Similarity Structure and Load-Factor Control of Enthalpy Sources in Unsteady Compressible Magnetohydrodynamic Boundary Layers

1. Introduction

1.1. Background and Prior Work

The boundary layer of an electrically conducting fluid in an applied magnetic field has a long history. In the incompressible case the Lorentz force enters the momentum balance as a linear Rayleigh-type drag and produces the Hartmann layer, with thickness ${\mathit{Ha}}^{-1}$ and a characteristically full velocity profile [10,16,23,25,26,33]. Similarity solutions for magnetohydrodynamic (MHD) Falkner–Skan and stagnation-point flow were given by Neuringer and McIlroy [27] and have been extended in many directions since [1,8,28].

The compressible problem is also classical, and it is important to state clearly what is already known. Rossow [32] treated the conducting flat plate with a transverse field and recognised that the results depend on whether the field is fixed relative to the plate or to the fluid — the distinction that is formalised below as the load factor. Bleviss [3] analysed magnetogasdynamic Couette flow, and Bush [5] gave what is essentially the steady limit of the system studied here: compressible flat-plate boundary-layer flow with an applied magnetic field, including the coupling of Joule heating to the thermal field. Resler and Sears [29] set out the magneto-aerodynamic framework, including the accelerator/generator distinction. More recent work has addressed adverse pressure gradients and mass transfer [42,43], stability [7,40], rarefaction [22], the validity of the low-$\mathrm{Rm}$ assumption for hypersonic control [11], and the rigorous well-posedness of the compressible MHD Prandtl system [18].

We therefore do not claim the steady effects of a magnetic field on a compressible boundary layer as new. That Joule dissipation raises the wall temperature, and that the transformed skin friction ceases to be Mach-independent once a field acts, are consequences that the 1957–1962 literature had the machinery to reach, and our steady computations should be read as recovering and quantifying them within a unified framework rather than discovering them.

1.2. What Is Missing, and What This Paper Adds

What that literature does not contain is the unsteady similarity structure. The unsteady compressible boundary layer has a rich theory of its own — the small-time Rayleigh–Stokes phase, the approach to the steady state, and Stewartson’s obstruction associated with the history terms [30,31,34,35,41] — and a similarity framework in diffusion-time variables has recently been given by Sun [37], building on Ma and Hui [24]. Applying that framework to a conducting fluid [14] exposes structure that the steady theory cannot see, because the obstruction, the relaxation behaviour and the history of the recovery factor are all intrinsically time-dependent.

Two reductions are combined here. The first casts the unsteady incompressible MHD problem in diffusion-time similarity form, in which the reduced momentum equation acquires a linear damping term and the unsteady terms group into $f_{\eta \tau }+c\tau (f_{\tau }{f}_{\eta \eta }-f_{\eta }{f}_{\eta \tau })$, so that the coefficient $1-c\tau f_{\eta }$ can change sign. The second applies the same ansatz to the compressible problem after a Dorodnitsyn transformation. Their union is the subject of this paper.

Combining them also brings into focus a control parameter that the steady literature used but did not exploit systematically in a boundary-layer thermal context: the load factor k of the external circuit. Our main structural finding is that k cancels from the momentum equation and acts only on the enthalpy equation, which turns it into a clean thermal-control knob.

Contributions.

1.

A closed unsteady master system (§Section 4) containing both parent problems as exact limits, with the field orientation chosen so that no electrostatic potential problem arises.

2.

A load-factor decomposition theorem (§Section 3): the combined Lorentz work and Joule dissipation equal $\mathbf{J}·\mathbf{E}$ exactly; k cancels from momentum and survives only in the enthalpy source, whose sign structure $k(k-f_{\eta })$ permits net extraction of enthalpy from the layer.

3.

A complete classification of magnetic Crocco integrals (§Section 6.1): survival only at $k=0$ (classical $g\equiv 1$, exact for all $\tau$ and all N) and at $k=1$ (with the selected wall $T_w/T_e=2-\Theta$).

4.

A kinematic change-of-type theorem (§Section 5.1) with thermal amplification $1/\mathrm{Pr}$, and an incompatibility theorem (§Section 5.2) between exact similarity and unconditional well-posedness.

5.

An exact recovery identity (§Section 6.3) and a load-factor map of the thermal state (§Section 8.3), including an optimum cooling load factor $k_{\mathrm{opt}}\simeq 0.26$ predicted analytically and confirmed numerically.

2. Formulation

2.1. Governing Equations and the Low-$\mathrm{Rm}$ Approximation

Consider two-dimensional unsteady flow of a compressible, electrically conducting, calorically perfect gas over a wall at $y=0$, with an external inviscid stream $U\left(x\right)$ and edge conditions ${\rho }_e\left(x\right)$, $T_e\left(x\right)$, $p_e\left(x\right)$. Let $\sigma$ be the electrical conductivity, $\mu$ the dynamic viscosity, k the thermal conductivity, $c_p$ the specific heat, and $\mathrm{Pr}=\mu c_p/k$.

We work throughout at low magnetic Reynolds number,

$$\mathrm{Rm}={\mu }_0\sigma UL\ll 1,$$

(1)

which is the relevant regime for liquid metals, seeded combustion gases and moderately ionised hypersonic flows [25,38]. The induced magnetic field is then negligible compared with the applied field, the magnetic field is prescribed rather than solved for, and Ohm’s law in the quasi-static form reads

$$\mathbf{J}=\sigma \left(\mathbf{E}+\mathbf{u}\times \mathbf{BJ}\right),$$

(2)

with $\mathbf{E}=-\nabla \varphi$ and $\nabla ·\mathbf{J}=0$.

2.2. Orientation of the Applied Field

Two orientations are compatible with a two-dimensional boundary layer: a wall-normal field $\mathbf{B}=B_0\left(x\right){\widehat{\mathbf{J}}}_y$ and a spanwise field $\mathbf{B}=B_0\left(x\right){\widehat{\mathbf{e}}}_z$. They are not equivalent, and the choice materially affects how much work is required to close the momentum equation.

Wall-normal field (adopted here).

With $\mathbf{B}=B_0{\widehat{\mathbf{e}}}_y$ and $\mathbf{u}=(u,v,0)$, the motional electromotive force is $\mathbf{u}\times \mathbf{B}=-u{B}_0{\widehat{\mathbf{e}}}_z$, so the current is purely spanwise, $\mathbf{J}=J_z{\widehat{\mathbf{e}}}_z$ with

$$J_z=\sigma \left(E_z-u{B}_0\right).$$

(3)

Because nothing depends on z, the solenoidality condition reduces to $\partial J_z/\partial z\equiv 0$, which is satisfied identically. No electrostatic potential problem arises: $E_z$ is set by the external circuit. Following standard magneto-aerodynamic practice [29,38] we parametrise that circuit by the load factor

$$k=-{\displaystyle \frac{E_z}{U{B}_0}},\mathrm{so}\mathrm{that}{J}_z=\sigma \left(E_z+u{B}_0\right)=\sigma B_0(u-kU),$$

(4)

whence the streamwise Lorentz force density is

$$F_x={(\mathbf{J}\times \mathbf{B})}_x=-\sigma B_0^2(u-kU).$$

(5)

Three cases are distinguished classically: $k=0$ is the short-circuited layer, in which no electric field is applied and the force opposes the motion everywhere; $0<k<1$ is generator operation; $k=1$ is the matched or unit-load condition, for which the free-stream current $J_z{|}_{u=U}=\sigma B_{0}U(1-k)$ vanishes and the force acts only inside the layer, accelerating the retarded fluid; and $k>1$ is accelerator operation. Rossow’s distinction [32] between a field fixed to the plate and one fixed to the fluid is the special case $k=0$ versus $k=1$. Much of the compressible literature [5,22] works at $k=1$; we retain general k throughout, because §Section 3 shows it to be the natural thermal-control parameter.

Spanwise field (for comparison).

With $\mathbf{B}=B_0{\widehat{\mathbf{e}}}_z$ one finds $\mathbf{u}\times \mathbf{B}=(v{B}_0,-u{B}_0,0)$, so the current has components in both x and y. Solenoidality is no longer automatic and one must solve ${\nabla }^2\varphi =\nabla ·(\mathbf{u}\times \mathbf{B})$ subject to wall and far-field electrical boundary conditions. For perfectly insulating walls the net cross-stream current must vanish, and after elimination of $\varphi$ the streamwise force reduces to the same functional form as (5) but with an orientation-dependent constant absorbed into the interaction parameter. Everything that follows therefore applies to both cases with N rescaled; we adopt the wall-normal field because it requires no auxiliary elliptic solve.

Remark 1.

The conductivity of a real gas depends strongly on temperature. Our baseline takes $\sigma ={\sigma }_r$ constant, appropriate to a liquid metal or a seeded gas at fixed ionisation fraction. A one-parameter generalisation $\sigma /{\sigma }_r={(T/T_r)}^s$ is accommodated at no cost: every occurrence of ω below is replaced by ${\omega }^{1+s}$ (see Remark 4).

