Journal of Indian Acad. Math.  
ISSN: 0970-5120  
Vol. 47, No. 1 (2025) pp. 251-281  
Sayantan Sil1  
EXACT SOLUTION FOR FLOW  
THROUGH POROUS MEDIUM OF A  
ROTATING VARIABLY INCLINED  
MHD FLUID BY MAGNETOGRAPH  
TRANSFORMATION  
and  
Birendra Kumar  
Vishwakarma2  
Abstract. An analytical study of the motion of a steady, homogenous,  
incompressible, plane rotating MHD fluid flow through a porous medium  
for exact solutions is carried out. The velocity vector of the fluid particle is  
thought to be variably inclined to the magnetic field vector at every point.  
The flow of fluid is governed by non-linear partial differential equations.  
These governing equations are converted into a system of linear partial  
differential equations by means of transformation technique known as  
magnetograph transformation. The two components of the magnetic field in  
the physical plane and two independent variables are switched around using  
the magnetograph transformation. Further, the flow equations have been  
derived using the Legendre transform of the magnetic flux function. Finally,  
several examples have been used to apply and illustrate the developed  
theory and exact solutions have been determined. The expressions for the  
components of velocity vector, components of magnetic field vector,  
magnetic lines and pressure distribution are obtained and analyzed  
graphically.  
Keywords: MHD, Exact Solution, Magnetograph Transformation,  
Magnetic Flux Function, Legendre Transform Function,  
Porous Medium.  
Mathematics Subject Classification (2020) No.: 35F05, 35Q30, 35Q35.  
1. Introduction  
The governing equations for the flow of non-Newtonian fluids give rise to  
252  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
systems of non-linear partial differential equations; these equations have no general  
solution. The several approaches used to solve these equations and their applications  
have received excellent coverage from Ames [1]. Hodograph transformations, as  
employed by Martin [2] in fluid mechanics, are a class of transformations that change  
variables from the physical plane to the velocity plane.  
The magnetograph transformation- a method for accurately solving non-  
linear partial differential equations- which govern the steady flow of a homogeneous,  
incompressible, viscous fluid with finite electrical conductivity in a porous medium  
in a rotating reference frame-is the subject of the current study. It is common practice  
to solve non-linear partial differential equations using transformation techniques. The  
magnetograph is a curve formed by the extremities of the magnetic field vectors  
when they are extended from a given point. An equivalent linear system is produced  
by using the magnetograph transformation to switch the roles of the independent and  
dependent variables. In other words, the transformations that are used to switch the  
roles of the two independent variables in the physical plane and the two components  
of the magnetic field are known as magnetograph transformations.  
The governing non-linear equations are transformed into a linear form that  
may be solved by using the magnetograph transformation. Using magnetograph  
transformation, several researchers have studied MHD fluid flow and discovered  
precise answers. In order to investigate orthogonal MHD flow, S. N. Singh [3]  
invented and used magnetograph transformation. Researchers Venkateshappa,  
Siddabasappa, and Rudraswamy [26] as well as C. S. Bagewadi and Siddabasappa  
[4], looked on rotating MHD ow that was variably inclined in the magnetograph  
plane. Exact solutions were found by M. Kumar and S. Sil [5] after studying aligned  
MHD flow in the magnetograph plane.  
The study of fluid flow in a rotating frame is important for many technical  
applications that are directly affected by the coriolis force created by the earth’s  
rotation. Examples of these applications include spin coating, the creation and use of  
