Transport Processes

• A1.18

  Transport Processes

  Part II, 2001

  (i) The diffusion equation for a spherically-symmetric concentration field $C(r, t)$ is

  $$C_{t}=\frac{D}{r^{2}}\left(r^{2} C_{r}\right)_{r},$$

  where $r$ is the radial coordinate. Find and sketch the similarity solution to (1) which satisfies $C \rightarrow 0$ as $r \rightarrow \infty$ and $\int_{0}^{\infty} 4 \pi r^{2} C(r, t) d r=M=$ constant, assuming it to be of the form

  $$C=\frac{M}{(D t)^{a}} F(\eta), \quad \eta=\frac{r}{(D t)^{b}},$$

  where $a$ and $b$ are numbers to be found.

  (ii) A two-dimensional piece of heat-conducting material occupies the region $a \leqslant r \leqslant$ $b,-\pi / 2 \leqslant \theta \leqslant \pi / 2$ (in plane polar coordinates). The surfaces $r=a, \theta=-\pi / 2, \theta=\pi / 2$ are maintained at a constant temperature $T_{1}$; at the surface $r=b$ the boundary condition on temperature $T(r, \theta)$ is

  $$T_{r}+\beta T=0,$$

  where $\beta>0$ is a constant. Show that the temperature, which satisfies the steady heat conduction equation

  $$T_{r r}+\frac{1}{r} T_{r}+\frac{1}{r^{2}} T_{\theta \theta}=0,$$

  is given by a Fourier series of the form

  $$\frac{T}{T_{1}}=K+\sum_{n=0}^{\infty} \cos \left(\alpha_{n} \theta\right)\left[A_{n}\left(\frac{r}{a}\right)^{2 n+1}+B_{n}\left(\frac{a}{r}\right)^{2 n+1}\right]$$

  where $K, \alpha_{n}, A_{n}, B_{n}$ are to be found.

  In the limits $a / b \ll 1$ and $\beta b \ll 1$, show that

  $$\int_{-\pi / 2}^{\pi / 2} T_{r} r d \theta \approx-\pi \beta b T_{1}$$

  given that

  $$\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}}=\frac{\pi^{2}}{8} .$$

  Explain how, in these limits, you could have obtained this result much more simply.

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• A3.16

  Transport Processes

  Part II, 2001

  (i) Incompressible fluid of kinematic viscosity $\nu$ occupies a parallel-sided channel $0 \leqslant y \leqslant h_{0},-\infty<x<\infty$. The wall $y=0$ is moving parallel to itself, in the $x$ direction, with velocity $\operatorname{Re}\left\{U e^{i \omega t}\right\}$, where $t$ is time and $U, \omega$ are real constants. The fluid velocity $u(y, t)$ satisfies the equation

  $$u_{t}=\nu u_{y y}$$

  write down the boundary conditions satisfied by $u$.

  Assuming that

  $$u=\operatorname{Re}\left\{a \sinh [b(1-\eta)] e^{i \omega t}\right\}$$

  where $\eta=y / h_{0}$, find the complex constants $a, b$. Calculate the velocity (in real, not complex, form) in the limit $h_{0}(\omega / \nu)^{1 / 2} \rightarrow 0$.

  (ii) Incompressible fluid of viscosity $\mu$ fills the narrow gap between the rigid plane $y=0$, which moves parallel to itself in the $x$-direction with constant speed $U$, and the rigid wavy wall $y=h(x)$, which is at rest. The length-scale, $L$, over which $h$ varies is much larger than a typical value, $h_{0}$, of $h$.

  Assume that inertia is negligible, and therefore that the governing equations for the velocity field $(u, v)$ and the pressure $p$ are

  $$u_{x}+v_{y}=0, p_{x}=\mu\left(u_{x x}+u_{y y}\right), p_{y}=\mu\left(v_{x x}+v_{y y}\right)$$

  Use scaling arguments to show that these equations reduce approximately to

  $$p_{x}=\mu u_{y y}, \quad p_{y}=0$$

  Hence calculate the velocity $u(x, y)$, the flow rate

  $$Q=\int_{0}^{h} u d y$$

  and the viscous shear stress exerted by the fluid on the plane wall,

  $$\tau=-\left.\mu u_{y}\right|_{y=0}$$

  in terms of $p_{x}, h, U$ and $\mu$.