2.3. Boundary-Layer Equations with Electromagnetic Terms

At high Reynolds number the usual scaling gives $\partial p/\partial y=0$, so $p=p_e\left(x\right)$ across the layer. Evaluating the momentum equation at the edge, where $u=U$, gives

$${\rho }_{e}U{\displaystyle \frac{\mathrm{d}U}{\mathrm{d}x}}=-{\displaystyle \frac{\mathrm{d}{p}_e}{\mathrm{d}x}}-\sigma B_0^{2}U(1-k),$$

(6)

so for $k\ne 1$ the outer stream itself carries current and the pressure gradient is modified. Only at the matched condition $k=1$ is the ordinary compressible Bernoulli relation recovered unchanged. It is essential to retain (6); doing so produces the cancellation of Theorem 1. The equations are

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial \left(\rho u\right)}{\partial x}}+{\displaystyle \frac{\partial \left(\rho v\right)}{\partial y}}=0,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & \rho \left({\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+v{\displaystyle \frac{\partial u}{\partial y}}\right)=-{\displaystyle \frac{\mathrm{d}{p}_e}{\mathrm{d}x}}+{\displaystyle \frac{\partial }{\partial y}}\left(\mu {\displaystyle \frac{\partial u}{\partial y}}\right)-\sigma B_0^2(u-kU),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & \rho \left({\displaystyle \frac{\partial H}{\partial t}}+u{\displaystyle \frac{\partial H}{\partial x}}+v{\displaystyle \frac{\partial H}{\partial y}}\right)={\displaystyle \frac{\partial }{\partial y}}\left[{\displaystyle \frac{\mu }{\mathrm{Pr}}}{\displaystyle \frac{\partial H}{\partial y}}+\mu \left(1-{\displaystyle \frac{1}{\mathrm{Pr}}}\right)u{\displaystyle \frac{\partial u}{\partial y}}\right]+\underset{\mathrm{Lorentz}\mathrm{work}}{\underbrace{u{F}_x}}+\underset{\mathrm{Joule}}{\underbrace{{\displaystyle \frac{J_z^2}{\sigma }}}},\hfill \end{array}$$

(9)

where $H=c_{p}T+\frac{1}{2}{u}^2$ is the total enthalpy. Boundary conditions are $u=v=0$ at $y=0$ with either $T=T_w$ or $\partial T/\partial y=0$, and $u\to U$, $H\to H_e$ as $y\to \infty$.

3. The Load-Factor Decomposition Theorem

The two electromagnetic terms in () are usually carried separately, and the load factor is usually thought of as affecting both the momentum and the energy balance. Both impressions are misleading.

Theorem 1

(Load-factor decomposition). For any load factor k, with the outer pressure field determined consistently by (6):

(i)

the total electromagnetic acceleration appearing in the momentum equation is

$${\displaystyle \frac{1}{\rho }}\left[-{\displaystyle \frac{\mathrm{d}{p}_e}{\mathrm{d}x}}-{\rho }_{e}U{\displaystyle \frac{\mathrm{d}U}{\mathrm{d}x}}\right]+{\displaystyle \frac{F_x}{\rho }}={\displaystyle \frac{\sigma B_0^2}{\rho }}(U-u),$$

(10)

which isindependent of k;

(ii)

the total electromagnetic contribution to the total-enthalpy equation is

$$\mathcal{P}=u{F}_x+{\displaystyle \frac{J_z^2}{\sigma }}=\mathbf{J}·\mathbf{E}=\sigma B_0^{2}kU(kU-u),$$

(11)

which does depend on k, and changes sign across the layer at $u=kU$.

Proof. (i) From (6), $-\mathrm{d}{p}_e/\mathrm{d}x-{\rho }_{e}U\mathrm{d}U/\mathrm{d}x=\sigma B_0^{2}U(1-k)$. Adding $F_x=-\sigma B_0^2(u-kU)$ and dividing by $\rho$,

$${\displaystyle \frac{\sigma B_0^2}{\rho }}\left[U(1-k)-(u-kU)\right]={\displaystyle \frac{\sigma B_0^2}{\rho }}\left[U-kU-u+kU\right]={\displaystyle \frac{\sigma B_0^2}{\rho }}(U-u).$$

The terms in k cancel identically: the current carried by the outer stream contributes a streamwise pressure gradient that exactly offsets its own contribution to the force inside the layer.

(ii) With $E_z=-kU{B}_0$ and $J_z=\sigma B_0(u-kU)$, $J_z^2/\sigma =\sigma B_0^2{(u-kU)}^2$ and $u{F}_x=-\sigma B_0^{2}u(u-kU)$, so

$$\mathcal{P}=\sigma B_0^2(u-kU)\left[(u-kU)-u\right]=-\sigma B_0^{2}kU(u-kU),$$

which equals $J_z{E}_z=\sigma B_0(u-kU)·(-kU{B}_0)$. The factor $(kU-u)$ vanishes at $u=kU$ and changes sign there. □

This is the organising result of the paper. The physical reading is that $\mathbf{J}·\mathbf{E}$ is the rate at which the external circuit delivers power to the fluid, and the load factor controls that delivery without altering the mechanical problem at all. Three regimes follow.

Corollary 1

(Short-circuited layer, $k=0$). When no electric field is applied, $\mathcal{P}\equiv 0$: the Lorentz force extracts mechanical energy at exactly the rate at which Joule dissipation returns it as heat, and the total-enthalpy equation is untouched by the field even though the momentum equation is not.

Corollary 2

(Generator operation, $0<k<1$). The source is positive where $u<kU$, i.e. in the inner part of the layer, andnegativewhere $u>kU$, i.e. in the outer part. The layer therefore delivers net electrical power to the circuit over part of its thickness, and its enthalpy is correspondingly depressed. §Section 8.3 shows this to be a large effect.

Corollary 3

(Matched and accelerator operation, $k\ge 1$). For $k\ge 1$ we have $u\le U\le kU$ throughout, so $\mathcal{P}\ge 0$ everywhere: the layer acts as a distributed motor and heater simultaneously. At $k=1$, $\mathcal{P}=\sigma B_0^{2}U(U-u)$.

4. Similarity Reduction and the Master System

4.1. Dorodnitsyn Variables

Introduce the Howarth–Dorodnitsyn normal coordinate and the Chapman–Rubesin approximation [6],

$$\overline{y}={\int }_0^y{\displaystyle \frac{\rho }{{\rho }_r}}\mathrm{d}{y}^{\prime },\rho \mu =C{\rho }_r{\mu }_r,$$

(12)

with reference state $({\rho }_r,{\mu }_r,T_r)$ and C the Chapman–Rubesin constant. Since $p=p_e$ across the layer, ${\rho }_r/\rho =T/T_r$ exactly. Defining the compressible stream function through $\rho u={\rho }_r\partial \psi /\partial \overline{y}$ and V the transformed normal velocity, (7)–() become

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u}{\partial t}}+u{\displaystyle \frac{\partial u}{\partial x}}+V{\displaystyle \frac{\partial u}{\partial \overline{y}}}={\displaystyle \frac{T}{T_e}}U{\displaystyle \frac{\mathrm{d}U}{\mathrm{d}x}}+C{\nu }_r{\displaystyle \frac{{\partial }^{2}u}{\partial {\overline{y}}^2}}-{\displaystyle \frac{\sigma B_0^2}{{\rho }_r}}{\displaystyle \frac{T}{T_r}}(u-U),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial H}{\partial t}}+u{\displaystyle \frac{\partial H}{\partial x}}+V{\displaystyle \frac{\partial H}{\partial \overline{y}}}=C{\nu }_r{\displaystyle \frac{\partial }{\partial \overline{y}}}\left[{\displaystyle \frac{1}{\mathrm{Pr}}}{\displaystyle \frac{\partial H}{\partial \overline{y}}}+\left(1-{\displaystyle \frac{1}{\mathrm{Pr}}}\right)u{\displaystyle \frac{\partial u}{\partial \overline{y}}}\right]+{\displaystyle \frac{\sigma B_0^2}{{\rho }_r}}{\displaystyle \frac{T}{T_r}}U(U-u).\hfill \end{array}$$

(14)

Note that both the pressure-gradient term and the electromagnetic terms acquire the same temperature weighting: every body-force-like term in a compressible boundary layer is multiplied by the local density ratio ${\rho }_e/\rho =T/T_e$ (equivalently ${\rho }_r/\rho =T/T_r$). This is the single structural statement from which everything below follows.

