Exact Solutions of Equations for the Strongly-Conductive and Weakly-Conductive Magnetic Fluid Flow in a Horizontal Rectangular Channel ()
1. Introduction
The first classical study of electro-magnetic channel flow was carried out by Hartmann in the 1930s [1]. Hartmann’s well-known exact solution can be applied to very closely related problems in magneto-hydrodynamics (MHD) to appreciably simplify physical problems and give insights into new physical phenomena.
In magnetic fluids, the fluid dynamic phenolmena with magnetic induction create new difficulties for the solution of the problems under consideration. The classical Hartmann flow can be further generalized to include arbitrary electric energy extraction from or addition to the flow. In general, classical MHD flows are dealt with using the exact solution of the Couette flow which is presented when the magnetic Prandtl number is unity [1].
The exact solutions of appropriately simplified physical problems provide estimates for the approximate solutions of complex problems. In view of its physical importance, the flow in a channel with a considerable length, rectangular, two-dimensional, and unidirectional cross section, which is assumed steady, pressure-driven of an incompressible Newtonian liquid, is the simplest case to be considered. In such a flow, taking into account the symmetrical planes 
and 
and an exact solution is obtained by using the separation of variables. The solution indicates that, when the width-to-height ratio increases, the velocity contours become flatter away from the two vertical walls and that the flow away from the two walls is approximately one-dimensional (the dependence of 
on 
is weak) [2].
If all walls are electrically insulating, 
, Shercliff (1953) has investigated principle sketch of the phenomenology of Magnetohydrodynamics (MHD) channel flow of rectangular cross-section with Hartmann walls and side walls [3]. For perfectly conducting Hartmann walls, 
, Hunt (1965) gave velocity profile and current paths for different Hartmann number. They found that the current density is nearly constant in most of channel cross sections, the velocity distribution is flat, and the thickness of the side layers decreases with increasing intensity of
, i.e. increasing Hartmann number [3]. Recently, Carletto, Bossis and Ceber defined the ratio magnetic energy of two aligned dipoles to the thermal energy
, and their theory well predicts the experimental results in a constant unidirectional field [4]. Further results can be found in reference [5].
Other flow configurations in basic MHD may include Hele-Shaw cells. Wen et al. [6,7] were motivated to visualize the macroscopic magnetic flow fields in a square Hele-Shaw cell with shadow graphs for the first time, taking advantage of its small thickness and corresponding short optical depth. Examples of applications of MHD include the chemical distillatory processes, design of heat exchangers, channel type solar energy collectors and thermo-protection systems. Hence, the effects of combined magnetic forces due to the variations of magnetic fields on the laminar flow in horizontal rectangular channels are important in practice [8–10].
In the present study, we consider the characteristics of magnetic fluids in a horizontal rectangular channel under the magnetic fields and use the flow equations with a conductivity coefficient. The exact solutions of the strongly-conductive and weakly-conductive magnetic fluids are considered using the series expansion technique in order to obtain the relationship between the flow and magnetic induction. Also, a quadratic function on flow and magnetic induction is studied to verify the characteristic of flow field using the obtained solutions.
2. The Exact Solution of the Magnetic Fluid Equations
The configuration of the flow geometry is illustrated in Figure 1. The problem considered in this study is an incompressible steady flow in the positive x-direction with a magnetic field applied in the positive z-direction. The cross-section of the channel is given by the flow region 2
and 2
while the channel length is 2
. The system of basic magnetic fluid equations is given as follows [5]
(1)
(2)
Figure 1. Illustration of flows in a rectangular channel
The Maxwell’s equations in their usual form
,
,
(3)
with the relation equations and the Ohm’s law given as
(4)
(5)
where 
is the permeability of free space, 
is the magnetic susceptibility (H/m),
is the conductivity, 
is the magnetic induction,
is the magnetic field (A/m), and 
is the magnetization (A/m).
We choose the 
axis such that the velocity vector of the fluid is 
and from the continuity Equation (1), we have
. We also choose 
=
, where 
is a constant representing magnetic induction.
Applying the Maxwell’s equation 
and
, we have
. To simplify our presentations, the following assumptions are made for related variables:
,
(6)
,
(7)
(8)
,
(9)
And
, Equation (3) is satisfied. As there is no excess charge in the fluid, then, by using (5), 
is obtained as follows
(10)
(11)
The magnetic fluid boundary conditions considered here are
at
, 
at
, 
We shall also assume that all quantities are independent of time
, that is to say, the fluid we consider here is in a steady state.
2.1 The Strongly-Conductive Fluid
The magnetic fluid is called strongly-conductive if the term 
appears [8]. Under the condition of strongly conductive, the coefficient 
is much larger than the Kelvin force density 
so that 
is considered while 
is neglected. Using the steady-state assumption, i.e., 
, Equation (2) can be written as follows
(12)
(13)
(14)
where
. Note that Sutton and Sherman [1] gave an incorrect result in their equation (10.85) which should be the above Equation (14).
For Hartmann flow, it is feasible to replace 
by 
for a simple model, then 
=
where
.