  Now assume that $h=h_{0}(1+\epsilon \sin k x)$, where $\epsilon \ll 1$ and $k h_{0} \ll 1$, and that $p$ is periodic in $x$ with wavelength $2 \pi / k$. Show that

  $$Q=\frac{h_{0} U}{2}\left(1-\frac{3}{2} \epsilon^{2}+O\left(\epsilon^{4}\right)\right)$$

  and calculate $\tau$ correct to $O\left(\epsilon^{2}\right)$. Does increasing the amplitude $\epsilon$ of the corrugation cause an increase or a decrease in the force required to move the plane $y=0$ at the chosen speed $U ?$

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• A4.19

  Transport Processes

  Part II, 2001

  Fluid flows in the $x$-direction past the infinite plane $y=0$ with uniform but timedependent velocity $U(t)=U_{0} G\left(t / t_{0}\right)$, where $G$ is a positive function with timescale $t_{0}$. A long region of the plane, $0<x<L$, is heated and has temperature $T_{0}\left(1+\gamma(x / L)^{n}\right)$, where $T_{0}, \gamma, n$ are constants $[\gamma=O(1)]$; the remainder of the plane is insulating $\left(T_{y}=0\right)$. The fluid temperature far from the heated region is $T_{0}$. A thermal boundary layer is formed over the heated region. The full advection-diffusion equation for temperature $T(x, y, t)$ is

  $$T_{t}+U(t) T_{x}=D\left(T_{y y}+T_{x x}\right),$$

  where $D$ is the thermal diffusivity. By considering the steady case $(G \equiv 1)$, derive a scale for the thickness of the boundary layer, and explain why the term $T_{x x}$ in (1) can be neglected if $U_{0} L / D \gg 1$.

  Neglecting $T_{x x}$, use the change of variables

  $$\tau=\frac{t}{t_{0}}, \quad \xi=\frac{x}{L}, \quad \eta=y\left[\frac{U(t)}{D x}\right]^{1 / 2}, \quad \frac{T-T_{0}}{T_{0}}=\gamma\left(\frac{x}{L}\right)^{n} f(\xi, \eta, \tau)$$

  to transform the governing equation to

  $$f_{\eta \eta}+\frac{1}{2} \eta f_{\eta}-n f=\xi f_{\xi}+\frac{L \xi}{t_{0} U_{0}}\left(\frac{G_{\tau}}{2 G^{2}} \eta f_{\eta}+\frac{1}{G} f_{\tau}\right)$$

  Write down the boundary conditions to be satisfied by $f$ in the region $0<\xi<1$.

  In the case in which $U$ is slowly-varying, so $\epsilon=\frac{L}{t_{0} U_{0}} \ll 1$, consider a solution for $f$ in the form

  $$f=f_{0}(\eta)+\epsilon f_{1}(\xi, \eta, \tau)+O\left(\epsilon^{2}\right) .$$

  Explain why $f_{0}$ is independent of $\xi$ and $\tau$.

  Henceforth take $n=\frac{1}{2}$. Calculate $f_{0}(\eta)$ and show that

  $$f_{1}=\frac{G_{\tau} \xi}{G^{2}} g(\eta)$$

  where $g$ satisfies the ordinary differential equation

  $$g^{\prime \prime}+\frac{1}{2} \eta g^{\prime}-\frac{3}{2} g=\frac{-\eta}{4} \int_{\eta}^{\infty} e^{-u^{2} / 4} d u .$$

  State the boundary conditions on $g(\eta)$.

  The heat transfer per unit length of the heated region is $-\left.D T_{y}\right|_{y=0}$. Use the above results to show that the total rate of heat transfer is

  $$T_{0}[D L U(t)]^{1 / 2} \frac{\gamma}{2}\left\{\sqrt{\pi}-\frac{\epsilon G_{\tau}}{G^{2}} g^{\prime}(0)+O\left(\epsilon^{2}\right)\right\}$$

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• A1.18

  Transport Processes

  Part II, 2002

  (i) Material of thermal diffusivity $D$ occupies the semi-infinite region $x>0$ and is initially at uniform temperature $T_{0}$. For time $t>0$ the temperature at $x=0$ is held at a constant value $T_{1}>T_{0}$. Given that the temperature $T(x, t)$ in $x>0$ satisfies the diffusion equation $T_{t}=D T_{x x}$, write down the equation and the boundary and initial conditions satisfied by the dimensionless temperature $\theta=\left(T-T_{0}\right) /\left(T_{1}-T_{0}\right)$.

