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程序代写案例-MATH45132

时间：2021-04-26

MATH45132 Coursework

submit as a single pdf to Blackboard by the deadline on 29/04/2021

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 ]

学霸联盟

submit as a single pdf to Blackboard by the deadline on 29/04/2021

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 ]

学霸联盟