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Example ACSE-2 Autumn 2019

Imperial College London 2019

Please answer all questions

Total time for the exam: 90 minutes

Calculators will be provided

Total marks= 80

(1) Give a mathematical definition of the Taylor series. Using an expansion about the

point x=0 use Taylor series to estimate the value of exp(x) when x=0.1 accurate

to 6 significant figures. [6 marks]

(2)

(a) Consider the ordinary differential equation system

= , ℎ = −1 33 −1

The solution takes the form: () = () , where is a constant

(vector) of integration. Using matrix diagonalization to compute this matrix

exponential, show that the solution can be written as

() = 12 (! + !) 11 (2) + 12 (! − !) −11 (−4)

[10 marks]

(b) What choice of initial condition for this problem yields a solution that

tends to zero as t tends to infinity? [4 marks]

(3) Define the vector one, two and max norms mathematically. Consider a vector

with two components, plot the shapes mapped out in 2D by all vectors with unit

norm, i.e. the “unit circle”, using each of these three norms. [6 marks]

(4) Consider a plane of reflection that passes through the origin. Let be the unit

normal vector to the plane and let r be the position vector for a point in space.

(a) Show that the reflected vector for r is given by · = − 2( · ),

where T is the transformation that corresponds to the reflection. [6 marks]

(b) Let = + + . Find the matrix of T. [5 marks]

(5) Given a vector field = !!! + !!! + !!!. For the point x=(1,1,0), find the

following: [7 marks]

2

i. ∇

ii. ∇ ∙

iii. ∇×

iv. The differential dv for = (! + ! + !)/√3

(6)

(a) Given the following strain tensor ε =

5 4 04 −1 00 0 3 ×10!! in a point

i. Sketch and describe how two lines originally in ! and ! direction,

respectively, would be deformed by this strain field. [5 marks]

ii. Find the principal strains [6 marks]

iii. Find the principal strain directions and use these to sketch how a

sphere would deform in this strain field [9 marks]

(7)

Consider a case with material flowing in a channel of a width 2b, driven by a

pressure drop along the channel of ΔP over every distance L. Take x1 to be the

direction of flow and x2 the direction across the width of the channel, i.e.

ranging from -b on the bottom to b on the top. The channel can be considered

infinite in x3 direction.

For this case, the Navier-Stokes equation simplifies to: ∇ · = ∇

where τ is the deviatoric stress tensor and p is pressure. Assume that there is

linear relation between deviatoric stress τ tensor and strain rate D: τ =2ηD,

where η is viscosity.

(a) Write out the relevant components of the force balance for this case, i.e. for

the non-zero components of the stress divergence and pressure gradient.

[4 marks]

(b) Show that the following is the velocity profile: ! = 12 ∆ ! − !!

[5 marks]

(c) Derive equations to describe the pathlines of the flow, i.e. relating position

of a fluid particle x(t) to its initial position ξ and time t. Sketch the pathlines

of 3 particles with positions chosen to illustrate the character of the flow.

[7 marks]