## Indicators

• Similars in SciELO

## Print version ISSN 0035-001X

### Rev. mex. fis. vol.65 n.3 México May./Jun. 2019  Epub Apr 30, 2020

#### https://doi.org/10.31349/revmexfis.65.239

Research

Analysis of the oscillatory liquid metal flow in an alternate MHD generator

aInstituto de Energías Renovables, Universidad Nacional Autónoma de México, Apartado Postal 34, Temixco, Morelos, 62580 México.

bInstituto de Ciencias Marinas y Pesquerías, Universidad Veracruzana, Hidalgo 617, Col. Rio Jamapa, Boca del Rio, Veracruz, 94290, México.

Abstract

The zero-mean oscillatory flow of a liquid metal in an alternate magnetohydrodynamic electric generator is explored analytically. The flow, confined in a two-dimensional insulating wall duct under a transverse magnetic field, is driven by an externally imposed oscillatory pressure gradient. The flow behaviour is analyzed in two different regions. First, asymptotic solutions for low and high oscillating frequencies in the uniform magnetic field region far from the magnet edges, are used to explore the phase lag produced by the Lorentz force between the velocity and the axial pressure gradient. In addition, the entrance flow region where the oscillatory fluid motion interacts with the non-uniform magnetic field is studied. A perturbation analysis of the boundary layer flow in this region reveals that non-linear effects lead to the appearance of steady streaming vortices superimposed on the harmonic flow. The influence of these vortices on the performance of the generator is analyzed.

Keywords: MHD generator; liquid metal oscillatory flow; steady streaming; perturbation analysis

PACS: 47.65.-d; 47.15.Rq; 47.15Cb

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

u=0, (1)

ut+RRω2(u)u=-p+1Rω2u+Ha2Rωj×B, (2)

×E=-Rω2RBt,×B=Rmj, (3)

B=0,j=E+u×B (4)

where the flow velocity \u, the pressure p, the magnetic field B, the electric field E and the current density j are normalized by Uo=G/ρω, Gh, Bo, UoBo and σUoBo, respectively. Here, G and ω are the amplitude and frequency of the imposed oscillatory pressure gradient, ρ and σ are the mass density and the electrical conductivity of the fluid, h is the distance between the walls transverse to the magnetic field, and Bo is the maximum strength of the applied field, respectively. The coordinates (x,y,z) and time t, are normalized by h and 1/ω, respectively.

Further, the dimensionless parameters Rω=ωh2/ν, R=Gh3/ρν, Ha=Bohσρν, are the frequency parameter (or oscillation Reynolds number), the amplitude parameter and the Hartmann number, respectively, where ν is the kinematic viscosity. Assuming that the physical and geometrical properties of the system remain unchanged, these dimensionless parameters express, correspondingly, the influence of the oscillation frequency, the amplitude of the pressure gradient, and the magnetic field strength. In turn, Rm=μ0σUoh is the magnetic Reynolds number that gives an estimation of the induced magnetic field compared with the applied field [12], where μ0 is the magnetic permeability of vacuum.

The oscillatory motion of the fluid inside the magnetic field induces an electric current density in the spanwise (z) direction. The current, in turn, interacts with the applied field originating a braking Lorentz force in the axial x-direction. Usually, in liquid metal MHD flows the low magnetic Reynolds number approximation holds, which means that the magnetic field induced by the fluid motion is much smaller than the applied field and can be neglected [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, w/h>>1 (see Fig. 1) so that the conducting walls (electrodes) are located at distant positions z=±zo, connected to an external electrical circuit. Under this approximation, we can consider that the oscillatory flow is two-dimensional, confined between the insulating walls transverse to the magnetic field (see Fig. 2). Since the current is induced in the direction perpendicular to the plane of motion, there must exist an electric field, Ez, the value of which depends on the external electrical load. As the magnetic field remains unperturbed, Faraday’s law of induction reduces to ×E=0 and the electric field becomes potential. In fact, it can be shown that under the present assumptions Ez is spatially constant and it is at most a function of time [16].