4.2. Diffusion-Time Similarity Ansatz

Let $\delta \left(x\right)$ be a diffusion length scale and set

$$\eta ={\displaystyle \frac{\overline{y}}{\delta \left(x\right)}},\tau ={\displaystyle \frac{C{\nu }_{r}t}{{\delta }^2\left(x\right)}},\psi =U\delta f(\eta ,\tau ),H=H_{r}g(\eta ,\tau ),$$

(15)

so that $u=U{f}_{\eta }$. With $H_r=H_e$ and $U_{\mathrm{ref}}=U$ define

$$\Theta ={\displaystyle \frac{T_r}{T_e}}=1+{\displaystyle \frac{\gamma -1}{2}}{M}_e^2,E={\displaystyle \frac{U^2}{H_r}}={\displaystyle \frac{2(\Theta -1)}{\Theta }},$$

(16)

and the reduced static temperature

$$\begin{array}{|c|}\hline \omega (\eta ,\tau )={\displaystyle \frac{T}{T_r}}=g-\frac{1}{2}E{f}_{\eta }^2,{\displaystyle \frac{T}{T_e}}=\Theta \omega =\Theta g-(\Theta -1)f_{\eta }^2.\\ \hline\end{array}$$

(17)

The geometric coefficients are

$$a={\displaystyle \frac{\delta }{C{\nu }_r}}{\displaystyle \frac{\mathrm{d}\left(U\delta \right)}{\mathrm{d}x}},b={\displaystyle \frac{{\delta }^2}{C{\nu }_r}}{\displaystyle \frac{\mathrm{d}U}{\mathrm{d}x}},c=2(a-b),$$

(18)

and the magnetic interaction parameter is

$$\begin{array}{|c|}\hline N={\displaystyle \frac{\sigma B_0^2{\delta }^2}{C{\mu }_r}}={\displaystyle \frac{\sigma B_0^2{\delta }^2}{{\rho }_{r}C{\nu }_r}}={\mathit{Ha}}_{\delta }^2,\\ \hline\end{array}$$

(19)

a squared Hartmann number built on the diffusion thickness. Since $\rho \mu =C{\rho }_r{\mu }_r$, an equivalent form is $N=\sigma B_0^2{\delta }^2{\rho }_r/\left(\rho \mu \right)$, which may be evaluated anywhere across the layer.

4.3. The Master System

Carrying out the reduction yields the coupled pair

with $\omega$ given by (17). Boundary conditions for a stationary wall beneath a stream are

$$f(0,\tau )=f_{\eta }(0,\tau )=0,f_{\eta }(\infty ,\tau )=1,g(\infty ,\tau )=1,$$

(22)

together with either $g(0,\tau )=g_w$ (isothermal) or, for an adiabatic wall, $g_{\eta }(0,\tau )=E{f}_{\eta }(0,\tau )f_{\eta \eta }(0,\tau )=0$.

Remark 2

(Parent problems recovered). Setting $M_e=0$ gives $\Theta =1$, $E=0$, $\omega \equiv g\equiv 1$, and (Section 4.3) collapses to the incompressible unsteady MHD equation $f_{\eta \eta \eta }+af{f}_{\eta \eta }+b(1-f_{\eta }^2)+N(1-f_{\eta })=f_{\eta \tau }+c\tau (\dots )$. Setting $N=0$ recovers the unsteady compressible system exactly. The joint system contains both.

Remark 3

(Structure of the source). By Theorem 1 the electromagnetic terms in (Section 4.3)–(Section 4.3) are

$${\mathcal{S}}_{\mathrm{mom}}=N\omega (1-f_{\eta }),{\mathcal{S}}_{\mathrm{ener}}=NE\omega k(k-f_{\eta }),$$

(23)

so that ${\mathcal{S}}_{\mathrm{mom}}$ is common to every circuit while ${\mathcal{S}}_{\mathrm{ener}}$ carries all the k-dependence. Only at $k=1$ are the two proportional, ${\mathcal{S}}_{\mathrm{ener}}=E{\mathcal{S}}_{\mathrm{mom}}$; this is the lever used in §Section 6.1. At $k=0$ the enthalpy source vanishes identically.

Remark 4

(Temperature-dependent conductivity). If $\sigma ={\sigma }_r{(T/T_r)}^s$ then $\sigma /\rho \propto {\omega }^{1+s}$ and the substitution $\omega \mapsto {\omega }^{1+s}$ in $\mathcal{S}$ carries every result below through unchanged in form. The baseline $s=0$ is used in all computations.

4.4. Admissible Similarity Families

Similarity requires a, b, c, N, $\Theta$ and $\mathrm{Pr}$ to be independent of x. For a power-law stream $U=C_1{x}^m$ with

$$\delta \left(x\right)={\left[{\displaystyle \frac{2C{\nu }_{r}x}{(1+m)U}}\right]}^{1/2},$$

(24)

the definitions (18) give the standard Falkner–Skan gauge

$$a=1,b={\displaystyle \frac{2m}{1+m}}=\beta ,c={\displaystyle \frac{2(1-m)}{1+m}}=2(1-\beta ).$$

(25)

Constancy of N requires $B_0\delta =\mathrm{const}$, hence

$$B_0\left(x\right)\propto x^{(m-1)/2},$$

(26)

uniform only at the plane stagnation point $m=1$. Constancy of $\Theta$ requires a constant edge Mach number, which for an isentropic outer flow with $U=C_1{x}^m$ holds exactly only for $m=0$. These two requirements pull in opposite directions, a tension made precise in §Section 5.2.

Figure 1. Configuration and energy pathways. (a) A wall-normal applied field $\mathbf{B}=B_0\left(x\right){\widehat{\mathbf{e}}}_y$ drives a purely spanwise current $J_z=\sigma B_0(u-kU)$, where $k=-E_z/\left(U{B}_0\right)$ is the load factor of the external circuit. Because $\nabla ·\mathbf{J}=\partial J_z/\partial z\equiv 0$ identically in two dimensions, no electrostatic potential problem arises. The matched condition $k=1$ is drawn: the current then vanishes in the free stream and the Lorentz force $F_x=-\sigma B_0^2(u-U)$ is positive, accelerating the retarded fluid and producing the characteristically full Hartmann profile. (b) The two electromagnetic channels sum exactly to the power $\mathbf{J}·\mathbf{E}$ delivered by the circuit, shown here for $k=1$. By Theorem 1 the load factor cancels from the momentum balance and governs only this energy exchange, which reverses sign over the outer layer when $0<k<1$ and vanishes identically at $k=0$.

Figure 1. Configuration and energy pathways. (a) A wall-normal applied field $\mathbf{B}=B_0\left(x\right){\widehat{\mathbf{e}}}_y$ drives a purely spanwise current $J_z=\sigma B_0(u-kU)$, where $k=-E_z/\left(U{B}_0\right)$ is the load factor of the external circuit. Because $\nabla ·\mathbf{J}=\partial J_z/\partial z\equiv 0$ identically in two dimensions, no electrostatic potential problem arises. The matched condition $k=1$ is drawn: the current then vanishes in the free stream and the Lorentz force $F_x=-\sigma B_0^2(u-U)$ is positive, accelerating the retarded fluid and producing the characteristically full Hartmann profile. (b) The two electromagnetic channels sum exactly to the power $\mathbf{J}·\mathbf{E}$ delivered by the circuit, shown here for $k=1$. By Theorem 1 the load factor cancels from the momentum balance and governs only this energy exchange, which reverses sign over the outer layer when $0<k<1$ and vanishes identically at $k=0$.

5. Structural Consequences

5.1. Change of Type Is Purely Kinematic

Group the time derivatives in (Section 4.3)–(Section 4.3):

$$\begin{array}{cc}\hfill \left(1-c\tau f_{\eta }\right)f_{\eta \tau }+c\tau f_{\eta \eta }{f}_{\tau }& =f_{\eta \eta \eta }+af{f}_{\eta \eta }+b\Theta (g-f_{\eta }^2)+\mathcal{S},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \left(1-c\tau f_{\eta }\right)g_{\tau }+c\tau f_{\tau }{g}_{\eta }& =\frac{1}{\mathrm{Pr}}{g}_{\eta \eta }+af{g}_{\eta }+\left(1-\frac{1}{\mathrm{Pr}}\right)E{\left(f_{\eta }{f}_{\eta \eta }\right)}_{\eta }+E\mathcal{S}.\hfill \end{array}$$

(28)

Both equations carry the same coefficient

$$1-\Lambda ,\Lambda (\eta ,\tau )=c\tau f_{\eta }={\displaystyle \frac{2(1-m)}{1+m}}\tau f_{\eta }={\displaystyle \frac{(1-m)}{(1+m)}}{\displaystyle \frac{2C{\nu }_{r}t}{{\delta }^2}}{\displaystyle \frac{u}{U}}.$$

(29)

Theorem 2

(Kinematic threshold). The master system is well posed as an initial-value problem in τ so long as $\Lambda <1$ throughout the layer. Since $0\le f_{\eta }\le 1$ with $f_{\eta }\to 1$ at the edge, the first loss of parabolicity occurs at the boundary-layer edge at

$${\tau }_{\mathrm{crit}}={\displaystyle \frac{1}{c}}={\displaystyle \frac{1+m}{2(1-m)}},$$

(30)

independently of N, $M_e$, $\mathrm{Pr}$, the wall thermal condition and the conductivity law. For $c\le 0$, i.e. $m\ge 1$, the threshold is never reached and the march is unconditionally well posed.