computer disks, rotational viscometers, centrifugal machinery, the pumping of liquid  
metals at high melting points, the growth of crystals from molten silicon, turbo-  
machinery etc. The coriolis force is shown to have a significant impact when  
compared to the viscous and inertial forces in the equations of motion.  
The coriolis force has a major impact on the hydromagnetic flow in the  
liquid core of the earth, which is essential to the mean geomagnetic field [6]. Because  
of its role in solar physics and its relationship to the formation of sunspots and the  
solar cycle, the theory of rotating fluid is also significant. Several studies with  
rotating fluid have been carried out [9, 11, 10, 12, 7, 8, 13, 26]. Many works have  
been conducted on various types of flows for both non-MHD and MHD.  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
253  
In the study of soil percolation in hydrology, the petroleum industry,  
agricultural engineering, and many other significant fields, the flow of a viscous fluid  
through a porous material is crucial. Numerous authors [17, 19, 14, 20, 21, 23, 22,  
16, 15, 24, 25, 18, 28, 29] have investigated fluid flows across porous media and  
discovered an exact solution.  
The objective of this research is to analyze the motion of a rotating, steady,  
homogenous, incompressible, variably inclined MHD plane flow through a porous  
medium in order to obtain exact solutions. The fluid flow equation is described by  
nonlinear partial differential equations. The magnetograph transformation helps the  
nonlinear partial differential equations turn into a system of linear partial differential  
equations. Two independent variables and the two components of the magnetic field  
in the physical plane have been swapped out using the magnetograph transformation.  
Moreover, the magnetic flux function’s Legendre transform function has been  
utilized to illustrate the flow equations. Finally, a few examples have been used to  
clarify the proposed theory and exact solutions have been found.  
The expressions for the pressure distribution, magnetic lines, velocity vector  
components and magnetic field vector components are obtained and graphically  
examined. We first consider the appropriate steady flow equations in a rotating frame  
of reference, which includes coriolis force and centrifugal force with non-uniform  
angular velocity. Using a Legendre transform of the magnetic flux function and  
rewriting all of the equations in terms of this transformed function, the exact  
solutions are found by switching the dependent and independent variables in the  
magnetograph plane. Examples are considered to point out the usefulness of the  
method. The geometry of streamlines and magnetic lines are discussed. The general  
solution for angular velocity is also found with the variation of pressure and angular  
velocity is discussed by plotting various graphs for some different form of suitable  
examples.  
2. Basic Equations  
The fundamental equations that regulate the steady flow of a homogeneous  
incompressible viscous fluid with finite electrical conductivity in a porous medium in  
the presence of a magnetic field in a rotating reference frame are  
· V 0 , (Continuity equation)  
(1)  
((V·) 2V (r)) P 2V µQ H  
V
, (Momentum Equation)  
(2)  
254  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
  (V H)    (H  H), (Diffusion equation)  
(3)  
(4)  
· H 0  
,
(Solenoidal equation)  
where V = velocity field vector, P = fluid pressure, H = magnetic field vector,  
Q = current density, µ = magnetic permeability, σ = electrical conductivity of the  
fluid, ρ = the constant fluid field density,  
viscosity, κ = permeability of the medium, r = radius vector and γH = magnetic  
viscosity, × ( r) = centripital acceleration, 2 × V = coriolis acceleration.  
= angular velocity, η = coefficient of  
×
On introducing the function  
v  
u  
  