  Use dimensional analysis to show that the lengthscale of the region in which $T$ is significantly different from $T_{0}$ is proportional to $(D t)^{1 / 2}$. Hence show that this problem has a similarity solution

  $$\theta=\operatorname{erfc}(\xi / 2) \equiv \frac{2}{\sqrt{\pi}} \int_{\xi / 2}^{\infty} e^{-u^{2}} d u$$

  where $\xi=x /(D t)^{1 / 2}$.

  What is the rate of heat input, $-D T_{x}$, across the plane $x=0 ?$

  (ii) Consider the same problem as in Part (i) except that the boundary condition at $x=0$ is replaced by one of constant rate of heat input $Q$. Show that $\theta(\xi, t)$ satisfies the partial differential equation

  $$\theta_{\xi \xi}+\frac{\xi}{2} \theta_{\xi}=t \theta_{t}$$

  and write down the boundary conditions on $\theta(\xi, t)$. Deduce that the problem has a similarity solution of the form

  $$\theta=\frac{Q(t / D)^{1 / 2}}{T_{1}-T_{0}} f(\xi)$$

  Derive the ordinary differential equation and boundary conditions satisfied by $f(\xi)$.

  Differentiate this equation once to obtain

  $$f^{\prime \prime \prime}+\frac{\xi}{2} f^{\prime \prime}=0$$

  and solve for $f^{\prime}(\xi)$. Hence show that

  $$f(\xi)=\frac{2}{\sqrt{\pi}} e^{-\xi^{2} / 4}-\xi \operatorname{erfc}(\xi / 2)$$

  Sketch the temperature distribution $T(x, t)$ for various times $t$, and calculate $T(0, t)$ explicitly.

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• A3.16

  Transport Processes

  Part II, 2002

  (i) A layer of fluid of depth $h(x, t)$, density $\rho$ and viscosity $\mu$ sits on top of a rigid horizontal plane at $y=0$. Gravity $g$ acts vertically and surface tension is negligible.

  Assuming that the horizontal velocity component $u$ and pressure $p$ satisfy the lubrication equations

  $$\begin{aligned} &0=-p_{x}+\mu u_{y y} \\ &0=-p_{y}-\rho g, \end{aligned}$$

  together with appropriate boundary conditions at $y=0$ and $y=h$ (which should be stated), show that $h$ satisfies the partial differential equation

  $$h_{t}=\frac{g}{3 \nu}\left(h^{3} h_{x}\right)_{x},$$

  where $\nu=\mu / \rho$.

  (ii) A two-dimensional blob of the above fluid has fixed area $A$ and time-varying width $2 X(t)$, such that

  $$A=\int_{-X(t)}^{X(t)} h(x, t) d x$$

  The blob spreads under gravity.

  Use scaling arguments to show that, after an initial transient, $X(t)$ is proportional to $t^{1 / 5}$ and $h(0, t)$ is proportional to $t^{-1 / 5}$. Hence show that equation $(*)$ of Part (i) has a similarity solution of the form

  $$h(x, t)=\left(\frac{A^{2} \nu}{g t}\right)^{1 / 5} H(\xi), \quad \text { where } \quad \xi=\frac{x}{\left(A^{3} g t / \nu\right)^{1 / 5}}$$

  and find the differential equation satisfied by $H(\xi)$.

  Deduce that

  $$H= \begin{cases}{\left[\frac{9}{10}\left(\xi_{0}^{2}-\xi^{2}\right)\right]^{1 / 3}} & \text { in }-\xi_{0}<\xi<\xi_{0} \\ 0 & \text { in }|\xi|>\xi_{0}\end{cases}$$

  where

  $$X(t)=\xi_{0}\left(\frac{A^{3} g t}{\nu}\right)^{1 / 5}$$

  Express $\xi_{0}$ in terms of the integral

  $$I=\int_{-1}^{1}\left(1-u^{2}\right)^{1 / 3} d u$$

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• A4.19

  Transport Processes

  Part II, 2002

  (a) A biological vessel is modelled two-dimensionally as a fluid-filled channel bounded by parallel plane walls $y=\pm a$, embedded in an infinite region of fluid-saturated tissue. In the tissue a solute has concentration $C^{o u t}(y, t)$, diffuses with diffusivity $D$ and is consumed by biological activity at a rate $k C^{o u t}$ per unit volume, where $D$ and $k$ are constants. By considering the solute balance in a slice of tissue of infinitesimal thickness, show that

