5 10/11 W

1. 6.1. Use the appendices. Based on the results, discuss whether this is incompressible flow, and in what direction the viscous stresses on the surface of the sphere are. Also state in which direction the inviscid stress on the surface is.

2. Noting that in the above flow, the pressure is given by $p-p_\infty=-3\mu U\cos(\theta)r_0/2r^2$, evaluate the pressure and shear stresses on the surface. Then find and integrate the components of the pressure and viscous stresses in the axial direction (the line $\theta=0$) on the surface of the sphere to find the viscous drag force on the sphere. Thus recover the Stokes formula for the drag of a sphere at low Reynold number, $F_{\rm {D}}=6\pi\mu{}r_0U$. Note that the surface element on a spherical surface of radius $r_0$ is given by $r_0^2\sin\theta{\rm d}\theta{\rm d}\phi$.

3. 6.2 Discuss your result in view of the fact, as stated in (6.1), that the Reynolds number must be small for Stokes flow to be valid. So what about the dynamic pressure, (as produced by the kinetic energy of the fluid particles), in Stokes flow?

4. 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?