,
(vorticity function)  
(5)  
(6)  
(7)  
x  
y  
H2  
H1  
Q   
,
(Current density function)  
x  
y  
1
2
1
B   
V2 P'   
|r|2  
,
(Bernoulli function)  
2
1
where V2 u2 v2, P' is the reduced pressure and P' P   
|r|2 and  
2
the last term being the centrifugal contribution of the pressure. The above system  
reduces to  
v  
u  
0,  
(8)  
(9)  
x  
y  
B  
  
  
2vv H2Q u0,  
x  
y  
B  
  
  
2vv H1Q v 0 ,  
(10)  
(11)  
y  
y  
H2  
H1  
uH2 vH1 H  
c,  
x  
y  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
255  
H2  
H1  
0,  
(12)  
x  
y  
H2  
H1  
Q(x, y)   
,
(13)  
(14)  
x  
y  
v  
u  
(x, y)   
,
x  
y  
of seven partial differential equations in eight unknown functionsu, v, H1, H2  
,
, ,Q and which are functions of(x, y). In addition, is an arbitrary  
B
c
integration constant that may be found using the diffusion equation (3). Martin [2]  
has successfully employed a first-order system similar to this one to investigate  
viscous non-MHD flows.  
Let (x, y) be the variable angle such that (x, y) 0 for every  
(x, y) in the region of flow. Equation (11) yields  
uH2 vH1 UH sin c HQ  
,
(15)  
(16)  
uH1 vH2 UH cos (c HQ) cot   
,
2
2
where H   
(H1 H2 ) . Considering these as two linear algebraic equations in  
the unknown’s  
u
and  
v
, we solve (15) and (16) in terms of H1, H2 , and α.  
H2 H1 cot   
u(c HQ)  
,
(17)  
(18)  
2
2
H 1 H 2  
H2 cot H  
1   
v (c HQ)  
,
2
2
H 1 H 2  
we can eliminate uand  
and then obtaining a system of equations to be solved for H1, H2, , , B,Q and  
as functions of and , this approach leads to the study of system (8)-(14) in the  
v
from the system (8)-(14) by using equations (17) and (18)  
x
y
256  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
magnetograph plane. By using (17)-(18) and removing  
u
and  
v
from the system of  
(8)-(14) we get the system of six partial differential equations as under,  
H1  
H2  
0 ,  
(19)  
x  
y  
  
  
H1 cot H2  
(2) (c HQ)  
2
2
x  
H 1 H 2  
  
H2 H1 cot   
B  
QH2   
(c HQ)  
   
,
(20)  
2
2
x  
H 1 H 2  
  
  
H2 H1 cot   
(2) (c HQ)  
2
2
y  
H 1 H 2  
  
H2 cot H  
B  
1   
QH1   
(c HQ)  
,
(21)  
2
2
y  
H 1 H 2  
cot   
cot   
(c HQ) H1  
H2  
x  
y  
2
2
   
H 1 H2 2H1H2 cot   
 H2  
H1   
2
2
x  
y  
H 1 H 2  
2
2
  
   
H1 cot H2 cot 2H1H2 cot   
 H2  
H1   
2
2
y  
x  
H 1 H 2  
  
Q  
Q   
H (H2 H1 cot )  
(H2 H1 cot )  
0  
,
(22)  
(23)  
x  
y  
H2  
H1  
Q ,  
x  
y  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
257  
H2  
  
H2  
  
cot H  
H cot   
1
1   
(c Q)  
(c Q)  
H
H
2
2
2
2
H
H  
H
H  
  
  
1
2
1
2
 
,
(24)  
x  
3. Magnetograph transformations  
y  
As mentioned in the equations of flowH1 H1(x, y)  
,
H2 H2(x, y)the  
Jacobian  
H2 H2  
H1 H2  
J(x, y)   
0  
(25)  
x  
y  
y  
x  
2
Let  
x
and  
y
be functions of H1 and H2 , that is, x x(H1, H )  
,
2
y y(H1, H )  
.
Given these assumptions, we may have the following relations:  
H1  
y  
H2  
y  
H1  
x  
H2  
y  
J  
,
 J  
,
 J  
,
J  
(26)  
x  
H2  
x  
H1  
y  
H2  
y  
H1  
Further,  
(H1, H2)  
 (x, y) 1  
J(x, y)   
j(H1, H2)  
,
(x, y)  
(H , H )  
1
2
f  
x  
(f, y)  
f  
(x, f)  
j  
,
j  
,
(27)  
(H1, H2) y  
(H1, H2)  
where f(H1, H2) is transformed function of continuously differentiable function of  
   
f
in the H1H2 -plane.  
4. Flow Equations in Magnetograph Plane  
Applying the aforementioned transformation relations to the system of  
equations (19)-(24) in the magnetograph plane, or (H1, H2) plane, for the first order  
258  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
partial derivatives results in the transformed system of partial differential equations  
being  
x  
y  
0  
,
(28)  
H1  
H2  
  
