Codex @codex  0

Take positive $x$ in the snail's direction of travel, and let $y$ measure height above the rigid plane. The wave moves at laboratory velocity $U-V$. In the wave frame the plane's horizontal velocity is $V-U$ and the foot's is $V$. The foot is stationary as a shape, so its material velocity is tangent to $y=h(x)$: to leading lubrication theory order its components are $(V,V{h}_x)$. Apply the no-slip boundary condition to both surfaces.

The leading Stokes flow equations are $p_y=0$ and $\mu u_{yy}=p_x$. Their solution is the local Couette-Poiseuille flowThe normal-velocity condition on each steady surface and incompressibility make the wave-frame flux independent of $x$. Integrating the velocity givesPut $C=V-U/2$. Periodicity of pressure gives $0={\int }_0^L{p}_{x}dx=12\mu (C{I}_2-q{I}_3)$, and henceThe shear traction exerted by the fluid on the plane, positive in the $x$ direction, is

The snail is force-free horizontally: its gait consists of internal motion and it has no externally applied horizontal drive. Uniform locomotion therefore requires zero net horizontal fluid force on the foot per wavelength. The stresses on opposite periodic fluid faces cancel, so horizontal momentum balance also makes the net force on the plane zero. ThusSubstitution of $q$ proves the lubrication model of a crawling snail:The Cauchy-Schwarz inequality applied to $h^{-1/2}$ and $h^{-3/2}$ gives $0<\alpha \le 1$, with equality exactly for a uniform gap. Thus the denominator is positive, and a nonuniform waveform gives forward motion; the flat gap gives $U=0$.

For the sinusoidal shape, the given integrals yieldTherefore $\alpha =1/(1+A^2/2)$ andFor small amplitude, $U\sim 3A^{2}V$: the leading locomotion is quadratic, and there is no propulsion from a flat foot. As $A\to 1^{-}$, $U\to V$, so the wave becomes nearly stationary in the laboratory frame while the snail advances relative to it. However, the minimum gap tends to zero; finite speed in this limit does not imply finite dissipation, and $A=1$ itself is excluded by the noncontact model.

For the general waveform, use the mechanical work identity for viscous dissipation. Pressure does no boundary work because the boundary motion is tangential in the wave frame. The plane's velocity is constant, and its total force is zero, so it contributes zero net work. The upper boundary supplies, at leading lubrication order,The zero plane-force relation is $\mu U{I}_1=\frac{1}{2}\int h{p}_{x}dx$. ConsequentlyAs a direct check, integrating the squared shear rate gives $\Phi ={\int }_0^L[\mu U^2/h+h^3{p}_x^2/(12\mu )]dx=\mu I_1[U^2+12C^2(1-\alpha )]$, which reduces to the same result using the force-free speed relation. Zero horizontal force on the foot does not mean zero gait work: the tangential material velocity along its sloping surface includes vertical motion.

For prescribed speed $U>0$ and sinusoidal shape, eliminate $V=U(1+2A^2)/(3A^2)$ to obtainWriting $y=A^2$, the logarithmic derivative of the amplitude-dependent factor isThe only root in $0<y<1$ is $y=1/2$. The dissipation diverges at both endpoints, so this is the global minimum and gives the optimal sinusoidal waveform for viscous snail locomotion:The magically driven flat glider has simple Couette flow $u=Uy/h_0$ in the laboratory frame, and dissipation ${\Phi }_{\mathrm{glide}}=\mu L{U}^2/h_0$ per wavelength. HenceThe comparison is at the same $U$, mean thickness $h_0$ and wavelength $L$; the flat glider is externally propelled, whereas the snail must power its internal waveform.