4 HW 4

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Write the momentum equation in the $x$-direction for the same cylindrical coordinates finite volume as used in class. Approximate the time derivative of the $x$-momentum inside the control volume. Angle $\theta$ is the angle between $\vec r$ and the positive $x$-axis, positive counterclockwise. Remember that the unknowns are the cylindrical velocity components $v_r$ and $v_\theta$. The $x$-component of velocity $v_x$ is not, and must be rewritten in terms of the chosen unknowns.

2. Continuing the previous question, approximate the integral giving the net flow of $x$-momentum out of the control volume. Use similar techniques as in class.

3. Continuing the previous question, approximate the $x$-components of the surface and gravity forces. Assume that the stress tensor in cylindrical coordinates at the finite volume center points can be computed. (This would be done using the formulae as found in Appendix B.) Note that the $x$-component of the force is needed. I think it is easier to take $x$-components of the cylindrical-coordinate forces than to transform the cylindrical tensor into a Cartesian one.

4. A cylinder of radius $R$ is surrounded by a fluid. The cylinder is rotating with angular velocity $\Omega$. (a) If the fluid is inviscid, what is the fluid velocity boundary condition at the surface of the cylinder, in terms of cylindrical coordinates? (b) Same question, but viscous fluid. (c) Is it possible for the fluid outside the cylinder to be at rest?

5. Consider the following velocity field:
   
   $$u = C ( x^3 - 3 x y^2) \qquad v = C ( y^3 - 3 x^2 y)$$
   
   
   where $C$ is a constant that only depends on time. This flow field applies for $y>0$; at $y=0$ the flow meets a stationary solid surface along the $x$-axis. Based on the boundary condition, would this be a viscous flow, an inviscid one, or an impossible one (that enters the solid wall)? Now write the complete three-dimensional viscous stress tensor, assuming a Newtonian fluid.

6. Continuing the previous question, assume that the fluid has a constant density $\rho$ and viscosity $\mu$ and that the pressure is given as
   
   $$- p = {\textstyle\frac{1}{2}} \rho C^2 (x^3-3xy^2)^2 + {\t... ...textstyle\frac{1}{4}} \rho \dot C(x^4-6x^2y^2+y^4) + \rho g y$$
   
   
   Note the leading minus sign. Also, gravity $g$ is in the minus $y$ direction and $\dot C$ means the time derivative of $C$. Write out the conservative continuity and $x$- and $y$ momentum equations given in class. Then plug in the given velocity and pressure and the stress tensor and so show that the equations of viscous incompressible flow are satisfied. Note: I think it is quickest not to multiply out products but just use the product rule of differentiation.