Phys 311 - Assignment 6

Due Thursday, Oct 23

1. Bernoulli applied.
   1. Explain, in qualitative terms, how wind blowing across water can generate waves. [Hint: think about the Kelvin-Helmholtz instability.]
   2. If you ever need to paint a large area I recommend renting a paint sprayer. How does a paint sprayer work?
   3. If a strong wind blows past (i.e., parallel to) a window, the glass may break outward. Why?
   4. If you take a shower in a bathtub with a shower curtain you might have noticed that the bottom of the curtain flutters inward during the shower. Why?

2. Lift. In class we derived the velocity field for nonviscous irrotational flow past a cylinder:

   

   

   By integrating the pressure around the cylinder we were able to show that there was no drag on the cylinder. Now suppose that there is some net circulation of the fluid about the cylinder (due to rotation of the cylinder, say), which can be superimposed on the irrotational flow. The circulating flow can be represented as a vortex at the center of the cylinder, which has a velocity field

   

   Therefore, our new velocity field has a radial component given by Eq.(1), and an angular component given by the sum of Eq.(2) and Eq. (3).

   1. Show that the superimposed velocity field still has a net circulation for a path which encloses the cylinder.
   2. Calculate the fluid velocity on the surface of the cylinder, and from Bernoulli's Law the pressure distribution on the surface of the cylinder. From the pressure find the lift and drag on the cylinder. You should find that the drag is still zero while the magnitude of the lift (per unit length of the cylinder) is . How does the direction of lifting force depend upon the sense of circulation of the fluid?
   3. The conclusion here is that a nonviscous fluid can exert a lifting force on the cylinder but not a drag force. Is this paradoxical? [Hint: think about the work done by the lifting force.]
   4. Compute the cartesian components of the velocity and make a couple of plots, using MAPLE, for suitable values of the dimensionless parameter
3. Reynolds number. Think of four fluid flows which are of interest to you, and estimate the Reynolds number for each. For example, if you are a tennis player, try estimating the Reynolds number for one of your serves; if you are interested in biology, estimate the Reynolds number for a paramecium swimming in water. Do flows with high or low Reynolds number prevail?

4. Stokes' formula.
   1. The drag on a sphere which is moving slowly (so that the flow is laminar) in a viscous fluid is given by Stokes' formula, which can be derived from the Navier-Stokes equation. Derive Stokes' formula (up to a constant) from dimensional reasoning. What is the constant [just look it up in the notes]?
   2. If in addition to the viscous force the sphere is acted upon by gravity, what is the terminal velocity of the sphere?
   3. Estimate the terminal velocity of a raindrop in air. Is your estimate consistent with your assumptions?

5. Poiseuille flow. In class we considered Poiseuille flow in a channel. Here I want you to derive and use the analogous result for a pipe.
   1. We want to calculate the flow rate Q (the volume of fluid passing through a cross section of the pipe per unit time) as a function of the radius a of the pipe, the pressure difference across the length of the pipe, the length l of the pipe, and the viscosity of the fluid flowing in the pipe. Use dimensional analysis to deduce the form of the result at low Reynolds numbers, when is proportional to . [Hint: the length l can only enter in the combination , since this is a pressure gradient.] Compare with the exact result [given in the notes].
   2. Is your result applicable to a garden hose (which could be 10 m long and have radius 1 cm, with a pressure overhead of 1 atm at most)?
   3. Is your result applicable to a watery solution flowing in an hypodermic needle? You are expected to supply numbers for the needle and the pressure that can be reasonably applied by a hand-operated syringe.
   4. Is your result applicable to blood flow in the veins? The viscosity of blood is given in the notes and the blood pressure you may remember from your last medical checkup (it was quoted to you in torr, i.e., mm of Hg).