Warning 6.11.1.
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Exercises 6.11 Problems set 8 to hand in
Please hand in the problems in this Section for Set 8 (the final set of problems, due in the week commencing 8 Dec).
1.
A tethered sphere of radius \(a\) is surrounded by a flowing incompressible Newtonian fluid of density \(\rho\) and shear viscosity \(\mu\text{.}\) In the far field, the fluid velocity is uniform and equal to \(U\) (right to left).
(a)
Write down an expression for the Reynolds number \(\mathrm{Re}\) of the fluid.
(b)
Describe what you can infer about the Reynolds number in each case (a), (b) and (c) from these pictures. Explain your reasoning.
2. Channel flow.
A Newtonian viscous fluid flows steadily in the \(+x\)-direction in two dimensions between plates at \(y=\pm d\text{.}\) You may neglect gravity in this problem.
(a)
Write down the boundary conditions at each plate.
(b)
Make the assumption that the flow is fully developed and use the continuity equation to show that \(v=0\) everywhere.
(c)
Simplify the Navier–Stokes equations using the assumptions and \(v=0\) to show that
\begin{equation*}
\mu\pd{^2u}{y^2}=-G,
\end{equation*}
where \(G\) is a constant that you should specify.
(d)
Solve this applying the boundary conditions to find the flow profile.
(e)
Find the stress exerted by the fluid on both plates. If each plates has (large) area \(A\text{,}\) what is the \(x\)-component of the tethering force required to keep each plate fixed in place?
3. Wavy-walled pipe.
A long cylindrical pipe with slowly varying radius \(R(z)\) is aligned with its axis pointing in the \(z\)-direction. A much narrower cylinder of constant radius \(a\) is placed coaxially within the larger cylinder. The narrow cylinder has porous walls and emits fluid at a constant velocity \(U\) perpendicular to its surface. A fluid flows steadily in the gap between the two cylinders.
In this question, you will work in cylindrical polar coodinates \((r,\theta,z)\text{,}\) centred on the axis of the pipe.
Despite the fact that there is flow out from the inner cylinder in the radial direction, you may assume that the flow in the annular gap between the cylinders is unidirectional along the pipe in the \(+z\)-direction. (In reality there will be a thin layer next to the inner cylinder in which this is not true.) Furthermore, you may assume that the flow is uniform across the cross-section, that is \(\bu=(u(z),0,0)\text{.}\)
(a)
Consider a narrow slice of the cylinder between \(z\) and \(z+\de z\text{.}\) Find an expression for the volumetric flux of fluid through the surfaces \(z\) and \(z+\de z\text{.}\)
(b)
Now find an expression for the volumetric flux of fluid through the section of the inner cylinder between \(z\) and \(z+\de z\text{.}\)
(c)
Now use the principle of mass conservation to show that
\begin{equation*}
\dd{u}{z}=\frac{2(aU-RR'u)}{R^2-a^2}.
\end{equation*}
Note that you can actually solve this equation by instead considering the volume between \(z=0\) and \(z=z_0\text{.}\) This volume has four surfaces: the inner cylinder surface \(r=a\text{,}\) the outer cylinder surface \(r=R(z)\text{,}\) and the two cross-sectional surfaces at \(z=0\) and \(z=z_0\text{.}\) The inner cylinder surface gives a volume flux in of \(2\pi aUz_0\text{,}\) the outer cylinder surface gives no volume flux, the surface \(z=0\) gives a volume flux in of \(\pi(R(0)^2-a^2)u(0)\text{,}\) and the surface \(z=z_0\) gives a volume flux out of \(\pi(R(z_0)^2-a^2)u(z_0)\text{.}\) Equating total volume flux in and volume flux out gives
\begin{equation*}
2\pi aUz_0+\pi(R(0)^2-a^2)u(0)=\pi(R(z_0)^2-a^2)u(z_0),
\end{equation*}
and hence
\begin{equation*}
u(z_0)=\frac{2aUz_0+(R(0)^2-a^2)u(0)}{R(z_0)^2-a^2}.
\end{equation*}
This is true for all \(z_0\text{,}\) hence (setting \(z=z_0\))
\begin{equation*}
u=\frac{2aUz+(R(0)^2-a^2)u(0)}{R(z)^2-a^2}.
\end{equation*}
This was not asked for in this question.
(d)
Write down an expression for the Reynolds number of this flow, explaining your reasoning. You may assume that the pipe has length \(L\) and that \(R_0\) is a typical value of \(R(z)\text{.}\) Your answer should be in terms of the problem parameters \(\rho\text{,}\) \(\mu\text{,}\) \(a\text{,}\) \(U\text{,}\) \(L\) and \(R_0\text{.}\)
(e)
State in a sentence the conditions under which it might be appropriate to assume that the velocity is uniform across the cross section. You do not need to do a calculation.
(f)
In reality, in a sentence, what would you observe in the flow near the outer wall?