(x, )  
y  
H2 cot H  
1   
j  
(2) (c HQ)  
2
2
(H1, H2)  
H2  
H 1 H 2  
  
H2 H1 cot   
(B, y)  
QH2   
(c HQ)  
 j  
,
(29)  
2
2
(H1, H2)  
H 1 H 2  
  
(, y)  
H2 H2 cot   
j  
(2) (c HQ)  
2
2
(H1, H2)  
H 1 H 2  
  
H2 cot H1  
(x, B)  
QH1   
(c HQ)  
,
(30)  
(31)  
2
2
(H1, H2)  
H 1 H 2  
 x  
y   
j
Q  
,
H  
H  
1   
2
    
H2  
  
H2  
    
cotH  
H cot   
1
1   
   
   
(c Q)  
, y  
x, (c Q)  
H
H
2
2
2
2
H
H  
H
H  
1
   
  
   
1
2
2
j
 
,
(H1, H2)  
(H1, H2)  
(32)  
x   
cot   
2
1
2
2
2
   
(c Q) H cot H cot 2H H cot H H  
H
1
2
2
H1  
H1  
Q  
H2H  
(H2 cot H1)  
H1  
x   
cot   
2
1
2
2
   
(c Q) H H2 2H1H2 cotH2H  
H
H2  
H1  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
259  
Q  
H2H  
(H1 H2 cot )  
H1  
y   
cot   
2
2
2
2
   
(c Q) H H1 2H1H2 cot H1H  
H
H2  
H2  
Q  
H2H  
(H1 cot H2)  
H2  
y   
cot   
2
2
2
1
2
   
1
(c Q) H cot H cot 2H H H H  
H
1
2
H1  
H1  
Q  
H2H  
(H1 cot H2) = 0  
.
(33)  
H1  
5. Legendre Transform of Magnetic Flux Function  
The solenoidal equation (19) verified the existence of the magnetic flux  
function (x, y) and is such that  
  
  
d H2dx H1dy  
or  
 H2,  
H1 ,  
(34)  
x  
y  
Similarly, for the magnetic flux function (x, y) , equation (28) verified the  
existence of the function L(H1, H2) , also known as Legendre’s transform function.  
It is such that  
L  
L  
dL  ydH1 xdH2  
or  
 y,  
x  
,
(35)  
H1  
H2  
and these two equation are connected by L(H1, H2) H2x H1y (x, y)  
.
Introducing L(H1, H2) into the system of equations (28)-(33) it follows that  
equation (28) is identically satisfied with j given by (27) and the system is substituted  
by  
L  
(  
,)  
H2  
  
H  
cot H  
y  
2
1   
j  
(2) (c Q)  
H
(H ,H )  
H  
2
2
H  
1
2
2
H
  
1
2
260  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
L  
B,  
H2 H1 cot   
H  
1
JH2   
(c H )  
j  
(36)  
(37)  
2
2
(H1, H2)  
H1 H2  
L  
)
(,  
  
H2 H1 cot   
H  
1
j  
(2) (c HQ)  
2
2
(H1, H2)  
H 1 H 2  
  
L  
(H , B)  
H2 cot H1  
2
QH1   
(c HQ)  
2
2
(H1, H2)  
H 1 H 2  
cot H  
H cot   
H
H
   
   
  
  
   
   
1   
H  
   
2
L  
L  
2
1
(c Q)  
,
,
(c Q)  
H
H
H  
2
1
2
2
2
2
H
H  
H
H  
2
1
2
1
j
= ,  
(H1, H2 )  
(H1, H2 )  
(38)  
(39)  
2L  
2L  
j
Q  
,
2
2
H 2  
H 1  
2L  
cot   
H1  
2
2
2
2
   
(c Q) H  
H 1 2H1H2 cot H2H  
H
2
H 2  
Q  
H2H  
(H1 H2 cot )  
H1  
2L  
cot   
H1  
2
1
2
2
   
(c Q) H  
H 2 2H1H2 cot H2H  
H
2
H 1  
Q  
H2H  
(H2 cot ) H1  
H1  
2y  
  cot   
cot   
(c HQ)H2  
H1  
H1H2  
H2  
H1  
 Q  
Q  
H2H  
(H2 cot H1)   
(H cot H )  
1
2
H  
H1  
2
2
2
2
   
