Navier-slip Taylor swimming sheet ($ϵ=k{y}_0,\delta =k\gamma$)

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.