The first classical study of electro-magnetic channel flow was carried out by Hartmann in the 1930s [
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 [
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) [
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 [
, 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 [
, and their theory well predicts the experimental results in a constant unidirectional field [
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.
and 2
while the channel length is 2
. The system of basic magnetic fluid equations is given as follows [
,
,
(3)
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).
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.
and
, we have
. To simplify our presentations, the following assumptions are made for related variables:
,
(6)
,
(7)
,
(9)
, Equation (3) is satisfied. As there is no excess charge in the fluid, then, by using (5), 
is obtained as follows
at
, 
at
, 
, that is to say, the fluid we consider here is in a steady state.
appears [
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)
. Note that Sutton and Sherman [
by 
for a simple model, then 
=
.
is taken to be 
if the gravitational field is neglected, and
, where 
is the viscosity of magnetic fluid. Combining Equations (12)–(14) yields
,
(17)
is defined as
.
is obtained by expressing 
over the range 
as a cosine Fourier series,
is a constant. The solution for 
is then written
(21)
and 
are given as
is just the same function as 
in which 
and 
are displaced by 
and
, respectively, which are given by
(23)
is not time-dependent, the term 
will disappear as shown in equations (104) [
with
, we have 
= 
as well as in Subsection 2.1. Then, as the axial pressure gradient 
is taken to be
, where
is a constant. Let
, 
(28)
as in (17) is also obtained
and 
are as follows
is the same function as 
in which 
and 
are given as follows
(for
). We find that 
which is obviously independent of the fluid viscosity.
, Equations (18) and (19) are reduced to
at 
(33)
at 
(34)
at
,
(35)
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 [
,
(38)
,
. 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
.
, 0.2 and 1.0, where 
is the conductivity coefficient. The velocity contours are displayed in 
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[
and the Rayleigh number
. Our analysis shows that the flow field changes with different conductivity coefficients.
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 [
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-
than that near the boundary walls 
for a given
.
, the Hartmann number 
and conductivity coefficient 
are introduced in our work. The flow and magnetic induction
. 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.
and 
and the variation technique may be tried. Firstly, consider the following integral [
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:
, then
. According to 
and 
in the Equation (17), we only considered the function of
, 
and 
in order to minimize the special function as follows
and 
are considered as the function of
, 
and
, 
, 
, and
.
, 
, 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 
, 
, and 
as
, where 
is the velocity composition of the flow field and the magnetic induction.
has the same variation characteristic as the following function
(44)
on 
is given by
(45)
is a nonlinear function of
, and is very complex to study the variation characteristics of 
by using the method of mathematical analysis.
is determined by flow and magnetic induction as a function of
.
, 
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)
is the kinetic energy, 
is the magnetic energy.
=
+ 
(47)
and 
are nonlinear functions of
, it is very complex to study the variational characteristics of 
by using the method of mathematical analysis.
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.
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
.
, 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
.
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].
, 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.
, 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.