## Abstract

Magnetohydrodynamic flow of an incompressible fluid through a plane channel with slowly varying walls and a magnetic field applied transverse to the channel is investigated in the high Reynolds number limit. It is found that the magnetic field can first influence the hydrodynamic flow when the Hartmann number reaches a sufficiently large value. The magnetic field is found to suppress the steady and unsteady viscous flow near the channel walls unless the wall shapes become large.

## 1. Introduction

The flow of an incompressible electrically conducting fluid through a plane channel is considered when the channel walls are insulated and the applied magnetic field is directed normal to the channel. In particular, this study examines the response of a viscous channel flow to slowly varying wall distortions, which can take the form of humps or troughs, oscillating actuators, or various types of expansions or contractions. It is well known that an applied transverse magnetic field will tend to suppress the Poiseuille flow in a channel, and if the magnetic field strength is large enough then the channel velocity can become small. Most asymptotic studies of channel flows have addressed situations where the applied field is strong enough to reduce the channel velocity to a level where a Stokes flow prevails (see [1,2]).

The hydrodynamic flow, on the other hand, is characterized by a competition between viscosity and inertia, often resulting in flow separation near the channel walls. In fact, a strong near-wall response lies at the heart of the high Reynolds number theory of viscous flow through a slowly varying channel. In the hydrodynamic flow of Smith [3,4], the velocities in the core of the channel are *O*(1), and the flow near the channel wall has a sensitive nonlinear response to changes in the wall shape.

This type of flow is quite different from large magnetic field magnetohydrodynamic (MHD) flows, where the channel core velocity is suppressed, and the flow near the wall is more benign. This study takes a preliminary step to reconcile these two situations by examining how a slowly varying hydrodynamic channel flow begins to change under the action of an applied transverse magnetic field. Sufficiently large magnetic fields are found to suppress the hydrodynamic flow near the wall when the heights of the wall shapes remain fixed. However, a variety of nonlinear wall responses can be recovered as the wall distortion becomes large.

## 2. Governing equations

An incompressible electrically conducting fluid with density *ρ* flows through a channel under the action of an applied streamwise pressure gradient *g*. Spatial coordinates within the channel are non-dimensionalized by the channel width *L*. Pressure is non-dimensionalized by *gL*, and velocities are non-dimensionalized by the characteristic velocity (see Smith [3,4]). Time is non-dimensionalized using the flow time *L*/*V* _{0}. A magnetic field of strength *B*_{0} is applied perpendicular to the upstream channel walls, as shown in figure 1 in non-dimensional form, and the magnetic field is non-dimensionalized by this value. The electric field is non-dimensionalized by *V* _{0}*B*_{0}, the current density by *σV* _{0}*B*_{0} and the charge density by *ϵ*_{0}*V* _{0}*B*_{0}/*L*, where *ϵ*_{0} is the fluid permittivity, *μ*_{0} is the fluid permeability and *σ* is the conductance, all assumed to be constant.

Given the above non-dimensionalization and the assumption that fluid velocities are at levels appropriate for a hydrodynamic Poiseuille flow, the flow is governed by the incompressible mass conservation equation (see [2])
2.1
the incompressible Navier–Stokes equations
2.2
and the magnetic induction equation
2.3
where *Re*=*ρV* _{0}*L*/*μ* is the Reynolds number, *Rm*=*μ*_{0}*σV* _{0}*L* is the magnetic Reynolds number and is the Hartmann number. In all cases considered here, the magnetic Reynolds number is sufficiently close to *O*(1) values. No-slip conditions are applied along oscillating lower and upper channel walls in the form *u*(*x*,*f*(*x*,*t*),*t*)=0 and *v*(*x*,*f*(*x*,*t*),*t*)=*f*_{t}, where the fluid velocity components in the (*x*,*y*) directions are (*u*,*v*), and the lower and upper wall shapes are *f*=*f*_{L}(*x*,*t*) and *f*=1−*f*_{U}(*x*,*t*), respectively. The channel walls are assumed to be insulated, and a known constant magnetic field is applied at the walls with zero tangential component and a normal component given by *B*_{y}(*x*,*f*(*x*,*t*),*t*)=1. When *Ha*∼*O*(1), the flow far upstream in the straight channel is given by the Hartmann solution, which is *v*=0 and
2.4

