1 HW 1

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. What is the key number that determines whether the continuum assumption is valid for a gas? Explain its definition in detail. So for a body moving through air at standard sea-level conditions, when is the continuum approximation valid?

2. Couette flow is the viscous flow in the gap between two horizontal plates, the top one of which is moving in the positive $x$-direction with some velocity $U$. If $y$ is distance measured from the bottom plate at $x=0$, for laminar flow the fluid velocity is given by
   
   $$\vec v= \hat\imath u \qquad u = \frac{U}{h} y$$
   
   
   where $h$ is the distance between the plates. Sketch the plates and the fluid velocity vectors at some line of constant $x$ (giving the velocity profile). For a simple two-dimensional unidirectional flow like this,
   
   $$\tau_{yx} = \mu \frac{\partial u}{\partial y}$$
   
   
   where the constant $\mu$ is called the kinematic viscosity of the fluid. Also, because the positive and negative $z$ directions are equivalent in this problem, shear forces in the $z$-direction can only be zero, so
   
   $$\tau_{yz} = \tau_{xz} = 0$$
   
   
   At a point at a position 0.75 $h$ above the bottom plate, the stress tensor is given by
   
   $$\bar{\bar\tau} = \left( \begin{array}{ccc} -p & \mu U/h & 0 \\ \mu U/h & -p & 0 \\ 0 & 0 & -p \end{array} \right)$$
   
   
   Here $p$ is the pressure (which is an inviscid effect). Fully explain every term in this stress tensor. Draw a little cube around the given point. Clearly show all stress components on the surfaces of this little cube, in terms of the quantities above, after drawing a magnified cube if needed.

3. Going back to the previous question, suppose there is a little area $A$ going through the considered point above, parallel to the $z$-axis, but rotated 30$^\circ$ counterclockwise from the positive $y$-axis. (So take a small surface normal to $\hat\imath$ and then rotate it 30 degrees around the $z$-direction.) Find the stress force per unit area $\vec R$ acting on that area $A$. Then find the components of the stress force normal and tangential to area $A$. Comment on the tangential component of the pressure force and the normal component of the viscous stress force.

4. Going back to the second-last question, suppose you rotate the coordinate system $xyz$ 45$^\circ$ counterclockwise around the $z$-direction to get an $x'y'z'$ coordinate system. Find the stress tensor in this rotated coordinate system. Is the $x'y'z'$ coordinate system the principal axis system? Draw again a little cube around the given point with the stresses on its surfaces. But this time show the cube and correct stress components aligned with the new coordinate system.