We now restrict to the region where the applied field is uniform so that, in dimensionless terms, B=y^. In this region the flow is fully developed, therefore, u=u(y,t)x^ and the Navier-Stokes equation reduces to

ut=-px+1Rω2uy2-Ha2Rω(Ez+u). (5)

We disregard transient solutions and consider that the harmonic axial pressure gradient that drives the flow is given by the real part of -p/x=eit. Assuming that the axial velocity component and the electric field are also harmonic functions of time, u=u0(y)eit and Ez=-Keit, with K a constant known as the load factor [17], a solution to the Eq.(5) that satisfies the no-slip boundary conditions can be found, namely,

u(y,t)=Umeitλcoshλ-coshλyλcoshλ-sinhλ, (6)

where λ=Ha2+iRω and Um is the dimensionless spatial average of the velocity profile in the cross-section. From this solution it is possible to establish a model of the alternate MHD generator that allows to assess the electrical performance of the device [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 Rω1 and Rω1, which correspond to the low and high frequency oscillatory motions, respectively.

3.1 Low-frequency solution: Rω1

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 Rω takes very small values, it is convenient to use the rescaled variables u^=u/Rω and E^z=Ez/Rω, so that Eq.(5) becomes

Rωu^t=-px+2u^y2-Ha2(E^z+u^). (7)

Substituting the harmonic pressure gradient and assuming solutions given as the real part of the expressions u^=g^(y)eit and E^z=-K^eit, an equation for the function g^(y) is found. We expand this function as a perturbation series on the small parameter Rω, namely,

g^(y)=g^o(y)+Rωg^1(y)+O(Rω2), (8)

and solve the corresponding equations with no-slip boundary conditions at each order on the parameter Rω. After taking the real part, the final result is

u^(y,t)=u^op{[1-coshHaycoshHa]cost+Rω[12Ha(ysinhHaycoshHa-tanhHacoshHaycoshHa)+1Ha2(1-coshHaycoshHa)]sint}+O(Rω2), (9)

where u^op=Ha-2+K^. At zero-order in Rω, a quasi-steady Hartmann flow in phase with the pressure gradient oscillation, is obtained. As usual, the profile is flattened as Ha increases [12]. An out of phase contribution is also found at O(Rω), but it is modulated by terms of O(Ha-1) and O(Ha-2) which become negligible the higher the Ha values are. When the Hartmann number is very small, i.e.Ha0, a purely hydrodynamic flow is recovered

u^(y,t)=(1-y2)2cost+Rω24(1-y2)(5-y2)sint, (10)

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

θ=arctan-Rω12HaysinhHaycoshHa-tanhHacoshHaycoshHa+1Ha21-coshHaycoshHa1-coshHaycoshHa (11)

Note that when Ha0 viscosity originates a non-zero phase angle, namely, θ=-arctan[Rω(5-y2)/12]. In turn, when Ha, the phase angle reduces to zero indicating that the flow is frozen by the strong magnetic field interaction.

3.2 High-frequency solution: Rω1

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 u=g(y)eit and Ez=-Keit, assuming that -p/x=eit. Hence, we get the equation

ig-1=1Rωd2gdy2-Ha2(g-K). (12)

We now look for a solution g(y) as an expansion in the small parameter Rω-1, namely,

g(y)=go(y)+1Rωg1(y)+O(Rω-2). (13)

Here, we assume that Ha2=γRω, where γ is a positive real number. Then in the limit Rω, from Eq.(12) and (13), the first order solution in the core is go=(1+γK)/(γ+i)=(1+γK)(γ-i)/(γ2+1). Therefore, the core velocity field is

uc=(1+γK)(γ2+1)(γcost+sint)+O(Rω-2). (14)

This represents a uniform time-periodic flow that lags behind the imposed pressure gradient according to the value of γ, where the phase angle between the pressure gradient and the core velocity is θc=-arctan[1/γ]. For γ1, a purely hydrodynamic flow is obtained. In this case, the Lorentz force is negligible and there is a lag of -π/2 in the motion of the core with respect to the pressure gradient. In turn, if γ=1, the Lorentz force is of the same order of magnitude as the inertial acceleration, and the core flow presents a phase difference of -π/4 with respect to the pressure gradient. When γ1, the Lorentz force is dominant, therefore, the phase lag is negligible and the core follows the pressure gradient oscillation.