The axial pressure gradient 
is taken to be 
if the gravitational field is neglected, and
, where 
is the viscosity of magnetic fluid. Combining Equations (12)–(14) yields
(15)
(16)
Let
,
(17)
then Equations (15) and (16) are reduced to
(18)
(19)
where the Hartmann number 
is defined as
.
The solution for 
is obtained by expressing 
over the range 
as a cosine Fourier series,
(20)
where 
is a constant. The solution for 
is then written
(21)
where 
and 
are given as
(22)
It should be pointed out that the solution for 
is just the same function as 
in which 
and 
are displaced by 
and
, respectively, which are given by
(23)
2.2 The Weakly-Conductive Fluid
If the fluid is weakly-conductive and the field 
is not time-dependent, the term 
will disappear as shown in equations (104) [8]. Considering Equation (2) through (4) and making use of the assumptions mentioned above, the following relations are obtained,
(24)
(25)
(26)
Replacing 
with
, we have 
= 
as well as in Subsection 2.1. Then, as the axial pressure gradient 
is taken to be
, where
(27)
where 
is a constant. Let
, 
(28)
Thus, the solution of Equation (28) for 
as in (17) is also obtained
(29)
where 
and 
are as follows
(30)
Note that the solution for 
is the same function as 
in which 
and 
are given as follows
(31)
For a weak-conductive fluid, its solution is simply the solution of the conductive fluid with a conductive coefficient 
(for
). We find that 
which is obviously independent of the fluid viscosity.
2.3 Unidirectional Two Dimensional Flow without a Magnetic Field
Here we only consider a unidirectional two dimensional flow without a magnetic field, so that
, Equations (18) and (19) are reduced to
(32)
The boundary conditions are as follows
at 
(33)
at 
(34)
at
,
(35)
In order to obtain an exact solution of Equation (32), comparing with our above results, we have
(36)
where 
is a constant. The problem consisting Equation (32) and its conditions are solved similarly using the separation of variables, which has the solution as follows [9]
(37)
where
,
(38)
2.4 The Solutions, Flow Field and Discussions
In the present study, the flow fields and their associated functions are presented in the flow region with
,
. Since 
ranges from 10 to 100 in most practical problems, the initial magnetic induction is taken to be 
(kg.s-2.A-1), 
(H/m), 
and the constant
.
Figure 2 depicts the solutions for
, 0.2 and 1.0, where 
is the conductivity coefficient. The velocity contours are displayed in Figure 2(a) for different values of the conductivity coefficient. It is shown that the velocity gradients become larger gradually near four vertical walls as the conductivity coefficient increases for different magnetic fluids. On the contrary, in the region of (0,0), the velocity gradients lower gradually as the conductivity coefficient increases. For the case of a constant Hartmann number, the magnetic fluid are shown in Figure 2(b) for different values of the conductivity coefficient, and as indicated, the strength of magnetic induction dampens horizontal away from the plane 
and the walls 
as the conductivity coefficient increases. On the contrary, near the walls
, the magnetic induction becomes gradually low as the conductivity coefficient increases. For conductivity coefficient
, the flow contours are similar to those in the reference[10] for the magnetic Rayleigh number 
and the Rayleigh number
. Our analysis shows that the flow field changes with different conductivity coefficients.
Figure 3 shows the velocity fields for steady, unidirectional flows in a rectangular channel and 
is a cosine Fourier series of 
and conductivity coefficient
. The velocity contours are similar to those given by Papanastasiou et al. for the width-to-height ratio 1:1 [2].
In Figures 4(a) and 4(b), the development of the velocity profile in 
and 
directions are shown for various values of the conductivity coefficient
. For the symmetry, we only consider two cases: (a)
,
; and (b)
,
. Several interesting observations are readily made from the results. The cooperate process of 
and 
is shown in the above analysis. In order to clearly show the self-governed process of 
and
, the contours of the velocity versus coordinate
, and the velocity versus coordinate 
are given. It is clear that the velocity gradients increase quickly near the boundary walls 
and 
as 
is increased. On the other hand, the exact solutions are multiple hyper-cosine functions of
, and cosine functions of
. Therefore, the velocity gra-
dient is larger near the boundary walls
than that near the boundary walls 
for a given
.
For a given Hartmann number, comparing Figure 2 of this paper with Figures 3.2, 3.3 and 3.4 of Reference [3], current density distribution magnetic fluid and magnetohydrodynamics (MHD) are the same. Comparing Figure 2 of this paper with Figure 3.1 of Reference [3], the velocity profile of the phenomenology is also same for magnetic fluid and MHD. Of course, magnetic fluid and MHD have different equations and formulations of the Hartmann number. For magnetic fluid, the constitutive equation is
, the Hartmann number 
and conductivity coefficient 
are introduced in our work. The flow and magnetic induction
change with different
. Differently, in MHD, Shercliff and Hunt considered the induction equation 
and used only the linear constitutive equation 
with Hartmann number
. They gave velocity profile and current paths for different Hartmann number.
