Question 1 -

When heat is generated inside a medium, the temperature Θ(r, t) at position r and time t satisfies the modified heat equation

∂Θ/∂t = D∇²Θ + H,

where D is the thermal diffusivity, and H is proportional to the rate of heat production.

A high-power electrical circuit has a conductor in the form of a long, thick hollow cylindrical tube, with inner and outer radii R₁ and R₂, respectively. Heat is generated within the conductor, where the temperature Θ(r, t) satisfies the differential equation given above. The inner and outer surfaces are cooled by a fluid maintained at constant temperature Θ₀.

(a) If the temperature is in a steady state and depends only on the distance r from the central axis of the tube, use cylindrical coordinates (r, θ, z) to write down an ordinary differential equation for Θ(r) that is valid in the region R₁ < r < R₂, and state the boundary conditions.

(b) Show that the general solution of the ordinary differential equation obtained in part (a) for the region R₁ < r < R₂ is

Θ(r) = -(H/D)(r²/4) + Alnr + B,

where A and B are constants.

(c) Use the boundary conditions to determine the temperature throughout the cylindrical conductor, and hence show that

Θ(r) = Θ₀ + H/4D[(R₂² - r²) + (R₂² - R₁²/ln(R₂/R₁))ln(r/R₂)].

Question 2 -

Consider a system where the concentration of particles is given by the time-dependent scalar field

c(x, y, z, t) = K/2t(y + x² + z²),

where (x, y, z) are Cartesian coordinates, t > 0 is time and K is a constant.

(a) Find the number of particles N(t) inside the rectangular cuboid with sides defined by the planes x = -1, x = 1, y = 0, y = 4, z = 0, z = 2.

(b) Calculate the vector field F = ∇c.

(c) Calculate the surface integral

Φ_S = ∫_S F·dA,

where S is the surface of the cuboid defined in part (a). (Hint: Take the normal vector to each surface to be pointing outwards from the cuboid.)

(d) Verify that Gauss's theorem is satisfied by the field F for the cuboid with boundary S.

(e) Find the value of the surface integral of F over the surface of an ellipsoid centred at the origin of volume V =4/3 πabc, where a, b and e are constants.

(f) If J = - D∇c, is there any choice of the constant D for which the continuity equation is satisfied?

Question 3 -

Calculate the Fourier transform of χ(½(x-2)) + χ(½(x-4)),  where x is the top-hat function.

Question 4 -

Consider the differential equation

d²y/dx² - 2y = e^-|x|.

(a) Take the Fourier transform of this differential equation to show that

y^~(k) = √(2/π)(1/(2+k²) - 1/(1+k²)),

where y^~(k) is the Fourier transform of y(x).

(b) Hence calculate y(x).

Question 5 -

A rectangular lamina of width a and height b is sandwiched between two slabs of ice, so that the lower and upper sides, of width a, are at a temperature Θ = 0. The other two sides are thermally isolated. Fix coordinates so that the origin is the lower left corner, as shown in the figure below.

Model this situation as a temperature Θ(x, y, t) that satisfies the two-dimensional diffusion equation with thermal diffusivity D, namely

∂Θ/∂t = D∇²Θ, 0 < x < a, 0 < y < b,

together with Dirichlet boundary conditions on the upper and lower sides given by

Θ(x, 0) = 0, Θ(x, b) = 0, 0 < x < a,

and Neumann boundary conditions on the other two sides given by

∂Θ/∂x(0, y) = 0, ∂Θ/∂x(a, y) = 0,  0 < y < b.

(a) Show that applying the method of separation of variables using Θ(x, y, t) = V(x, y)T(t) gives the solution T(t) = Ce^μDt and the eigenproblem ∇²V(x, y) = μV(x, y), where C and μ are constants.

(b) Applying the method of separation of variables again, using V(x, y) = X(x)Y(y), gives the two further eigenproblems X" = λX and Y" = (μ - λ)Y.

Translate the boundary conditions for Θ into boundary conditions for X(x) and Y(y).

