= Solution
Write $P(x)$ for the outer <pressure> extrapolated to the wall. The normal balance in the thin <boundary layer> is $p_y=\overline F_y$, giving
$$
p(x,y)=P(x)-\frac{C(x)^2}{4\mu_0}e^{-2y/\delta}.
$$
The spatially varying <magnetic pressure> deficit drives a tangential <pressure gradient>. For fixed forcing as $\omega\to\infty$, layer inertia is small compared with <viscous dissipation>; explicitly the ratio is $U\delta^2/(\nu b)$. The leading tangential balance is
$$
\rho\nu u_{yy}=p_x=-\frac{CC'}{2\mu_0}e^{-2y/\delta}.
$$
The outer gradient $P'$ contributes only at higher inner order. Matching the leading outer shear gives $u_y\to0$, and the physical <no-slip boundary condition> gives $u(x,0)=0$. Two integrations yield the <magnetic skin-layer streaming slip>:
$$
\boxed{u(x,y)=U(x)(1-e^{-2y/\delta}),\qquad U(x)=\frac{\delta^2CC'}{8\mu_0\rho\nu}=\frac{\delta^2}{16\mu_0\rho\nu}\frac{d(C^2)}{dx}.}
$$
For the present field amplitude,
$$
\boxed{U(x)=-\frac{\mu_0J^2b^2\delta^2}{4\pi^2\rho\nu}\frac{x}{(x^2+b^2)^3}.}
$$
Using <incompressibility> and $v(x,0)=0$ also gives
$$
v(x,y)=-U'(x)\left[y-\frac\delta2(1-e^{-2y/\delta})\right].
$$
Hence $v/u=O(\delta/b)$ in the layer, except at symmetry zeros where a componentwise ratio is inappropriate. In the matching region $\delta\ll y\ll b$, $u\sim U$ and $v\sim-yU'$, up to the smaller displacement term $\delta U'/2$.
\b[The effective outer boundary conditions are tangential slip $u_{\rm bulk}(x,0)=U(x)$ and zero leading normal velocity $v_{\rm bulk}(x,0)=0$.] These describe the bulk flow extrapolated through the unresolved <magnetic skin layer>; the actual wall remains at rest. For $x>0$ the slip is negative, and for $x<0$ it is positive. Fluid converges along the wall toward the wire, turns upward near $x=0$, and returns outward farther above the wall.
The following original <streamline> sketch uses the small-<Reynolds number> bulk <Stokes flow> for this slip. It illustrates that circulation without claiming that the unspecified bulk <Reynolds number> fixes a unique complete flow. With $X=x/b$, $Y=y/b$, $Z=X+iY$, a dimensionless <stream function> is
$$
\Psi=Y\operatorname{Re}\left[\frac1{4(Z+i)^3}-\frac{i}{8(Z+i)^2}\right].
$$
Its factor in brackets is <holomorphic> in $Y>0$, so $\Psi$ solves the <biharmonic equation>; its wall derivative is $\Psi_Y(X,0)=-X/(1+X^2)^3$. Thus it satisfies both effective wall conditions and gives an exact creeping-flow illustration of the derived slip.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-36-streamlines.png]
{title=Bulk Stokes-flow streamlines driven by magnetic skin-layer slip, converging along the wall and rising above the wire}
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