Proof. 

The threshold is determined solely by the sign of the coefficient of the $\tau$-derivative, which is $1-c\tau f_{\eta }$ in both equations. Neither N nor $\Theta$, E, $\mathrm{Pr}$ nor the boundary data appear in that coefficient. The supremum of $f_{\eta }$ over the layer is 1, attained as $\eta \to \infty$. □

Although the threshold is untouched, the severity of the ill-posedness is not.

Proposition 1

(Thermal amplification). Linearising about a state with $\Lambda >1$ and inserting a short-wave disturbance $\propto e^{ik\eta +\lambda \tau }$ into the principal parts gives

$${\lambda }_{\mathrm{mom}}={\displaystyle \frac{k^2}{\Lambda -1}},{\lambda }_{\mathrm{therm}}={\displaystyle \frac{k^2}{\mathrm{Pr}(\Lambda -1)}}={\displaystyle \frac{{\lambda }_{\mathrm{mom}}}{\mathrm{Pr}}}.$$

(31)

For a gas with $\mathrm{Pr}<1$ the thermal field is the more violently ill posed of the two, diverging a factor $1/\mathrm{Pr}$ faster; for air, $1/0.72\simeq 1.39$.

This is confirmed dramatically in §Section 8.4: past the threshold the wall shear reaches $O\left({10}^{11}\right)$ while the wall heat flux reaches $O\left({10}^{55}\right)$.

Remark 5.

Theorem 2 corrects a natural expectation. One might suppose that a strong magnetic field, which suppresses the velocity defect and makes the profile fuller, would postpone the change of type. It does not: because the fieldraises$f_{\eta }$ throughout the interior, points inside the layer reach $\Lambda =1$ earlierthan they otherwise would. The field does not move the first crossing (which occurs at the edge, where $f_{\eta }=1$ regardless), but it makes the ill-posed region invade the interior faster.

5.2. An Incompatibility Theorem

Theorem 3

(No free lunch). Within the power-law family (23), exact constancy of the compressibility parameter Θ and unconditional well-posedness of the τ-march cannot hold simultaneously.

Proof. 

Exact constancy of $\Theta =1+\frac{\gamma -1}{2}{M}_e^2$ requires a constant edge Mach number. For an isentropic outer flow driven by $U=C_1{x}^m$, the edge temperature satisfies $T_e+U^2/\left(2c_p\right)=\mathrm{const}$, so $M_e$ is constant only if U is, i.e. $m=0$. By (23), $m=0$ gives $c=2>0$ and hence ${\tau }_{\mathrm{crit}}=1/2<\infty$. Conversely, unconditional well-posedness requires $c\le 0$, i.e. $m\ge 1$, for which U varies and $M_e$ cannot be constant. The two sets are disjoint. □

The practical consequence is that one must choose. The flat plate $m=0$ is the only exactly similar compressible member, and it is precisely the case in which Stewartson’s obstruction at $\tau =1/2$ appears; the stagnation point $m=1$ marches forever but requires the local-similarity interpretation for $\Theta$. We exploit both: §Section 8.5 uses $m=1$ for clean transients, and §Section 8.6 uses a global chart to solve $m=0$ despite the obstruction. Physically, requiring $B_0\propto x^{(m-1)/2}$ from (24) adds a third constraint, uniform field being admissible only at $m=1$.

6. Exact Solutions and Integrals

6.1. The Magnetic Crocco Integral

Write both transport equations with the common operator

$$\mathcal{L}\left[\phi \right]={\phi }_{\eta \eta }+af{\phi }_{\eta }-{\phi }_{\tau }-c\tau \left(f_{\tau }{\phi }_{\eta }-f_{\eta }{\phi }_{\tau }\right),$$

(32)

noting that $\mathcal{L}\left[f_{\eta }\right]$ reproduces exactly the non-source terms of (Section 4.3). At $\mathrm{Pr}=1$ the dissipation term in (Section 4.3) vanishes and the pair reads

$$\mathcal{L}\left[f_{\eta }\right]+b\Theta (g-f_{\eta }^2)+\mathcal{S}=0,\mathcal{L}\left[g\right]+E\mathcal{S}=0.$$

(33)

Theorem 4

(Classification of magnetic Crocco integrals). Let $\mathrm{Pr}=1$ and $b=0$, and let $N>0$. Then $g=A+B{f}_{\eta }$ solves (Section 4.3) for all τ if and only if the load factor takes one of two values:

(a)

$k=0$, with $B=0$, giving theclassicalintegral

$$g(\eta ,\tau )\equiv 1,$$

(34)

valid for arbitrarily strong fields and compatible with the adiabatic wall at rest;

(b)

$k=1$, with $B=E$, giving

$$g(\eta ,\tau )=1-E+E{f}_{\eta }(\eta ,\tau ),$$

(35)

which is compatible only with theisothermalwall

$${\displaystyle \frac{T_w}{T_e}}=\Theta (1-E)=2-\Theta =1-{\displaystyle \frac{\gamma -1}{2}}{M}_e^2,$$

(36)

requiring $M_e<\sqrt{2/(\gamma -1)}$ ($M_e<\sqrt{5}\approx 2.236$ for air).

For every other load factor no linear enthalpy–velocity relation exists.

Proof. 

Since $\mathcal{L}$ is linear and $\mathcal{L}\left[1\right]=0$, substituting $g=A+B{f}_{\eta }$ gives $\mathcal{L}\left[g\right]=B\mathcal{L}\left[f_{\eta }\right]=-B{\mathcal{S}}_{\mathrm{mom}}$ when $b=0$. Requiring $\mathcal{L}\left[g\right]+{\mathcal{S}}_{\mathrm{ener}}=0$ therefore demands

$$NE\omega k(k-f_{\eta })=BN\omega (1-f_{\eta })\mathrm{for}\mathrm{all}\eta ,\tau .$$

(37)

Cancelling $N\omega >0$, the polynomials $Ek(k-f_{\eta })$ and $B(1-f_{\eta })$ in $f_{\eta }$ must agree identically. Matching the coefficient of $f_{\eta }$ gives $-Ek=-B$, and matching the constant term gives $E{k}^2=B$. Hence $Ek=E{k}^2$, so $k\in \{0,1\}$. For $k=0$, $B=0$ and $g\equiv A$; the far-field condition gives $A=1$. For $k=1$, $B=E$ and $A=1-E$, whence $T_w/T_e=\Theta g\left(0\right)=\Theta (1-E)=2-\Theta$ using $E=2(\Theta -1)/\Theta$. □

Case (a) is striking and easy to overlook. At $k=0$ the magnetic field alters the velocity profile substantially — our computations give $f_{\eta \eta }\left(0\right)$ rising from $0.469600$ at $N=0$ to $1.912727$ at $N=4$ — yet the total enthalpy remains exactly uniform, $g\equiv 1$, so that at $\mathrm{Pr}=1$ the adiabatic wall recovers precisely the free-stream stagnation temperature and $r=1$, just as in a non-conducting gas. We verify this to machine precision ($max|g-1|=0$ to double precision at $N=4$). The classical Crocco result is thus not destroyed by a magnetic field; it is destroyed by the applied electric field, and only when $k\notin \{0,1\}$.

Two consequences are worth stating. First, when (33) holds the system decouples: g is known algebraically in terms of $f_{\eta }$, and (Section 4.3) becomes a single closed PDE

$$f_{\eta \eta \eta }+af{f}_{\eta \eta }+N\left(1-E+E{f}_{\eta }-\frac{1}{2}E{f}_{\eta }^2\right)(1-f_{\eta })=f_{\eta \tau }+c\tau \left(f_{\tau }{f}_{\eta \eta }-f_{\eta }{f}_{\eta \tau }\right),$$

(38)

an exact unsteady compressible MHD solution family for arbitrary N, $\Theta$ and $\tau$. Second, at $k=1$ the circuit selects the wall temperature. Without a field any linear Crocco relation is admissible and the adiabatic case $g\equiv 1$ is the celebrated one; at unit load factor only the single cooled wall (34) survives, and the adiabatic wall — which satisfies $g_{\eta }\left(0\right)=0$ and hence would require $B=0$ — is not compatible. That incompatibility is the first sign of the effect developed in §Section 6.3.

6.2. Small-Time Limit and the Recovery Factor

As $\tau \to 0$ the diffusion terms are $O\left({\tau }^{-1}\right)$ while $af$, b and N terms are $O\left(1\right)$, so the field is not felt at leading order. With $\zeta =\eta /\left(2\sqrt{\tau }\right)$ the momentum equation reduces to $U^{\prime \prime }+2\zeta U^{\prime }=0$, giving the impulsive-start Rayleigh solution $f_{\eta }=erf\zeta$. Writing $\theta =T/T_r=g-\frac{1}{2}E{f}_{\eta }^2$, the enthalpy equation reduces to a remarkably compact form.

