Viscous energy dissipation in slender channels with porous or semipermeable walls

We study the viscous dissipation in pipe flows in long channels with porous or semipermeable walls, taking into account both the dissipation in the bulk of the channel and in the pores. We give simple closed-form expressions for the dissipation in terms of the axially varying flow rate $Q\left(x\right)$ and the pressure $p\left(x\right)$, generalizing the well-known expression $\dot{W}=Q\mathrm{\Delta }p=R{Q}^2$ for the case of impenetrable walls with constant $Q$, pressure difference $\mathrm{\Delta }p$ between the ends of the pipe and resistance $R$. When the pressure $p_0$ outside the pipe is constant, the result is the straightforward generalization $\dot{W}=\mathrm{\Delta }\left[(p-p_0)Q\right]$. Finally, applications to osmotic flows are considered.

Figure 1

A tube of length $L$ and circular cross section of radius $r_0$ is filled with a liquid of viscosity $\eta$ and density $\rho$. Liquid is injected from the surrounding medium (arrows) and creates a bulk flow of speed $u$ along the positive $x$ direction of the tube.

Figure 2

A sketch of the flow in the membrane pores showing a tube with pores of radius $a$ and length $d$.