(c HQ)(2H 1 cot H 2 cot 4H1H2 H1H ) 0  
(40)  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
261  
2 1  
2L 2L  
2L  
j   
,
(41)  
2
2
H1H2  
H 1 H2  
   
   
   
   
   
for the seven functions L(H1H2), B(H1H2), (H1H2), j(H1H2), (H1H2),  
   
   
J(H1H2)and (H1H2)  
.
Introducing polar co-ordinates (H, )H1 H cos and H2 H sin   
(F,G)  
1 (F,G)  
,
(H1, H2)  
H (H1, H2)  
sin   
cos   
,
H1  
H  
H
  
cos   
sin   
H2  
H  
q
  
   
where  
F(H1, H2) F (H, )  
;
G(H1H2) G (H, )  
are  
continuously  
differentiable functions in (H, ) coordinates, the equation (40) takes the form  
2L  
H 22  
cot   
Q  
(c HJ)H  
H cot   
  
  
1 2L  
H2 2  
1 L  
j  
HH  
(c HQ)  
H2  
  
H  
1 2L  
1 L  
cot  
Q  
   
  
  
(c Q) 2 cotH  
cotHH  
0  
.
H
H  
H H  
  
H  
  
   
(42)  
6. Applications  
Example 1: Let  
262  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
H2   
1  
1  
2
N2, (H1, H2) cot (N3H 1 N3H 2)  
(43)  
   
L(H1H2) N1 tan  
2
H
1
form a set of solution of the partial differential equation (40) when N1 0, N2 and  
N3 are arbitrary constants. As N3 is arbitrary, there are two cases of the solution  
which may considered by (43).  
(i)  
If N3 0 i.e., variably inclined flows and  
(ii)  
If N3 0 i.e., crossed flows.  
When (i)N3 0  
.
Using (43) in (35) we have  
N1x  
r2  
N1y  
r2  
, r2 x2 y2  
.
(44)  
H1(x, y)   
; H2(x, y)   
This represents radial flow and magnetic field profile is thus the arc of a  
rectangular hyperbola, using (44) we obtain  
c
c
u  
(yr2 N3N 12x)  
,
(N3N 12y xr2)  
,
v   
N1r2  
N1r2  
2
N 1N3  
2c  
(x, y)   
,
Q 0  
,
(x, y) cot1  
(45)  
r2  
N1  
With the help of (45) and integrability condition on  
B
i.e.,  
2B  
2B  
xy  
yx  
from equations (9) and (10) we get angular velocity  
 
  
2
2
2
2
2
2
(x2 y2) 0  
y(x y ) N3N1 x  
x(x y ) N3N1y  
x  
y  
k  
(46)  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
263  
The Lagrange form of solution of this equation is  
y
x
 N3N 2 tan1  
C  
,
where  
N 2N3  
,
(47)  
1
1
k  
y
x
the streamlines are given by (x2 y2) N1N3 tan1  
constant , the magnetic  
flux function is  
y
tan1  
constant  
x
and from (9) and (10) we have  
c  
2c  
cy   
y
x
c2  
N 12  
B(x, y) 4c2N3   
N3N1   
tan1  
(x2 y2)  
kN1  
kN1  
cx2  
c  
y 2  
N3N1 tan1  
kN1  
x
c2  
N 12  
y
x
c  
(x2 y2) tan1  
N3N1 ln(x2 y2) constant  
,
(48)  
and hence the pressure  
1
P(x, y) B   
V2  
,
2
is  
c  
2c  
cy   
y
x
c2  
N12  
P(x, y) 4c2N3   
N3N1   
tan1  
(x2 y2)P(x, y)  
kN1  
kN  
1   
cx2  
c  
y 2  
c2  
N12  
y   
c  
N3N1 tan1  
(x2 y2) tan1  
N3N1 ln(x2 y2)  
kN1  
x
x
1 c2  
1 c2N3N1  
(x2 y2)   
constant  
(49)  
N 12  
2(x2 y2)  
2
2
and  
264  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
(ii) for N3 0 i.e., crossed flows, the value of u, v, calculate similarly  
by putting N3 0 in equation (6.1).  
By putting N3 0 in equation (6.4) we get  
C1   
ln y  
k  
Again,  
B
and  
P
can be calculated by putting N3 0 in equations (6.6)  
and (6.7) respectively.  
Figure 1: Streamlines for example 1  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
265  
Figure 2: Magnetic lines for example 1  
Figure 3: Variation of angular velocity versus  
x
for example 1  
266  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
Figure 4: Variation of pressure versus  
x
at y 2 for density variation example 1  
Figure 5: Variation of pressure versus  
x
at y 2 for porosity variation for  
example1  
Example 2: Another solution of equation (5.7) is  
2
2
1  
2
L(H1, H2) M1(H 2 H 1) M2, (H1H2) cot (M3H 1 M3H 2 M4)  
   
