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© 2020 The Korean Society of Rheology and Springer 287
Korea-Australia Rheology Journal, 32(4), 287-299 (November 2020)DOI: 10.1007/s13367-020-0027-0
www.springer.com/13367
pISSN 1226-119X eISSN 2093-7660
Mathematical model on magneto-hydrodynamic dispersion in a porous medium
under the influence of bulk chemical reaction
Ashis Kumar Roy1,*, Apu Kumar Saha
2, R. Ponalagusamy
3 and Sudip Debnath
4
1Department of Science & Humanities, Tripura Institute of Technology, Agartala, Tripura-799009, India2Department of Mathematics, National Institute of Technology, Agartala, 799046, Tripura, India
3Department of Mathematics, National Institute of Technology, Tiruchirappalli, 620015, Tamil Nadu, India4Center for Theoretical Studies, Indian Institute of Technology Kharagpur, 721302, West Bengal, India
(Received May 15, 2020; final revision received July 26, 2020; accepted September 21, 2020)
The mathematical model of hydrodynamic dispersion through a porous medium is developed in the pres-ence of transversely applied magnetic fields and axial harmonic pressure gradient. The solute introduce intothe flow is experienced a first-order chemical reaction with flowing liquid. The dispersion coefficient isnumerically determined using Aris’s moment equation of solute concentration. The numerical techniqueemployed here is a finite difference implicit scheme. Dispersion coefficient behavior with Darcy number,Hartmann number and bulk flow reaction parameter is investigated. This study highlighted that the depen-dency of Hartmann number and Darcy number on dispersion shows different natures in different ranges ofthese parameters.
Keywords: Darcy number, Hartmann number, Taylor-Aris dispersion, bulk flow reaction
1. Introduction
In fluid dynamics, dispersive mass transfer is the move-
ment of mass through convection and molecular diffusion
from high concentrated region to a less concentrated
region. For non-uniform velocity, the tracer material
induces a concentration gradient in the transverse direc-
tion that leads to a transverse diffusion along with axial
diffusion and convection, and the spreading of the tracer
as a result of all these three factors is termed as Taylor-
Aris dispersion. This theory was discovered by Taylor
(1953), who calculated the effective diffusion coefficient
of a passive solute injected into a laminar flow through a
straight capillary tube, followed by Aris (1956), who
developed a new methodology viz., method of moment to
study the same. Ananthakrishnan (1965) studied the Tay-
lor-Aris dispersion numerically and found that the theory
provides a good explanation of the dispersion mechanism
after times of solute injection, where a denotes
radius of the conduct and D is the constant molecular dif-
fusivity of the solute. Barton (1983) overcame this lim-
itation of Taylor-Aris dispersion by resolving technical
difficulties in Aris methodology. By devolving a new
technique (General dispersion model) to estimate the
effective dispersion coefficient, Sankarasubramanian and
Gill published a series of articles (1971, 1972, 1973). The
authors also considered a first-order reaction at the wall
(1973) for which a new transport coefficient appeared for
the first time through their investigation. Over the last
seven decades, the subject has gained considerable atten-
tion due to its widespread application in chemical engi-
neering (Balakotaiah et al., 1995), biomedical engineering
(Fallon et al., 2009), environmental sciences (Chatwin and
Allen, 1985), physiological fluid dynamics (Grotberg et
al., 1994), etc. On the progress of numerical process to
solving the partial differential equations, researchers
(Mazumder and Das, 1992; Mazumder and Paul, 2008)
are motivated and encouraged to look into the time-based
behavior of Taylor-Aris dispersion coefficient applying
the above two techniques in different geometries, e.g.,
channel (Bandyopadhyay and Mazumder, 1999; Mazum-
der and Paul, 2008), pipe (Mazumder and Das, 1992; Ng,
2006) and annular region (Mondal and Mazumder, 2005;
Paul and Mazumder, 2009; Paul, 2010). Some of the
researchers have also attempted to directly solve the con-
vection-diffusion equation, either numerically (Ananthakrish-
nan et al., 1965; Baily and Gogarty, 1962) or semi
analytically (Ng, 2004; Paul, 2009).
