MTH

EXAM CODES: MTH3360
TITLE OF PAPER: Fluid Dynamics
EXAM DURATION: 3 hours writing time
READING TIME: 10 minutes
THIS PAPER IS FOR STUDENTS STUDYING AT: ( tick where applicable)
 Berwick  Clayton  Malaysia  Off Campus Learning  Open Learning
 Caulfield  Gippsland  Peninsula  Enhancement Studies  Sth Africa
 Parkville  Other (specify)
During an exam, you must not have in your possession, a book, notes, paper, electronic device/s, calculator,
pencil case, mobile phone, smart watch/device or other material/item which has not been authorised for the
exam or specifically permitted as noted below. Any material or item on your desk, chair or person will be
deemed to be in your possession. You are reminded that possession of unauthorised materials, or attempting to
cheat or cheating in an exam is a discipline offence under Part 7 of the Monash University (Council) Regulations.
No exam paper or other exam materials are to be removed from the room.

Attempt all questions, showing all working. Each question is worth 20 marks.
Complete sections A and B in separate booklets with A or B clearly marked on the front of
each booklet.
A page of formulae is at the back of the exam. You may remove this page from the exam
booklet.
AUTHORISED MATERIALS
OPEN BOOK  YES  NO
CALCULATORS  YES  NO
SPECIFICALLY PERMITTED ITEMS  YES  NO

Part A

Problem 1.

Consider a layer of incompressible fluid of constant thickness $h$, density $\rho$ and viscosity $\mu$ on a conveyor belt inclined at an angle $\alpha$ to the horizontal. Assume that $x$ is directed down the belt, $z$ is directed perpendicular to the belt and that the flow is independent of the direction $y$. The conveyor belt moves with velocity $U_{c}$ in the $x$ direction, with $U_{c}$ positive if the motion is downslope and negative if the motion is upslope. The fluid flows steadily under the combined effects of gravity and the moving conveyor belt.
(a) Sketch the above configuration. $[2$ marks]
(b) Write down the governing equations in component form. Assuming that the flow is unidirectional, indicate which terms are zero, briefly justifying your answer. $[6$ marks]
(c) Write down appropriate boundary conditions on the conveyor belt at $z=0$ and the free surface at $z=h . \quad[4$ marks $]$
(d) Hence find expressions for the pressure and the velocity. [4 marks]
(e) For what conveyor belt velocities $U_{c}$ is the fluid everywhere moving up the slope? $\quad[2$ marks $]$
(f) What is the force per unit area that must be exerted on the conveyor belt in the $x$ -direction to keep the belt moving? $[2$ marks $]$

Problem 2.

Consider steady Stokes flow in the absence of gravity around a stationary solid sphere of radius $a$ in an infinite incompressible fluid of viscosity $\mu$. Far from the sphere, the fluid is moving with constant speed $U$.
(a) Using dimensional analysis, find the functional form of the drag force on the sphere. [Hint: the dimensions of $\mu$ are M/LT.] $[4$ marks $]$
(b) Is the flow fore-aft symmetric? Justify your answer without calculation. marks]
(c) Spherical polar coordinates are defined so that the $\phi=0$ direction is aligned with the far-field flow as shown above. Assuming that the flow is independent of $\theta$ and that $v_{\theta}=0$ everywhere, a streamfunction $\psi(r, \phi)$ can be defined with
$v_{r}=\frac{1}{r^{2} \sin \phi} \frac{\partial \psi}{\partial \phi} \quad$ and $\quad v_{\phi}=-\frac{1}{r \sin \phi} \frac{\partial \psi}{\partial r}$
Verify that the continuity equation is satisfied exactly in this case. [Hint: the governing equations in spherical polar coordinates are given on the final page of this exam booklet.] [2 marks]
(d) Write down boundary conditions for the velocity components $v_{r}$ and $v_{\phi}$ on the surface of the sphere and hence boundary conditions for $\psi$ on $r=a$.
[3 marks]
(e) Write down conditions on the velocity components $v_{r}$ and $v_{\phi}$ far away from the sphere. Hence find a condition on $\psi$ for $r \gg a .$
(f) It can be shown that the general solution of Stokes’ equations for $\psi$ is
$$
\psi=\left(A r^{4}+B r^{2}+C r+\frac{D}{r}\right) \sin ^{2} \phi
$$
Use the boundary conditions on $\psi$ to find the values of the constants $A, B, C$ and $D .$
(g) Is the assumption of Stokes flow valid far from the sphere? Briefly justify your answer (you may state your justification without calculation). $\quad[2$ marks $]$

Problem 3.

(a) Explain what it means for a velocity field to be irrotational. $\quad[1$ marks]
(b) Calculate the vorticity for the velocity field $(0,0, x y t)$. $[2$ marks $]$
(c) Find the vortex lines for this field. $[3$ marks]
(d) A vortex tube consists of all the vortex lines through a closed curve $C$. Consider a vortex tube with arbitrary end caps. The volume enclosed by the vortex tube and arbitrary end caps is $V$ and the surface is $S$, which is split into the portion $S_{\text {tube }}$ forming the tube and $S_{\text {top }}$ and $S_{\text {bottom }}$ for the two end caps.
i. Sketch the tube and various surfaces. $[1$ marks]
ii. Briefly explain why
$$
\iiint_{V} \boldsymbol{\nabla} \cdot \boldsymbol{\omega} \mathrm{d} V=0
$$
(You may state $\boldsymbol{\nabla}$ identities without proof.) $[2$ marks $]$
iii. Using this result, show that
$$
\iint_{S_{\text {top }}} \boldsymbol{\omega} \cdot \mathbf{n} \mathrm{d} S+\iint_{S_{\text {bottom }}} \boldsymbol{\omega} \cdot \mathbf{n} \mathrm{d} S=0
$$
justifying your reasoning carefully. $[5$ marks $]$
iv. Define the circulation around a closed curve $C^{\prime} . \quad[1$ marks $]$
v. Using the results above, show that the circulation around the curve $C_{\text {top }}$ bounding the surface $S_{\text {top }}$ equals the circulation around $C_{\text {bottom }}$ bounding the surface $S_{\text {bottom }}$, justifying your reasoning carefully. On your sketch in part (i), you may wish to indicate directions in which the curves are traversed. $\quad[5$ marks]