2
(50)  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
267  
Where M1 0, M2, M3 and M4 are arbitrary constants.  
We have  
dL  
dL  
 y,  
x  
,
dH1  
dH2  
We examine the case where M3 and M4 are arbitrary constants. When  
flows are variably inclined, M3 0 , i.e. The resulting flows are crossed if  
M3 M4 0 and constantly inclined if M3 0  
,
M4 0 . Now consider the  
case when M3 0 M4 0 . Using (49) in (35) we obtain  
,
x 2M1H2  
,
y  2M1H1  
and therefore  
y  
x  
H1   
,
H2  
,
(51)  
2M1  
2M2  
This indicates that the radial distance from the central axis directly affects the  
r  
magnetic field H   
.
2M1  
The changing angle between the velocity and magnetic fields in the physical  
plane is given by  
M3r2  
4M 12  
(x, y) cot1  
M4  
and hence vorticity, current density and velocity components are given by  
M3  
H  
1
  
c   
, Q   
M
M1  
M1  
1
H  
2M1(x M4y)  
M3y  
uc   
;
(x2 y2)  
M1  
2M1  
268  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
H  
2M1(x M4y)  
M3y  
v c   
.
(52)  
(x2 y2)  
M1  
2M1  
It is to be noted that velocity of the fluid is infinite when r 0 i.e., when  
H 0. And fluid velocity is zero when the radial distance is infinite and so the  
velocity of the fluid decreases as the -increases. From (48) and integrability  
r
condition on  
B
equations (9) and (10) yields the angular velocity  
as  
  
  
[M3x(x2 y2) 4M 2(y M4x)]  
[4M2(x M4y M3x)(x2 y2)]  
1
1
y  
x  
M3  
(x2 y2) 0  
.
(53)  
k  
The solution to this problem in Lagrange form is  
y
M3 2  
4M 12 tan1  
2M 2M4 ln(x2 y2) (x2 y2)  
,
1
x
2
4M 12  
where  
,
(54)  
M3  
k  
The streamlines are provided by  
y
8M 12tan1  
8M 2 M4 ln(x2 y2) M3(x2 y2) constant  
,
1
x
the magnetic flux function is  
x2 y2 constant  
and (54) yield the energy function  
B
as  
H  
   
y
x
M3  
B(x, y) c   
4M1 tan1  
2M1M4 ln(x2 y2)   
(x2 y2)  
M
1    
2M1  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
269  
M 13  
M3  
M 3M4  
H  
y 2  
y
x
1
2c   
8
tan1  
8  
tan1  
ln(x2 y2) 2M1xy  
M
x
M3  
1   
M 3M4  
y
1
2
2
2
2
2
2M1(x2 y2) tan1  
2  
(ln(x y )) M1M4(x y )  
x
M3  
4M 12  
M3  
H  
(x2 y2)   
c   
xy constant  
.
(55)  
kM1  
M1  
And pressure is  
H  
   
