4 9/25 W

1. 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, $y$ across the gap measured from the middle, and $h$ is half the duct height. Neatly sketch the duct and its velocity profile. Find the viscous stress tensor for this flow, assuming a Newtonian fluid, using Table C3 in the book.

   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 aligned with the axis), 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 $p$.)

   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).

   Note: This is essentially question 5.3 from the book, but do not assume that the pressure is 5; just leave it as $p$.

2. The two-dimensional Poiseuille flow of the last question had 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, $y$ across the gap measured from the middle, and $h$ is half the duct height. 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 found in the last question. 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.

3. 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.)

4. 6.1. Use the appendices. Based on the results, confirm that this is incompressible flow, and discuss 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.

5. 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 of the sphere. Then for a small surface area element of the sphere find the pressure and viscous stress forces on the element. Evaluate the components of these forces in the axial direction (the line $\theta=0$) and then integrate these forces over all surface elements to find the total drag force exerted by the fluid 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 $r_0^2\sin\theta{\rm d}\theta{\rm d}\phi$.

6. 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?