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EML 5709 Homework 1 Spring 1996

1. At sea level, the number of air molecules per cubic meter and the free path length is meter. For what sort of bodies can we describe the motion of the air as a continuum instead of as the motion of individual molecules? How about at 150 kilometers, where and meter? How about the starship Enterprise?
2. The velocity of a fluid such as air or water is given by

   

   Find and draw the surface forces acting on a small cube of air located at . Does it depend on Stokes' hypothesis?

3. Assume that the temperature in a fluid such as air or water varies with position as

   

   Find the heat fluxes through the sides of a small cube of air located at . Are the heat fluxes through, for example, the left and right hand surfaces of the cube exactly the same?

4. In questions 2 and 3, are the velocity and temperature fields steady or unsteady? Does the temperature of the fluid at vary with time? What are the Eulerian and Lagrangian time derivatives of the temperature at ? What is the acceleration of the fluid at in the x-direction? Does the fluid at this position experience a net force per unit volume?
5. Make question 1.6 in the book.
6. A jet-engine can be roughly modelled as a piece of circular pipe of radius R with its axis along the z-direction. At one end, air enters the engine with density and uniform velocity , at the other end the air exits with velocity . The velocity field inside the pipe is steady. Draw the air which was at time t within the engine at time t+Dt. The mass of this air did not change during time interval Dt, neglecting the added fuel. Explain from the figure why this means that the density at the entrance of the engine must be different from that at the exit. Assume the density is constant at both the entrance and at the exit. Show that the momentum of the air has changed during the time interval Dt, so that the engine produces a thrust force. Compare the results you get using this Lagrangian approach with the control volume expressions derived in class.
7. Repeat question 6 for the case that the engine itself moves toward the left with a velocity . Show that the force exerted on the air in the jet engine, (which is the negative of the thrust force), is now given by

   

   In other words, the relative velocity is used in , but the absolute velocity in the momentum per unit volume .

8. From the energy equation derived in class, show that the sum of kinetic energy and enthalpy per unit mass of air that passes through a constriction in a pipe is unchanged. To do so, ignore gravity, assume that the viscous stresses on the air entering and leaving the pipe can be ignored (although they are important inside the pipe), that there is no heat conduction through the surface of the pipe, and that the flow inside the pipe is steady. Take the flow velocity and thermodynamic properties to be constant at both the entrance and the exit. Discuss each term in the energy equation.
9. 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 and k>0 depend on temperature. Ignore gravity.
10. Assume in addition to question 8 that the pressure p, density , viscosity , and heat conduction coefficient k are constant, and that v and w are zero. Write the five resulting equations for u and T. Based on the results, explain why you expect this class will be a fun continuation of Analysis in Mechanical Engineering.
11. A velocity field is locally approximated by u = x, v=y, w=z. Consider the fluid that is initially at x=y=z=1. How long does it take for the density of this fluid to decrease by a factor 8? Where is the fluid when that happens?
12. Consider a little cube of fluid . Show that the force due to shear stresses on the outside faces of the cube is . So is the force per unit volume. What is the force per unit volume for inviscid fluids? Express using nabla. Write Newton's second law per unit volume for an inviscid fluid. Give the name of this equation.
13. The air inside a balloon is expanding uniformly in all directions. Give the rate of strain tensor and the viscous stress tensor.
14. Show that the dissipation for a Newtonian fluid satisfying Stokes' hypothesis with is always positive.
15. Consider an incompressible fluid extending over entire space. Ignore gravity. Show that if the flow is inviscid, the total kinetic energy of the fluid is preserved.
16. Show that in the absence of heat conduction, the entropy of a Newtonian fluid increases with time.
17. A metal sphere has been dropped in the sea and is sinking at a speed V. Give the boundary conditions for the flow of the sea water around this sinking sphere in a spherical coordinate system with origin at the center of the sphere. What would the inviscid boundary conditions be?

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