1. Introduction

In the last decade, the interest on magnetohydrodynamic (MHD) electrical generators
has been renewed due to its potential applications as converters of acoustic ^{[1},^{2 ]} and
ocean wave energy . A MHD generator is a device that converts the kinetic energy
^{[3},^{4
]} of an electrically conducting fluid, for instance a liquid metal, into
electrical energy through the interaction with a magnetic field. A common MHD
generator consists of a duct with a rectangular cross-section immersed in a static
magnetic field that is transverse to a pair of insulating walls. The walls parallel
to the applied field are electrical conductors (electrodes). When a conducting fluid
flows inside the duct, its motion within the imposed magnetic field induces an
electrical current perpendicular to both the fluid motion and the applied field that
can be extracted through the electrodes connected to an external load. If the fluid
motion is unidirectional, a DC current is induced while, if the fluid moves in
oscillatory motion, an AC current is generated. In this way, the kinetic energy of
the fluid is converted directly into electrical energy without the need of
mechanical parts. DC MHD generators were the first to be developed, particularly at
high temperatures using plasma as a working fluid ^{[5]}. In the late eighties a proposal was made at Los Alamos National
Laboratory to transform acoustic power into electrical power through a liquid metal
MHD acoustic transducer ^{[6]}. In these devices,
the thermoacoustic effect is used to generate an oscillatory motion of a conducting
fluid in a duct immersed in an applied transverse magnetic field ^{[2},^{7},^{8 ]}. Alternate liquid metal MHD generators were
later proposed to convert the oscillatory motion of ocean waves into electricity
^{[9}-^{11]}.

A full evaluation of the performance of liquid metal MHD generators should rely on a
detailed analysis of the dynamics of the oscillatory flow interacting with a
magnetic field. In contrast with steady MHD duct flows that have been widely studied
experimentally and theoretically ^{[12]},
oscillatory MHD flows have been much less explored ^{[13]}. In this paper, we use a two-dimensional model of an MHD generator
to investigate analytically the laminar liquid metal flow created by an oscillatory
pressure gradient (for instance, produced by either thermoacoustic effect or ocean
waves) imposed at the extremes of the generator duct. We analyze the flow in two
different regions. First, we consider the region far from the edges of the generator
where the magnetic field is uniform. With the aim of understanding the interplay of
the imposed pressure gradient and the braking Lorentz force created by the
interaction of the induced current with the applied magnetic field, asymptotic
solutions are derived in the limits of low and high oscillating frequencies. The
second analyzed flow region is where the applied magnetic field is non-uniform, that
is, the region where the flow enters or leaves the applied magnetic field. Several
studies have addressed the MHD unidirectional flow in this region (see for instance
^{[14},^{15]}), however, it appears that the oscillatory MHD duct flow in a
fringing magnetic field has not been previously considered. Assuming high
oscillation frequencies, we pay a particular attention to the behavior of the
oscillatory boundary layers immersed in the spatially varying magnetic field. These
layers, which are a combination of the Stokes and Hartmann layers, determine the
flow dynamics to a large extent. Using a perturbation solution, it is found that
non-linear effects give rise to steady streaming vortices in the fringing magnetic
field. The effect that these vortices may have on the overall performance of the
generator is discussed.

2. Oscillatory flow in an MHD generator

We consider the oscillatory flow of a liquid metal in a duct of rectangular cross-section under a transverse magnetic field. The walls perpendicular to the applied field are electrical insulators, while those parallel to the field are perfect conductors connected to an external load (see Fig. 1). The oscillatory flow is driven by a zero mean, time-periodic pressure gradient imposed at the extremes of the duct.

The system of equations that govern the unsteady flow of an incompressible, electrically conducting viscous fluid in the presence of a magnetic field are the continuity equation, Navier-Stokes equation, Faraday’s law of induction, Ampère’s law, Gauss’s law for the magnetic field and Ohm’s law which, respectively, can be conveniently written in the following dimensionless form

where the flow velocity \

Further, the dimensionless parameters ^{[12]}, where

The oscillatory motion of the fluid inside the magnetic field induces an electric
current density in the spanwise (^{[12]}. Hence, the magnetic field is uncoupled
from the fluid motion and governed by the magnetostatic equations.