Proposition 2.

At leading order as $\tau \to 0$,

$${\theta }^{\prime \prime }+2\mathrm{Pr}\zeta {\theta }^{\prime }=-\mathrm{Pr}E{\left(U^{\prime }\left(\zeta \right)\right)}^2=-{\displaystyle \frac{4\mathrm{Pr}E}{\pi }}{e}^{-2{\zeta }^2},U=erf\zeta ,$$

(39)

with $\theta (\infty )=1/\Theta$. For an adiabatic wall at rest, ${\theta }^{\prime }\left(0\right)=0$, and the quadrature evaluates in closed form to

$$\begin{array}{|c|}\hline r_{\tau \to 0}={\displaystyle \frac{4}{\pi }}\sqrt{{\displaystyle \frac{\mathrm{Pr}}{2-\mathrm{Pr}}}}arctan\sqrt{{\displaystyle \frac{2-\mathrm{Pr}}{\mathrm{Pr}}}}\\ \hline\end{array}$$

(40)

for the recovery factor defined by $T_{aw}/T_e=1+r(\Theta -1)$.

Proof. 

Substituting $g=\theta +\frac{1}{2}E{U}^2$ into the leading-order enthalpy balance and using $U^{\prime \prime }=-2\zeta U^{\prime }$ collapses the dissipation terms to $-\mathrm{Pr}E{\left(U^{\prime }\right)}^2$, giving (37). Integrating with ${\theta }^{\prime }\left(0\right)=0$,

$$\theta \left(0\right)={\displaystyle \frac{1}{\Theta }}+{\displaystyle \frac{4\mathrm{Pr}E}{\pi }}{\int }_0^{\infty }{\int }_0^{\zeta }{e}^{-\mathrm{Pr}{\zeta }^2-(2-\mathrm{Pr})s^2}\mathrm{d}s\mathrm{d}\zeta .$$

Polar substitution $\zeta =\varrho cos\phi$, $s=\varrho sin\phi$ over $0<\phi <\pi /4$ gives the double integral as $\frac{1}{2}{\left[\mathrm{Pr}(2-\mathrm{Pr})\right]}^{-1/2}arctan\sqrt{(2-\mathrm{Pr})/\mathrm{Pr}}$. Using $T_w/T_e=\Theta \theta \left(0\right)$ and $\Theta E=2(\Theta -1)$ produces (38); the factors of $\Theta$ cancel identically, so $r_{\tau \to 0}$ is independent of Mach number. □

Two points follow. Expression (38) coincides with the value known for the impulsively moved plate in a gas at rest, so the same closed form governs the opposite configuration treated here. And, by Proposition 2, $r_{\tau \to 0}$ is independent of N: however strong the field, the recovery factor begins its history at the field-free value, $0.88550$ for $\mathrm{Pr}=0.72$. Everything the field does to r, it does later. Our computations give $0.88563$ (Table 7), a $0.015\%$ check on the global solver.

6.3. An Exact Steady Identity at $\mathrm{Pr}=1$

Theorem 5

(Recovery identity). For the steady flat plate ($b=0$, $a=1$, ${\partial }_{\tau }\equiv 0$) at unit load factor $k=1$, with $\mathrm{Pr}=1$ and an adiabatic wall at rest,

$$\begin{array}{|c|}\hline g_w=1+E\left[f_{\eta \eta }\left(0\right)Q-1\right],r=2f_{\eta \eta }\left(0\right)Q-1,\\ \hline\end{array}Q={\int }_0^{\infty }exp\left[-{\int }_0^{\eta }f\left(s\right)\mathrm{d}s\right]\mathrm{d}\eta ,$$

(41)

and $f_{\eta \eta }\left(0\right)Q>1$ whenever $N>0$, so that $r>1$.

Proof. 

The steady momentum equation gives $N\omega (1-f_{\eta })=-(f^{\prime \prime \prime }+f{f}^{\prime \prime })$, so the energy equation becomes $g^{\prime \prime }+f{g}^{\prime }=E(f^{\prime \prime \prime }+f{f}^{\prime \prime })$. Setting $\psi =g-1-E{f}_{\eta }$ yields ${\psi }^{\prime \prime }+f{\psi }^{\prime }=0$, whose solutions are spanned by 1 and $q\left(\eta \right)={\int }_0^{\eta }{e}^{-F}$, $F={\int }_0^{\eta }f$. Hence $g=1+E{f}_{\eta }+C_1+C_{2}q$. The condition $g(\infty )=1$ gives $C_1=-E-C_{2}Q$; the adiabatic condition $g^{\prime }\left(0\right)=E{f}^{\prime \prime }\left(0\right)+C_2{q}^{\prime }\left(0\right)=0$ with $q^{\prime }\left(0\right)=1$ gives $C_2=-E{f}^{\prime \prime }\left(0\right)$. Evaluating at $\eta =0$ where $f_{\eta }=q=0$ gives $g_w=1+C_1=1-E+E{f}^{\prime \prime }\left(0\right)Q$, which is (39); the recovery factor follows from $T_w/T_e=\Theta g_w=1+r(\Theta -1)$ and $\Theta E=2(\Theta -1)$.

For the inequality, integrate ${\left(f^{\prime \prime }{e}^F\right)}^{\prime }=-N\omega (1-f_{\eta })e^F$ to get $f^{\prime \prime }{e}^F=f^{\prime \prime }\left(0\right)-N{\int }_0^{\eta }\omega (1-f_{\eta })e^F$. Using ${\int }_0^{\infty }{f}^{\prime \prime }\mathrm{d}\eta =1$,

$$1=f^{\prime \prime }\left(0\right)Q-N{\int }_0^{\infty }{e}^{-F\left(\eta \right)}{\int }_0^{\eta }\omega (1-f_{\eta })e^{F\left(s\right)}\mathrm{d}s\mathrm{d}\eta ,$$

and the double integral is strictly positive because $\omega >0$ and $f_{\eta }<1$. Hence $f^{\prime \prime }\left(0\right)Q=1+N\mathcal{J}$ with $\mathcal{J}>0$. □

At $N=0$ the identity degenerates gracefully: the Blasius relation $f^{\prime \prime }=f^{\prime \prime }\left(0\right)e^{-F}$ gives $Q=1/f^{\prime \prime }\left(0\right)$ exactly, so $r=1$, recovering the classical $\mathrm{Pr}=1$ result. Our computations return $f^{\prime \prime }\left(0\right)Q=0.99999983$ at $N=0$ and reproduce (39) to six decimal places for all N and $M_e$ tested (Table 4).

Corollary 4

(Adiabatic walls are singular in MHD). Because Joule dissipation is a volumetric source that an adiabatic wall cannot reject, the wall enthalpy must rise until convection alone removes the deposited power. Combining $g_w=1+E(f^{\prime \prime }\left(0\right)Q-1)$ with the Hartmann estimate $f^{\prime \prime }\left(0\right)\sim \sqrt{N{g}_w}$ and $Q\to \sqrt{\pi /2}$ gives the self-consistent strong-field balance $g_w\sim \frac{\pi }{2}{E}^{2}N$ and $r\sim \pi EN$: the adiabatic wall temperature grows without bound with field strength. The constant-property assumptions underlying the model fail long before this limit, but the trend is real and is the dominant design consideration for MHD flow-control surfaces.

7. Numerical Method

Three solvers are used, each validated against the other two where they overlap.

(i)

Steady two-point boundary-value solver.

The steady limit of (Section 4.3)–(Section 4.3),

$$\begin{array}{cc}\hfill f^{\prime \prime \prime }+af{f}^{\prime \prime }+\beta \Theta (g-f^{\prime 2})+N\omega (1-f^{\prime })& =0,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{Pr}}{g}^{\prime \prime }+af{g}^{\prime }+\left(1-\frac{1}{\mathrm{Pr}}\right)E{\left(f^{\prime }{f}^{\prime \prime }\right)}^{\prime }+NE\omega k(k-f^{\prime })& =0,\hfill \end{array}$$

(43)

is written as a five-dimensional first-order system (eliminating $f^{\prime \prime \prime }$ from the dissipation term) and solved by collocation with adaptive mesh refinement to a tolerance of ${10}^{-10}$. Solutions are continued in N from $N=0$, each solution seeding the next; without continuation the solver readily jumps to spurious branches with $T<0$ at moderate N, and all results reported here were checked for pointwise positivity of $\omega$.

(ii)

Unsteady marching solver.

Equations (Section 4.3)–(Section 4.3) are marched in $\xi =ln\tau$ using second-order backward differentiation (implicit Euler on the first step) and second-order central differences in $\eta$, with Picard sweeps and tridiagonal solves alternating between $u=f_{\eta }$ and g to a tolerance of ${10}^{-12}$. The adiabatic wall condition $g_{\eta }\left(0\right)=E{f}_{\eta }\left(0\right)f_{\eta \eta }\left(0\right)=0$ is imposed with a ghost node, preserving second-order accuracy. The march is started at ${\tau }_0={10}^{-3}$ from the leading-order similarity state of §Section 6.2, obtained by solving (37) to machine precision; the resulting $O\left({\tau }_0\right)$ start-up error is verified by $f_{\eta \eta }\left(0\right)\sqrt{\pi {\tau }_0}\to 1$, which returns $1.005571$ on $N_{\eta }=2400$ and $1.001412$ on $N_{\eta }=4800$.