For discussion purposes, these velocity profiles are plotted in figure 2. The wall shear of the Hartmann flow is defined to be λ=*U*′_{0}(0)=1/2. Solutions for the magnetic field may be found in Müller & Bühler [2]. When *Ha*∼*O*(1), equation (2.4) may be used as the upstream flow in a slowly varying channel, a situation that allows a direct interaction between the applied magnetic field and the viscous flow throughout the channel. In particular, when *Re*≫1, (*x*,*y*,*t*)=(*Re* *X*,*y*,*Re* *T*) and (*u*,*v*,*p*)∼(*U*,*Re*^{−1}*V*,*P*)+⋯ , then the flow is known to be governed by the following boundary layer problem (see Genin & Podshibyakin [5], which is also documented in Blūms *et al.* [6]), i.e. *U*_{X}+*V* _{y}=0 and
2.5
where the average of the streamwise velocity across the channel is defined to be
2.6

These equations follow from equations (2.1)–(2.3) providing that *Rm* is not too large or too small (specifically, *Ha*^{2} *Re*^{−1}≪*R*_{m}≪*Re*). The above equations exhibit the well-known features found in the Hartmann solution (and evident in figure 2), including the fact that the applied magnetic field decelerates the flow near the centre of the channel where the Poiseuille flow velocity exceeds the mean. However, when the length of the channel wall variations become shorter than *O*(*Re*) a magnetic field with finite Hartmann number can no longer influence the flow, and the Hartmann number must become large to maintain an interaction between the magnetic field and the hydrodynamic flow.

On streamwise scales shorter than *O*(*Re*) and longer than *O*(*Re*^{1/7}), the hydrodynamic solution has been given by Smith [3,4]. In this study, we assume that
2.7
where Δ≪1. The results discussed in this paper follow from equation (2.5) as and are expected to hold for a reasonably wide range of long scale channel solutions at Hartmann numbers slightly larger than *O*(1). In this situation, the primary response of the flow occurs in viscous wall layers of thickness *O*(Δ^{1/3}) situated near the upper and lower channel walls. The core of the channel is a linearized rotational inviscid region which has the form of the main deck of a triple-deck (see Smith [3]). This core region is controlled by the following equations *u*_{X}+*v*_{y}=0, and
2.8
which have the well-known solution
2.9

Matching of the core flow to the viscous wall layers serves to fix the displacement function *A*(*X*,*T*) (which becomes the average of the channel wall shapes). The solutions within the wall layers remain unchanged when the Hartmann layers are thicker than the wall layers, so long as the wall shape and the shear stress of primary viscous flow at the wall remain unchanged, which is the case for the Hartmann flow (2.4) at all values of *Ha*. This behaviour can be seen in figure 2, i.e. the velocity profiles all approach the wall with the same slope. Other well-known properties of the Hartmann flow are also evident in figure 2. As *Ha* increases, the flow in the core of the channel approaches a constant velocity *U*_{0}∼1/2 *Ha*^{−1}, which is becoming small, and Hartmann layers of thickness *O*(*Ha*^{−1}) are developing near the channel walls.

Some trial and error reveals that the first breakdown of the hydrodynamic solution as occurs in the core of the channel when the Hartmann number rises to the level 2.10

This behaviour may be inferred from the solution (2.9). Because the wall shear λ and the wall shapes within the viscous wall layers do not change with increasing Hartmann number, the pressure also does not change. However, the *U*_{0} velocity (2.4) in the core perturbation velocity solution (2.9) is becoming small, which means that the convective terms on the left-hand side of the core momentum equation (2.8) will eventually fall to the level of the streamwise pressure gradient, which is the largest neglected term on the right-hand side of this equation. This happens when condition (2.10) is satisfied.

### (a) The channel core

Throughout the channel, (*x*,*t*)=(*Re* Δ*X*,*Re* Δ^{2/3}*T*). In the channel core, *y*∼*O*(1), and the expansions are given by
where *P*_{0} is the local pressure, and the mass and momentum conservation equations become
and

Given the assumption that the upstream flow decays to a uniform Hartmann flow in the farfield, the solution in the core is found to be and .

