Exercises 6.10 Problems set 7 to hand in

Please hand in the problems in this Section for Set 7 (due in the week commencing 1 Dec).

1. Analysis of flow in a pipe.

It might help to try Exercise 6.9.6 before attempting this problem.

A viscous incompressible fluid flows steadily along an infinitely long pipe of radius \(a\) under a constant pressure gradient \(P=-\de p/\de z\text{.}\) Work in cylindrical polar coordinates \((r,\theta,z)\) with corresponding velocity components \((u_r,u_\theta,u_z)\) with the axis of the pipe aligned with \(r=0\text{.}\)

(a)

Write down the boundary conditions to be applied.

(b)

The velocity is steady and therefore independent of \(t\text{.}\) Explain why it is also reasonable to assume that the velocity profile is independent of \(\theta\) and \(z\text{,}\) and also that the \(\theta\)-component of the velocity is zero.

(c)

Use the continuity equation to show that the \(r\)-component of the velocity is also zero (\(u_r=0\)).

In cylindrical polar coordinates, the continuity equation for an incompressible fluid is

\begin{equation*} \frac{1}{r}\pd{}{r}(ru_r)+\frac{1}{r}\pd{u_\theta}{\theta}+\pd{u_z}{z}=0. \end{equation*}

(d)

Hence the only non-zero component of the velocity is the \(z\)-component, \(u_z\text{,}\) and this only depends on \(r\text{,}\) \(u_z(r)\text{.}\) Check that the profile

\begin{equation*} u_z=\frac{P}{4\mu}\left(a^2-r^2\right) \end{equation*}

satisfies the boundary conditions.

Note that we also need to check that the velocity is smooth at \(r=0\text{.}\) This is easier to see if we rewrite the velocity profile in Cartesian coordinates. Converting to Cartesian coordinates, the flow is \(u=(0,0,u_z)\) with

\begin{equation*} u_z=\frac{P}{4\mu}\left(a^2-(x^2+y^2)\right), \end{equation*}

which is smooth at \(r=0\text{.}\) Note, however, that for example the profile

\begin{equation*} u_z=\frac{P}{4\mu}\left(a-r\right) \end{equation*}

would not be smooth at \(r=0\text{.}\)

(e)

The velocity profile increases as \(P\) increases and decreases as \(\mu\) increases. Explain why this is to be expected. Thus, although you cannot yet prove this is the correct velocity profile (you will soon be able to do so), this profile does look reasonable from what you already know.

(f)

Calculate the three components of the stress vector, the force per unit area that the fluid exerts, on the inside of the pipe. You may assume that the pressure equals \(p_0\) at \(z=0\text{.}\) You may also assume that the rate-of-strain tensor in cylindrical polar coordinates is given by (6.4.7), and that, being a Newtonian fluid, the stress tensor is given by (6.4.2).

(g)

You should have found that the normal stress is radially outward, while the shear stress points along the pipe in the direction of increasing \(z\text{.}\) Explain why you would expect these directions.

2. Rotating flow.

Based on Exercise 6.6 in [4].

Consider an incompressible Newtonian fluid flowing with velocity \(\bu=\bOmega\times\bx\text{,}\) where \(\bOmega\) is a constant vector. Since \(\bOmega\) is constant, for simplicity you may assume that \(\bOmega=\Omega\bk\text{.}\)

(a)

Calculate the velocity components in Cartesian coordinates in the case \(\bOmega=\Omega\bk\text{.}\)

(b)

Show that this flow is incompressible.

(c)

Find the components of the stress tensor.

(d)

Without doing a calculation, explain why you would have expected this answer.

3. Flow outside a rotating cylinder.

Based on problem 2.8 in [4].

An incompressible Newonian fluid fills \(\mathbb{R}^3\text{.}\) An infinite solid cylinder of radius \(a\) is placed in the fluid with its axis on the line \(r=0\) in cylindrical polar coordinates.

(a)

The cylinder rotates at constant angular velocity \(\Omega\) about its axis. In the resulting flow, explain why it is reasonable to assume that \(p=p(r)\) and \(\bu=(u_r(r),u_\theta(r),0)\text{.}\)

(b)

Hence perform a calculation to show that \(u_r=0\text{.}\)

You may use the continuity equation in cylindrical polar coordinates:

\begin{equation*} \frac{1}{r}\pd{}{r}(ru_r)+\frac{1}{r}\pd{u_\theta}{\theta}+\pd{u_z}{z}=0. \end{equation*}

(c)

The Navier–Stokes equations in cylindrical polar coordinates are

\begin{align*} &\rho\left(\pd{u_r}{t}+u_r\pd{u_r}{r}+\frac{u_{\theta}}{r}\pd{u_r}{\theta}+u_z\pd{u_r}{z}-\frac{u_{\theta}^2}{r}\right)\\ =&-\pd{p}{r} +\mu\left(\nabla^2u_r-\frac{u_r}{r^2}-\frac{2}{r^2}\pd{u_\theta}{\theta}\right) +\rho f_r,\\ &\rho\left(\pd{u_{\theta}}{t}+u_r\pd{u_{\theta}}{r}+\frac{u_{\theta}}{r}\pd{u_{\theta}}{\theta}+u_z\pd{u_{\theta}}{z}+\frac{u_ru_{\theta}}{r}\right)\\ =&-\frac{1}{r}\pd{p}{\theta} +\mu\left(\nabla^2u_\theta+\frac{2}{r^2}\pd{u_r}{\theta}-\frac{u_{\theta}}{r^2}\right) +\rho f_{\theta},\\ &\rho\left(\pd{u_z}{t}+u_r\pd{u_z}{r}+\frac{u_{\theta}}{r}\pd{u_z}{\theta}+u_z\pd{u_z}{z}\right)\\ =&-\pd{p}{z} +\mu\nabla^2u_z+\rho f_z, \end{align*}

where the Laplacian is defined by its action on a function \(A(r,\theta,z)\text{:}\)

\begin{equation*} \nabla^2A=\frac1r\pd{}{r}\left(r\pd{A}{r}\right)+\frac1{r^2}\pd{^2A}{\theta^2}+\pd{^2A}{z^2}, \end{equation*}

and \(\bf=(f_r,f_\theta,f_z)\) represents the body forces per unit mass acting on the fluid, which you may neglect in this question.

Show that the flow

\begin{equation*} \bu=\left(0,\frac{a^2\Omega}{r},0\right) \end{equation*}

satisfies the Navier–Stokes equations and boundary conditions.

(d)

If the fluid and cylinder are initially at rest, and the cylinder suddenly starts rotating with angular velocity \(\Omega\) at time \(t=0\text{,}\) explain qualitatively how you would expect the flow to evolve with time.

Prev Top Next