The transport of species in capillary blood is an obvious
concern for biomechanics, as such studies facilitate the
diagnosis and cure of different cardiovascular diseases by
the physiologists. The existence of hemoglobin (an iron
compound) in RBC has prompted several researchers
(Mekheimer and El Kot, 2007; Midya et al., 2003; Motta
et al., 1998) to concentrate on the biofluids flows in the
context of a magnetic field. It is important to note that the
involvement of an external magnetic field significantly
impacted the biological structures (Rao and Deshikachar,
2013). Bhargava et al. (2007) testified that the magnetic
field can act as a control mechanism to regulate blood
flow in many clinical uses. Haldar and Ghosh (1994) stud-
20.5( / )a D
*Corresponding author; E-mail: rk.ashis10@gmail.com
Ashis Kumar Roy, Apu Kumar Saha, R. Ponalagusamy and Sudip Debnath
288 Korea-Australia Rheology J., 32(4), 2020
ied the impact of the magnetic field on the blood circu-
lation in arteries and veins by treated blood as a Newtonian
fluid. It is reported that cell separation, provocation of
occlusion of the feeding vessels of cancer tumors, and pre-
vention of bleeding during surgeries are some of the major
uses of magnetic devices (Voltairas et al., 2002). An exter-
nally applied transverse magnetic field is observed to
decrease the blood flow rate and blood velocity in arteries
significantly. Therefore, it is very much essential to inves-
tigate the consequence of the magnetic field on the blood
flow through arteries. Moreover, When the plaques depos-
ited on the inner walls of the artery (circular tube) and the
establishment of arterial clots in the lumen of the vessel
pave the way for blood flow; in this case, the lumen of the
artery containing the cholesterol, thrombus, and fatty
plaques embodies the porous medium, and thus the flow
of blood through the arteries can be considered as an
equivalent to a flow in fictitious porous media (Dash et
al., 1996). The pulsatile blood flow through a non-uniform
(constricted) porous artery treating blood as a Newtonian
fluid in the presence of body acceleration is analyzed by
El-Shahed et al. (2003). Mehmood et al. (2012) consid-
ered the unsteady two-dimensional flow of blood (New-
tonian fluid) in a constricted artery occupied with a porous
medium. The analysis exposed that the velocity profile in
the constricted area of the tube depends upon the perme-
ability of the porous medium, and the smaller permeability
extremely attenuates the bloodstream. Das and Saha
(2009) have examined the pulsatile MHD flow of blood
through a porous artery in the presence of a periodic body
force by supposing blood as an electrically conducting
incompressible Newtonian fluid by adopting the tech-
niques of Laplace and Hankel transforms. Knowing the
importance of flow through a porous medium subject to a
magnetic field, a sufficiently useful investigation has
recently been conducted to explore the influence of Darcy
number and magnetic parameters on flow properties. All
the study reveals that the presence of a magnetic field
decreases the blood velocity (Ponalagusamy and Priyad-
harshini, 2017; Rao and Deshikachar, 2013).
The solute dispersion of passive tracer for channel flow
in the presence of a uniform transverse magnetic field was
studied by Gupta and Chatterjee (1968) using both Tay-
lor's theory and Aris analysis. Annapurna and Gupta
(1979) re-investigated the problem using a generalized
dispersion model in order to estimate the dispersion coef-
ficient, which is valid for all time. Both studies disclose
the fact that the coefficient of dispersion decreases as the
Hartmann number increases. As already discussed, Mag-
neto Hydrodynamic concepts are widely applicable in the
field of Biomechanics; however, this scenario is rarely
studied in the case of species transport.
Other factors in species transport in blood flow are bulk
flow reaction and flow pulsation because, very often, sol-
ute reacts with a flowing stream, and it is common prac-
tice to consider reactive solute while studying dispersion
in blood flow. Also, the heart pumps periodically, result-
ing in blood flowing from the heart to the entire body
through various blood vessels. These two factors play
essential roles in species transport. Gupta and Gupta
(1972) and Roy et al. (2017) are few among others who
shed some light on the impact of irreversible chemical
reaction on the dispersion process and found that reaction
rate reduces the effective dispersion coefficient.