### (b) The Hartmann layers

The leading-order flow (2.4) produces Hartmann layers near the upper and lower walls. Near the lower wall and 2.11 where 2.12 The solution in the lower Hartmann layer takes the standard form The upper Hartmann layer is similar, but with and 2.13

Note that in the upper Hartmann layer is also given by (2.12). The solution in the upper Hartmann layer has the same form as the solution in the lower Hartmann layer, but with a different displacement function, i.e. Matching the two Hartmann layers to the flow in the channel core gives 2.14

### (c) The viscous wall layers

The viscous wall layers are unchanged from the hydrodynamic solution of Smith [3,4]. Including a Prandtl transposition to create a body fitted structured grid, the lower wall layer has *x*=*Re* Δ*X*, *t*=*Re* Δ^{2/3}*T* and *y*=Δ^{1/3}(*Y* +*hF*_{L}), with expansions given by
and
where the lower and upper wall shapes are
The lower wall layer is governed by an unsteady boundary layer problem
2.15
and
2.16

Matching the viscous wall layer and the lower Hartmann layer produces the standard matching condition
2.17
where λ=1/2. The upper wall layer is also controlled by an unsteady boundary layer problem of the form of (2.15) and (2.16), but with and a sign change in the *v*-velocity expansion
Matching the upper wall layer to the upper Hartmann layer, gives
2.18

The remainder of the problem follows the argument of Smith [3,4]. The lower and upper wall layers are driven into motion solely by the effective displacement functions in (2.17) and (2.18), which are *A*+*hF*_{L} and *B*+*hF*_{U}. Different values of these two functions produce different pressures *P*(*X*,*T*). However, because the pressure gradient across the channel must be zero, the two viscous wall layer pressures must be the same, and so the forcing functions must be the same, i.e. *hF*_{L}+*A*=*hF*_{U}+*B*. This equation is combined with (2.14) to give the displacement function of the lower wall layer
2.19
which reduces to Smith's [3] solution when *H*=0. A similar expression may be found for the upper wall layer displacement function, *B*. In summary, the final equations in the lower wall layer are (2.15) through (2.17) together with the no-slip conditions *U*(*X*,0,*T*)=*V* (*X*,0,*T*)=0 and equation (2.19) for the displacement function. The streamwise wall shear stress is given by *τ*_{x}=∂*U*/∂*Y* (*X*,0,*T*).

As is the case with Smith's [3] original problem, any solution in the upper wall layer is a mirror image of the solution in the lower wall layer. Symmetry or asymmetry of the flow is dictated by the choice of the lower and upper wall shapes. A symmetric channel geometry has *F*_{U}=*F*_{L}. Irrespective of the geometry, the flow solutions in the lower and upper wall layers are the same for both steady and unsteady flows. These properties are contained in the original formulation of Smith [3,4]. From here on, attention is focused solely on the solutions within the lower wall layer for symmetric channels.

The wall layer equations (2.15) through (2.17) and (2.19) are solved using a fully implicit second-order accurate space marching scheme in the *X*-direction with Newton linearization, and a block tridiagonal inversion in *Y* at each *X* gridline.

As discussed above, the grid is generated using a Prandtl transposition. The boundary conditions (2.17) and (2.19) are directly coupled to the finite difference solution. Reverse flow regions are computed using an upwind difference scheme, and repeated iteration in the *X*-direction is used to converge the upwinded solution. A fully implicit Crank–Nicholson version of this algorithm is used for the unsteady solutions. The grid in figures 3 and 4 is −2≤*X*≤4 with 151 points and 0≤*Y* ≤8 with 121 points. In addition, for each value of *H* in figure 3, the hump height is increased from *h*=0 to *h*=4 in 20 increments in order to ensure stability of the solutions at larger hump height.

A comparison between the wall layer solution with *H*=0 and the solution of Smith [3] is shown in figure 2 for the hump when *X*>0, with *h*=2.5. Typical streamfunction solutions for the steady MHD wall layer problem are shown in figure 3 for the Gaussian hump , with *h*=4. Applying a magnetic field suppresses steady flow separation. When the strength of the magnetic field is increased to a sufficiently large level at fixed hump height, the wall layer is driven towards a pure shear flow at all points along the hump, which is close to the situation shown in figure 3 when *H*=2.

A similar result holds for unsteady flows, as typified by the solutions shown in figure 4 for the oscillating Gaussian hump , where , *h*=1, λ=2*π*/*ω* and *ω*=1. Figure 4 shows plots of *τ*_{x} as a function of *X* for different times *T*, varying over six cycles of with 60 points per cycle (resulting in a total of 360 temporal points over 37.7 time units). Approximately three cycles in each plot are at full amplitude (i.e. *g*(*T*)≃1). The hump height and frequency of oscillation are selected to produce attached unsteady flow. As seen in these figures, applying a magnetic field also suppresses the unsteady flow and drives the shear stress towards the undisturbed value of 1/2.