Let us now consider the boundary layer flow. We introduce the stretched variable Y=Rω1/2(1+y) in the bottom boundary layer, hence at the wall, Y=0 and u(0,t)=0. Then Eq.(12) becomes

d2gbdY2-igb+1=γ(gb-K). (15)

for the corresponding function gb in the boundary layer. Expressing gb as a series like (13), the solution of Eq. (15) that satisfies no-slip boundary conditions and that matches with the core flow (gbY1=(1+γK)/(γ2+1)) within an error of order O(Rω-1), leads to the boundary layer flow

u(Y,t)=(1+γK)(γ2+1)×{[γ-exp(-αY)(γcosβY-sinβY)]cost+[1-exp(-αY)(cosβY+γsinβY)]sint}+O(Rω-2), (16)

where

αβ=γ2+1±γ21/2, (17)

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

θb=arctan{-[1-exp(-αY)(cosβY+γsinβY)]γ-exp(-αY)(γcosβY-sinβY)} (18)

Again, in-phase and out-of-phase contributions are obtained in the boundary layer, the structure of which depends on the value of γ. Provided γ1, a purely hydrodynamic (Stokes) boundary layer is found [18]. When γ=1, a mixture of Stokes and Hartmann layers results. Finally, in the case γ1, magnetic forces dominate and a Hartmann layer oscillating in phase with the pressure gradient (u^(1/Rω)[1-e-γY]cost) is obtained.

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 u vs. -p/x [19]. If we define X=-p/x and Y=u, we get for either the core or the boundary layer flows

a2b2+1X2-2afb2XY+Y2f2b2=1, (19)

where f(γ)=(1+γK)/(γ2+1). For the core flow, a=γ and b=1, while for the boundary layer flow, we have a=γ-exp(-αY)(αcosβY-sinβY) and b=1-exp(-αY)(cosβY+γsinβY. In Figs. 3, Eq. (19) is plotted during a whole cycle for the case Rω=30, K=0.8, and different γ values. Some interesting information can be extracted from these plots, particularly because they clearly compare the velocity amplitude before and after the pressure gradient inversion. In fact, the vertical coordinate axis indicates the precise moment at which the pressure gradient is inverted. In the second and fourth quadrants, the pressure gradient acts in favor of the fluid motion, while in the first and third quadrants it acts against the fluid motion. Figure 3a shows the curves corresponding to the core flow. As it was shown, in the laminar hydrodynamic regime (γ=0, i.e.Ha=0) the core flow presents a phase difference of -π/2 with respect to the pressure gradient when Rω1, and the corresponding curve is a circle. For increasing values of γ, the curve is distorted and rotated clockwise as a result of stronger magnetic interaction which changes the phase difference between the velocity and the pressure gradient. When the Lorentz force is of the same order of magnitude as the inertial acceleration (γ=1), a tilted ellipse is obtained while in the case γ1 (Ha), no phase difference exists between the flow and the pressure gradient, therefore, the curve reduces to a straight line. The corresponding curves for the boundary layer flow are shown in Fig. 3b. Although similar ellipses are formed, note that they are not the same as in the core flow since, in addition to the magnetic interaction, viscosity also affects the phase difference between the velocity and the pressure gradient. In fact, no circle is formed even when Ha=0. However, when γ1 (Ha) a straight line is formed indicating that the phase difference disappears.

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 x distance to the edge of the magnet increases. Although the cross-section of the duct does not change, this can be considered as an entrance flow problem due the non-homogeneity of the magnetic field. Forced oscillations produced by the imposed pressure gradient in the outer flow produce an oscillatory flow in the Stokes-Hartmann boundary layer, however, due to the action of viscosity, the flow oscillations in this layer do not average to zero but a net steady flow is produced, known as steady streaming [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 B=By(x)y^ , where the variation of the field in the axial direction is given in dimensionless form as [15,27]

By(x)=11+e-x/x0. (20)