  $$C_{t}^{o u t}=D C_{y y}^{o u t}-k C^{\text {out }} .$$

  A steady concentration profile $C^{o u t}(y)$ results from a flux $\beta\left(C^{i n}-C_{a}^{o u t}\right)$, per unit area of wall, of solute from the channel into the tissue, where $C^{i n}$ is a constant concentration of solute that is maintained in the channel and $C_{a}^{\text {out }}=C^{\text {out }}(a)$. Write down the boundary conditions satisfied by $C^{\text {out }}(y)$. Solve for $C^{\text {out }}(y)$ and show that

  $$C_{a}^{o u t}=\frac{\gamma}{\gamma+1} C^{i n}$$

  where $\gamma=\beta / \sqrt{k D}$.

  (b) Now let the solute be supplied by steady flow down the channel from one end, $x=0$, with the channel taken to be semi-infinite in the $x$-direction. The cross-sectionally averaged velocity in the channel $u(x)$ varies due to a flux of fluid from the tissue to the channel (by osmosis) equal to $\lambda\left(C^{i n}-C_{a}^{\text {out }}\right)$ per unit area. Neglect both the variation of $C^{i n}(x)$ across the channel and diffusion in the $x$-direction.

  By considering conservation of fluid, show that

  $$a u_{x}=\lambda\left(C^{i n}-C_{a}^{o u t}\right)$$

  and write down the corresponding equation derived from conservation of solute. Deduce that

  $$u\left(\lambda C^{i n}+\beta\right)=u_{0}\left(\lambda C_{0}^{i n}+\beta\right),$$

  where $u_{0}=u(0)$ and $C_{0}^{i n}=C^{i n}(0)$.

  Assuming that equation $(*)$ still holds, even though $C^{\text {out }}$is now a function of $x$ as well as $y$, show that $u(x)$ satisfies the ordinary differential equation

  $$(\gamma+1) a u u_{x}+\beta u=u_{0}\left(\lambda C_{0}^{i n}+\beta\right)$$

  Find scales $\hat{x}$ and $\hat{u}$ such that the dimensionless variables $U=u / \hat{u}$ and $X=x / \hat{x}$ satisfy

  $$U U_{X}+U=1$$

  Derive the solution $(1-U) e^{U}=A e^{-X}$ and find the constant $A$.

  To what values do $u$ and $C_{i n}$ tend as $x \rightarrow \infty$ ?

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• A1.18

  Transport Processes

  Part II, 2003

  (i) A solute occupying a domain $V_{0}$ has concentration $C(\boldsymbol{x}, t)$ and is created at a rate $S(\boldsymbol{x}, t)$ per unit volume; $\boldsymbol{J}(\boldsymbol{x}, t)$ is the flux of solute per unit area; $\boldsymbol{x}, t$ are position and time. Derive the transport equation

  $$C_{t}+\nabla \cdot \boldsymbol{J}=S$$

  State Fick's Law of diffusion and hence write down the diffusion equation for $C(\boldsymbol{x}, t)$ for a case in which the solute flux occurs solely by diffusion, with diffusivity $D(\boldsymbol{x})$.

  In a finite domain $0 \leqslant x \leqslant L, D, S$ and the steady-state distribution of $C$ depend only on $x ; C$ is equal to $C_{0}$ at $x=0$ and $C_{1} \neq C_{0}$ at $x=L$. Find $C(x)$ in the following two cases: (a) $D=D_{0}, S=0$, (b) $D=D_{1} x^{1 / 2}, S=0$,

  where $D_{0}$ and $D_{1}$ are positive constants.

  Show that there is no steady solution satisfying the boundary conditions if $D=$ $D_{1} x, S=0 .$

  (ii) For the problem of Part (i), consider the case $D=D_{0}, S=k C$, where $D_{0}$ and $k$ are positive constants. Calculate the steady-state solution, $C=C_{s}(x)$, assuming that $\sqrt{k / D_{0}} \neq n \pi / L$ for any integer $n$.

