Math 705: Fall 2002

HOMEWORK .

Notes and exercises

• 9/9/02 lecture notes [PDF] by Hao Lu and Fabian Waleffe.
• 9/11/2002: See B3.1 (i.e. Batchelor 3.1) for evolution of line and surface elements.
  1. Assume that the scalar function f(x,t) is conserved along particle trajectories, i.e. Df/Dt=0. The gradient of f, grad f, is a vector. Does it evolve like a line element, a surface element or neither?
  2. We have calculated the time rate of change of the integral over a material volume. Calculate the time rate of change of the circulation over a closed material curve. The circulation is the line integral of v.dx where v is the velocity vector.
  3. Sketch the streamlines and calculate the vorticity and rate-of-strain tensor for (a) a rigid body rotation u= (a y, -ax, 0), (b) a shear flow u= (a y, 0 , 0), (c) a stagnation point flow u= (a x, -ay, 0). (Note: (x,y,z) are Cartesian coordinates). Does vorticity imply rotation?
• 9/13/2002: Conservation of mass and momentum. Concept of stress tensor.
• 9/16/2002: Symmetry of stress tensor. Indicial notation. Governing equations.
  1. Derive the equation for angular momentum from Newton's law as we derived for a material volume, conservation of mass and the Reynolds transport theorem.
  2. Deduce symmetry of the stress tensor from the angular momentum equation as in Batchelor 1.3 and Acheson exercise 6.14.
  3. In the `cheap derivation' of the stress tensor symmetry we assumed that the resultant of the forces on the sides of our little triangular region applies at the center of each side and that the moment (torque) of those forces about the middle point was of higher order. Justify.
  4. Use conservation of mass to show that the left-hand side of the momentum equation can be rewritten as: rho Dv/Dt.
• W 9/18/2002: Derivation of momentum equation using fixed control volume. Macroscopic momentum flux: rho vv, a 2nd order tensor. Quick look at kinetic theory of gases, distribution function f(x,v,t), definition of macroscopic density, momentum and stresses, Boltzmann's equation, Maxwell-Boltzmann equilibrium distribution.
• F 9/20/2002: Parametrization of the stress tensor in terms of the velocity gradient: Newtonian fluid. Navier-Stokes equations for an incompressible flow.
  1. Using the Newtonian flow expression for the stress tensor, determine the stress on a surface element with normal n in terms of velocity derivatives and the pressure.
  2. Show that the divergence of the Newtonian stress tensor becomes the Laplacian of velocity for an incompressible flow.
  3. Show that the flows in exercise 3 of 9/11/2002 satisfy the Navier-Stokes equations and determine the pressure for each flow. (these flows extend to infinity where the velocity is infinite, therefore they are physically valid only in a local sense).
• M 9/23/2002: Poisson equation for the pressure. No-slip boundary condition. Plane Couette flow.
  1. For flow in a closed domain D, with the velocity prescribed for all times on the boundary of D, determine the boundary condition for the pressure. (Hint: take the dot product of the momentum equation with the normal to the boundary and deduce a Neumann boundary condition for the pressure. Hence the pressure is known up to an arbitrary constant.)
  2. Consider incompressible flow between two infinite parallel plane separated by a distance H. The velocity is zero then at time t=0 the upper plane moves parallel to itself at constant velocity U. Find a solution of the Navier-Stokes equation in the form (u(y,t),0,0), where y is perpendicular to the planes and u is in the direction of the wall motion. What is the stress on the wall?
  3. Same as previous problem but adapted to flow between two concentric cylinder. At time 0, the outer cylinder impulsively starts rotating at constant speed. What is the stress on the outer cylinder? the inner cylinder?
• W 9/25/2002: Impulsively started shear flow in a bounded channel (problem 2 of 9/23/2002). Convective and diffusive time scales, non-dimensionalization, Reynolds number. Semi-infinite domain: self-similar solution.
• F 9/27/2002: Impulsively started wall in a semi-infinite domain. Self-similar solutions.
  1. Show that the (heat) equation u_t= u_yy admits self-similar solutions of the form u(y,t)=t^-mu f(eta) where eta=y/(t)^1/2 and mu is any constant. (Note that "t" stands here for "nu t", i.e. the kinematic viscosity has been absorbed into "t" which now has units of length-squared!) Each value of mu leads to a different self-similar solution. (These various solutions can be obtained by taking derivatives of the fundamental Gaussian solution exp(-eta²/4)/ sqrt(4 pi t) that corresponds to mu=1/2).
  2. A deeper look: Do the change of variables u(y,t)=v(eta,t).
     • What is the equation for v(eta,t)? Note that this equation is an advection-diffusion equation for the passive scalar v . That means that in the (t,eta) plane, v is given by a balance between advection by the "velocity field" (t,-eta/2) [a stagnation-point type "flow"] and diffusion in the eta direction, and v (which happens to be a velocity in the (x,y) plane) does not influence the "velocity" (t,-eta/2) in the (t,eta) plane, so it is a "passive scalar" (as opposed to an "active scalar" that influences the flow). Note that the flow is `squeezing' v toward eta=0, while diffusion wants to spread v in the eta direction.
     • Separate variables and show that the v(eta,t) equation admits solutions of the form t^-mu f(eta). Show that the f(eta) equation is a Sturm-Liouville problem. The boundary conditions are that f(eta) -> 0 "sufficiently fast" as eta -> +/- infinity (how fast is that?). Deduce that the eigenvalues mu are positive (and they form a countable infinite set, etc...)
     • Find a solution for mu=1/2. Show that the n-th derivative of that fundamental solution is a solution of the same equation but with mu=(n+1)/2. This give you all the solutions as derivatives of a Gaussian. Therefore each of these solutions has the form of a polynomial times a Gaussian.
     • So, let f(eta)=g(eta) exp(-eta²/4) and derive the equation for g(eta). Change variables: eta= 2 xi, and show that g(xi) satisfies Hermite's equation g'' - 2 xi g' + L g =0, where the primes now denote derivatives with respect to xi, not eta. Relate L to mu. Show that Hermite's equation has polynomial solutions (called Hermite polynomials of course) by looking for a Taylor series solution and showing that it terminates when L=2n (n a positive integer).
  3. Take a good look at all exercises in Acheson Chapter 2 and chapter 6.
• M 9/30/2002: Rotational form of NSE, vorticity equation, 2D-3C flows, 2D-2C flows.
  1. Find a 2D-3C solution of the NSE of the form [u(y),-a y, a z] with u(y) --> +/- U as y --> +/- infinity, where "a" is a positive constant. Note the similarity with the v(eta,t) equation above. Your solution is a stretched vortex sheet. Vortex stretching (and/or vorticity advection) balances diffusion. Sketch the flow. This is the planar analog of the axisymmetric Burgers' vortex (Acheson sect. 5.9). All of these simple flows are useful in a local sense.