Lubrication model of a crawling snail ($U/V=6(1-\alpha )/(4-3\alpha ),\alpha =I_2^2/(I_1{I}_3)$)

For a periodic thin mucus gap $h(x)$ translating as a backward wave relative to a force-free foot, put $I_j={\int }_0^L{h}^{-j}dx$. In the wave frame, the plane and foot have horizontal velocities $V-U$ and $V$. A Couette-Poiseuille flow with periodic pressure has flux $q=(V-U/2)I_2/I_3$. Zero net plane force gives the displayed crawling speed. The leading viscous dissipation per wavelength is $\Phi =2\mu UV{I}_1$.

For $h=h_0[1-A\sin (2\pi x/L)]$, the crawling speed is $U/V=3A^2/(1+2A^2)$. At prescribed $U$, eliminate $V$ to obtain $\Phi =2\mu L{U}^2(1+2A^2)/(3h_0{A}^2\sqrt{1-A^2})$. The derivative in $A^2$ vanishes only at $A^2=1/2$, and the dissipation diverges at both endpoints. The displayed optimum still dissipates $8\sqrt{2}/3$ times the simple Couette flow dissipation of an externally driven flat foot moving at speed $U$ across a uniform film of thickness $h_0$.