MATH225 HOMEWORK ASSIGNMENT 3. NOVEMBER 2022
Please attempt all FOUR questions, included in this assignment. The total
mark available in this assignment is 100.
1. Consider a fluid flow characterised by the velocity field
V = ? × ((2x + 3y)(xi + yj)).
(a) Evaluate V and, hence, describe the streamlines for this flow.
[13 marks]
(b) Show that the fluid flow with the above velocity field is incompressible.
[7 marks]
(c) Evaluate the acceleration of a fluid particle within the flow.
[5 marks]
2. Consider a two-dimensional fluid flow, described in polar coordinates
(r, θ). Let a fluid particle within the flow move along the trajectory
described by
r = 2, θ = ωt, with ω being a positive dimensional constant.
Here, t is time.
(a) For this particular motion, verify that at any time t, the position
vector r(t) of the particle, and the corresponding velocity vector
v =
d
dtr(t), are orthogonal. Show that |v| = 2ω at any time.
[8 marks]
1
(b) Show that the acceleration of the particle a =
d
dtv(t) has the form
a = Cer.
Find the value of the coefficient C above.
[8 marks]
(c) Evaluate the limit
lim
t→+∞
|r × v|.
[9 marks]
3. Consider a fluid flow in spherical coordinates (ρ, φ, θ).
(a) Assuming that the velocity field V is characterised by the velocity
potential Φ(ρ), derive the representation for V in terms of Φ and
show that
V × eρ = 0.
[12 marks]
(b) For a particular case when the velocity potential has the form
Φ(ρ, φ, θ) = ?ρ
?1
,
show that when ρ 6= 0 the velocity field is given by
V = ρ
?2
eρ,
and evaluate div V.
[13 marks]
4. Consider a fluid flow characterised by the velocity field in cylindrical
coordinates (r, θ, z)
V(r, θ, z) = 2r cos(2θ)er ? 2r sin(2θ)eθ + 6zez.
(a) Verify that the vector field V is conservative, i.e. ? × V = 0.
[10 marks]
(b) Derive the representation for the velocity potential for this fluid
flow.
[15 marks]