(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 $x \rightarrow \infty$ ?