Taylor swimming sheet

The Taylor swimming sheet is an infinite two-dimensional microswimmer whose prescribed traveling deformation drives Stokes flow. A transverse wave $y_s=ϵ\sin (x-t)$ of small amplitude swims opposite to its direction of propagation, with speed $U={ϵ}^2/2+O({ϵ}^4)$ when lengths are scaled by inverse wavenumber and velocities by wave speed. Its biharmonic stream function for planar Stokes flow has first-order part ${\psi }_1=(1+y)e^{-y}\sin (x-t)$.

A small transverse Taylor swimming sheet in a Brinkman medium obeys $({\nabla }^2-A^2){\nabla }^2\psi =0$ with $A=\alpha /k$. Its no-slip material velocity is imposed on the moving sheet. The matrix's preferred frame and the mean force-free condition must be included when finding the second-order swimming speed.

For $s=\sqrt{1+(\alpha /k{)}^2}$, the leading work per projected area on one side of a transverse Brinkman swimming sheet is $\mu b^{2}k{\omega }^{2}s(s+1)/2$. Two fluid sides double it. Work includes viscous dissipation and drag against the matrix, giving a greater fixed-stroke cost than in the unscreened fluid.

The leading fixed-stroke speed of a transverse Brinkman swimming sheet is enhanced over the pure-fluid value by $\sqrt{1+(\alpha /k{)}^2}$. It follows from the mean displaced-boundary velocity and zero mean traction. The expansion is at fixed screening parameter and does not assert an enhancement at fixed available motor power.

With $s=\sqrt{1+A^2}$, a dimensionless decaying first-order transverse sheet has $f=(s{e}^{-y}-e^{-sy})/(s-1)$, $f(0)=1$, $f^{\prime }(0)=0$, and pressure $-s(s+1)e^{-y}\cos (x-t)$. The coincident-root limit is $(1+y)e^{-y}$, the ordinary transverse mode of a Taylor swimming sheet.

A transversely waving Taylor swimming sheet can have a Navier slip boundary condition relating surface tangential velocity to shear and a slip length $\gamma$. The small-amplitude parameters are $ϵ=k{y}_0$ and $\delta =k\gamma$. The first-order velocity is unchanged by slip, but evaluation of shear at the displaced surface changes the second-order mean velocity.

At second order, the displaced-boundary Navier slip boundary condition gives $({\psi }_2{)}_Y-\delta [({\psi }_2{)}_{YY}-({\psi }_2{)}_{XX}]=(1+2\delta ){\sin }^2(X-\tau )$ at $Y=0$. Its mean shear is zero in a bounded half-space flow, so the Navier-slip Taylor swimming sheet dimensionless speed is $U^{\ast }={ϵ}^2(1+2\delta )/2+O({ϵ}^4)$. The sheet travels opposite to the wave. This fixed-slip asymptotic expansion predicts a speed enhancement factor $1+2\delta$.

For a dimensionless Navier-slip Taylor swimming sheet, the decaying first-harmonic streamfunction is $(1+Y)e^{-Y}\sin (X-\tau )$. Its tangential surface velocity and surface shear both vanish at $Y=0$. Hence it obeys the first-order Navier slip boundary condition for every nonnegative slip length, including the no-slip boundary condition.

For tangential displacement $ϵa\sin (k(x-t))$, the first-order decaying streamfunction is $-kay{e}^{-ky}\cos (k(x-t))$. It obeys tangential material velocity $-ka\cos (k(x-t))$ and zero normal velocity on the reference boundary. Its second-order mean swimming coefficient is $-k^2{a}^2/2$.

For normal displacement $ϵb\sin (k(x-t))$, the first-order decaying streamfunction is $b(1+ky)e^{-ky}\sin (k(x-t))$. It obeys zero tangential material velocity and normal velocity $-kb\cos (k(x-t))$ on the reference boundary. Its second-order mean swimming coefficient in the convention of far-field velocity $U{\mathbf{e}}_x$ is $k^2{b}^2/2$.

A Taylor expansion of the no-slip boundary condition about the flat sheet makes the second-order mean tangential velocity $⟨u_2(0)⟩=-⟨y_1{\partial }_y{u}_1(0)⟩$. The mean mode of Stokes flow is linear in height when no pressure gradient is imposed. In an unbounded fluid, bounded velocity excludes mean shear; in a confined fluid, the force-free condition excludes it. The remaining mean velocity is uniform and equals the swimming speed in the sheet frame. Thus the leading speed can be found from the first-order field without solving the oscillatory second-order field.

The second-order mean no-slip boundary condition correction contains products of first-order deformation modes and velocity gradients. Distinct integer wavenumbers have zero averaged mixed products. Their separate mean longitudinal and transverse contributions therefore add, even though mixed products create nonzero oscillatory flow modes.

When longitudinal and transverse sheet waves share unit phase speed but have different wavenumbers $k_{\parallel },k_{\perp }$, their second-order swimming coefficient is $(k_{\perp }^2{b}^2-k_{\parallel }^2{a}^2)/2$. Thus their leading propulsion cancels at $b/a=k_{\parallel }/k_{\perp }$. This statement concerns the small-amplitude leading order.

For a transverse Taylor swimming sheet below a flat no-slip boundary condition at height $d$, the first-order amplitude $f(y)$ satisfies $({\partial }_y^2-1{)}^{2}f=0$, $f(0)=1$, $f^{\prime }(0)=f(d)=f^{\prime }(d)=0$. Writing $\Delta ={\sinh }^{2}d-d^2$ givesThe mean boundary velocity determines Taylor-sheet swimming speed, yieldingThis is greater than the unbounded value $1/2$ for every $d>0$. In a narrow gap it scales as $3/d^2$, with the small-amplitude calculation requiring $ϵ\ll d$ as well as $ϵ\ll 1$. Taylor's swimming sheet near a soft boundary recovers this rigid-wall limit while studying how compliance changes propulsion.