-16 \ln 2}{18}\right) \rho V^{2}$

[Hint: Note that in cylindrical polar coordinates $(r, \phi, z)$, the curl of a vector field $\mathbf{A}(r, \phi)=[a(r, \phi), b(r, \phi), c(r, \phi)]$ is

$\boldsymbol{\nabla} \times \mathbf{A}=\left[\frac{1}{r} \frac{\partial c}{\partial \phi},-\frac{\partial c}{\partial r}, \frac{1}{r}\left(\frac{\partial(r b)}{\partial r}-\frac{\partial a}{\partial \phi}\right)\right]$

Paper 2, Section I, B

commentConsider the steady two-dimensional fluid velocity field

$\mathbf{u}=\left(\begin{array}{l} u \\ v \end{array}\right)=\left(\begin{array}{ll} \epsilon & -\gamma \\ \gamma & -\epsilon \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)$

where $\epsilon \geqslant 0$ and $\gamma \geqslant 0$. Show that the fluid is incompressible. The streamfunction $\psi$ is defined by $\mathbf{u}=\boldsymbol{\nabla} \times \boldsymbol{\Psi}$, where $\boldsymbol{\Psi}=(0,0, \psi)$. Show that $\psi$ is given by

$\psi=\epsilon x y-\frac{\gamma}{2}\left(x^{2}+y^{2}\right)$

Hence show that the streamlines are defined by

$(\epsilon-\gamma)(x+y)^{2}-(\epsilon+\gamma)(x-y)^{2}=C$

for $C$ a constant. For each of the three cases below, sketch the streamlines and briefly describe the flow. (i) $\epsilon=1, \gamma=0$, (ii) $\epsilon=0, \gamma=1$, (iii) $\epsilon=1, \gamma=1$.

Paper 3, Section II, B

commentA bubble of gas occupies the spherical region $r \leqslant R(t)$, and an incompressible irrotational liquid of constant density $\rho$ occupies the outer region $r \geqslant R$, such that as $r \rightarrow \infty$ the liquid is at rest with constant pressure $p_{\infty}$. Briefly explain why it is appropriate to use a velocity potential $\phi(r, t)$ to describe the liquid velocity u.

By applying continuity of velocity across the gas-liquid interface, show that the liquid pressure (for $r \geqslant R$ ) satisfies

$\frac{p}{\rho}+\frac{1}{2}\left(\frac{R^{2} \dot{R}}{r^{2}}\right)^{2}-\frac{1}{r} \frac{d}{d t}\left(R^{2} \dot{R}\right)=\frac{p_{\infty}}{\rho}, \quad \text { where } \dot{R}=\frac{d R}{d t} .$

Show that the excess pressure $p_{s}-p_{\infty}$ at the bubble surface $r=R$ is

$p_{s}-p_{\infty}=\frac{\rho}{2}\left(3 \dot{R}^{2}+2 R \ddot{R}\right), \quad \text { where } \ddot{R}=\frac{d^{2} R}{d t^{2}}$

and hence that

$p_{s}-p_{\infty}=\frac{\rho}{2 R^{2}} \frac{d}{d R}\left(R^{3} \dot{R}^{2}\right)$

The pressure $p_{g}(t)$ inside the gas bubble satisfies the equation of state

$p_{g} V^{4 / 3}=C$

where $C$ is a constant, and $V(t)$ is the bubble volume. At time $t=0$ the bubble is at rest with radius $R=a$. If the bubble then expands and comes to rest at $R=2 a$, determine the required gas pressure $p_{0}$ at $t=0$ in terms of $p_{\infty}$.

[You may assume that there is contact between liquid and gas for all time, that all motion is spherically symmetric about the origin $r=0$, and that there is no body force. You may also assume Bernoulli's integral of the equation of motion to determine the liquid pressure

$\frac{p}{\rho}+\frac{\partial \phi}{\partial t}+\frac{1}{2}|\nabla \phi|^{2}=A(t)$

where $\phi(r, t)$ is the velocity potential.]