Here x0 is a positive constant whose magnitude governs the magnetic field gradient. Figure 4 shows the magnetic field distribution for different vales of x0. We can observe that By0 as x- and By1 as x. Although this field is not curl-free, it is a reasonable approximation that take into account the streamwise variation of the magnetic field [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 U(x,t)=U0(x)eit. Owing to continuity, the component of the velocity in the perpendicular direction to the wall is -(y+c)(dU/dx)eit where c is a complex constant. The governing equations of the outer flow in dimensionless form are

Ut+sUUx=-Nwpx-NwJzBy, (21)

Jz=Ez+UBy,    Jzz=0, (22)

where the outer velocity U, the pressure p, the outer current density Jz and the electric field Ez are normalized by U , σUB02h, σUB0 and UB0h, respectively. Here U is the amplitude of the outer flow velocity. Likewise, the coordinates (x, y, z) and time t are normalized by h and 1/ω, respectively. Further, the inverse of the Strouhal number, ϵs, and the oscillation interaction parameter, Nω, [26] are given by

ϵs=Uωh,Nω=Ha2Rω=σB02ρω. (23)

These dimensionless parameters estimate, respectively, the ratio of the amplitude of the oscillation to the characteristic length h and the ratio of the magnetic to the inertial forces. Equation (21) corresponds to the Euler equation while Equations (22) express the Ohm’s law and conservation of current in the outer flow. In order to guarantee that boundary-layer separation will not arise, the small amplitude of oscillation approximation is assumed, that is ϵs1.

From Eqs. (21) and (22), the explicit form of the function U0(x) can be determined at the lowest order in ϵs, namely,

U0(x)=Nω(1+ByK)i+NωBy2 (24)

where the harmonic variation of the electric field was assumed.

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

ux+vy=0 (25)

ut+ϵs(uux+vuy)=-Nωpx+1Rω2uy-NωjzBy (26)

jz=Ez+uBy,jzz=0, (27)

where U and σUB0 have been used to normalize the velocity components in x- and y-directions (u and v) and the inner current density, jz, respectively. In order to ensure the validity of the boundary-layer approximation for the inner flow, it is assumed that Rω1. The boundary conditions to be satisfied by the inner flow are

u(x,0,t)=0, (28)

v(x,0,t)=0, (29)

u(x,y,t)U(x,t);asy, (30)

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 ϵs. From the incompressible condition, the velocity components are

u=ψy,v=-ψx, (31)

where ψ is the stream function that can be expressed in the form

ψ(x,y,t)=ψ0(x,y,t)+ϵsψ1(x,y,t)+O(ϵs2). (32)

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 ψ0 satisfies the equation

2ψ0ty-1Rω3ψ0y3+NωBy2ψ0y=Ut+NωBy2U, (33)

with boundary conditions

ψ0y(x,0,t)=0, (34)

ψ0x(x,0,t)=0, (35)

ψ0y(x,y,t)U(x,t);asy. (36)

Assuming that

ψ0(x,y,t)=U0(x)ξ0(x,y)eit, (37)

the function ξ0 satisfies

3ξ0y3-ξ0yiRω+Ha2By2=-(Ha2By2+iRω), (38)

with boundary conditions

ξ0(x,0)=ξ0(x,0)y=0,ξ0(x,y)y1asy. (39)

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

ξ0(x,y)=y-1α+iβ(1-e-(α+iβ)y), (40)

where

α(x)=Ha4By4+Rw2+Ha2By2212β(x)=Ha4By4+Rw2-Ha2By2212

From the solution(40) it is possible to estimate the thickness of the Stokes-Hartmann boundary layer, namely, δ1/[α(x)2+β(x)2]1/2. Notice that due to the streamwise variation of the magnetic field, the layer thickness is not uniform in this region. The layer is much thinner where the magnetic field is strong (By1). If Ha2Rω, the layer thickness is of the order of the Hartmann layer, namely, δHa-1 [12]. Far enough from the magnet edges, say x<10x0, the magnetic field vanishes so that Ha=0, and the velocity components reduce to

u0=U0(x)eit(1-e-(1+i)η),v0=-dU0dxeitη-12(1-i)1-e-(1+i)η),

which coincide with the ordinary hydrodynamic limit [21], where η=yRω/2. In that region, the layer thickness reduces to that of the Stokes layer, that is δ1/Rω1/2.

