5 10/02 W

1. As seen in class, the second law requires that the dissipation for a Newtonian fluid may not be negative. Examine what constraints this puts on the values of $\mu$ and $\lambda$. To do so, first write out the strain rate tensor and then the compressible Newtonian stress tensor in terms of the strain rates only. (So write ${\rm {div}} \vec{v}$ in terms of the strain rates.) Then note that $\tau_{ij}s_{ij}$ simply means multiplying all corresponding components of the two tensors together and then adding all 9 terms together (much like taking a dot product between vectors). Then explain why $\mu$ must be positive (or at least not negative) because otherwise, say, a Couette flow field in which only $s_{12}=s_{21}$ is nonzero would violate the second law. Then argue that with $\mu$ positive, the worst-case scenario for negative entropy generation occurs when all off-diagonal ($i\ne{}j$) strain rates are zero. So you can from now on limit your considerations to only the terms involving diagonal ($i=j$) strain rates. (But that is expected, since you can always switch to principal axes where there are no off-diagonal terms.) For the diagonal terms the following trick works: your terms should include what can be considered the dot product between the vectors $\vec{v}_1=(s_{11},s_{22},s_{33})$ and $\vec{v}_2=(1,1,1)$. You should know that $\vec{v}_1\cdot\vec{v}_2=\vert\vec{v}_1\vert\vert\vec{v}_2\vert\cos(\theta)$. Here $\cos^2(\theta)$ is no bigger than one, and it is one only if the two vectors are parallel. (In general this is known as the Cauchy-Schwartz inequality.) From that argue that $\lambda$ may not be more negative than $-\frac23\mu$. In the marginal case of Stokes' hypothesis that $\lambda$ is $-\frac23\mu$, there is one particular straining in which the dissipation, though not negative, is zero. Show that that corresponds to a uniform expansion or compression in all directions. Apparently, such an expansion is perfectly reversible according to Stokes, unlike, say, a unidirectional expansion in the $x$-direction only. What do you think of that?

2. Write down the worked-out mathematical expressions for the integrals requested in question 5.1. This is a good exercise in identifying various surface and volume integrals in integral conservation laws. Explain their physical meaning, if any. Don't worry about actually doing the integrations. However, show integrands and limits completely worked out.

   Take the surfaces $S_{I}$, $S_{II}$, $S_{III}$, and $S_{IV}$ to be one unit length in the $z$-direction. (To figure out the correct direction of the normal vector $\vec n$ at a given surface point, note that the control volume in this case is the right half of the region in between two cylinders of radii $r_0$ and $R_0$ and of unit length in the $z$-direction. The vector $\vec n$ is a unit normal vector sticking out of this control volume.)

3. 5.14. (The class question but with the nozzle turning the flow 120 degrees.) Find both the horizontal and vertical components of the force. Make sure that you clearly define what control volume you are using, as there is no unique choice.