3. Flow in the uniform magnetic field region

We now assume that the aspect ratio of the generator is very large, that is, ^{[16]}.

We now restrict to the region where the applied field is uniform so that, in
dimensionless terms,

We disregard transient solutions and consider that the harmonic axial pressure
gradient that drives the flow is given by the real part of ^{[17]}, a solution to the Eq.(5) that satisfies the no-slip boundary conditions can be found,
namely,

where ^{[17]}.

In the present work the attention is focused in the interplay of inertia and the
braking Lorentz force. The explicit form of the velocity profile (6) is, however, not particularly
insightful. In order to get a better understanding of the physical behavior of this
oscillatory MHD flow, we look for asymptotic solutions in the limits

3.1 Low-frequency solution:

In the low frequency limit it is possible to obtain a regular asymptotic solution
in the flow domain ^{[18]}. Since we are
interested in the limit when

Substituting the harmonic pressure gradient and assuming solutions given as the
real part of the expressions

and solve the corresponding equations with no-slip boundary conditions at each
order on the parameter

where ^{[12]}. An out of
phase contribution is also found at *i.e.*

which shows an in-phase Poiseuille flow contribution. The phase angle between the pressure gradient and the velocity is given by

Note that when

3.2 High-frequency solution:

At high frequencies a uniform asymptotic solution for the whole domain does not
exist. Therefore, matching asymptotic solutions in the core and the boundary
layer has to be sought. For the core, we start from Eq.(5) and introduce the variables

We now look for a solution

Here, we assume that

This represents a uniform time-periodic flow that lags behind the imposed
pressure gradient according to the value of

Let us now consider the boundary layer flow. We introduce the stretched variable

for the corresponding function

where

The phase angle between the boundary layer and the pressure gradient is given by

Again, in-phase and out-of-phase contributions are obtained in the boundary
layer, the structure of which depends on the value of ^{[18]}. When

An illustrative way of visualizing the phase lag produced by the Lorentz force
between the velocity and the pressure gradient is by noticing that these
quantities satisfy the parametric equations of an ellipse in the plane ^{[19]}. If we define

where *i.e.*

4. Flow in the non-uniform magnetic field region

In this section, we address the oscillatory flow of the liquid metal close to the
edges of the magnets where the magnetic field is non-uniform. In this region, the
transverse magnetic field varies from its maximum strength to zero as the ^{[21},^{22]}. The steady streaming is induced by the
non-linear Reynolds stresses in the boundary layer that appear due to the axial
dependence of the streamwise velocity, produced in this case by the existence of the
non-uniform field. In hydrodynamic flows, steady streaming appears, for instance, at
the entrance of a rigid tube when a zero-mean oscillatory flow is imposed ^{[23]} or in the classic problem of oscillating
bluff bodies ^{[22},^{24]}. The persistence of the steady streaming beyond the
boundary layer is one of the distinctive aspects of this class of oscillatory flows
^{[25]}. The action of a uniform transverse
magnetic field on the steady streaming produced by an oscillatory laminar boundary
layer close to an insulating curved wall was previously studied using a perturbation
expansion taking the inverse of the Strouhal number as a small parameter ^{[26]}. Following a similar procedure, we explore
here the appearance of steady streaming in the boundary layers of the MHD generator
promoted by the non-uniformity of the applied transverse magnetic field.

As in Sec. 3, we consider the oscillatory motion of the liquid metal limited by two
infinite insulating plane walls at rest under a transverse magnetic field. We are
now focused on the region close to the edges of the magnets, so the transverse
magnetic field is expressed in the form ^{[15},^{27]}

Here ^{[15},^{27]}.