## 3. The wall layer as

The computations discussed above show that the flow within the viscous wall layer can be suppressed by an applied magnetic field. The implications of this observation are readily seen when using the two-dimensional *u*-matching condition (2.17) of the wall layer
In order for *U* to approach the shear flow λ*Y* as *H* becomes large, the pressure must balance the wall shape, or
3.1
where
3.2
The velocity response is found to be linear, with
3.3

The velocities (*U*_{1},*V* _{1}) satisfy linearized wall layer equations but with prescribed pressure *P*_{0} from (3.2). The boundary condition becomes , with the *u*-matching condition providing the pressure *P*_{1} for the higher-order wall layer problem.

For larger values of *h*, the wall layers can respond nonlinearly. In particular, the above solution indicates that when then the leading-order pressure and velocities in (3.1) and (3.3) become *O*(1). Formally, the pressure and velocity expansions become
3.4
whereas the displacement function becomes *A*∼*H*^{2}*A*_{0}+*A*_{1}+⋯ . The leading-order nonlinear problem is given by equations (2.15) and (2.16) but with all variables having subscript 0. The *u*-matching condition gives
3.5
and as , which produces the boundary condition as . The displacement relation (2.19) then gives
3.6
and
3.7

Note that equation (3.6) for *A*_{0} provides the solution for the pressure when combined with equation (3.5), whereas equation (3.7) for *A*_{1} gives the higher-order pressure *P*_{1}. Figure 5 shows this behaviour in the wall layer solutions. Figure 5*a* shows the computed wall layer pressures as the Hartmann number increases (with pressure plotted in the form *H*^{2}*P*/*h*) for the hump , with *h*=6 and *H*=1,2,4,8,16 approaching the limit solution (3.2). Figure 5*b* shows the computed wall shear for the hump , with and a sequence of solutions that have and *H*=1,2,4,8 and 16, producing the following sequence of hump heights: *h*=0.4,1.6,6.4,25.6 and 102.4. The wall layer solutions of figure 5*a* are seen to approach the computed limit solution (3.4) with , which was selected to give an attached flow close to separation. This limit solution comprises the boundary layer equations (2.15) and (2.16), but with the 0 subscript corresponding to the first term in (3.4). The pressure in the boundary layer is set from , the wall boundary conditions are no-slip conditions, *U*_{0}(*X*,0,*T*)=*V* _{0}(*X*,0,*T*)=0, and the matching condition of the lower wall layer with the lower Hartmann layer is as .

## 4. Nonlinear inviscid Hartmann layers

As *A*(*X*,*T*) and the height of the wall geometry become large, the above solution persists until the point where the Hartmann layers become nonlinear (i.e. when the non-uniformities in the expansions (2.11) and (2.13) are reached). This happens when (figure 6). At this point, the wall geometry heights have increased to the height of the Hartmann layers, i.e. . In all of the layers, *x*=*Re* Δ*X* and *t*=*Re* Δ^{2/3}*T*. The core of the channel remains unchanged and has the solution
4.1

In the lower Hartmann layer, and the wall shape scales to the same level, namely . Within the lower Hartmann layer 4.2 The equations within the lower Hartmann layer are found to be and . These equations have the solution and . Because the Hartmann layer is inviscid, a tangency boundary condition is used to provide the lower wall boundary condition 4.3

When substituted into this boundary condition, the lower Hartmann layer expansions (4.2) yield the following boundary condition for both steady and unsteady flows

Combining this equation with and upstream decay gives the solution . The coordinate transformation *ξ*=*X* and then gives the solution , where . The *v*-velocity solution becomes . It should be noted that because , a wall layer of the form discussed in the previous section is preserved.

The solutions in the upper Hartmann layer have a similar form, but with and
This produces the equations and , which have the solutions and , with . These solutions also make use of an upper wall tangency condition which provides the solution . Matching the *v*-velocities of the two Hartmann layers with the channel core yields . Combining this equation with and provides the pressure solution
4.4

The lower wall layer is similar to the one discussed previously, but with an expansion that accounts for a coordinate system in the wall layer that is fitted to the wall geometry . The vertical coordinate within the lower wall layer is given by and the expansions are

The higher-order terms in the expansions are required to consistently account for the effect of the large wall geometry. Given the above expansions, the leading-order equations within the wall layer become *U*_{X}+*V* _{Y}=0 and
4.5
with the boundary conditions *U*(*X*,0,*T*)=*V* (*X*,0,*T*)=0. Matching with the lower Hartmann layer gives as . The pressure appearing in equation (4.5) is given by (4.4). A similar wall layer problem is found near the upper wall.

