4 10/04 W

1. Use the expression derived in class to ballpark the kinematic viscosity $\nu$ of standard air. From that ballpark the kinematic viscosity $\mu$. Use the data from Appendix A in the book and the posted solutions of homework 1. Compare with the exact values.

2. Write out the continuity equation
   
   $$\frac{1}{\rho} \frac{{\rm D}\rho}{{\rm D}t} + {\rm div}\vec v = 0$$
   
   
   in cylindrical and spherical coordinates, for both a compressible and an incompressible fluid. (Here incompressible means that the density of individual fluid particles is constant, not that all fluid particles must have the same density. Usually, when people say incompressible they mean that the density is the same everywhere. But looking in the sea, different regions have different density, because of different salt, but the individual particles are still pretty much incompressible.) Note that
   
   $$\frac{\rm D}{{\rm D}t} = \frac{\partial}{\partial t} + \vec v \cdot \nabla$$
   
   
   and use the appendices. Next assume that $\vec v$ and $\rho$ only depend on $r$ and $t$ (so the flow is cylindrically or spherically symmetric). How do the equations simplify? In the incompressible case, you should see that there is a quantity that must be a constant (at least for any given time) in each flow. What is it? In the compressible steady flows, there is also a quantity that must be constant. What is that? What happens to the radial velocity when going to large $r$?

3. Two-dimensional Poiseuille flow (in a duct instead of a pipe) has the velocity field
   
   $$\vec v = {\widehat \imath}v_{\rm max} \left(1-\frac{y^2}{h^2}\right)$$
   
   
   Here $x$ is along the centerline of the duct and $h$ is half the duct width. Neatly sketch the duct and its velocity profile. Find the strain rate tensor of this flow, and from that the viscous stress tensor, assuming a Newtonian fluid. Compare with the direct expression for the stress tensor given in Appendix C. Evaluate the viscous stress tensor at $y/h=\frac12$. Draw a little cube of fluid at that position in the duct (in cross-section), and sketch all viscous stresses acting on that cube. In a different color, also sketch the inviscid pressure forces acting on it. (assume the pressure has some value $p$.) Also write out the total stress tensor, (including pressure), as given by
   
   $$T_{ij} = -p \delta_{ij} + \tau_{ij}$$
   
   
   Here $\delta_{ij}$ is called the Kronecker delta or unit matrix, it is 1 if $i=j$ and zero otherwise.

4. (5.3) Next assume that the little cube is rotated counter-clockwise over a 30 degree angle around the $z$-axis. Find the total stresses $\sigma$ (including pressure) normal and $\tau$ tangential on the now oblique front surface of the little cube. To do so, first find a unit vector $\vec n$ normal to the surface. Then find the vector stress on the surface using $\vec R=\bar{\bar\tau}\vec n$. Then find the components of $\vec R$ in the direction of $\vec n$ (so normal to the surface), and normal to $\vec n$ (so tangential to the surface).

5. 5.6. $Z$ is the height $h$. The final sentence is to be shown by you based on the obtained result. Hints: take the curl of the equation and simplify. Formulae for nabla are in the vector analysis section of math handbooks. If there is a density gradient, then the density is not constant. And neither is the pressure. $T_{ij}$ is the book’s notation for the complete surface stress, so the book is saying there is no viscous stress. (That is self-evident anyway, since a still fluid cannot have a strain rate to create viscous forces.)