Brinkman swimming sheet

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.