Note that the near-wall viscous response is effectively the same as that found at lower wall heights and Hartmann numbers. The match with the wall layer at lower Ha is achieved, because the wall layer problem of the previous section, and the nonlinear Hartmann layer and wall layer combination of this section are both governed by the same set of equations which are controlled by the same single parameter, which is the scaled form of *h*/*Ha*^{2}. In both problems, the scaling of this parameter is the same, namely *h*/*Ha*^{2}∼*O*(Δ^{2/3}). This implies a simple continuation of the solution of the last section through the nonlinear inviscid Hartmann layer stage.

## 5. Nonlinear viscous Hartmann layers

The above solution persists until a non-uniformity is reached in the core expansions (4.1). This happens when the wall geometries increase to the channel width, and the Hartmann layers collapse into the wall layers. At this point, (*x*,*t*)=(*Re* Δ*X*,*Re* Δ^{2/3}*T*), and the Hartmann number becomes (figure 7). The expansions in the channel core are found to be

In the core of the channel, the upstream *u*-velocity asymptotes to the Hartmann solution (2.4), which becomes . Using these expansions, the equations of the inviscid core are found to be and
where is the average of across the channel. Because the flow is inviscid, the tangency condition (4.3) must be used at the wall. When the above expansions are used in this boundary condition, it is found that all terms must be retained, and the lower wall boundary condition takes the form
where the wall shapes (*f*_{L},*f*_{U}) are now *O*(1). Again, a similar equation holds at the upper wall. Because the leading-order upstream flow is constant, a one-dimensional solution may be found for the channel core, i.e. . The solution for is found to be
with the pressure gradient given by
5.1

The viscous layer takes the form of a Prandtl-transposed boundary layer, with *y*=*f*_{L}(*X*,*T*)+Δ^{1/3}*Y* and the expansions
The boundary layer is governed by the equations *U*_{X}+*V* _{Y}=0 and
5.2
Note that the magnetic force enters directly into the boundary layer equations at this point in a manner similar to (2.5), a feature that has been absent in all of the other wall layers. The no-slip conditions become *U*(*X*,0,*T*)=*V* (*X*,0,*T*)=0, and matching with the inviscid core gives as . The pressure gradient is determined from (5.1) and that solution is consistent with the value determined from the limit of equation (5.2). The initial condition for the boundary layer is found from the Hartmann solution (2.4), and is as . A similar boundary layer problem is found to hold at the upper wall. At large-scaled Hartmann numbers, the boundary layer equations (5.2) limit to a Stokes layer with , and the solution , where *C* is the scaled average velocity

This final limit approaches the low-speed viscous flow properties that are typically observed in large Hartmann number MHD flows.

## 6. Conclusion

A high Reynolds number asymptotic structure has been developed for steady and unsteady two-dimensional laminar MHD flows in channels with slowly varying wall geometries of length *O*(*Re* Δ), where *Re*≫1 and Δ≪1. At sufficiently low Hartmann numbers, the hydrodynamic wall layer solution is recovered. When the Hartmann number increases to the level *Ha*∼*O*(Δ^{−1/6}), the magnetic field is found to suppress the viscous response near the wall through a modification of the displacement effect in the channel core. When *Ha*≫*O*(Δ^{−1/6}), a strong nonlinear viscous response near the wall can be maintained providing that the wall geometry height also increases. This viscous layer takes the form of a wall layer with a prescribed pressure that is set by the wall geometries through the linear core and Hartmann layer solutions. When *Ha*∼*O*(Δ^{−2/9}), the Hartmann layers become nonlinear but remain inviscid, with viscous effects coming into play in a thin viscous sublayer. The viscous response near the wall when *Ha*∼*O*(Δ^{−2/9}) remains unchanged from the form seen when Δ^{−1/6}≪*Ha*≪Δ^{−2/9}. As the Hartmann number continues to increase, the channel core eventually becomes nonlinear when the Hartmann number reaches the level *Ha*∼*O*(Δ^{−1/3}). At this point, the wall geometry height is comparable to the channel width, and the Hartmann layers merge with the viscous wall layers. When the Hartmann number increases above *O*(Δ^{−1/3}), a Stokes-layer flow develops within the viscous wall layers.

## Footnotes

One contribution of 15 to a Theme Issue ‘Stability, separation and close body interactions’.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.