(iii)

Williams–Rhyne global solver.

For $c>0$ the march is obstructed at ${\tau }_{\mathrm{crit}}=1/c$. We therefore adopt the Williams–Rhyne chart [41]

$$s=1-e^{-\tau },\zeta ={\displaystyle \frac{\eta }{2\sqrt{s}}},f(\eta ,\tau )=2\sqrt{s}F(\zeta ,s),$$

(44)

in which both $s=0$ and $s=1$ are characteristic, so that the whole history is obtained as one boundary-value problem in the strip $(\zeta ,s)\in [0,{\zeta }_{\infty }]\times [0,1]$ with no marching at all. The transformed system is

$$\begin{array}{cc}\hfill F^{\prime \prime \prime }+P{F}^{\prime \prime }+4sb\Theta (G-F^{\prime 2})+4sN\omega (1-F^{\prime })& =\left(d_1-d_2{F}^{\prime }\right)F_{,s}^{\prime },\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill \frac{1}{\mathrm{Pr}}{G}^{\prime \prime }+P{G}^{\prime }+\left(1-\frac{1}{\mathrm{Pr}}\right)E{\left(F^{\prime }{F}^{\prime \prime }\right)}^{\prime }+4sNE\omega k(k-F^{\prime })& =\left(d_1-d_2{F}^{\prime }\right)G_{,s},\hfill \end{array}$$

(46)

with $P=2(1-s)\zeta +KF-d_2{F}_{,s}$, $K=4s-2c\tau (1-s)$, $d_1=4s(1-s)$, $d_2=4s(1-s)c\tau$ and $\tau =-ln(1-s)$. Both $d_1$ and $d_2$ vanish at $s=0$ and $s=1$. The limits are carried automatically: at $s=0$ one obtains $F^{\prime \prime \prime }+2\zeta F^{\prime \prime }=0$, i.e. $F^{\prime }=erf\zeta$; at $s=1$, $K\to 4$, $P\to 4F$, and the steady system (40) is recovered under $f=2F$, $\eta =2\zeta$. The discrete residual on a $201\times 101$ grid (${\zeta }_{\infty }=8.5$) is driven below ${10}^{-8}$ by a Newton–Krylov iteration with LGMRES inner solves, continued in N.

Cross-validation.

Table 1 collects checks against independently known values. The agreement with both parent theories is to five or six significant figures throughout.

The Hartmann-asymptote entry deserves a word: the incompressible strong-field expansion predicts $f_{\eta \eta }\left(0\right)\simeq N^{1/2}+(a/12+2b/3)N^{-1/2}$, which for $a=1$, $b=0$, $N=100$ gives $10+1/(12·10)=10.008333$, matched to seven digits. This is a stringent test of the momentum discretisation in the regime where the Hartmann layer is thinnest.

8. Results

Throughout, $\gamma =1.4$ and $\mathrm{Pr}=0.72$ unless stated otherwise.

8.1. Steady Structure and the Loss of Mach-independence

Figure 2 and Table 2 present the central steady result. In the absence of a field the transformed flat-plate momentum equation is exactly Blasius, independent of $M_e$: this Mach-independence in the Dorodnitsyn plane is one of the most useful features of classical compressible boundary-layer theory. The magnetic field destroys it.

The reason is structural. In the transformed momentum equation the Lorentz term is $N\omega (1-f_{\eta })$, and $\omega =T/T_r$ is not unity in a compressible gas. The sign of the effect is set by a competition, which explains the non-monotonic behaviour in Fig. Figure 2(c):

• Weak field. The force acts across the whole layer, where $\omega$ ranges from $g_w=1$ at the wall to $1/\Theta =0.238$ (at $M_e=4$) at the edge. The layer-averaged $\omega$ is therefore below unity, the effective Lorentz force is weaker than in an incompressible fluid at the same N, and the wall shear falls below its incompressible value.
• Strong field. The force concentrates in a Hartmann layer at the wall, where $\omega \ge 1$; moreover Joule dissipation raises the interior temperature substantially. The peak of $\omega$ grows from $1.000$ at $N=0$ to $1.0753$, $1.3340$ and $1.4148$ at $N=1$, 25 and 100 (with $maxg$ rising from $1.042$ to $2.020$). The effective force is now stronger than incompressible and the shear exceeds it.

The crossover occurs near $N\approx 1$. The strong-field prefactor in Fig. Figure 2(d) makes the mechanism quantitative: it converges to $1.0008$ at $M_e=0$, confirming the incompressible asymptote, but to $1.065$, $1.113$ and $1.130$ at $M_e=2$, 4, 6, tracking the growth of $\sqrt{max\omega }$ rather than the wall value $\sqrt{{\omega }_w}=1$. In words:

Joule heat lightens the gas; a lighter gas feels a larger Lorentz acceleration per unit mass; a larger acceleration steepens the wall shear, which increases the current defect and hence the Joule heat.

This positive feedback loop has no counterpart in constant-density MHD, where $\omega \equiv 1$ by construction. It is the essential new mechanism of the joint problem.

8.2. Recovery Factors Above Unity

For a non-conducting gas the adiabatic-wall recovery factor is bounded by simple classical values: $r=\sqrt{\mathrm{Pr}}\approx 0.8485$ for the steady flat plate, and $r=1$ exactly at $\mathrm{Pr}=1$, the statement that an adiabatic wall recovers the free-stream stagnation temperature and no more. Joule dissipation breaks this bound, because it injects energy that did not come from the kinetic energy of the stream.

Figure 3. Recovery factor of the steady adiabatic flat plate. (a) r against N for $M_e=2,4,6$; the inset resolves the crossing of $r=1$ near $N\simeq 0.06$. (b) Verification of the exact identity of Theorem 5: thick pale curves are $2f_{\eta \eta }\left(0\right)Q-1$, open symbols are r computed directly from the wall temperature; the two agree to six decimals. (c) r against $\mathrm{Pr}$ at $M_e=4$: the black curve is the small-time closed form (38), the dashed curve the classical steady $\sqrt{\mathrm{Pr}}$, and the coloured curves the steady values at increasing N. (d) The resulting adiabatic wall temperature (logarithmic scale); dotted lines mark the field-free values, exceeded by an order of magnitude by $N=3$.

Figure 3. Recovery factor of the steady adiabatic flat plate. (a) r against N for $M_e=2,4,6$; the inset resolves the crossing of $r=1$ near $N\simeq 0.06$. (b) Verification of the exact identity of Theorem 5: thick pale curves are $2f_{\eta \eta }\left(0\right)Q-1$, open symbols are r computed directly from the wall temperature; the two agree to six decimals. (c) r against $\mathrm{Pr}$ at $M_e=4$: the black curve is the small-time closed form (38), the dashed curve the classical steady $\sqrt{\mathrm{Pr}}$, and the coloured curves the steady values at increasing N. (d) The resulting adiabatic wall temperature (logarithmic scale); dotted lines mark the field-free values, exceeded by an order of magnitude by $N=3$.

Table 3 shows r passing through unity near $N\simeq 0.06$ for every Mach number, and reaching an order of magnitude by $N=3$. The corresponding adiabatic wall temperatures are severe: at $M_e=6$ and $N=1$ the wall sits at 35 times the edge static temperature, against $7.1$ without a field. Since r is defined against the stagnation-temperature rise, values $r>1$ mean literally that the surface is hotter than the stagnation temperature of the oncoming stream—an outcome impossible in non-conducting aerodynamics and controlled entirely by the interaction parameter.

Table 4 verifies Theorem 5 numerically. The agreement between the geometric quantity $2f_{\eta \eta }\left(0\right)Q-1$, computed from the momentum solution alone, and the thermodynamic quantity r, computed from the wall temperature, is to six decimal places over the whole range.

Table 4. Verification of the exact identity $r=2f_{\eta \eta }\left(0\right)Q-1$ of Theorem 5; steady flat plate, $\mathrm{Pr}=1$, adiabatic wall.

