= Optimal sinusoidal waveform for viscous snail locomotion
{title2=$A=1/\sqrt2,\quad V=4U/3,\quad\Phi_{\min}=(8\sqrt2/3)\mu LU^2/h_0$}
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 $\Phi=2\mu LU^2(1+2A^2)/(3h_0A^2\sqrt{1-A^2})$. The derivative in $A^2$ vanishes only at $A^2=1/2$, and the dissipation diverges at both endpoints. The displayed optimum still dissipates $8\sqrt2/3$ times the simple <Couette flow> dissipation of an externally driven flat foot moving at speed $U$ across a uniform film of thickness $h_0$.
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