  Now let

  $$C(x, 0)=C_{0} \frac{\sin \alpha(L-x)}{\sin \alpha L}$$

  where $\alpha=\sqrt{k / D_{0}}$. Find the equations, boundary and initial conditions satisfied by $C^{\prime}(x, t)=C(x, t)-C_{s}(x)$. Solve the problem using separation of variables and show that

  $$C^{\prime}(x, t)=\sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi x}{L} \exp \left[\left(\alpha^{2}-\frac{n^{2} \pi^{2}}{L^{2}}\right) D_{0} t\right]$$

  for some constants $A_{n}$. Write down an integral expression for $A_{n}$, show that

  $$A_{1}=-\frac{2 \pi C_{1}}{\alpha^{2} L^{2}-\pi^{2}},$$

  and comment on the behaviour of the solution for large times in the two cases $\alpha L<\pi$ and $\alpha L>\pi$.

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• A3.16

  Transport Processes

  Part II, 2003

  (i) When a solid crystal grows into a supercooled infinite melt, latent heat must be removed from the interface by diffusion into the melt. Write down the equation and boundary conditions satisfied by the temperature $\theta(\boldsymbol{x}, t)$ in the melt, where $\boldsymbol{x}$ is position and $t$ time, in terms of the following material properties: solid density $\rho_{s}$, specific heat capacity $C_{p}$, coefficient of latent heat per unit mass $L$, thermal conductivity $k$, melting temperature $\theta_{m}$. You may assume that the densities of the melt and the solid are the same and that temperature in the melt far from the interface is $\theta_{m}-\Delta \theta$, where $\Delta \theta$ is a positive constant.

  A spherical crystal of radius $a(t)$ grows into such a melt with $a(0)=0$. Use dimensional analysis to show that $a(t)$ is proportional to $t^{1 / 2}$.

  (ii) Show that the above problem should have a similarity solution of the form

  $$\theta=\theta_{m}-\Delta \theta(1-F(\xi))$$

  where $\xi=r(\kappa t)^{-1 / 2}, r$ is the radial coordinate in spherical polars and $\kappa=k / \rho_{s} C_{p}$ is the thermal diffusivity. Recalling that, for spherically symmetric $\theta, \nabla^{2} \theta=\frac{1}{r^{2}}\left(r^{2} \theta_{r}\right)_{r}$, write down the equation and boundary conditions to be satisfied by $F(\xi)$. Hence show that the radius of the crystal is given by $a(t)=\lambda(\kappa t)^{1 / 2}$, where $\lambda$ satisfies the equation

  $$\int_{\lambda}^{\infty} \frac{e^{-\frac{1}{4} u^{2}}}{u^{2}} d u=\frac{2}{S \lambda^{3}} e^{-\frac{1}{4} \lambda^{2}}$$

  and $S=L / C_{p} \Delta \theta$.

  Integrate the left hand side of this equation by parts, to give

  $$\frac{\sqrt{\pi}}{2} \lambda e^{\frac{1}{4} \lambda^{2}} \operatorname{erfc}\left(\frac{1}{2} \lambda\right)=1-\frac{2}{S \lambda^{2}} .$$

  Hence show that a solution with $\lambda$ small must have $\lambda \approx(2 / S)^{\frac{1}{2}}$, which is self-consistent if $S$ is large.

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• A4.19

  Transport Processes

  Part II, 2003

  A shallow layer of fluid of viscosity $\mu$, density $\rho$ and depth $h(x, t)$ lies on a rigid horizontal plane $y=0$ and is bounded by impermeable barriers at $x=-L$ and $x=L$ $(L \gg h)$. Gravity acts vertically and a wind above the layer causes a shear stress $\tau(x)$ to be exerted on the upper surface in the $+x$ direction. Surface tension is negligible compared to gravity.

  (a) Assuming that the steady flow in the layer can be analysed using lubrication theory, show that the horizontal pressure gradient $p_{x}$ is given by $p_{x}=\rho g h_{x}$ and hence that

  $$h h_{x}=\frac{3}{2} \frac{\tau}{\rho g} .$$

  Show also that the fluid velocity at the surface $y=h$ is equal to $\tau h / 4 \mu$, and sketch the velocity profile for $0 \leqslant y \leqslant h$.