${\mathit{M}}_{\mathit{e}}$	N	${\mathit{f}}_{\mathit{\eta }\mathit{\eta }}\left(0\right)$	Q	$2{\mathit{f}}_{\mathit{\eta }\mathit{\eta }}\left(0\right)\mathit{Q}-1$	r (direct)	${\mathit{g}}_{\mathit{w}}$
2	$0.00$	$0.469600$	$2.129471$	$1.000000$	$1.000000$	$1.000000$
2	$0.25$	$0.716200$	$1.976949$	$1.831784$	$1.831785$	$1.369682$
2	$0.50$	$0.989969$	$1.864185$	$2.690971$	$2.690972$	$1.751543$
2	$1.00$	$1.570608$	$1.714499$	$4.385614$	$4.385616$	$2.504718$
4	$0.00$	$0.469600$	$2.129471$	$1.000000$	$1.000000$	$1.000000$
4	$0.25$	$0.758877$	$1.961746$	$1.977447$	$1.977447$	$1.744722$
4	$0.50$	$1.172255$	$1.814156$	$3.253306$	$3.253308$	$2.716806$
4	$1.00$	$2.130200$	$1.630329$	$5.945851$	$5.945853$	$4.768269$

Table 4. Verification of the exact identity $r=2f_{\eta \eta }\left(0\right)Q-1$ of Theorem 5; steady flat plate, $\mathrm{Pr}=1$, adiabatic wall.

${\mathit{M}}_{\mathit{e}}$	N	${\mathit{f}}_{\mathit{\eta }\mathit{\eta }}\left(0\right)$	Q	$2{\mathit{f}}_{\mathit{\eta }\mathit{\eta }}\left(0\right)\mathit{Q}-1$	r (direct)	${\mathit{g}}_{\mathit{w}}$
2	$0.00$	$0.469600$	$2.129471$	$1.000000$	$1.000000$	$1.000000$
2	$0.25$	$0.716200$	$1.976949$	$1.831784$	$1.831785$	$1.369682$
2	$0.50$	$0.989969$	$1.864185$	$2.690971$	$2.690972$	$1.751543$
2	$1.00$	$1.570608$	$1.714499$	$4.385614$	$4.385616$	$2.504718$
4	$0.00$	$0.469600$	$2.129471$	$1.000000$	$1.000000$	$1.000000$
4	$0.25$	$0.758877$	$1.961746$	$1.977447$	$1.977447$	$1.744722$
4	$0.50$	$1.172255$	$1.814156$	$3.253306$	$3.253308$	$2.716806$
4	$1.00$	$2.130200$	$1.630329$	$5.945851$	$5.945853$	$4.768269$

8.3. Load-factor Control of the Thermal State

Theorem 1 says the load factor is a pure thermal actuator: it leaves the momentum problem untouched and acts only on the enthalpy source. Figure 4 and Table 5 map the consequence.

Four features stand out.

The optimum is analytically predictable.

The enthalpy source $k(k-f_{\eta })$ is, at each station, minimised over k at ${\partial }_k[k^2-k{f}_{\eta }]=2k-f_{\eta }=0$, i.e. at $k=f_{\eta }/2$. Since $f_{\eta }$ sweeps $[0,1]$ across the layer, a single load factor cannot be optimal everywhere, and the best compromise should lie near half the layer-averaged velocity ratio, i.e. $k\approx 0.25$–$0.3$. The computed optima are $k_{\mathrm{opt}}=0.24$–$0.28$ across a sixteen-fold range of N (Table 5), in good agreement, and the near-invariance of $k_{\mathrm{opt}}$ with N follows because $f_{\eta }$ enters the criterion but N does not.

The cooling is large.

At $N=4$ the adiabatic wall temperature falls from $T_{aw}/T_e=3.713$ without a field to $2.075$ at $k=0.24$. Since $T_{aw}$ sets the equilibrium load on a thermal protection system, a $44\%$ reduction is of direct engineering interest, and it is achieved not by removing heat but by converting part of the layer’s enthalpy into electrical power delivered to the external circuit — the boundary layer operating as a distributed generator. This is the practically useful face of Corollary 2.

The heating and cooling regimes are separated sharply.

The crossing $r=1$ occurs at $k_{*}=0.61$–$0.73$, drifting slowly downward with N. Below $k_{*}$ the wall is cooler than the stagnation temperature; above it, hotter. Note that $k_{*}<1$: the matched condition $k=1$, which much of the compressible MHD literature adopts by default, sits well inside the heating regime and is close to the worst case rather than a neutral choice.

There is a runaway boundary.

Beyond $k_{max}$ no steady solution with $T>0$ throughout could be continued, and $k_{max}$ contracts from $1.70$ at $N=0.25$ to $1.08$ at $N=4$. This is the finite-k expression of the adiabatic singularity noted after Theorem 5: an adiabatic wall cannot reject volumetric heat, so beyond a critical delivery rate the balance between Joule deposition and convective removal cannot be struck. We do not claim $k_{max}$ is a sharp turning point of the exact problem — resolving that would require arclength continuation — only that our continuation fails there and that the failure moves systematically with N.

8.4. The Change of Type, Numerically

Figure 5 tests the theory directly. Below the threshold the solutions are grid-converged: at $\tau =0.45$ the wall shear agrees to six significant figures on meshes of 1600, 3200 and 6400 points. Above it the solutions diverge at mesh-dependent times, the finer meshes surviving slightly longer—the signature of an ill-posed problem rather than a coding error, since refinement does not converge but merely postpones. The threshold itself is insensitive to N and $M_e$: repeating the experiment at $N=0$, $M_e=0$ gives the same ${\tau }_{\mathrm{crit}}=0.5$, confirming Theorem 2. The disparity between panels (a) and (b) confirms Proposition 1: with $\mathrm{Pr}=0.72$ the thermal growth rate exceeds the momentum growth rate by a factor $1.39$, which compounded over the post-threshold interval produces the enormous separation observed.

8.5. Transients at the Stagnation Point

Figure 6 uses the stagnation point, where $c=0$ makes the march unconditionally well posed, to display complete transients. Three features are worth noting.

First, all wall-shear histories collapse onto ${\left(\pi \tau \right)}^{-1/2}$ at small $\tau$, independently of N: the field is not felt until $N\tau =O\left(1\right)$, as predicted in §Section 6.2. Second, the shear is non-monotonic: it undershoots its steady value before recovering, a feature inherited from the non-magnetic stagnation transient and amplified by the field. Third, and most important, the approach to the steady state is now controlled by the thermal field rather than the momentum field.

Table 6 makes this quantitative. In the incompressible problem the magnetic field monotonically accelerates the approach to the steady state, with ${\tau }_{1\%}\sim 1.8/N$, because magnetic damping adds a relaxation rate to viscous diffusion. Here the behaviour is non-monotonic: ${\tau }_{1\%}$ first rises, peaking near $N\approx 1$ at more than twice its field-free value, before falling again at large N. The explanation is that an adiabatic wall must accumulate the Joule heat until convection can balance it, and that thermal accumulation is slow. At small N the extra heat load grows faster than the magnetic damping rate and the system takes longer to equilibrate; at large N the Hartmann damping eventually dominates and the classical acceleration returns. In compressible MHD it is thermal, not momentum, relaxation that sets the settling time over a wide intermediate range of field strengths—a point of direct practical relevance for pulsed or switched magnetic actuation.

8.6. Global Histories in the Williams–Rhyne Chart

The flat plate is the only exactly similar compressible member of the family (Theorem 3) and is precisely the case forward marching cannot complete. Figure 7 and Table 7 give the global solutions.

The table isolates the paper’s two main numerical claims in a single place. The column $F^{\prime \prime }(0,0)$ is constant at $1.130158$ (against the exact $2/\sqrt{\pi }=1.128379$, a $0.16\%$ discretisation error) for every N and $M_e$, confirming that neither field nor compressibility is felt at the impulsive start. The column $r\left(0\right)$ is likewise constant at $0.88563$ against the closed form $0.88550$. At the other end, $F^{\prime \prime }(0,1)$ is identical for $M_e=0,2,4$ when $N=0$—Mach-independence—but differs by $6\%$ between $M_e=0$ and $M_e=4$ at $N=0.25$ and by $26\%$ at $N=0.5$. Meanwhile $r\left(1\right)$ crosses unity between $N=0.05$ and $N=0.10$, consistent with the steady computation of Table 3.

Table 7. Williams–Rhyne global solutions, $\beta =0$, adiabatic wall, $\mathrm{Pr}=0.72$; $s=0$ is the impulsive start and $s=1$ the steady state. Note that $F^{\prime \prime }(0,0)$ and $r\left(0\right)$ are independent of both N and $M_e$, and that $F^{\prime \prime }(0,1)$ is Mach-independent at $N=0$ only.

