Devaraj van der Meer, Gerrit de Bruin, November 2, 2006
Exam Advanced Fluid Mechanics (357001)
November 3, 2006
9:00 – 12:30
This exam is an ‘open book’ exam, i.e., you can use your course book, lecture notes and
your own notes in solving the problems that constitute this exam. There are five problems
of varying length and difficulty. The number of points that can maximally be earned by
giving a correct answer is indicated at the end of each question. There are 100 points in
total. Good luck !
Problem 1 – Outflow of a vertical pipe
Water flows through a pipe in a gravitational field as shown in
the accompanying figure. Neglect the effects of viscosity and
surface tension. The velocity of the fluid at z = 0, where the
water leaves the pipe, is uniform and has a magnitude u0. The
cross-sectional area of the orifice is A0.
a) Solve the appropriate conservation equations for the
variation of the cross-sectional area of the fluid column
A(z) after the water has left the pipe. [5 pts.]
Problem 2 – Irrotational vortices
Two two-dimensional irrotational vortices of strength Γ1 and Γ2 respectively interact with
each other through their mutual velocity fields. The vortices are initially at a distance d
from each other.
In answering the first three questions (a, b1, and b2) exclude the special cases Γ1 = Γ2 and
Γ1 = – Γ 2 .
a) Argue that, at a distance very far away from the vortex pair, the flow field of the
two vortices will look like that of a single vortex. Calculate the strength of this
vortex. [5 pts.]
b1) In the laboratory frame there must be a point in which the velocity field vanishes.
Calculate the distances of the two vortices to this point in terms of Γ1, Γ2 and d.
[6 pts.]
b2) Describe the motion of the two vortices for the case that both vortices are corotating (i.e. in the same direction, e.g., Γ1 > 0 and Γ2 > 0) and for the case that
they are counter-rotating (e.g., Γ1 > 0 and Γ2 < 0). [3 pts.]
Now consider the special case: Γ1 = – Γ2 ≡ Γ, a so-called counter-rotating vortex pair.
c1) Describe the motion of such a counter-rotating vortex pair as they approach a wall
that is parallel to the line connecting the two vortices. [2 pts.]
Exam Adv. Fluid Mech.,
Friday November 3, 2006, 9:00 – 12:00
page 1 of 3
Devaraj van der Meer, Gerrit de Bruin, November 2, 2006
c2) Calculate both components of the velocities of each of the vortices, i.e., both the
component parallel to as the component perpendicular to the wall. In your
calculation let s be the distance of the vortex pair to the wall. [6 pts.]
d) Write down the equations of motion for each vortex and show that in the limit of
d >> s they lead to the following equation of motion for the distance d between
the vortices:
2
∂ 2d  Γ  1
= 
∂t 2  π  d 3
[6 pts.]
(Remark: This equation can be solved analytically, from which the trajectory of
the vortices can be calculated.)
Problem 3 – Waves on a fluid-fluid interface
Two immiscible inviscid fluids at rest are positioned on top of each other such that the
upper halfspace z > 0 is covered with fluid 1 with density ρ1 and the lower halfspace z < 0
with fluid 2 (density ρ2). The surface tension of the two-fluid interface is equal to σ.
a) For a (small) disturbance η of the interface, give the dynamic boundary condition
at the interface. Take its surface tension into account, and formulate the above
boundary condition in terms of the flow potentials φ1 and φ2 of the fluids and the
disturbance. [8 pts.]
Using an ansatz for small disturbances of the interface of the form
η ( x, t ) = ηˆ eik ( x −ct )
one can show that from the kinematic boundary conditions the potentials in the upper and
lower fluid must satisfy:
ϕ1 = icηˆ e− kz eik ( x−ct ) ; ϕ2 = −icηˆ ekz eik ( x−ct )
b) Show that the dynamic boundary condition of a) leads to the following relation:
ρ − ρ2 g
σk
c2 = − 1
+
ρ1 + ρ 2 k ρ1 + ρ 2
Carefully indicate the different approximations that you make and why they are
valid [7 pts.]
c) Determine the condition on k for which traveling waves are possible. Are
traveling waves ever possible if ρ1 > ρ2 ? [7 pts.]
Exam Adv. Fluid Mech.,
Friday November 3, 2006, 9:00 – 12:00
page 2 of 3
Devaraj van der Meer, Gerrit de Bruin, November 2, 2006
Problem 4 – Rotating sphere in a viscous fluid
A rigid sphere of radius a is rotating with a constant angular velocity Ω in a fluid which is
at rest at infinity. Assume that the Reynoldsnumber
Re =
a 2Ω
ν
≪1 .
is very small. This problem is conveniently treated in spherical polar coordinates where
the axis of rotation coincides with the z-axis.
a) Based on symmetry, argue that none of the flow variables u and p can depend on
the azimuthal coordinate φ, most notably, the pressure gradient vanishes in this
direction. Also argue that the only non-zero flow component is the one in
azimuthal direction. [5 pts.]
b) Show that, using the separation ansatz uϕ = f ( r )sin θ , the appropriate equation for
the flow field leads to
d  2 df 
r
−2f =0
dr  dr 
[8 pts.]
c) Show that the general solution of the above equation is a linear combination of
two terms of the form r n, and using the appropriate boundary conditions show that
Ωa 3
uϕ = 2 sin θ
r
[7 pts.]
d) Calculate the torque that the sphere exerts on the liquid. [7 pts.]
Problem 5 – Perturbation expansion
The temperature distribution in a certain flow problem is described by a differential
equation, which in non-dimensional form and together with the boundary conditions is
given by
d2 3
(Y ) = ε Y ; with: Y (0) = 1 ;
dx 2
dY
dx
=0 .
x =1
(Here the parameter Y is the square root of the temperature Y = T . Knowledge of this
fact is not necessary to solve the below problems.)
a) Assuming that ε is a small parameter, solve this boundary value problem using a
regular perturbation expansion up to 1st order in ε. [12 pts.]
b1) If ε is very large (i.e., ε >> 1) one can try to reformulate the problem such that it
can still be treated within the framework of a perturbation expansion in a small
parameter. Give such an alternative formulation of the problem. [3 pts.]
b2) State whether this alternative formulation is regular or singular and prove your
statement. [3 pts.]
Exam Adv. Fluid Mech.,
Friday November 3, 2006, 9:00 – 12:00
page 3 of 3