Paper 4, Section II, B

commentConsider a layer of fluid of constant density $\rho$ and equilibrium depth $h_{0}$ in a rotating frame of reference, rotating at constant angular velocity $\Omega$ about the vertical $z$-axis. The equations of motion are

$\begin{aligned} \frac{\partial u}{\partial t}-f v &=-\frac{1}{\rho} \frac{\partial p}{\partial x} \\ \frac{\partial v}{\partial t}+f u &=-\frac{1}{\rho} \frac{\partial p}{\partial y} \\ 0 &=-\frac{\partial p}{\partial z}-\rho g \end{aligned}$

where $p$ is the fluid pressure, $u$ and $v$ are the fluid velocities in the $x$-direction and $y$ direction respectively, $f=2 \Omega$, and $g$ is the constant acceleration due to gravity. You may also assume that the horizontal extent of the layer is sufficiently large so that the layer may be considered to be shallow, such that vertical velocities may be neglected.

By considering mass conservation, show that the depth $h(x, y, t)$ of the layer satisfies

$\frac{\partial h}{\partial t}+\frac{\partial}{\partial x}(h u)+\frac{\partial}{\partial y}(h v)=0 .$

Now assume that $h=h_{0}+\eta(x, y, t)$, where $|\eta| \ll h_{0}$. Show that the (linearised) potential vorticity $\mathbf{Q}=Q \hat{\mathbf{z}}$, defined by

$Q=\zeta-\eta \frac{f}{h_{0}}, \text { where } \zeta=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$

and $\hat{\mathbf{z}}$ is the unit vector in the vertical $z$-direction, is a constant in time, i.e. $Q=Q_{0}(x, y)$.

When $Q_{0}=0$ everywhere, establish that the surface perturbation $\eta$ satisfies

$\frac{\partial^{2} \eta}{\partial t^{2}}-g h_{0}\left(\frac{\partial^{2} \eta}{\partial x^{2}}+\frac{\partial^{2} \eta}{\partial y^{2}}\right)+f^{2} \eta=0$

and show that this equation has wave-like solutions $\eta=\eta_{0} \cos [k(x-c t)]$ when $c$ and $k$ are related through a dispersion relation to be determined. Show that, to leading order, the trajectories of fluid particles for these waves are ellipses. Assuming that $\eta_{0}>0, k>0$, $c>0$ and $f>0$, sketch the fluid velocity when $k(x-c t)=n \pi / 2$ for $n=0,1,2,3$.

Paper 1, Section I, A

commentA two-dimensional flow is given by

$\mathbf{u}=(x,-y+t)$

Show that the flow is both irrotational and incompressible. Find a stream function $\psi(x, y)$ such that $\mathbf{u}=\left(\frac{\partial \psi}{\partial y},-\frac{\partial \psi}{\partial x}\right)$. Sketch the streamlines at $t=0$.

Find the pathline of a fluid particle that passes through $\left(x_{0}, y_{0}\right)$ at $t=0$ in the form $y=f\left(x, x_{0}, y_{0}\right)$ and sketch the pathline for $x_{0}=1, y_{0}=1 .$

Paper 1, Section II, A

commentStarting from the Euler momentum equation, derive the form of Bernoulli's equation appropriate for an unsteady irrotational motion of an inviscid incompressible fluid.

Water of density $\rho$ is driven through a horizontal tube of length $L$ and internal radius $a$ from a water-filled balloon attached to one end of the tube. Assume that the pressure exerted by the balloon is proportional to its current volume (in excess of atmospheric pressure). Also assume that water exits the tube at atmospheric pressure, and that gravity may be neglected. Show that the time for the balloon to empty does not depend on its initial volume. Find the maximum speed of water exiting the pipe.

Paper 2, Section I, A

commentAn incompressible, inviscid fluid occupies the region beneath the free surface $y=\eta(x, t)$ and moves with a velocity field determined by the velocity potential $\phi(x, y, t) .$ Gravity acts in the $-y$ direction. You may assume Bernoulli's integral of the equation of motion:

$\frac{p}{\rho}+\frac{\partial \phi}{\partial t}+\frac{1}{2}|\nabla \phi|^{2}+g y=F(t)$

Give the kinematic and dynamic boundary conditions that must be satisfied by $\phi$ on $y=\eta(x, t)$.

