MATH3102 Final Assignment, Semester 2, 2020 November 3, 2020 1. (15 marks) The vertical displacement u(x) of an elastic string of length l satisfies the boundary value problem τ d 2u dx2 + µu = p, for 0 < x < l where u(0) = 0, u(l) = U and p is a constant that has the dimensions of force per length.
(a) What are the dimensions for the constants τ and µ?
(b) Show how it is possible to nondimensionalize this problem so it takes the form d 2v ds2 + αv = β, for 0 < s < 1 with v(0) = 0, v(1) = 1. State what α and β are. 2. (20 marks) Find a composite expansion for small for the solution of the boundary value problem y00 + y 0 + y = 0, for 0 < x < 1, where y(0) = 1 and y(1) = 2.
Include a detailed step-by-step explanation with the calculation of the composite solution. Plot the exact analytical solution together with the composite expansion for a few representative values of to illustrate how the approximate solution approaches the exact solution as becomes smaller. 3. (15 marks)
Consider a traffic flow with initial density distribution at t = 0 described by a step function ρL = ρM/4 for x < 0, and ρR = ρM/2 for x > 0, where the constant ρM is the maximum density of cars along the road.
(a) Explain what will be the density distribution ρ(x, t) at times t > 0 if the car velocity is described by the Grenshields law: v(ρ) = vM 1 − ρ ρM .
(b) Explain how is the solution different if the constitutive law is modified to v(ρ) = vM 1 − ρ 2 ρ 2 M . 1 4. (15 marks) Consider the spring-dashpot system sketched below.
(a) Find a differential equation which relates the force F and displacement u.
(b) Use this result to formulate a constitutive law for the stress T of a 1D visco-elastic material.
(c) Find the associated relaxation function. 5. (20 marks)
Consider the following adaption of the Poiseuille problem: a cylinder of length L centered along the z-axis has radius R2. It is filled with two incompressible viscous fluids. In terms of cylindrical coordinates fluid 1 occupies the space with radius r < R1 and fluid 2 occupies the space with radius R1 < r < R2. Assume that the pressure at z = 0 is p0 and at z = L it is p1. Assume that there are no body forces and that the flow is steady and rotationally symmetric and in z-direction, i.e. (cylindrical coordinates) vr = 0, vθ = 0 and vz = vz(r, z) as well as p = p(r, z) for the pressure. Use the notation v1 := vz of the first fluid and v2 := vz of the second fluid. Also assume bounded velocities, no-slip at the boundary r = R2 and continuity of velocity and stress at the interface r = R1. 6. (15 marks)
List the steps we took in the lecture to derive the Navier-Stokes equation. List them in consecutive order starting with the formulation of the balance law for momentum until the formulation of the compressible and incompressible Navier-Stokes equations.
For every step: Describe the modelling assumption/physical law/mathematical argument that is employed in ONE sentence and provide the mathematical description of the stress (either ~t, T or ∇ · T) resulting in this step. Where this makes sense, you may also earn marks if you add a description (words and/or mathematics) of the key mathematical argument used in this step. 2