MATH45132 Coursework
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Problem 1
We consider an inviscid model to describe the capillary instability of a fluid column of density
ρ1 surrounded by a fluid of density ρ2. The interface between the fluids is represented by
the surface r = f(θ, z, t), where (r, θ, z) are standard cylindrical coordinates relative to an
orthonormal basis (er, eθ, ez). The notation v = (u, v, w) = uer + veθ + wez will be used
for the velocity field. The problem is governed by Euler equations
∇ · v = 0 , vt + (v · ∇)v = −
∇p
ρ
for r 6= f(θ, z, t) ,
applied on both sides of the interface. At the interface, the usual kinematic and dynamic
conditions
u(r = f±) =
Df
Dt
, [ p ]f
+
r=f−
= −γ∇ · n ,
are to be used; n is a unit vector normal to the interface pointing in the direction of
increasing r and γ is the surface tension.
We want to examine the stability of the stationary solution (v, p, f) = (v¯, p¯, f¯) given by
v¯ = 0 , f¯ = a and p¯ = p¯(r) ,
corresponding to a cylindrical interface with constant radius a.
1. Determine p¯(r) and confirm that it is a piecewise constant function.
[2 marks ]
2. Introducing perturbations of small amplitude ǫ such that
(v, p, f) = (v¯, p¯, f¯) + ǫ(v′, p′, f ′) ,
write down the linearised equations for v′ ≡ (u′, v′, w′) and p′. In particular, show
that p′ is governed on each side of the interface by the Laplace equation ∇2p′ = 0.
[1 mark ]
3. Determine the conditions that must be satisfied by the perturbations at r = a.
[2 marks ]
4. Show that the linear stability problem can be written in terms of p′ alone and fully
specify the p′-problem.
[2 marks ]
5. Considering normal modes of the form p′ = pˆ(r) exp(st+ ikz + inθ) with k real and
n integer, write down the equation satisfied by pˆ(r) and all its auxiliary conditions.
[1 mark ]
6. Determine the general solution for pˆ(r) which remains bounded in the domain, and
express it in terms of the modified Bessel functions In and Kn.
[1 mark ]
7. Determine the dispersion relation between s, k, n and the parameters of the problem.
[2 marks ]
8. Discuss the physical implications of the dispersion relation paying special attention
to the two limiting cases corresponding to ρ2 ≪ ρ1 and ρ2 ≫ ρ1.
[1 mark ]
Problem 2
The stability of an interface to perturbations of the form a(k)es(k)t+ikx is found to obey the
following dispersion relation:
(α + 1)S2 + 2α(1 + βk)S − α(α − 1)
(
1−
2αβ
α− 1
k −
γ
αk
)
= 0 where S =
s
k
The parameters are allowed to take real values such that α > 0, β ≥ 0 and −∞ < γ <∞.
1. In the special case where β = 0 and γ = 0, discuss the stability in terms of the
parameter α. Show that a bifurcation occurs when α exceeds a critical value αc an
determine whether it is a ‘zero-eigenvalue’ or a Hopf bifurcation.
[2 marks]
2. Assuming that α > 1 but taking γ = 0, determine the range of wavenumbers k
corresponding to stable modes in terms of β.
[1 mark]
3. Assuming that α > 1 but taking β = 0, determine the range of wavenumbers k
corresponding to stable modes in terms of γ. Can damped oscillatory modes exist in
this case?
[2 marks]
4. Assuming that α > 1 and keeping all other terms, show that the interface is stable
(to perturbations of all wavenumbers k) if γβ exceeds a critical value depending on α
that you are asked to determine.
[3 marks]
Hint: You may find it convenient to use the fact that the roots of the real-coefficient second
order equation s2 − Ts + D = 0 both have negative real parts if and only if T < 0 and
D > 0.
Note: Physically, the problem corresponds to an interface propagating vertically towards a medium
with density ρ1 away from a medium with density ρ2, with α ≡ ρ1/ρ2. The precise definitions of
β and γ are not needed here. Suffice it to say that γ is a non-dimensional measure of gravity
effects, with γ > 0 corresponding to a downward propagation, γ < 0 to an upward propagation, and
γ = 0 to a neglect of gravity. The parameter β accounts for the dependence of the interface normal
propagation speed on its local curvature, and is set to zero when such dependence is neglected.
[Total: 20 marks. Coursework will count 20% of the course unit ]