In the absence of waves, the fluid has constant uniform velocity $U$ in the $x$ direction. Derive the linearised form of the boundary conditions for small amplitude waves.

Assume that the free surface and velocity potential are of the form:

$\begin{aligned} \eta &=a e^{i(k x-\omega t)} \\ \phi &=U x+i b e^{k y} e^{i(k x-\omega t)} \end{aligned}$

(where implicitly the real parts are taken). Show that

$(\omega-k U)^{2}=g k$

Paper 3, Section II, A

commentA layer of incompressible fluid of density $\rho$ and viscosity $\mu$ flows steadily down a plane inclined at an angle $\theta$ to the horizontal. The layer is of uniform thickness $h$ measured perpendicular to the plane and the viscosity of the overlying air can be neglected. Using coordinates $x$ parallel to the plane (in steepest downwards direction) and $y$ normal to the plane, write down the equations of motion and the boundary conditions on the plane and on the free top surface. Determine the pressure and velocity fields and show that the volume flux down the plane is

$\frac{\rho g h^{3} \sin \theta}{3 \mu}$

Consider now the case where a second layer of fluid, of uniform thickness $\alpha h$, viscosity $\beta \mu$ and density $\rho$, flows steadily on top of the first layer. Explain why one of the appropriate boundary conditions between the two fluids is

$\mu \frac{\partial}{\partial y} u\left(h_{-}\right)=\beta \mu \frac{\partial}{\partial y} u\left(h_{+}\right),$

where $u$ is the component of velocity in the $x$ direction and $h_{-}$and $h_{+}$refer to just below and just above the boundary respectively. Determine the velocity field in each layer.

Paper 4, Section II, A

commentThe axisymmetric, irrotational flow generated by a solid sphere of radius $a$ translating at velocity $U$ in an inviscid, incompressible fluid is represented by a velocity potential $\phi(r, \theta)$. Assume the fluid is at rest far away from the sphere. Explain briefly why $\nabla^{2} \phi=0$.

By trying a solution of the form $\phi(r, \theta)=f(r) g(\theta)$, show that

$\phi=-\frac{U a^{3} \cos \theta}{2 r^{2}}$

and write down the fluid velocity.

Show that the total kinetic energy of the fluid is $k M U^{2} / 4$ where $M$ is the mass of the sphere and $k$ is the ratio of the density of the fluid to the density of the sphere.

A heavy sphere (i.e. $k<1$ ) is released from rest in an inviscid fluid. Determine its speed after it has fallen a distance $h$ in terms of $M, k, g$ and $h$.

Note, in spherical polars:

$\begin{gathered} \boldsymbol{\nabla} \phi=\frac{\partial \phi}{\partial r} \mathbf{e}_{\mathbf{r}}+\frac{1}{r} \frac{\partial \phi}{\partial \theta} \mathbf{e}_{\theta} \\ \nabla^{2} \phi=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \phi}{\partial \theta}\right) \end{gathered}$

Paper 1, Section I, A

commentViscous fluid, with viscosity $\mu$ and density $\rho$ flows along a straight circular pipe of radius $R$. The average velocity of the flow is $U$. Define a Reynolds number for the flow.

The flow is driven by a constant pressure gradient $-G>0$ along the pipe and the velocity is parallel to the axis of the pipe with magnitude $u(r)$ that satisfies

$\frac{\mu}{r} \frac{\mathrm{d}}{\mathrm{d} r}\left(r \frac{\mathrm{d} u}{\mathrm{~d} r}\right)=-G,$

where $r$ is the radial distance from the axis.

State the boundary conditions on $u$ and find the velocity as a function of $r$ assuming that it is finite on the axis $r=0$. Hence, show that the shear stress $\tau$ at the pipe wall is independent of the viscosity. Why is this the case?

Paper 1, Section II, A

commentConsider inviscid, incompressible fluid flow confined to the $(x, y)$ plane. The fluid has density $\rho$, and gravity can be neglected. Using the conservation of volume flux, determine the velocity potential $\phi(r)$ of a point source of strength $m$, in terms of the distance $r$ from the source.