4.2 Second order approximation

The equation for the second order approximation ψ1 to order ϵ has the form

2ψ1ty-1Rω3ψ1y3+NωBy2ψ1y=UUx-ψ0y2ψ0xy+ψ0x2ψ0y2. (41)

Note that the products of the harmonic functions and derivatives on the right-hand side of (41) introduce terms proportional to sin2t and cos2t, as well as steady-state terms. This means that convective non-linear terms give rise to steady state terms that contribute to the steady streaming flow. In order to solve Eq.(41) we assume that

ψ1(x,y,t)=U0dU0dxξ1t(x,y)e2it+ξ1s(x,y), (42)

where the real part of U0(x) must be taken. The equation satisfied by ξ1t is

3ξ1ty3-(Ha2By2+2iRω)ξ1ty=-Rω21-(ξ0y2+ξ0y2ξ0xyU0(x)U0'(x))+ξ02ξ0y2+U0(x)U0'(x)ξ0x2ξ0y2 (43)

with boundary conditions

ξ1t(x,0)=ξ1ty(x,0)=0andξ1ty0asy. (44)

The solution is given in the form

ξ1t(x,y)=Rω4U0'(x)κt1e-λy+κt2e-γy+κt3e-2γy+κt4 (45)

with γ=α(x)+iβ(x) and λ=αt(x)+iβt(x), where

αt(x)=Ha4By4+4Rω2+Ha2By2212,βt(x)=Ha4By4+4Rω2-Ha2By2212.

The constants κt1, κt2, κt3, and κt4 in (45) are defined in the Appendix.

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

ξ1t(x,y)y=12-ie-(1+i)2η+ie-(1+i)η-(i-1)ηe-(1+i)η (46)

In turn, the boundary value problem satisfied by the steady state part, ξ1s, is

3ξ1sy3-λs2ξ1sy=Rω42-2ξ0yξ0y¯+ξ02ξ0y2¯+ξ0¯2ξ0y2+U0(x)U0'(x)(-ξ0y2ξ0xy¯-ξ0y¯2ξ0xy+2ξ0y2ξ0x¯+2ξ0y2¯ξ0x) (47)

ξ1s=ξ1s'=0,aty=0, (48)

ξ1s'0,asy, (49)

where conjugate complex quantities are denoted by an overbar and λs=Ha2By(x). The solution of equation (47) that satisfies the required boundary conditions is

ξ1s(x,y)=RωU0'(x)κs6(κs1e-2αy+2κs2e-λsy-e-αy(κs4eiβy+κs5e-iβy)+κs3), (50)

where constants κsj(j=1 to 6) are defined in the appendix. It can be shown that taking the limit of vanishing magnetic field, Eq (50) and its derivative reduce to the corresponding expressions for the hydrodynamic flow [21,26].

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

u1s=ϵsU0dU0dxξ1sy,

where ξ1s/y can be obtained from Eq.(50). In the hydrodynamic case (i.e. vanishing magnetic field) it results that, as the distance from the wall tends to infinity, u1s does not tend to zero. As a matter of fact, in this case it is not possible to satisfy simultaneously the condition u1s0 as η as well as the non-slip condition at the wall [21,25]. Therefore, the condition at infinity must be relaxed, imposing that u1s remains finite as the distance from the wall tends to infinity.

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 ξ1s/y tends to the hydrodynamic limit as y, notice that dU/dx0 when the magnetic field is negligible. Therefore, the steady streaming disappears in the purely hydrodynamic region. Evidently, dU/dx is also zero in the uniform magnetic field region.

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 ξ1s/y as a function of the y-coordinate at a fixed position within the fringing field (x=0.5) for increasing values of Nω, with Rω=10, K=0.8 and x0=0.5. It can be observed that as Nω increases ξ1s/y0 and therefore the steady streaming becomes weaker as the strength of the field grows. This means that the disturbance created by streaming vortices at the extremes of the generator should not affect its performance drastically.