  (b) In the case in which $\tau$ is a constant, $\tau_{0}$, and assuming that the difference between $h$ and its average value $h_{0}$ remains small compared with $h_{0}$, show that

  $$h \approx h_{0}\left(1+\frac{3 \tau_{0} x}{2 \rho g h_{0}^{2}}\right)$$

  provided that

  $$\frac{\tau_{0} L}{\rho g h_{0}^{2}} \ll 1$$

  (c) Surfactant at surface concentration $\Gamma(x)$ is added to the surface, so that now

  $$\tau=\tau_{0}-A \Gamma_{x},$$

  where $A$ is a positive constant. The surfactant is advected by the surface fluid velocity and also experiences a surface diffusion with diffusivity $D$. Write down the equation for conservation of surfactant, and hence show that

  $$\left(\tau_{0}-A \Gamma_{x}\right) h \Gamma=4 \mu D \Gamma_{x}$$

  From equations (1), (2) and (3) deduce that

  $$\frac{\Gamma}{\Gamma_{0}}=\exp \left[\frac{\rho g}{18 \mu D}\left(h^{3}-h_{0}^{3}\right)\right]$$

  where $\Gamma_{0}$ is a constant. Assuming once more that $h_{1} \equiv h-h_{0} \ll h_{0}$, and that $h=h_{0}$ at $x=0$, show further that

  $$h_{1} \approx \frac{3 \tau_{0} x}{2 \rho g h_{0}}\left[1+\frac{A \Gamma_{0} h_{0}}{4 \mu D}\right]^{-1}$$

  provided that

  $$\frac{\tau_{0} h_{0} L}{\mu D} \ll 1 \quad \text { as well as } \quad \frac{\tau_{0} L}{\rho g h_{0}^{2}} \ll 1$$

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• A1.18

  Transport Processes

  Part II, 2004

  (i) In an experiment, a finite amount $M$ of marker gas of diffusivity $D$ is released at time $t=0$ into an infinite tube in the neighbourhood of the origin $x=0$. Starting from the one-dimensional diffusion equation for the concentration $C(x, t)$ of marker gas,

  $$C_{t}=D C_{x x}$$

  use dimensional analysis to show that

  $$C=\frac{M}{(D t)^{1 / 2}} f(\xi)$$

  for some dimensionless function $f$ of the similarity variable $\xi=x /(D t)^{1 / 2}$.

  Write down the equation and boundary conditions satisfied by $f(\xi)$.

  (ii) Consider the experiment of Part (i). Find $f(\xi)$ and sketch your answer in the form of a plot of $C$ against $x$ at a few different times $t$.

  Calculate $C(x, t)$ for a second experiment in which the concentration of marker gas at $x=0$ is instead raised to the value $C_{0}$ at $t=0$ and maintained at that value thereafter. Show that the total amount of marker gas released in this case becomes greater than $M$ after a time

  $$t=\frac{\pi}{16 D}\left(\frac{M}{C_{0}}\right)^{2} .$$

  Show further that, at much larger times than this, the concentration in the first experiment still remains greater than that in the second experiment for positions $x$ with $|x|>$ ${ }_{4} C_{0} D t / M$.

  [Hint: $\operatorname{erfc}(z) \equiv \frac{2}{\sqrt{\pi}} \int_{z}^{\infty} e^{-u^{2}} d u \sim \frac{1}{\sqrt{\pi} z} e^{-z^{2}}$ as $\left.z \rightarrow \infty .\right]$

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• A3.16

  Transport Processes

  Part II, 2004

  (i) Viscous, incompressible fluid of viscosity $\mu$ flows steadily in the $x$-direction in a uniform channel $0<y<h$. The plane $y=0$ is fixed and the plane $y=h$ has constant $x$-velocity $U$. Neglecting gravity, derive from first principles the equations of motion of the fluid and show that the $x$-component of the fluid velocity is $u(y)$ and satisfies

  $$0=-P_{x}+\mu u_{y y},$$

  where $P(x)$ is the pressure in the fluid. Write down the boundary conditions on $u$. Hence show that the volume flow rate $Q=\int_{0}^{h} u d y$ is given by

  $$Q=\frac{U h}{2}-\frac{P_{x} h^{3}}{12 \mu}$$

  (ii) A heavy rectangular body of width $L$ and infinite length (in the $z$-direction) is pivoted about one edge at $(x, y)=(0,0)$ above a fixed rigid horizontal plane $y=0$. The body has weight $W$ per unit length in the $z$-direction, its centre of mass is distance $L / 2$ from the pivot, and it is falling under gravity towards the fixed plane through a viscous, incompressible fluid. Let $\alpha(t) \ll 1$ be the angle between the body and the plane. Explain the approximations of lubrication theory which permit equations (1) and (2) of Part (i) to apply to the flow in the gap between the two surfaces.

