Similarity Structure of Unsteady Compressible Two-Fluid Plasma Boundary Layers

We construct the similarity structure of unsteady compressible two-fluid (Hall) plasma boundary layers with separate ion and electron temperatures. Combining the unsteady Howarth–Dorodnitsyn reduction of Stewartson \( [Q.~J.~Mech.~Appl.~Math.~\textbf{4}, 182 (1951)] \) with the diffusion-time similarity ansatz of Sun \( [Phys.~Fluids ~ \textbf{36}, 083616 (2024)] \), we obtain a closed system in \( (\eta,\tau) \) for the reduced stream function, flux function, out-of-plane field and velocity, and two temperatures. Three structural results organise the problem. First, the Hall transport operator is invariant under the Dorodnitsyn map: because the Hall coefficient \( \lambda=1/\mu_0 ne \) scales as the local temperature while the in-plane magnetic derivative \( B_\bot\!\cdot\!\nabla \) transforms with the inverse weight, the two cancel exactly, and the Hall term—together with the magnetic force per unit mass—takes precisely its incompressible form. The pressure gradient does not; it acquires the familiar factor \( \omega=T/T_r \). Second, variable density restores the electron-pressure (Biermann) term, which vanishes identically at constant density. We show it can never enter the in-plane flux equation, and that within a similarity solution at uniform pressure it still vanishes because all thermodynamic gradients are parallel; it survives only through the streamwise pressure gradient, where it reduces to the source \( \theta_{e,\eta}/2M^2 \) and generates a second, independent quadrupole. This Biermann quadrupole dominates the Hall quadrupole below \( M\simeq0.73 \) and exceeds it twelvefold at \( M^2=0.05 \). Third, exact Hall similarity requires a layer of constant thickness (\( m=1 \)) while exact compressible similarity requires constant edge Mach number (\( m=0 \)); the two are disjoint, so no exactly similar compressible two-fluid layer exists. The admissible families are a strongly heated stagnation-type layer, exact to all orders in \( \varepsilon_H \), and a compressible flat plate, exact in the thermodynamics and self-similar through \( O(\varepsilon_H) \), whose Hall subsystem remains a first-order pair with \( f\mapsto\omega f \). Further results: the change of type of the unsteady system is purely kinematic, both equations carrying \( 1-2\tau f_\eta, so \tau_{\rm crit}=1/2 \) independently of field strength, Mach number and wall condition; the Alfvénic degeneracy is not displaced by compressibility, because \( \omega\to1 \) at the layer edge where the transition is decided; an aligned frozen-in field destroys the Mach independence of the transformed flat-plate skin friction, by \( 38\% \) at \( M^2=0.7, M_e^2=16 \); a hot wall expels tangential flux, the wall field falling by a factor of \( 4.6 \) between \( M_e^2=0 \) and \( 16 \); and expansion cooling drives the electron temperature below its free-stream value when equipartition is weak. Numerical solutions reproduce Hiemenz, Blasius–Dorodnitsyn and the compressible recovery factor to six figures, and confirm the predicted \( O(\varepsilon_H^2) \) back-reaction over a sixteenfold range in \( \varepsilon_H \).