The steady part of the stream function, ψ1s(x,y)=U0(dU0/dx)ξ1s, is shown in Fig. 6 as a function of the x-coordinate for different values of the constant x0 that modulates the magnetic field gradient. The influence of the fringing region is clearly shown, the largest values of ψ1s occur when the field gradient is more pronounced. This shows that the stronger the magnetic field gradient the more confined the streaming vortices are and the more intense the flow is.

In Fig. 7, the streamlines in the fringing field region are displayed for the cases x0=0.5 and x0=1.5. Two steady recirculations are observed which extend, accordingly to the value of x0, across the zone where the magnetic field passes from a uniform value to zero.

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 (Rω1) and high (Rω1) oscillation frequencies and arbitrary Hartmann numbers. For small frequencies, a first order quasi-steady Hartmann flow in phase with the pressure gradient is obtained, while an out of phase contribution is found at O(Rω). For high frequencies a solution for the core and boundary layer was obtained. The core solution represent a uniform time periodic flow that lags from the imposed pressure gradient according to the strength of the magnetic field. When the magnetic field is negligible, a purely hydrodynamic flow is obtained and the lag between the core and the pressure gradient is -π/2. For very strong magnetic field, the lag is negligible and the core follows the pressure gradient oscillation. Out of phase and in phase contributions were also found in the boundary layer, where a purely hydrodynamic (Stokes) boundary layer is obtained for negligible field while a Hartmann layer, oscillating in phase with the pressure gradient, is obtained for strong fields. These results can be conveniently synthesized graphically.

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.

Acknowledgments

This work was supported by Centro Mexicano de Innovación en Energía Océano, CEMIE-Océano No. 249795 SENER-CONACYT and by CONACYT under Project 240785.

References

1. A. Alemany, A. Krauze, and M. A. Radi, Energy Proc. 6 (2011) 92-100. [ Links ]

2. S. M. H. Mirhoseini, A. Alemany , Magnetohydrodynamics 51 (2015) 519-530 [ Links ]

3. L. Hu, H. Kobayashi, Y. Okuno, Performance of a liquid metal MHD power generation system for various external forces Proc. 12th Int. Energy Conv. Eng. Conf ., Cleveland, OH, July 28-30, (2014), AIAA 2014-3558. [ Links ]

4. B. Liu, J. Li, Y. Peng, L. Zhao, R. Li, Q. Xia, C. Sha, Performance Study of Magnetohydrodynamic Generator for Wave Energy, Proc. 19th ISOPE Conf. Busan, Korea, June 15-20 (2014) 545. [ Links ]

5. V. R. Malghan, Energy Conv. Mgmt. 37 (1996) 569-590. [ Links ]

6. G.W. Swift, J. Acoust. Soc. Am. 83 (1988) 350-361. [ Links ]

7. A.A. Castrejón-Pita , G. Huelsz, Appl. Phys. Lett., 90, (2007) 174110. [ Links ]

8. C. Vogin andA. Alemany , Eur. J. Mech.-B/Fluids, 26 (2007) 479-493. [ Links ]

9. D. A. Altshuller, R. A. Koslover, Optimal Control of the Magnetohydrodynamic Ocean Wave Energy Converter: Theory. Proc. IEEE 2005 Int. Conf. Phys. Control, Saint Petersburg, Russia, Aug. 24-26 (2005) 126. [ Links ]

10. L. Zhao , Y. Peng , C. Sha, R. Li , Y. Xu, B. Liu andJ. Li , Effect of Liquid Metal Characteristics on Performance of LMMHD Wave Energy Conversion System Proc. 24th ISOPE Conf. Osaka, Japan, June 21-26, (2009) 308. [ Links ]

11. B. Liu , Y. Peng , L. Zhao , J. Li , R. Li , Y. Xu, C. Sha, Hydrodynamic modelling of heaving buoy wave energy conversion system with liquid metal magnetohydrodynamic generator, Proc. 21st ISOPE Conf., Maui, Hawaii, USA, June 19-24, (2011) 95. [ Links ]