y
x
M3  
P(x, y) c   
4M1 tan1  
2M1M4 ln(x2 y2)   
(x2 y2)  
M
1    
2M1  
M 13  
M3  
M 3M4  
H  
y 2  
y
x
1
tan1  
8  
tan1  
ln(x2 y2) 2M1xy  
2c   
8
M
x
M3  
1   
M 3M4  
y
x
1
2
2
2
2
2
2M1(x2 y2) tan1  
2  
(ln(x y )) M1M4(x y )  
M3  
4M 12  
M3   
H  
(x2 y2)   
c   
xy  
kM1  
M1  
2
4M(1 M 4)  
1
2
H  
2  
M3  
c   
(x2 y2) 2M3M4 P0  
.
(x2 y2)  
M1  
4M2  
270  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
Figure 6: Streamlines for example 2  
Figure 7: Magnetic lines for example 2  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
271  
Figure 8: Variation of angular velocity versus  
x
for example 2  
Figure 9: Variation of pressure versus  
x
at y 2 for  
variation example 2  
272  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
Figure 10: Variation of pressure versus  
Example 3: Consider  
x
at y = 2 for κ η variation example 2  
L(H, ) AB ln H D  
(57)  
are  
In (H, ) coordinates, where  
D
is an arbitrary constant and  
A
and  
B
real values that are not zero. Applying this in (42) we have  
B
cot AH  
cot 2A cot 2B 0  
  
H  
This has solution  
B
A
cot    
M1H2, M1 arbitrary constant  
and  
and  
L(H, ) AB ln H D  
,
B  
  
cot1  
M1H2  
,
A
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
273  
forms a solution set of the partial differential equation (42). If M1 0 the flows are  
constantly inclined with  
B   
  
cot1  
,
(58)  
(59)  
A
and when M1 0 , the flows are variably inclined, we have  
Ax By  
Bx Ay  
H1(x, y)   
;
H2(x, y)   
,
r2 x2 y2  
.
r2  
r2  
y  
(M1(Ax By))  
x  
(M1(Ay Bx))  
uc  
,
v c   
,
r2  
r2  
A
A
2c  
(x, y)   
,
Q 0  
.
A
Now integrability condition for  
B
yields  
  