Two point sources each of strength $m$ are located at $\boldsymbol{x}_{+}=(0, a)$ and $\boldsymbol{x}_{-}=(0,-a)$. Find the velocity potential of the flow.

Show that the flow in the region $y \geqslant 0$ is equivalent to the flow due to a source at $\boldsymbol{x}_{+}$and a fixed boundary at $y=0 .$

Find the pressure on the boundary $y=0$ and hence determine the force on the boundary.

[Hint: you may find the substitution $x=a \tan \theta$ useful for the calculation of the pressure.]

Paper 2, Section I, A

commentStarting from Euler's equation for the motion of an inviscid fluid, derive the vorticity equation in the form

$\frac{D \boldsymbol{\omega}}{D t}=\boldsymbol{\omega} \cdot \nabla \boldsymbol{u}$

Deduce that an initially irrotational flow remains irrotational.

Consider a plane flow that at time $t=0$ is described by the streamfunction

$\psi=x^{2}+y^{2} .$

Calculate the vorticity everywhere at times $t>0$.

Paper 3, Section II, A

commentA rigid circular cylinder of radius $a$ executes small amplitude oscillations with velocity $U(t)$ in a direction perpendicular to its axis, while immersed in an inviscid fluid of density $\rho$ contained within a larger concentric fixed cylinder of radius $b$. Gravity is negligible. Neglecting terms quadratic in the amplitude, determine the boundary condition on the velocity on the inner cylinder, and calculate the velocity potential of the induced flow.

With the same approximations show that the difference in pressures on the surfaces of the two cylinders has magnitude

$\rho \frac{\mathrm{d} U}{\mathrm{~d} t} \frac{a(b-a)}{b+a} \cos \theta$

where $\theta$ is the azimuthal angle measured from the direction of $U$.

Paper 4, Section II, A

commentThe equations governing the flow of a shallow layer of inviscid liquid of uniform depth $H$ rotating with angular velocity $\frac{1}{2} f$ about the vertical $z$-axis are

$\begin{aligned} \frac{\partial u}{\partial t}-f v &=-g \frac{\partial \eta}{\partial x} \\ \frac{\partial v}{\partial t}+f u &=-g \frac{\partial \eta}{\partial y} \\ \frac{\partial \eta}{t}+H\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right) &=0 \end{aligned}$

where $u, v$ are the $x$ - and $y$-components of velocity, respectively, and $\eta$ is the elevation of the free surface. Show that these equations imply that

$\frac{\partial q}{\partial t}=0, \quad \text { where } \quad q=\omega-\frac{f \eta}{H} \text { and } \omega=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}$

Consider an initial state where there is flow in the $y$-direction given by

$\begin{aligned} u &=\eta=0, \quad-\infty<x<\infty \\ v &= \begin{cases}\frac{g}{2 f} e^{2 x}, & x<0 \\ -\frac{g}{2 f} e^{-2 x}, & x>0\end{cases} \end{aligned}$

Find the initial potential vorticity.

Show that when this initial state adjusts, there is a final steady state independent of $y$ in which $\eta$ satisfies

$\frac{\partial^{2} \eta}{\partial x^{2}}-\frac{\eta}{a^{2}}= \begin{cases}e^{2 x}, & x<0 \\ e^{-2 x}, & x>0\end{cases}$

where $a^{2}=g H / f^{2}$.

In the case $a=1$, find the final free surface elevation that is finite at large $|x|$ and which is continuous and has continuous slope at $x=0$, and show that it is negative for all $x$.

Paper 1, Section I, B

commentInviscid fluid is contained in a square vessel with sides of length $\pi L$ lying between $x=0, \pi L, y=0, \pi L$. The base of the container is at $z=-H$ where $H \gg L$ and the horizontal surface is at $z=0$ when the fluid is at rest. The variation of pressure of the air above the fluid may be neglected.

Small amplitude surface waves are excited in the vessel.

(i) Now let $H \rightarrow \infty$. Explain why on dimensional grounds the frequencies $\omega$ of such waves are of the form

$\omega=\left(\frac{\gamma g}{L}\right)^{\frac{1}{2}}$

for some positive dimensionless constants $\gamma$, where $g$ is the gravitational acceleration.

