Home Work for MATH 6750: Fluid DynamicsHW 1. Due: Wednesday, 9/5/07Problem 1. Show that the pressure in the ocean is p = p0+ ρgh, where g is thegravity field, h is the dep th, and p0is the atmospheric pressure.Problem 2. Consider a fluid in a solid body rotation (e.g. fluid in a bucket on auniformly rotating turntable). Does the definition of a fluid imply that the stress in thefluid should be independent of the viscosity? Calculate the stress if the rotation speed ωand the gravity field g are known. Can you conclude that γ = µ without showing that thestress is a symmetric matrix?Problem 3.Consider a simp le sh ear flow, whenu =u(y)00.Calculate the stress tensor.HW 2. Due: Wednesday, 9/12/0 7Problem 1. Viscosity due to the exchange of momentum.Two railroad cars are moving in the same direction side by side on parallel tracks.Their speeds are slightly different: u1= 20m/sec, and u2= 21m/sec. In each car, a manis shoveling coal and pitching it across to the neighboring car. The rate of coal trans ferµ = 2kg/sec is th e same for each car (so that the net amount of coal on each car remainsconstant). Find the force on each car, caused by this exchange of coal.Problem 2..Consider the plane Coette flow; the upper plate is moving with the speed u0, whilethe lower plate is at rest, What force do we need to apply to the upper plate to move itwith the speed u0? The flu id viscosity µ, plate area A, and th e distance d between platesare given.Problem 3. Poiseuille flow.Consider a fluid flow in a long tube (say, of length l) of circular cross-section under theaction of a difference ∆p between the pressures imposed at the two ends of the tube.Find the dependence of the flux Q (of the fluid volume through any cross-section of thetube) vs. the pressure gradient G = ∆p/l.Find the total force exerted on the tube by the fluid.(The viscosity µ of the fl uid and the r ad ius a of the tube are known.)1HW 3. Due: Wednesday, 9/19/0 7Problem 1. Can normal stress be caused by the sheer viscosity µ ?When we pour a very viscous liquid, such as honey from a jar, the column of fluid obviouslydoes not accelerate as the local acceleration of gravity: A falling ball accelerates much morerapidly than the honey falling from the jar. The force that retards the honey is due to tothe normal viscous stress: If one were to cu t the column instantaneously, the two partswould separate because the normal viscous stress would no longer retard the lower p art.As a simple model of this phenomenon, consider the stationary flow with cylindricalsymmetryu =f(r)xf(r)yg(z), r =qx2+ y2.where f (r) and g(z) are some unknown functions. Using the incompressibility of thehoney, deduce thatu =−ax−ay2az.where a is some positive constant (which ch aracterizes the rate of expansion). Calculatethe stress tensor and show that there is the normal viscous stress, retarding the accelerationof the honey.Problem 2. Stress at a rigid boundary in an incompressible fluid.Consider the stress acting on an element of a rigid boundary from the fluid. Show thatif the fluid is incompressible, the normal component of the stress is determined only bythe pressure, it does not depen d on the viscosity (the viscosity determines only tangentialcomponents of the stress). [Use a curvilinear co-ordinate system of which one coordinatesurface coincides with the rigid boundary.]Problem 3.Assume that the temperature of the atmosphere varies with height z asT = T0+ Kz.1. Show that the pressure varies with height asp = p0T0T0+ KzgmKR.Here g is the gravity constant, m is the mole weight, and R is the universal gas constant.2. Suppose T0= 15 and K = −0.001 (T is in degrees Celsius and height z is in meters).Is this atmosphere stable?2HW 4. Due: Wednesday, 9/26/0 7Problem 1. The shape of the free surface in a cup of tea (when wemix sugar).Consider a bucket of water uniformly rotating with angular velocity ω in a uniformgravity field with free fall acceleration g. What is the shape of the free surface?What is wr ong with th e following approach?Consider the frame such that axis Z is directed up, against the gravity, and it is the axisof rotation.By Bernoulli’s theorem, p/ρ + u2/2 + gz is constant. So, the surfaces of constant pressurearez = constant −ω2(x2+ y2)2g.But this means that the water surface is at its highest in the middle.Problem 2.A mercury barometer in a non-moving f rame show s pressure h0mm Hg.Now suppose the barometer is fixed vertically in a frame of reference which has ahorizontal acceleration a. What is the reading of the barometer?What would be the reading if the barometer hung from a fixed point in that frame?Problem 3. Consider a container in the shape of part of a coneharea AWhen it is full of water, the water pressure on its base is ρgh, and the total force on itsbase is ρghA, which exceeds the weight of water in the container. Explain this paradox.3HW 5. Due: Wednesday, 10/3/0 7Problem 1. Uniqueness of potential flow.Consider potential flow u = ∇φ in a simply connected finite domain D.∆φ = 0 in D,u · n = 0 on the boundary of D.Show that the flow is unique.Problem 2. Rotating cylinder in a viscous incompressible fluid.Suppose a long cylinder of some radius a is rotating around its axis with angular veloc-ity Ω (consider 2D motion, assuming that the flow is independent of the third coordinatealong the length of the cylinder).1. Find the steady velocity field of the air.2. Sh ow that any initial velocity field with circular symmetry (satisfying the sameboundary conditions) would eventually evolve to th is steady field.3. Is this steady velocity field irrotational?4. Find its potential and the stream function.HW 6. Due: Wednesday, 10/17/07Problem 1. Guess ψ instead of φ? When we guessed the form of the velocitypotential for the fluid flow around a circle, we were arguing in the following way:1. The scalar quantity φ depends on two vectors r and U. So, it should depend ontheir dot product U · r = U cos θ.2. Since the velocity field u = ∇φ satisfies the linear equation ∆φ = 0, the potentialshould depend on U linearly. Thus, φ = f(r)U cos θ, where f (r) is u ndeterminedfunction.Why can’t we apply the same argument to the streamfunction, and guess that ψ =g(r)U cos θ where g(r) is another undetermined function?Problem 2. Flow at a wall angle. Find irrotational flow between two h alf-planeswith coinciding bou ndaries. (Suppose their common bound