We assume that as a result of the imposed pressure gradient, beyond the boundary
layer (outer flow) the fluid oscillates irrotationally with a zero-mean in the axial
direction so that the corresponding velocity component in dimensionless form can be
expressed as the real part of

where the outer velocity ^{[26]} are given by

These dimensionless parameters estimate, respectively, the ratio of the amplitude of
the oscillation to the characteristic length

From Eqs. (21) and (22), the explicit form of the function

where the harmonic variation of the electric field was assumed.

In turn, the inner layer flow is governed by the equations:

where

where (28) and (29) represent the no-slip condition of the velocity components at the wall and (30) is the matching condition for the inner and outer flows.

4.1 First order solution

We now look for a solution of the boundary layer problem as a perturbation
expansion in the small parameter

where

where the first and second approximations are denoted, respectively, by
subindexes 0 and 1. By eliminating the pressure gradient and current densities
in Eq.(26) with the substitution of Eqs.
(21), (22), (27), and using (31) and (32), we find that

with boundary conditions

Assuming that

the function

with boundary conditions

The solution of (38) that satisfies conditions (39) is

where

From the solution(40) it is
possible to estimate the thickness of the Stokes-Hartmann boundary layer,
namely, ^{[12]}. Far enough from the magnet edges,
say

which coincide with the ordinary hydrodynamic limit ^{[21]}, where

4.2 Second order approximation

The equation for the second order approximation

Note that the products of the harmonic functions and derivatives on the
right-hand side of (41) introduce
terms proportional to

where the real part of

with boundary conditions

The solution is given in the form

with

The constants

Solution (45) correctly recovers
the hydrodynamic solution ^{[21]} when the
magnetic field strength tends to zero. Particularly, the contribution to the
tangential velocity reduces to

In turn, the boundary value problem satisfied by the steady state part,

where conjugate complex quantities are denoted by an overbar and

where constants ^{[21},^{26]}.

In the boundary layer, the second order steady velocity component parallel to the wall is given by

where *i.e.* vanishing
magnetic field) it results that, as the distance from the wall tends to
infinity, ^{[21},^{25]}. Therefore,
the condition at infinity must be relaxed, imposing that

In this way, the steady streaming flow goes beyond the boundary layer,
penetrating into the potential flow. The finite velocity at the edge of the
boundary layer can be used as the inner boundary condition for the outer flow
^{[25]}. Although far form the magnet
edges

Unlike the hydrodynamic case, when a magnetic field is present the steady
solution (50) does satisfy the
vanishing of the streaming flow as the distance from the walls tends to infinity
^{[26]}. This means that the streaming
motion does not penetrate from the boundary layer into the potential flow. Figure 5 shows the contribution to the
tangential velocity

The steady part of the stream function,

In Fig. 7, the streamlines in the fringing
field region are displayed for the cases

5. Concluding remarks

We have explored the zero-mean oscillatory two-dimensional flow of a liquid metal in
an alternate MHD generator driven by an imposed harmonic pressure gradient. The
finite extension of the applied magnetic field transverse to the electrically
insulating duct walls was considered for the analysis of the flow behavior. In the
uniform field region, characteristic flows were explored through asymptotic
solutions for a small (

The analysis of the entrance oscillatory flow in the fringing field region at the edges of the MHD generator was carried out for high oscillation frequencies using a perturbation method, assuming the small amplitude of oscillation approximation. From the first order solution, the thickness of the boundary layer was estimated, and it resulted a combination of the Stokes and Hartmann layers, each of which are recovered in the corresponding limits. The second order solution revealed that, superimposed to the primary oscillatory flow, a secondary flow composed by a time periodic motion oscillating with twice the original frequency and a steady streaming contribution exist. A pair of steady streaming vortices emerges in the fringing field region as a consequence of non-linear effects caused by the spatial variation of the magnetic field. The extension and intensity of the vortices grow as the magnetic field gradient increases. Unlike the hydrodynamic case, these vortices do not penetrate into the potential flow but remain confined in the boundary layer and, moreover, their strength decreases as the magnetic field becomes stronger. Although the disturbance created by the steady streaming vortices is not expected to affect the performance of the MHD generator, one could conveniently consider a smooth magnetic field gradient for design purposes.