It is given that the velocity potential $\phi$ is of the form

$\phi(x, y, z) \approx C \cos (m x / L) \cos (n y / L) \mathrm{e}^{\gamma z / L}$

where $m$ and $n$ are integers and $C$ is a constant.

(ii) Why do cosines, rather than sines, appear in this expression?

(iii) Give an expression for $\gamma$ in terms of $m$ and $n$.

(iv) Give all possible values that $\gamma^{2}$ can take between 1 and 10 inclusive. How many different solutions for $\phi$ correspond to each of these values of $\gamma^{2} ?$

Paper 1, Section II, B

commentA spherical bubble in an incompressible fluid of density $\rho$ has radius $a(t)$. Write down an expression for the velocity field at a radius $R \geqslant a$.

The pressure far from the bubble is $p_{\infty}$. What is the pressure at radius $R$ ?

Find conditions on $a$ and its time derivatives that ensure that the maximum pressure in the fluid is reached at a radius $R_{\max }$ where $a<R_{\max }<\infty$. Give an expression for this maximum pressure when the conditions hold.

Give the most general form of $a(t)$ that ensures that the pressure at $R=a(t)$ is $p_{\infty}$ for all time.

Paper 2, Section I, D

commentA body of volume $V$ lies totally submerged in a motionless fluid of uniform density $\rho$. Show that the force $\mathbf{F}$ on the body is given by

$\mathbf{F}=-\int_{S}\left(p-p_{0}\right) \mathbf{n} d S$

where $p$ is the pressure in the fluid and $p_{0}$ is atmospheric pressure. You may use without proof the generalised divergence theorem in the form

$\int_{S} \phi \mathbf{n} d S=\int_{V} \boldsymbol{\nabla} \phi d V$

Deduce that

$\mathbf{F}=\rho g V \hat{\mathbf{z}},$

where $\hat{\mathbf{z}}$ is the vertically upward unit vector. Interpret this result.

Paper 3, Section II, $18 \mathrm{D}$

commentWater of constant density $\rho$ flows steadily through a long cylindrical tube, the wall of which is elastic. The exterior radius of the tube at a distance $z$ along the tube, $r(z)$, is determined by the pressure in the tube, $p(z)$, according to

$r(z)=r_{0}+b\left(p(z)-p_{0}\right)$

where $r_{0}$ and $p_{0}$ are the radius and pressure far upstream $(z \rightarrow-\infty)$, and $b$ is a positive constant.

The interior radius of the tube is $r(z)-h(z)$, where $h(z)$, the thickness of the wall, is a given slowly varying function of $z$ which is zero at both ends of the pipe. The velocity of the water in the pipe is $u(z)$ and the water enters the pipe at velocity $V$.

Show that $u(z)$ satisfies

$H=1-v^{-\frac{1}{2}}+\frac{1}{4} k\left(1-v^{2}\right)$

where $H=\frac{h}{r_{0}}, v=\frac{u}{V}$ and $k=\frac{2 b \rho V^{2}}{r_{0}}$. Sketch the graph of $H$ against $v$.

Let $H_{m}$ be the maximum value of $H$ in the tube. Show that the flow is only possible if $H_{m}$ does not exceed a certain critical value $H_{c}$. Find $H_{c}$ in terms of $k$.

Show that, under conditions to be determined (which include a condition on the value of $k)$, the water can leave the pipe with speed less than $V$.

Paper 4, Section II, D

Show that an irrotational incompressible flow can be determined from a velocity potential $\phi$ that satisfies $\nabla^{2} \phi=0$.

Given that the general solution of $\nabla^{2} \phi=0$ in plane polar coordinates is

$\phi=\sum_{n=-\infty}^{\infty}\left(a_{n} \cos n \theta+b_{n} \sin n \theta\right) r^{n}+c \log r+b \theta$

obtain the corresponding fluid velocity.

A two-dimensional irrotational incompressible fluid flows past the circular disc with boundary $r=a$. For large $r$, the flow is uniform and parallel to the $x$-axis $(x=r \cos \theta)$. Write down the boundary conditions for large $r$ and on $r=a$, and hence derive the velocity potential in the form

$$\varphi =U(r+\frac{{a}^{2}}{r})$$