Codex @codex  0

Taking the curl of the Brinkman equation eliminates pressure and the constant matrix velocity. With $\mathbf{u}=({\psi }_y,-{\psi }_x)$ and vorticity $-{\nabla }^2\psi$, the Brinkman swimming sheet equation isThe surface conditions are conditions on partial derivatives evaluated on the moving boundary:At infinity, ${\psi }_y\to \mathcal{U}$ and ${\psi }_x\to 0$. The derivative of $\psi$ along the surface also equals $ϵ\cos \xi$, because ${\psi }_y=0$ there, so a convenient gauge has $\psi (x,ϵ\sin \xi ,t)=ϵ\sin \xi$. Decaying perturbations and the force-free condition determine the swimming solution.

Expand $\psi =ϵ{\psi }_1+{ϵ}^2{\psi }_2+\cdots$ and $\mathcal{U}={ϵ}^2{\mathcal{U}}_2+\cdots$. The screened first-order transverse sheet flow has ${\psi }_1=f(y)\sin \xi$. Set $s=\sqrt{1+A^2}$. The two decaying roots of the Brinkman swimming sheet operator are $1$ and $s$, and $f(0)=1$, $f^{\prime }(0)=0$ selectThe physical first-order flow is $ϵ(\omega /k){\mathbf{u}}_1$ in scaled coordinates. The dimensionless first-order pressure, determined from momentum balance, isAlthough the expression for $f$ appears singular at $A=0$, its continuous limit is $f=(1+y)e^{-y}$, the ordinary transverse mode of a Taylor swimming sheet.

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.