7 HW 7

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. A rotating cylindrical axis of radius $r_0$ is enclosed within a stationary concentric pipe of inner radius $r_1$. The angular velocity of the axis is $\Omega_0$. There is fluid in the gap between the axis and pipe. Find the fluid velocity. Make the following assumptions:
   1. Incompressible fluid.
   2. Newtonian fluid.
   3. The velocity only depends on $r$, not $\theta$, $z$, or $t$.
   4. There is no velocity component in the axial direction.
   5. No gravity, (or alternatively, the pressure is the kinetic pressure).
   Show that given only the above assumptions, the pressure must be of the form
   
   $$p=p_0(r) + p_1(t) \qquad p_0(r) = \int_{r_1}^r \frac{\rho v_{\theta}^2}{\bar r} { \rm d}\bar r$$
   
   
   where $p_1(t)$ is the pressure on the pipe surface, and that $v_\theta$ must be of the form
   
   $$v_\theta = \Omega r + \frac{\Gamma}{2\pi r}$$
   
   
   where $\Omega$ and $\Gamma$ are integration constants. The first term above is our beloved solid body rotation with angular velocity $\Omega\hat\imath _z$ around the $z$-axis, while the second term is called an ideal vortex flow with circulation $\Gamma$. At every step that you make, list which one of the above assumptions, or earlier result, you are using.

2. Now put in the boundary conditions for $v_\theta$ to find $\Omega$ and $\Gamma$. Then find the moment around the $z$-axis that the pipe exerts on the fluid. Also find the power that the axis loses due to friction from the fluid.

3. Show the changes that occur in your analysis above if you do not use the kinetic pressure, but include $g_r$, $g_\theta$, and $g_z$ explicitly in the equations. Assume that the axis of the pipe is slanting downwards by an angle $\alpha$. Take the $\theta=0$ plane to be the plane sticking vertically upward from the axis. Warning: $g_r$ and $g_\theta$ are not constants, but depend on $\theta$. (I think it may be easiest first to define a Cartesian coordinate system where $x$ is inside the mentioned vertical plane, normal to $z$, and $y$ is sideways. Note that the height then does not depend on the sideways coordinate $y$. Figure out the height $h$ in terms of $x$ and $z$ by looking in the $x,z$ plane and convert that to cylindrical. Then you can find the gravity vector components by taking a gradient of $-gh$.) Show that the final result is the kinetic pressure as expected from your earlier solution. Do you now appreciate kinetic pressure?

4. A water pipe of radius $r_0$ is sticking straight up. Water is coming out of the top of the pipe and runs down the outer surface of this pipe as a thin sheet of water. You are to find the flow field in the sheet sufficiently far below the top end. Assume that the sheet is sufficiently thin compared to the radius of the pipe that you can approximate it as a sheet along a flat pipe surface, with $z$ the coordinate along the perimeter of the pipe, $y$ the distance from the pipe surface, and $x$ the downward coordinate. So $x,y,z$ can be approximated as Cartesian coordinates. Next make the following assumptions:
   1. Incompressible fluid.
   2. Newtonian fluid.
   3. The streamlines well below the top of the pipe go straight down. (What does that mean?).
   4. The velocity field is steady and independent of $z$, i.e. $\vec v=\vec v(x,y)$.
   5. The velocity field is steady, i.e. $\vec v=\vec v(x,y,z)$.
   6. The air exerts a constant atmospheric pressure $p_{\rm a}$ on the free water surface.
   7. However, the shear stress that it exerts on the surface may safely be ignored.
   Use only the above assumptions. At every step that you make, list which one of the above assumptions, or earlier result, you are using. Do not forget that since there is a free surface, you cannot use the kinetic pressure in this problem. You will need to include gravity explicitly.

5. Based on your solution above, answer a few physical questions:
   1. What is the vertical force per unit span in $z$ that the pipe surface exerts on the water?
   2. Explain why that force has this simple value in physical terms. Use a suitable control volume to do so.
   3. What is the volumetric flow rate $Q$?
   4. Suppose you increase the flow rate coming out of the top of the pipe by a factor 8. What happens to the maximum velocity in the sheet below? What happens to the sheet thickness? Which of the two changes most to accommodate the larger volumetric flow rate?
   5. Form the most meaningful Froude number to describe the sheet flow, following the ideas of, say, pipe flow, and relate it to other nondimensional numbers based on your solution. In particular, consider a Reynolds number based on $Q$ and $r_0$ as something your Froude number might depend upon.