1 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]