The primary objective of this article is to formulate a
dispersion model in blood flow through a porous medium
with a periodic pressure gradient under the presence of
chemically active solute at the bulk of the blood flow. An
external magnetic field is taken into account while pre-
paring the model and the porous media being considered
is homogeneous with constant permeability. The proposed
model may lead to the development of new diagnostic
tools for clinical purposes.
2. Mathematical Formulation
Let us consider contaminant transport in an unsteady,
fully devolved, unidirectional laminar flow of electrically
conducting liquid with conductivity , through a hori-
zontal tube of a radius . Also a uniform magnetic field
is applied normal to the fluid flow. A cylindrical coor-
dinate system is taken, as shown in Fig. 1, where the axial
and radial coordinates are represented in terms of and
, respectively (bar denotes dimensional quantity). The
problem has been fixed in the light of the following con-
siderations:
1. The tube is filled with isotropic porous media.
2. The boundary of the tube is impermeable.
3. The Newtonian fluid model is considered to represent
the blood characteristic. Blood is usually a non-New-
tonian fluid, and it follows Newtonian nature when
the shear rate exceeds 100 s1 (Anastasiou et al.,
2012; Berger and Jou, 2000; Pedley, 1980; Tu and
Deville, 1996). In large blood vessels like aorta,
where the shear rate is high enough, the impact of
non-Newtonian flow behavior is not important. Thus
the Newtonian assumption of blood is satisfactory
while flowing through large arteries like the aorta.
4. Fluid density and viscosity are constant.
5. The flow is driven by a periodic axial pressure gra-
dient given by (Debnath et al., 2018; Roy et al.,
2017; Roy et al., 2020; Wang and Chen, 2015)
(1)
where, and denote respectively the amplitude and
frequency of the pressure pulsation.
The governing equations for the case of magneto-hydro-
a
0B
z
r
( ) ( )
*1
1 ( ) ,i tpP Re e
z
ε
*
Pε
Mathematical model on magneto-hydrodynamic dispersion in a porous medium under the influence of bulk chemical…
Korea-Australia Rheology J., 32(4), 2020 289
dynamic fluid flow are (Ponalagusamy and Priyadarshini,
2017; Yadav et al., 2018)
(2)
(3)
and the convection-diffusion equation for reactive con-
taminant transport can be adopted generally as (Zeng and
Chen, 2011)
(4)
The variables and parameters are used in the above
equations are defined in Table 1.
The current density described in Eq. (3), obey Ohm's
law as
(5)
Also, we assumed that are constant.
The total magnetic field in Eq. (5) is the sum of the
induced magnetic field and the external magnetic field
. For small magnetic Reynolds numbers, the induced
magnetic field is negligible compared to the external mag-
netic field. Also, the electric field due to charging polar-
ization is assumed to be insignificantly small and hence
in Eq. (3) is simplified to . With all these
simplified assumptions, the governing equations of motion
(Eqs. (2) and (3)) for the fluid flow is reduced to
(6)
wherein denotes the axial velocity.
To find the solution of the flow problem, it is required
to specify the boundary condition. The boundary condi-
tion adopted in the present study is the usual no-slip
boundary condition i.e.,
(7)
Again, at the center of the pipe, the axial velocity is
maximum, i.e.,
(8)
Let, at the time a tracer of mass m with concen-
tration be released instantaneously in the flow as
mentioned above i.e.,
(9)
Transport Eq. (4) is also reduced to:
, (10)
with , , where
and are the axial and transverse diffusion coefficients
respectively. The following assumptions are made con-
cerning transport Eq. (10):
1. The boundary of the flow conduit is impermeable,
i.e., the solute cannot penetrate the wall boundary, so
,
D
Dt
u
2( ) ,pt K
uu u u u J B
1( ) ( ) .