  Deduce that, in the gap,

  $$P_{x}=\frac{6 \mu \dot{\alpha}}{x \alpha^{3}},$$

  where $\dot{\alpha}=d \alpha / d t$. By taking moments about $(x, y)=(0,0)$, deduce that $\alpha(t)$ is given by

  $$\frac{1}{\alpha^{2}}-\frac{1}{\alpha_{0}^{2}}=\frac{2 W t}{3 \mu L}$$

  where $\alpha(0)=\alpha_{0}$.

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• A4.19

  Transport Processes

  Part II, 2004

  (a) Solute diffuses and is advected in a moving fluid. Derive the transport equation and deduce that the solute concentration $C(\mathbf{x}, t)$ satisfies the advection-diffusion equation

  $$C_{t}+\nabla \cdot(\mathbf{u} C)=\nabla \cdot(D \nabla C)$$

  where $\mathbf{u}$ is the velocity field and $D$ the diffusivity. Write down the form this equation takes when $\nabla \cdot \mathbf{u}=0$, both $\mathbf{u}$ and $\nabla C$ are unidirectional, in the $x$-direction, and $D$ is a constant.

  (b) A solution occupies the region $x \geqslant 0$, bounded by a semi-permeable membrane at $x=0$ across which fluid passes (by osmosis) with velocity

  $$u=-k\left(C_{1}-C(0, t)\right)$$

  where $k$ is a positive constant, $C_{1}$ is a fixed uniform solute concentration in the region $x<0$, and $C(x, t)$ is the solute concentration in the fluid. The membrane does not allow solute to pass across $x=0$, and the concentration at $x=L$ is a fixed value $C_{L}$ (where $\left.C_{1}>C_{L}>0\right)$.

  Write down the differential equation and boundary conditions to be satisfied by $C$ in a steady state. Make the equations non-dimensional by using the substitutions

  $$X=\frac{x k C_{1}}{D}, \quad \theta(X)=\frac{C(x)}{C_{1}}, \quad \theta_{L}=\frac{C_{L}}{C_{1}},$$

  and show that the concentration distribution is given by

  $$\theta(X)=\theta_{L} \exp \left[\left(1-\theta_{0}\right)(\Lambda-X)\right]$$

  where $\Lambda$ and $\theta_{0}$ should be defined, and $\theta_{0}$ is given by the transcendental equation

  $$\theta_{0}=\theta_{L} e^{\Lambda-\Lambda \theta_{0}}$$

  What is the dimensional fluid velocity $u$, in terms of $\theta_{0} ?$

  (c) Show that if, instead of taking a finite value of $L$, you had tried to take $L$ infinite, then you would have been unable to solve for $\theta$ unless $\theta_{L}=0$, but in that case there would be no way of determining $\theta_{0}$.

  (d) Find asymptotic expansions for $\theta_{0}$ from equation $(*)$ in the following limits:

  (i) For $\theta_{L} \rightarrow 0, \Lambda$ fixed, expand $\theta_{0}$ as a power series in $\theta_{L}$, and equate coefficients to show that

  $$\theta_{0} \sim e^{\Lambda} \theta_{L}-\Lambda e^{2 \Lambda} \theta_{L}^{2}+O\left(\theta_{L}^{3}\right) .$$

  (ii) For $\Lambda \rightarrow \infty, \theta_{L}$ fixed, take logarithms, expand $\theta_{0}$ as a power series in $1 / \Lambda$, and show that

  $$\theta_{0} \sim 1+\frac{\log \theta_{L}}{\Lambda}+O\left(\frac{1}{\Lambda^{2}}\right)$$

  What is the limiting value of $\theta_{0}$ in the limits (i) and (ii)?

  (e) Both the expansions in (d) break down when $\theta_{L}=O\left(e^{-\Lambda}\right)$. To investigate the double limit $\Lambda \rightarrow \infty, \theta_{L} \rightarrow 0$, show that $(*)$ can be written as

  $$\lambda=\phi e^{\phi}$$

  where $\phi=\Lambda \theta_{0}$ and $\lambda$ is to be determined. Show that $\phi \sim \lambda-\lambda^{2}+\ldots$ for $\lambda \ll 1$, and $\phi \sim \log \lambda-\log \log \lambda+\ldots$ for $\lambda \gg 1$.

  Briefly discuss the implication of your results for the problem raised in (c) above.

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