Similarity Structure of Unsteady Compressible Two-Fluid Plasma Boundary Layers
Abstract
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…
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