Similarity Structure and Load-Factor Control of Enthalpy Sources in Unsteady Compressible Magnetohydrodynamic Boundary Layers
Abstract
Steady compressible magnetohydrodynamic (MHD) boundary layers have been studied since the work of Rossow \cite{Rossow1957}, Bleviss \cite{Bleviss1958} and Bush \cite{Bush1960}. What has not been developed is their \emph{unsteady} similarity structure. This paper supplies it, by combining the Howarth--Dorodnitsyn transformation with the diffusion-time similarity ansatz of Sun \cite{Sun2024}, and in doing so uncovers a control mechanism that the steady literature did not isolate. Taking the applied field normal to the wall, $\bm B = B_0(x)\hat{\bm e}_y$, makes the current solenoidal identically in two dimensions, so no auxiliary electrostatic problem arises. The external circuit is characterised by the classical load factor $k = -E_z/(UB_0)$ \cite{ReslerSears1958,Sutton1965}. The reduction yields a coupled pair of equations in $(\eta,\tau)$ for the reduced stream function $f$ and reduced total enthalpy $g$, governed by $(a,b,c;\Theta,\Pran,N,k)$ with $N=\sigma B_0^2\delta^2/(C\mu_r)$ a squared Hartmann number. Our central structural result is that \emph{the load factor cancels identically from the momentum equation once the outer pressure field is treated consistently, and survives only in the enthalpy equation}, where the combined Lorentz work and Joule dissipation reduce exactly to $\bm J\!\cdot\!\bm E = \sigma B_0^2kU(kU-u)$. The momentum source is $N\omega(1-f_\eta)$ for every $k$; the enthalpy source is $NE\,\omega\,k(k-f_\eta)$. Three consequences follow. (i) The linear…
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