${\mathit{M}}_{\mathit{e}}$	N	${\mathit{F}}^{\prime \prime }(0,0)$	${\mathit{F}}^{\prime \prime }(0,1)$	$\mathit{r}\left(0\right)$	$\mathit{r}\left(1\right)$
0	$0.00$	$1.130158$	$0.939370$	—	—
0	$0.25$	$1.130158$	$1.349913$	—	—
0	$1.00$	$1.130158$	$2.174772$	—	—
2	$0.00$	$1.130158$	$0.939370$	$0.88563$	$0.84775$
2	$0.05$	$1.130158$	$1.026391$	$0.88563$	$0.97394$
2	$0.10$	$1.130158$	$1.115365$	$0.88563$	$1.10088$
2	$0.25$	$1.130158$	$1.391897$	$0.88563$	$1.48394$
2	$0.50$	$1.130158$	$1.874120$	$0.88563$	$2.12047$
4	$0.00$	$1.130158$	$0.939370$	$0.88563$	$0.84775$
4	$0.10$	$1.130158$	$1.111520$	$0.88563$	$1.09702$
4	$0.25$	$1.130158$	$1.434203$	$0.88563$	$1.54594$
4	$0.50$	$1.130158$	$2.109260$	$0.88563$	$2.42984$

Table 7. Williams–Rhyne global solutions, $\beta =0$, adiabatic wall, $\mathrm{Pr}=0.72$; $s=0$ is the impulsive start and $s=1$ the steady state. Note that $F^{\prime \prime }(0,0)$ and $r\left(0\right)$ are independent of both N and $M_e$, and that $F^{\prime \prime }(0,1)$ is Mach-independent at $N=0$ only.

${\mathit{M}}_{\mathit{e}}$	N	${\mathit{F}}^{\prime \prime }(0,0)$	${\mathit{F}}^{\prime \prime }(0,1)$	$\mathit{r}\left(0\right)$	$\mathit{r}\left(1\right)$
0	$0.00$	$1.130158$	$0.939370$	—	—
0	$0.25$	$1.130158$	$1.349913$	—	—
0	$1.00$	$1.130158$	$2.174772$	—	—
2	$0.00$	$1.130158$	$0.939370$	$0.88563$	$0.84775$
2	$0.05$	$1.130158$	$1.026391$	$0.88563$	$0.97394$
2	$0.10$	$1.130158$	$1.115365$	$0.88563$	$1.10088$
2	$0.25$	$1.130158$	$1.391897$	$0.88563$	$1.48394$
2	$0.50$	$1.130158$	$1.874120$	$0.88563$	$2.12047$
4	$0.00$	$1.130158$	$0.939370$	$0.88563$	$0.84775$
4	$0.10$	$1.130158$	$1.111520$	$0.88563$	$1.09702$
4	$0.25$	$1.130158$	$1.434203$	$0.88563$	$1.54594$
4	$0.50$	$1.130158$	$2.109260$	$0.88563$	$2.42984$

Panel (a) of Fig. Figure 7 deserves particular attention: it maps $\Lambda =c\tau f_{\eta }$ over the whole strip and shows that a substantial part of the physically relevant history—everything to the right of the heavy contour, which the vertical dotted line at $\tau =1/c$ first touches—is inaccessible to any forward-marching scheme, however refined. The global chart obtains it as an ordinary boundary-value problem.

9. Discussion and Limitations

What is new, and what is not.

We separate these carefully. The steady effects of §Section 8.1–§Section 8.2 — loss of Mach-independence of the transformed skin friction, and Joule-driven wall heating — lie within reach of the classical steady analyses of Rossow [32] and Bush [5], and we present them as recovered and quantified within a unified framework, not as discoveries. Their value here is that the same solver reproduces both parent limits to six figures, which underwrites the unsteady results.

What we believe to be new is: (i) the load-factor decomposition (Theorem 1), namely that k cancels from momentum once the outer pressure field is treated consistently and survives only in the enthalpy source; (ii) the resulting classification of magnetic Crocco integrals into exactly two admissible load factors (Theorem 4), including the exact survival of $g\equiv 1$ at $k=0$ under arbitrarily strong fields; (iii) the load-factor map of the thermal state with its analytically predicted optimum $k=f_{\eta }/2$ (§Section 8.3); (iv) the kinematic change-of-type theorem with its $1/\mathrm{Pr}$ thermal amplification (§Section 5.1); (v) the incompatibility theorem (§Section 5.2); and (vi) the recovery identity $r=2f_{\eta \eta }\left(0\right)Q-1$ (§Section 6.3). Items (iv)–(vi) have no steady analogue at all. Underlying the compressible–magnetic coupling throughout is that the Lorentz acceleration per unit mass carries a factor T, so a temperature field partly generated by the field feeds back into the momentum balance.

Range of validity.

The results rest on four approximations. (a) Low $\mathrm{Rm}$. The induced field is neglected; for $\mathrm{Rm}\gtrsim 1$ the field must be solved for and the closure (4) fails. (b) Constant σ and Chapman–Rubesin C. This is the weakest assumption, and it is weakest precisely where we push hardest: at $M_e=4$–6 the gas conducts because it is hot, and $\sigma$ then varies by orders of magnitude across the layer. Remark 4 shows how a power law $\sigma \propto T^s$ is absorbed without changing the form of any result, and since $s>0$ the feedback loop of §Section 8.1 would strengthen; but the quantitative values at $N\gtrsim 1$ should be read as indicative of trend, not as predictions. A fully coupled ionisation model is the obvious next step. (b^′) Prescribed load factor. We treat k as an externally imposed constant. A real electrode arrangement fixes the terminal voltage, not k pointwise, and end effects near electrode edges are outside the boundary-layer framework. (c) Laminar flow. At the interaction parameters where $r>1$, the combination of strong wall heating and a Hartmann-thinned layer would in practice interact with transition; the field is known to be stabilising, but the laminar assumption is a genuine restriction. (d) The adiabatic idealisation. As the corollary to Theorem 5 shows, an adiabatic wall with volumetric heating is a singular limit whose temperature grows without bound with N. Real surfaces radiate and conduct; the adiabatic results should be read as an upper envelope, valuable because they bound the thermal load, not as a prediction of attainable wall temperature.

Relation to prior work.

The incompressible limit reproduces the known unsteady MHD reduction including its change of type and Hartmann asymptote; the field-free limit reproduces the unsteady compressible theory including the Crocco integral, the recovery factor (38) and the Williams–Rhyne structure. The contribution here is the coupling and its consequences, of which Theorem 1 (the exact source), Theorem 4 (the selected wall) and Theorem 5 (the recovery identity) are the analytical core.

10. Conclusions

We have formulated and solved the unsteady compressible MHD boundary layer at low magnetic Reynolds number, with a wall-normal applied field chosen so that the current is solenoidal identically, and with the external circuit characterised by its load factor k.

1.

Load-factor decomposition. Once the outer pressure field is determined consistently, k cancels identically from the momentum equation and survives only in the enthalpy equation, where the combined Lorentz work and Joule dissipation reduce exactly to $\mathbf{J}·\mathbf{E}=\sigma B_0^{2}kU(kU-u)$ (Theorem 1). The momentum source is $N\omega (1-f_{\eta })$ for every circuit; the enthalpy source is $NE\omega k(k-f_{\eta })$, which changes sign at $f_{\eta }=k$.

2.

Two admissible Crocco integrals, and no others. A linear enthalpy–velocity relation survives only at $k=0$, where the classical $g\equiv 1$ holds exactly for arbitrarily strong fields, and at $k=1$, where it holds only for the cooled wall $T_w/T_e=2-\Theta$ (Theorem 4). The classical Crocco result is destroyed not by the magnetic field but by the applied electric field.

3.

The load factor is a thermal actuator. Sweeping k moves the adiabatic recovery factor from $0.336$ — a $44\%$ reduction in adiabatic wall temperature at $M_e=4$, $N=4$ — through $r=1$ near $k_{*}\simeq 0.61$ and on to thermal runaway beyond $k_{max}$. The optimum cooling load factor $k_{\mathrm{opt}}\simeq 0.26$ is nearly independent of N and is predicted by the pointwise criterion $k=f_{\eta }/2$. The matched condition $k=1$ commonly adopted in the literature lies well inside the heating regime.

4.

The change of type is purely kinematic. Both equations share the factor $1-c\tau f_{\eta }$, so ${\tau }_{\mathrm{crit}}=1/c$ is independent of N, k, $M_e$, $\mathrm{Pr}$ and wall condition (Theorem 2), while the thermal mode diverges beyond it a factor $1/\mathrm{Pr}$ faster (Proposition 1), confirmed by a ${10}^{44}$ disparity in computed divergences.

5.

Exact similarity and unconditional well-posedness are incompatible within the admissible family (Theorem 3).

6.

An exact recovery identity$r=2f_{\eta \eta }\left(0\right)Q-1$ holds at $\mathrm{Pr}=1$, $k=1$ (Theorem 5) and is verified to six decimals.

7.

Steady effects recovered. The loss of Mach-independence of transformed flat-plate skin friction ($-2.4\%$ at $N=0.25$, $+12.9\%$ at $N=100$) and Joule-driven wall heating are quantified within the same framework; these are consistent with, rather than additional to, the classical steady literature.

Natural extensions include a temperature-dependent conductivity coupled to an ionisation model; three-dimensional and swept configurations, where the spanwise orientation requires the electrostatic problem to be solved; finite-$\mathrm{Rm}$ effects; arclength continuation to resolve the structure of the runaway boundary $k_{max}$; and the stability of these profiles, where the competition between Hartmann stabilisation [23,40] and the strongly modified thermal layer identified here — which the load factor can make either hotter or cooler than the field-free case — seems likely to be delicate [7].