WORKSHEET
Vector Algebra and Calculus
⃗ = 𝐴𝑥𝑖 + 𝐵𝑦𝑗 + 𝐶𝑧𝑘⃗, where A,B and C are constants. Derive
1. The velocity of a flow is 𝑉
⃗ and curl𝑉
⃗.
expressions for div 𝑉
2. The gradient of the fluid pressure, 𝑝(𝑥, 𝑦), is given by ∇𝑝 = −𝐴(𝑥𝑖 + 𝑦𝑗), where A is a
constant. Derive an expression for 𝑝(𝑥, 𝑦) − 𝑝(0,0).
3. The pressure field of an irrotational flow over a circular cylinder of radius a, expressed in
cylindrical coordinates, is 𝑝(𝑟, 𝜃, 𝑧) = 𝑝∞ −
𝜌𝑈 𝑎 2 𝑎 2
( ) [( 𝑟 )
2 𝑟
− 2𝑐𝑜𝑠(2𝜃)], where 𝑝∞ and U
are the pressure and flow speed far from the cylinder. Derive an expression for ∇𝑝 as a
function of r, 𝜃 and z.
4. Prove the following identities by using (i) vector notations and (ii) tensor notations.
     
   aA   a    A    a   A
  aA  a   A  A   a

 A2 
A A     A   A
 2 



     
     A      A   A

 

  A B  A   B  B  A  B   A  A B
2
Plane parallel steady flows
1. Couette flow
y
U
h
x
Consider a plane parallel flow between two plates, one of
which is stationary (the lower plate) and one moves in the
x-direction at a speed U (the upper plate). The thickness of
the layer is h.
(a) Determine the velocity profile across the layer.
(b) Determine the volumetric flow flux.
2.
liquid
air
U
𝑔
τ
h
Under the influence of gravity a viscous liquid (density ρ and viscosity η)
flows down a stationary vertical wall, forming a thin film of constant
thickness h. An upflow of air next to the film exerts an upward constant
y
shear stress τ on the surface of the liquid layer, as shown. The pressure in the flow is uniform.
Derive expressions for (a) the film velocity U(y) as a function of y and the flow parameters ρ, η, h
and τ, and (b) the shear stress τ that would result in a zero net volume flow rate in the film.
3.
A vertical tube of diameter 2mm has its upper end open to the atmosphere. At the lower end water
is maintained at a pressure of 104Nm-2 above atmospheric. What is the longest length of tube for
which this pressure will produce a flow through the tube? (Ignore surface tension.)
What volume of water will pass through the tube per second if the length is half this? (Assume that
the flow is laminar and ignore the entry length.)
4. Annular Poiseuille flow.
Fluid of constant density is forced between the rigid cylinders r=a and r=b by a pressure gradient A.
Solve the Navier-Stokes equations to find the velocity profile. Verify that the stresses on the walls
balance the pressure difference over a length L.
5. Flow down a slope.
A liquid of constant density flows down a plane which
slopes at angle α to the horizontal. The free surface has
no shear stress on it apart from a pressure p0, and is at
x
a uniform distance from the plane. For this flow you
U(y)
need to keep in the gravitational field, as it is now
α
dynamically active. Set up and solve equations for U(y),
and verify that the forces on a length L of the fluid layer are in equilibrium.
y
6.
Repeat the previous question for flow down a full pipe of circular section at angle α to the horizontal.
7. Consider two concentric cylinders with a Newtonian liquid of constant density, ρ, and
constant dynamic viscosity, η, contained between them. The outer pipe, with radius, R1, is
fixed while the inner pipe, with radius, R2, and mass per unit length, m, falls under the action
of gravity at a constant speed. There is no pressure gradient within flow and no swirl velocity
component. Determine the vertical speed, V, of the inner cylinder as a function of the
following (subset of) parameters: g, R1, R2, m, ρ, and η.
R1
R2
annular region filled with liquid of
density, ρ, and dynamic viscosity, η
fixed outer
cylinder with
radius R1
movable inner cylinder
with radius, R2, and
mass per unit length, m
V
Answer: V

R  g
mg  g 2
 
R 2  R12 
  R 2 ln 2 
R2 
R1  2
2R 2  4

Plane parallel unsteady flow
Fluid is at rest in a long channel with rigid walls 𝑦 = ±𝑎 when a pressure gradient –A is suddenly
imposed at t=0. Show that the velocity 𝑣𝑥 (𝑦, 𝑡)𝑖 satisfies the equation
𝜕𝑣𝑥 𝐴
𝜕 2 𝑣𝑥
= +𝜈
𝜕𝑡
𝜌
𝜕𝑦 2
for t>0, and state the boundary and initial conditions for this flow.
As 𝑡 → ∞ we expect to get the flow appropriate for a pressure gradient in a channel
𝑈(𝑦) = 𝐴
(𝑎2 − 𝑦 2 )
2𝜂
so seek a solution in the form
𝑣𝑥 (𝑦, 𝑡) = 𝑈(𝑦) + 𝑉(𝑦, 𝑡):
What equation and boundary values does V satisfy?
Find V. How long does it take for the flow U to be established?
Waves
Determine the damping coefficient for capillary waves.
Convection
1. Forced Convection
Consider convection in a circular pipe of radius a with the Poiseuille velocity profile. Suppose that
the temperature of the pipe wall increases linearly with distance along the pipe, and that, in
consequence, the same constant axial temperature gradient T x exists everywhere in the pipe.
Determine the radial temperature distribution. Determine also the heat transfer from the pipe wall.
2. Free Convection
Free convection in a layer between two vertical walls at different uniform temperatures of large
enough aspect ratio (h/d) can a have a region in the middle vertically in which the velocity and the
temperature depend only on the horizontal coordinate and not on the vertical. Show that, in this
region, the temperature profile is unaffected by the flow and determine the velocity profile. Express
the latter result in terms of the dependence of the Reynolds number, based on the maximum
velocity and the layer width, on the Grashof number.