12. U. Müller, L. Bülher, Magnetofluyddynamics in channels and containers, (Springer, 2001). [ Links ]

13. J. A. Rizzo-Sierra, O. I. López-Hernández, Rev. Mex. de Fís. 62 (2016) 369-380. [ Links ]

14. R. Moreau, S. Smolentsev and S. Cuevas, PMC Physics B 3 (2010) 3. [ Links ]

15. T. Zhou, H. Chen and Z. Yang, Fusion Eng. Design, 86 (2011) 2352-2357. [ Links ]

16. H. K. Moffatt and J. Toomre, J Fluid Mech. 30 (1967) 65-82. [ Links ]

17. G. Ibáñez, S. Cuevas and M. López de Haro, Energy Conv. & Management, 43 (2002) 350-361. [ Links ]

18. G. Leal, Advanced transport phenomena, Cambridge University Press, 2007. [ Links ]

19. H. Perales, Numerical study of oscillatroy flows in ducts, Doctoral thesis, Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, Mexico, April 2003 (in Spanish). [ Links ]

20. A. Yakhot and L. Grinberg, Phys. Fluids 15 (2003) 2081-2083. [ Links ]

21. H. Schlichting, Boundary layer theory, 7th ed. McGraw-Hill, New York, (1979) pp.428-432. [ Links ]

22. N. Riley, Ann. Rev. Fluid Mech 33 (2001) 43-65. [ Links ]

23. I. S. Goldberg, Z. Zhang and M. Tran, Phys. Fluids 11 (1999) 2957-2962. [ Links ]

24. A. Gopinath, Q. J. Mech. Appl. Math. 47 (1994) 461-480. [ Links ]

25. J. T. Stuart J. Fluid Mech. 24 (1966) 673-687. [ Links ]

26. S. Cuevas and E. Ramos. Phys. Fluids 9 (1997) 1430-1434. [ Links ]

27. A. Sterl. J. Fluid Mech. 216 (1990) 161-191. [ Links ]

Appendix

A.

The constants appearing in Eq.(45) are defined as follows:

κt1=ϑt1+ϑt2ϑt3,κt2=ϑt4+ϑt5ϑt6,κt3=U0α'+iβ'γ2λ2-8iαβ-4α2+4β2,

κt4=ϑt7γ2λ(γ+λ)2(2γ+λ),

where

ϑt1=2U0'[λ4-α27λ2+72β2+48iα3β+12α4+7λ2β2+12β4-2iα7λ2β+24β3],ϑt2=-2γU0λ2+10iαβ+5α2-5β2α'+iβ',ϑt3=λ3-4γ2λλ2-2iαβ-α2+β22,ϑt4=-2U0'[2λ2+4β2+α-8iβ+3yβ2+λ2y+α2(-4-3iyβ)-yα3+iλ2yβ+iyβ3],ϑt5=2U0α'+iβ'-2iβ+α(-2-2iyβ)-yα2+yβ2+λ2y,ϑt6=γλ2-2iαβ-α2+β22ϑt7=12α3U0'-12iβ3U0'+α2[2U0'(7λ+18iβ)-5U0α'+iβ']+β2-14λU0'+5U0α'+iβ'+2α[2U0'λ2+7iλβ-9β2-U0(2λ+5iβ)α'+iβ']-λ2U0α'+iβ'+4λβiλU0'+4λβU0β'-iα'.

The constants appearing in Eq. (50) are defined as follows:

κs1=λs-2α2λs2-β2+α4+λs2+β222ϑs1(γ-λs)2α2γγ¯2,κs2=ϑs2(γ-λs)2,κs3=(λs-2α)2(λs-α+iβ)2ϑs3α2γγ¯2,κs4=λs(γ+λs)2λs2-4α22ϑs4γγ¯2,κs5=λsλs2-4α22λs2+2iαβ-α2+β22ϑs5(γ-λs)2γγ¯2,κs6=4λs(γ+λs)2(λs-2α)2(λs+2α)2(λs+γ¯)2(λs-α+iβ)2,

where

ϑs1=2λs2αβ4U0'-yU0α'-λs2U0β4α'-4U0α6α'+2yββ'-8α5ββU0'-yU0α'+3U0β'+2U0λs2-2β2β'+U0α4α'λs2+24β2+2yβλs2-4β2β'+2U0α2β2λs2yββ'-2α'λs2-3β2+2α3ββλs2-4β2U0'-yU0α',ϑs2=(λs2-4α2)U0'[α62β2-31λs2+12α8+α427λs4-9λs2β2-34β4+λs2+β22λs4+λs2β2-2β4-α29λs6+20λs4β2+45λs2β4+26β6]+U0[α(20α8-α641λs2+108β2+α421λs4+λs2β2-156β4