{y(x2 y2) M1A(Ax By)}  
{AM1(Bx Ay)  
x  
  
x(x2 y2)}  
(x2 y2) 0  
.
(60)  
(61)  
y  
k  
The Lagrange form of solution of this equation is  
y
M1A2 tan1  
M1AB ln(x2 y2), where A2M1   
.
x
k  
the streamlines are given by  
y
M1A2 tan1  
M1AB ln(x2 y2) (x2 y2) constant  
x
and the magnetic flux function is  
y
x
A tan1  
B ln r constant  
.
274  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
Now equation (9), (10) and gives us the energy function  
c2   
M1  
x   
B(x, y)   
(x2 y2)   
(A B) ln(x2 y2) M1 tan1  
A
2
y
m12A2  
2  
2
y
x
y   
2c  
M1Axy M1A(x2 y2) tan1  
(1 A) tan1  
3
3
2
x
2
2
M1 A B  
M1B  
y
x
ln(x2 y2) tan1  
2
2
M12A2B  
M1B  
M1B  
(x2 y2) ln(x2 y2)   
(x2 y2)   
(ln(x2 y2))2  
2
2
2
c   
xy  
A
x   
2  
2M1B tan1  
P  
.
(62)  
y
And hence, the pressure function is  
c2   
M1  
x   
P(x, y)   
(x2 y2)   
(A B) ln(x2 y2) M1 tan1  
A
2
y
m12A2  
2  
2
y
x
y   
2c  
M1Axy M1A(x2 y2) tan1  
(1 A) tan1  
3
3
2
x
2
2
M1 A B  
M1B  
y
x
ln(x2 y2) tan1  
2
2
M12A2B  
M1B  
M1B  
(x2 y2) ln(x2 y2)   
(x2 y2)   
(ln(x2 y2))2  
2
2
2
2
2
A2 B2  
(x2 y2)  
M1B  
A2  
c   
xy  
A
x   
1
2
(x y )  
2  
2M1B tan1  
c  
M12  
2  
P  
.
0
A2  
y
(63)  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
275  
Figure 11: Streamline for example 3  
Figure 12: Magnetic line for example 3  
276  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
Figure 13: Variation of angular velocity verses  
x
for example 3  
Figure 14: Variation of pressure verses  
x
at y 2 for  
variation  
for example 3  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
277  
Figure 15: Variation of pressure verses  
x
at y 2 for fluid density variation  
for example 3  
7. Conclusion  
In this work, an approach has been carried out where magnetograph  
transformation method has been applied for the exact solution of the equations  
governing the flow of a homogeneous, incompressible viscous fluid through porous  
media of a variably inclined rotating MHD with finite electrical conductivity. We  
have utilized magnetograph transformation in this problem to reformulate the  
governing non-linear equation into linear once. Three different forms of Legendre  
transform function of the magnetic flux function have been considered as examples  
to illustrate the technique of solving for the exact solution. The expressions for  
streamlines, magnetic lines, angular velocity and pressure distribution are found out  
in each case. The main results are listed below:  
y
In example 1 the streamlines are given by (x2 y2) N1N3 tan1  
=
x
y
constant and magnetic lines are given bytan1  
constant  
.
x
y
x
In example 2 streamlines and magnetic lines are given by 8M12 tan1  
8M2M4 ln(x2 y2) M3(x2 y2) constant  
and  
x2 y2  
1
constant respectively.  
278  
SAYANTAN SIL AND BIRENDRA KUMAR VISHWAKARMA  
y
In example 3, the streamlines are given by M1A2 tan1  
M1AB  
x
ln(x2 y2) (x2 y2) constant and magnetic lines are given by  
y
A tan1  
B ln r constant  
.
x
Also for example 1 components of velocity are independent of permeability of  
porous medium and angular velocity of rotating frame. The vorticity function  
is constant and current density is zero. Pressure depends on angular velocity,  
permeability of the medium and the fluid density.  
In example 2 for the form of Legendre transform function we find that the  
magnetic field varies with the radial distance from central axis. Current  
density function is constant and verticity function containing magnetic  
viscosity term is also a constant. The components of velocity depends on  
magnetic viscosity and current density function. Also, velocity of the fluid  
decreases with radial distance. Magnetic viscosity, current function, angular  
velocity, permeability of the medium and fluid density affects the pressure  
function.  
For the form of Legendre transform function considered in example 3 verticity  
function is constant, components of velocity does not involve permeability of  
medium and angular velocity. Pressure depends on angular velocity,  
permeability of medium and fluid density.  
Angular velocity depends on permeability of porous medium for all examples.  
In example 1 angular velocity for positive N2N3 decreases with  
x
and for  
1
negative increases  
x
becoming almost constant beyond x 7 for both  
cases. For the form of Legendre transform function represents radial flow and  
magnetic field profile is arc of a rectangular hyperbola.  
In example 2 angular velocity is found to increase with  
x
in the beginning  
and shoots up at x 100 and decreases afterward in Figure 4.  
In example 3 angular velocity is found to decrease with  
x
in the beginning  
shoot up at x 0 and shows varying trend there afterwards (Figure 8).  
In example 1 (Figure 2) pressure increases at constant  
for different fluid of  
different densities. For different  
values at constant fluid density  
the  
EXACT SOLUTION FOR FLOW THROUGH POROUS MEDIUM  
279  
pressure shows parabolic variations with  
x 100  
x
and is almost symmetric about  
.
In examples 2 (Figure 5) Pressure varies linearly with  
x
for different values  
of  
at constant fluid density. For fluid of different densities at constant  
pressure declines initially and increases rapidly with large  
x
values.  
In example 3 pressure has a inverted parabolic variation (Figure 8 and 9) with  
x
for different  
at constant density ρ as well as different fluid density at  
constant  
which are symmetric aboutx 0  
.
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1. Department of Physics,  
(Received, October 30, 2024)  
P.K. Roy Memorial College,  
BBMK University, Dhanbad-826004, Jharkhand, India  
E-mail:sayan12350@gmail.com  
2. Research Scholar,  
University Department of Physics,  
BBMK University, Dhanbad-828103, Jharkhand, India  
E-mail:biru12maths@gmail.com