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In a thin magnetic skin layer next to a rigid no-slip boundary condition, vertical pressure balance gives $p=P-C^2{e}^{-2y/\delta }/(4{\mu }_0)$. With negligible inertia in the layer, the tangential Stokes flow balance is $\rho \nu u_{yy}=p_x$. Matching zero leading shear outside the layer gives $u=U(1-e^{-2y/\delta })$ and the displayed effective tangential slip. The physical wall still has zero velocity; the slip is a boundary condition for the outer flow extrapolated through the unresolved layer. It points toward larger squared boundary-field amplitude.

Let $\delta =\sqrt{2\eta /\omega }$ and $q=(1+i)/\delta$. Within the magnetic skin layer, differentiation across the layer dominates differentiation along the wall by the factor $b/\delta$. The leading resistive induction equation is thereforeBecause $\eta q^2=i\omega$ and $\mathrm{Re}q>0$, its decaying solution isThe solenoidal magnetic-field constraint determines the first smaller component,Corrections to $B_x$ from longitudinal diffusion have relative order $(\delta /b{)}^2$. Thus the zero normal boundary field from the perfect conductor calculation is a leading-order condition, not an additional exact condition at finite $\eta$. A smaller exterior-field correction matches $B_y$.

To leading order the electric current density is $j_z=qC{e}^{-qy}/{\mu }_0$. For two real harmonic fields, the period average of their product is half the real part of the product of one amplitude with the conjugate of the other. Thus the mean Lorentz force in a magnetic skin layer isIts normal component isThe first possible tangential contribution from the smaller $B_y$ vanishes: it is $-C{C}^{\prime }{e}^{-2y/\delta }\mathrm{Re}(q/q^{\ast })/(2{\mu }_0)$, and $q/q^{\ast }=i$. This cancellation matters in computing the tangential streaming flow.

Write $P(x)$ for the outer pressure extrapolated to the wall. The normal balance in the thin boundary layer is $p_y={\overline{F}}_y$, givingThe 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 isThe outer gradient $P^{\prime }$ contributes only at higher inner order. Matching the leading outer shear gives $u_y\to 0$, and the physical no-slip boundary condition gives $u(x,0)=0$. Two integrations yield the magnetic skin-layer streaming slip:For the present field amplitude,Using incompressibility and $v(x,0)=0$ also givesHence $v/u=O(\delta /b)$ 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 $v\sim -y{U}^{\prime }$, up to the smaller displacement term $\delta U^{\prime }/2$.

The effective outer boundary conditions are tangential slip $u_{\mathrm{bulk}}(x,0)=U(x)$ and zero leading normal velocity $v_{\mathrm{bulk}}(x,0)=0$. 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 isIts 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.