(c) The X(x) eigenproblem has been analysed in Unit 11, and the eigenfunctions were found to be

X_m(x) = cos(k_mx),  where k_m = mπ/a, m = 0, 1, 2, . . . ,

with corresponding eigenvalues λ = -k_m².

Find the eigenfunctions and eigenvalnes for the Y(y) eigenproblem.

(d) Use the eigenfunctions from part (c) to write down the eigenfunctions and eigenvalues for the V(x, y), eigenproblem.

(e) Write down a general solution for the temperature Θ(x, y, t).

Question 6 -

Consider the situation of an area of land being heated periodically at the surface. A model for the temperature Θ(x, t) of the soil at a distance x metres below the surface at time t is

∂Θ/∂t = D(∂²Θ/∂x²),

where Θ(0, t) has an annual sinusoidal variation about a mean temperature of -5^oC with amplitude 10^oC, and

∂Θ/∂x → 0 as x → ∞.

(a) Verify by substitution that the following function satisfies both the heat equation and the boundary conditions of the model for an appropriate value of k:

Θ(x, t) = -5 + 10e^-kx cos(ωt - kx),

where ω ≈ 2 x 10^-7 rod s^-1 is a constant that corresponds to the annual temperature variation. Calculate explicitly the value of k needed for this to be a solution of the model if the soil has diffusivity constant

D = 4 x 10^-7 m²S^-1.

(b) Find the depth at which there is permafrost (that is, where the ground is permanently below 0^oC).

Question 7 -

A monkey is walking randomly near a cliff. For simplicity, we consider the system to be one-dimensional, with the monkey following a random walk in 0 ≤ < ∞ and the cliff located at x = 0. In addition, the motion of the monkey is affected by wind that is flowing at constant speed v > 0, Under these conditions, the system can be modelled as a biased random walk with constant drift v and diffusion coefficient D in a semi-infinite domain, and it can be shown that the survival probability of this process is given by

S(x₀, t) = ½(1 - e^-vx_0/D) + ½[erf(x₀+vt/√(4Dt))] - e^-vx_0/D erf(-x₀+vt/√(4Dt))],

where x₀ > 0 is the initial position, and erf(x) is the error function.

Show that the first-passage probability density is

F(x₀, t) = x0/√(4πD)t^-3/2exp(-((x₀+vt)²/4Dt)).

(Hint! You may find the property d/dx erf(x) = 2/√π e^{-x^2} useful.)

(b) In the limit of large time, explain why the first-passage probability density obtained in part (a) can be approximated as

F(x₀, t) = (x₀exp(-x₀v/(2D))/√(4πD)) t^-3/2e^-t/τ,

where τ = 4D/v². Assuming that τ is large, use this expression to show that the probability P_c that the monkey hits the cliff at t ≥ τ is

P_c = (vx₀K/4D√π)exp(-x₀v/(2D)),

where

K = ₁∫^∞ x^-3/2e^-xdx

is a constant (you are not asked to evaluate this integral).

Question 8 -

An experiment where particles of mass m are suspended in a liquid bath shows that the velocity v of the particles fluctuates in time taking values in - ∞ < v < ∞ and is described by a stochastic process with drift

f(v) = -(γ/m)v,

where γ is a constant, and diffusion coefficient given by

D = kBTγ/m²,

where k_B is the Boltzmann constant, and T is the temperature of the liquid bath.

(a) Write down the Fokker-Planck equation for the probability density of the velocity fluctuations P(v, t), and the differential equation describing the stationary probability density function P_s(v).

(Hint: Treat the velocity v as the independent variable, and work with P(v, t) instead of P(x, t). So in the Fokker-Planck equation, the variable x is replaced by v.)

(b) Show that the normalised stationary probability density function is

P_s(v) = √(m/2πk_BT)exp(-(m/2k_BT)v²).

(c) What are the mean (v) and standard deviation σ = √(Var(v))? Find an approximate value of the Boltzmann constant if experimentally it is found that Var(v) = 8.095 x 10^-7 m² s^-2 for particles of mass m = 5 x 10^-15 kg in a liquid bath of temperature T = 293.15 K. Express the result to five significant figures.

Please answer all the questions.