+α2λs6+58λs4β2+13λs2β4-36β6-λs2+β2λs6+22λs4β2-λs2+β2(21λs2β4+8β6))α'+β(100α8-λs2(λs-β)(λs+β)λs2+β22α676β2-107λs2-α415λs4+69λs2β2+20β4+α223λs6+66λs4β2+α2(+39λs2β4+4β6))β'],ϑs3=24α10U0'+λsU0β4α'λs2+β22+2α944λsU0'-5U0α'+2αβ4λs2+β22U0α'2λs2+β2-λsU0'λs2+β2+α4(-52β6U0'-λs5U0α'+6λs3β22λsU0'+17U0α'+2λsβ468U0α'-59λsU0'-32λs4U0ββ'-44λs2U0β3β'+10U0β5β')+2α2(2λs5U0β2α'+λsβ611U0α'[x]-13λsU0'-2β8U0'+λs3β417U0α'-9λsU0'+λs2U0β5β'-U0β7β')+2α3β(βU0'λs5-32λs3β2-31λsβ4+U0α'16λs4+47λs2β2+9β4-2λsU0(λs4+2λs2β2-3β4)β')+α8(2U0'63λs2+2β2-U033λsα'+50ββ')+2α7(11λsU0'4λs2+β2-U074λsββ'+U03α'9β2-7λs2)+2α6(U0'15λs4+17λs2β2-34β4+U0λsα'(73β2-13λs2-β87λs2+19β2β'))+2α5(U0'2λs5+14λs3β2-63λsβ4U0'+U0α'-4λs4+84λs2β2+39β4-U04β13λs3+9λsβ2β'),

ϑs4=-yα6U0'+α5U0'(-4+2iyβ)+yU0α'-iβ'+α2[β(U0'4iλs2+β2λs2y+β(yβ+8i)+U0α'(2β(6-iyβ)-iλs2y))-U0β'-2iλs2+λs2yβ+2yβ3]+β2[β(U0'4iλs2+βλs2y+β(yβ+6i)-iU0α'λs2y+β(yβ+2i))-U0β'2iλs2+βλs2y+β(yβ+4i)]+αβ(λs2β2U0'-yU0α'-iβ'+2iyβ4U0'-4iλs2U0α'+β3-8U0'+yU0α'-iβ'+4U0β2β'-iα')+α4(U0'λs2y+β(-yβ+2i)+U0α'(2-iyβ)-β'(yβ+4i))+α3(U0α'-λs2y+2β(yβ+2i)+iβ'4iβ-2yβ2+λs2y+2U0'λs2-6β2+2iyβ3),ϑs5=-yα6U0'+α5(U0'(-4-2iyβ)+yU0(α'+iβ'))+α2(β(U0'(β(2λs2y+β(yβ-8i))-4iλs2)+U0α'(2β(6+iyβ)+iλs2y))-U0β'2iλs2+λs2yβ+2yβ3)+β2(βU0'βλs2y+β(yβ-6i)-4iλs2+U0α'β(2+iyβ)+iλs2y)-U0β'βλs2y+β(yβ-4i)-2iλs2)+αβ(λs2β2U0'-yU0α'+iβ'+4iλs2U0α'-2iyβ4U0'+β3-8U0'+yU0α'+iβ'+4U0β2β'+iα')+α4(U0'λs2y-β(yβ+2i)+U0α'(2+iyβ)+β'(-yβ+4i))+α3(2U0'λs2-6β2-2iyβ3-U0α'4iβ(x)-2yβ(x)2+λs2y+U0β'(-4β(x)+2iyβ(x)2-iλs2y)).

Received: October 15, 2018; Accepted: November 01, 2018