3. Two Class of Variational Functions on the Flow and Magnetic Induction
For many years the variation techniques have been effectively applied to problems in the theory of elasticity. However, they are rarely used in fluid dynamic problems. The great utility in elasticity problems are due to the fact that they can be conveniently applied to linear problems. This, of course, explains why they are not frequently used in fluid dynamics since most such problems are nonlinear [10,11].
For the conductive fluid problems of the type being considered here, we recall the governing Equations (15) and (16) which are linear for 
and 
and the variation technique may be tried. Firstly, consider the following integral [12]
(39)
where 
is some given function of
,
,
. Clearly, the value of the integral depends on the choice of the functions
, 
and
. Now, let us pose the following problem: to obtain functions
, 
and 
to minimize the value of
. As is well known from variational calculus, the necessary conditions that
, 
and 
for minimized 
are the Euler equations:
(40)
(41)
(42)
3.1 Decomposition and Composition Functions on the Flow and Magnetic Induction
Simply, let the parameters be fixed at
, then
. According to 
and 
in the Equation (17), we only considered the function of
, 
and 
in order to minimize the special function as follows
(43)
where 
and 
are considered as the function of
, 
and
, 
, 
, and
.
Let
, 
, then
. The expressions of 
and 
are called the velocity decompositions of the magnetic induction 
and the flow field 
with a variable coefficient 
of the flow and the magnetic induction, where
. From Figure 5, it is easy to show that
, 
, and 
as
, where 
is the velocity composition of the flow field and the magnetic induction.
It is also easy to know that 
has the same variation characteristic as the following function
(44)
The differential of 
on 
is given by
(45)
It is obvious that 
is a nonlinear function of
, and is very complex to study the variation characteristics of 
by using the method of mathematical analysis.
With Equation (45), the variation characteristics of the function 
is determined by flow and magnetic induction as a function of
.
3.2 A Total Energy Variational Function on the Flow and Magnetic Induction
Based on the above analysis of Equation (17), let
, 
with
, we call 
the velocity of magnetic fluid flow, which is equivalence to the velocity of magnetic fluid flow evoked by magnetic force. A total energy function is defined by
=
+ 
(46)
where 
is the kinetic energy, 
is the magnetic energy.
By the calculus of variations, we have
=
+ 
(47)
Figure 5. The sketch map of the composition function 
of flow and magnetic induction
As 
and 
are nonlinear functions of
, it is very complex to study the variational characteristics of 
by using the method of mathematical analysis.
3.3 Numerical Calculations and Discussions
As seen in Figures 2(a) and 2(c), in a region of (0,0), the contour of the function 
is more similar to that of the flow for
. Furthermore, the gradient of flow which is larger than that of magnetic induction, the distribution value of 
is largely affected depending on the flow. On the contrary, for
, the gradient of magnetic induction is larger than that of flow as 
is largely affected depending upon the gradient of the magnetic induction.
It is observed that the difference of flow and magnetic induction are almost the same for 
in a region of (0,0), where the distribution of the function 
is determined by the gradients of both the magnetic induction and the flow in this region. Furthermore, near 
and 
for any
, the difference of flow is acute singularly and the function 
is also changed singularly. It is noted that 
has only one limit point for
, and F has two limit points for
.
As seen in Figure 6, the gradient of the total energy is decided by the kinetic energy in the region (0,0) for different values of
, and near 
and 
for
. That is to say, the gradient of the magnetic energy is very large near 
for
, and its value is very small which does not affect the gradient of the total energy. Then, the gradient of the total energy will be affected by the magnetic energy near 
for 
and it will be affected by the magnetic energy near 
and 
for
.
4. Concluding Remarks
1) For magnetic fluid of this work and Magnetohydrodynamics (MHD) of Reference [3], the constitutive equa-
tions are different, the Hartmann numbers are also different. For magnetic fluid, conductivity coefficient 
is an important coefficient to analyze the flow and the current. For MHD, Hartmann number 
is the controlling coefficient, Shercliff and Hunt studied velocity profile and current paths for different Hartmann number [3–5].
2) For conductivity coefficient
, the velocity contours for steady unidirectional flow is shown in a rectangular channel with 
a cosine Fourier series function of
. Our result is in agreement with published findings.
3) A velocity decomposition and composition function
, and a total energy variational function
, on the flow and magnetic induction are considered. The variational characteristics of 
are analyzed only using the characteristics of the resultant flow field and the magnetic induction, and the number of its limit points changes as 
changes. It is shown in numerical simulations that the gradient of total energy 
is affected by the kinetic energy and the magnetic energy as 
changes.
4) Theoretically, the strongly-conductive and weaklyconductive magnetic fluid flows are studied on different conductivity coefficients which are independent of fluid viscosity in a horizontal rectangular channel.
5. Acknowledgments
This work was supported in part by a grant to the research Centre for Advanced Science and Technology at Doshisha University from the Ministry of Education, Japan; the National Natural Science of China (10771178, 50675185, 10676031); the Research Fund for the Doctoral Program of Higher Education (20070530003), Program for New Century Excellent Talents in University (NCET 06-0708)and the Scientific Research Foundation for the Returned Overseas Chinese Scholars of State Education Ministry of China. The authors thank readers for giving references [3–5] and suggestions on comparison with analytical solutions.