 ## MATH0087

### 1.  数学建模代考

Considerthe following model of a transverse magnetic electromagnetic field in an inhomogeneous two-dimensional electromagnetic medium occupying domain Ω due to distributed current source j(x, t) = (j1(x, t), j2(x, t), 0) in the in-plane direction, where x = (x1, x2) and t Magnetic field B is assumed to  be orthogonal  to the plane, i.e. B = B(x)k where k is the unit vector in the positive x3-direction. The magnetic field B is related to the electric field E(x) via Maxwell’s equations as follows

B E

(1) where positive smooth function η(x) is electric permittivity and positive constant µ0 is magnetic permeability of the medium.

(a)Showthat B(x, t) is a solution of an inhomogeneous wave equation (5 marks)

(b)Let the exterior boundary S of Ω be perfectly conducting, i.e. E× n = 0 on S where n is the outward unit normal to S. Let for the initial magnetic field at t = 0, B(x, 0) = B0(x), x Ω, and the initial electric field E at t = 0 be E0(x) = (E01(x), E01(x), 0). Derive the resulting initial boundary-value problem for B.(5 marks)

(c)Showthat the initial boundary-value problem of part (c) has no more than one solution.(10 marks)

### 2.Consider the following initial-value problem describing propagation of transverse- magnetic electromagnetic waves in an unbounded two-dimensional two-scale het- erogeneousmedium:  数学建模代考 Here x = (x1, x2) R2, the smooth functions η(x, z) > 0 and f (x, t, z), z  =  (z1, z2) R2, describing microscopic electric permittivity and sources due to cur- rents respectively, are both periodic in z with periodicity cell Q = [0, 1]2; constant µ0 > 0 is magnetic permeability of the medium.   The smooth functions η(x, z),     f (x, t, z), g(x) and h(x) are independent of ε, where ε > 0 is a small parameter.

You are given that the solution Bε(x, t) to (3)–(4) has an asymptotic approxi-mation in the two-scale form:

where B(1)(x, t, z) and B(2)(x, t, z) are periodic in z with the periodicity cell Q.

(a)Byequating the terms of the order ε1 in (3) show that where Nj(x, z), j = 1, 2, are independent of B0. Show that Nj(x, z) are the solutions to the “unit cell” problems, and formulate those problems.(5 marks)

(b)Byequating the terms of the order ε0 in (3), derive the equation for B0 (the homogenised equation). Find the components ηh (x) of the homogenised electric permeability matrix ηh(x), and the homogenised source f h(x, t) in terms of η(x, z), µ0, f (x, t, z) and Nj(x, z), j = 1, 2.(8 marks)

(c)Formulate the homogenised initial-value problem (but do not attempt to solveit).(2 marks)

(d)Whatwould be the problem with the above approach if the initial conditions were also two-scale,i. e. where g(x, z) and h(x, z) are periodic in z? Can there be any way for fixing it?(5 marks)

### 3.(a)Derive the first two terms in an asymptotic expansion of  数学建模代考 (5 marks)

(b)Apply Laplace’s Method from first principles to derive the leadingor- der term in the asymptotic expansion of You may re-use some of your answer to part (a) and standard identities involving the Gamma function without proof.(12 marks)

(c)Briefly explain why Integration by Parts should not be used when de- riving an asymptotic expansion of(6).(3 marks)

### 4.(a) Use Integration by Parts and Induction to prove thefollowing:  数学建模代考 where Γ denotes the Gamma function. You may use the identity Γ(1/2) = without proof.(6 marks)

(b)Use The Method of Steepest Descent to obtain a full asymptotic expansionof

clearly defining bn and cn for n N. You may use Watson’s Lemma and the following identity without proof: (c)Using an alternative method from the course notes, derive the leading order term in the asymptotic expansion of (7). You may quote general results from the notes  without  proof,  but  symbolic  software  solutions  will not be accepted.(5 marks)

### 5.(a) A power-law fluid has shear viscosity η= k |γ˙| n−1, where γ˙  is the shear rate, with k>0 and n> 0.

(i)The fluid is pumped along a straight channel of depth 2H and stress- freeside walls (so that there is no velocity variation in that direction), under a pressure gradient tt. Find the velocity profiles in the channel, and sketch them qualitatively for n < 1, n = 1 and n >1. (6 marks)

(ii)The power-law fluid is ‘matched’ with a Newtonian (n = 1) fluid in the sense that under gravity they flow out of a channel held vertically upright in the same time. Suppose the channel is now tilted so as to make an angle θ with the vertical. What is the ratio of the fluxes of the power-law and Newtonianfluids?(7 marks)

(b) The Oldroyd B model has shear stress    数学建模代考 where ηs and ηp are the solvent and polymer viscosities, respectively, λ

is  a  relaxation  time  and  γ˙  is  the  usual  rate  of  strain.   It  is  observed  that

the solvent and polymeric viscosities for a particular Oldroyd B fluid are identical, so ηs = ηp. An experimentalist is able to apply a simple fixed shear rate γ˙  to the fluid and measure both the shear stress response τ  and the normal stress difference N1. Show that and write down a similar expression for ηs = ηp.(7 marks)