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EML 5709 Homework 1 Spring 1995

  1. Find the free path length tex2html_wrap_inline33 between molecular collisions in the athmosphere at sea level under standard conditions. Compare with the typical dimension of a rocket ( tex2html_wrap_inline35 ). How many molecules in a cube of dimension tex2html_wrap_inline37 ? Repeat for an height of 150 km above sea level.
  2. Consider fluid in a gap between two horizontal plates. The bottom plate is at rest in the x,z-plane, while the top plate moves in the x-direction with speed U. The gap between the plates has size h and the plates have surface area S. Assume the fluid velocity in the gap changes linearly, tex2html_wrap_inline49 . Also, the thermodynamic pressure in the fluid equals the athmospheric pressure tex2html_wrap_inline51 and the fluid is Newtonian. Draw and list all forces acting on a little 3D cube of fluid in the gap. Also draw the fluid forces acting on the top and bottom plates themselves.
  3. Assume for the same flow that the bottom plate is at temperature tex2html_wrap_inline53 and the top plate at temperature tex2html_wrap_inline55 , and that the temperature varies linearly through the gap. Draw and list all heat flowing in or out a small 3D cube of fluid.
  4. Fluid mechanics is not the only field that has to deal with convective time derivatives. Write down the rate of change of potential energy with time for a particle of electric charge tex2html_wrap_inline57 that moves through an electro-static field of voltage V= x + 2y + 3z + 4t. Take the time to be t=2 and assume that the particle is at the origin at that time and moving with speed tex2html_wrap_inline63 .
  5. Make question 1.7 in the book.
  6. Consider an arbitrary region tex2html_wrap_inline65 fixed in space. At any time t, the net force tex2html_wrap_inline69 on this region can be found as the change in linear momentum of the Lagrangian region that coincides with tex2html_wrap_inline65 at the given time t. Repeating the derivation given in class, derive the relationship between the force tex2html_wrap_inline75 in x-direction and the time derivative of the linear momentum in x-direction contained in tex2html_wrap_inline65 . In particular, determine the integrals that correct for the fact that momentum can flow out of tex2html_wrap_inline65 through the outside surface.
  7. In numerical computations, it is often convenient to compute flows in moving coordinate systems. Repeat the derivation given in class to derive the equation of conservation of mass for a region whose outside surface has an arbitrary, given, variable surface velocity tex2html_wrap_inline85 .
  8. A fluid contains a pollutant. If the ratio of local pollutant mass to local total fluid mass is f(x,y,z,t), write the law of conservation of pollutant mass in integral form, assuming that the amount of pollutant mass in any given total mass of fluid is constant. Take tex2html_wrap_inline89 to be the total local fluid mass per unit volume.
  9. Convert the equation of the previous question to differential form.
  10. Write the five conservation equations in differential form for the case that all flow quantities depend on only one spatial dimension. (In other words, everything depends on y and t only.) Write out completely in terms of u, v, and w and thermodynamic quantities. Assume an isotropic Newtonian viscous fluid satisfying Fourier's law, where tex2html_wrap_inline101 and k>0 depend on temperature.
  11. The assumptions in the previous question would be appropriate for Couette flow in the gap between two very large plates. Assume that the flow is steady, that the bottom plate is at y=0 and is at rest, while the top plate moves in the x-direction with unit velocity. From the flow equations derived in the previous question, show that both v and w must be zero, while the pressure and the stress tex2html_wrap_inline111 are constant. In other words, only the temperature and u can vary accross the gap.
  12. From the results of the previous question, derive an equation involving the temperature, but not u. Reduce to a first order equation and solve. Show that if the Prandtl number tex2html_wrap_inline117 and the specific heat tex2html_wrap_inline119 are constant (such as for air under reasonable temperatures), the heat flux tex2html_wrap_inline121 is of the form tex2html_wrap_inline123 with A and tex2html_wrap_inline53 constants.
  13. In the ocean, the water density varies due to variations in salt content. Is the velocity solenoidal?

  14. Consider a little cube of fluid tex2html_wrap_inline129 . Show that the force due to shear stresses on the outside faces of the cube is tex2html_wrap_inline131 .

  15. Show that tex2html_wrap_inline133 represents a uniform expansion in all directions.
  16. If we look at a small fluid element is a suitably oriented coordinate system, the stress tensor will be diagonal (only principal stresses). Show that the rate of strain tensor is also diagonal in this system.
  17. Show that the dissipation for a Newtonian fluid satisfying Stokes' hypothesis with tex2html_wrap_inline101 is always positive.
  18. Show that in the absence of heat conduction, the entropy of the fluid can only increase.
  19. You want to compute the flow about a small ship bobbing around on waves. Discuss boundary conditions to impose on the water flow.

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Author: Leon van Dommelen