C kk C C C
t
u D
. J E u B
, , ,k D
B
1B
0B
( )J B 2
0B u
2
01
,u p u B
r u ut z r r r K
( , )u r t
0 at .u r a
0 at 0.u
r
r
t = 0
(0, , )C r z
2
( )(0, , ) , 0 .
m zC r z r a
a
2 effeff
2( , ) r
z
C C C D Cu r t D r C
t z r r rz
eff ( )/z z
D k D eff ( )/r r
D k D z
D
rD
Fig. 1. Flow Geometry.
Table 1. List of variable and parameters.
Symbols Name Unit
Time s
Superficial velocity m s1
Permeability m2
Porosity Dimensionless
Current density Cm2
Total magnetic field T (Tesla)
Superficial pressure including gravity Nm2
Solute concentration Kg m3
Concentration diffusivity m2 s2
Tortuosity Dimensionless
Concentration dispersivity tensor m2 s2
Bulk reaction rate s1
t
u
K
J
B
p
C
k
D
Ashis Kumar Roy, Apu Kumar Saha, R. Ponalagusamy and Sudip Debnath
290 Korea-Australia Rheology J., 32(4), 2020
(11)
2. Symmetry is assumed and thus
(12)
3. The total amount of solute is finite, and thus solute
cannot reach far away from the point of injection, i.e.,
(13)
The following dimensionless quantities are used:
(14)
where U is the characteristic velocity.
Using Eq. (14), the Momentum Eq. (6) with given pres-
sure gradient (Eq. (1)) is reduced to:
(15)
where is Darcy number, is
Hartmann number, and is Womersley num-
ber. is the steady part of the pressure gra-
dient, where is the amplitude of the oscillatory part of
the pressure gradient, and is Schmidt num-
ber.
The boundary condition Eqs. (7) and (8) becomes:
(16)
Similarly, the governing equation (Eq. (10)) and initial
and boundary conditions (Eqs. (9) and (11- 13)) can be
rewritten as:
(17)
(18)
(19)
(20)
(21)
here is the reaction rate,
represent the ratio of axial and radial diffusion coeffi-
cients. is the effective Péclet number
that measures the relative effect of the convection in
porous media against diffusion.
3. Velocity Distribution
To solve the BVP given in Eq. (15) and (16), we assume
a solution of the form:
(22)
Substituting Eq. (22) in Eqs. (15) and (16) and solving
we get
(23)
(24)
here J0 is the Bessel function of first kind of order zero
and .
When both the velocity component and
reduces to:
(25)
(26)
If the characteristic velocity U be chosen as axial veloc-
ity and , then the present model is similar
to the model of Mazumder and Das (1992), and both the
component of velocity merge with the velocities of
Mazumder and Das (1992).
4. Aris-Barton Approach
The pth order concentration moment of the tracer mate-
rial is defined as (Aris, 1956)
(27)
and the cross-sectional mean (denoted by an angle bracket)
of the pth concentration moment,
(28)
So using Eq. (27), transport Eq. (17) subject to initial
and boundary conditions (18-21) can be written as:
0 at .C
r ar
0 at 0.C
rr
( , , ) 0.C t r
eff 3
2, , , , ,
rD t r z C a u
t r z C ua a m Ua
2
2
1 11
1 ,
i Sct
a
u uF Re e r
Sc t r r r
M uD
ε
2/
aD K a
0/M B a
/a
* 2( )/F P a U
Fεeff
/ r
Sc D
0 at 0 ,
0 at 1 ,
u
r
r
u r
2
2
1Pe ( , ) ,D
C C C Cu r t r R C
t z r r r z
( )(0, , ) , (0 1),
zC r z r
0 at 0,C
rr
0 at 1,C
rr
( , , ) 0,C t r
2 eff/
ra D
eff eff/
D z rR D D
effPe ( / )r
Ua D
2
1
( , ) ( ) ( , ),
( , ) ( ) ( ) .
s o
i Sct
s
u r t u r u r t
u r t u r Re u r e
ε
0
20
( )( ) 1 ,
( )s
F J i ru r
J i
2 2
0
1 2 2 2 2
0
( )( ) 1 ,
( )
F J i i ru r
i J i i
2 1 2 )(a
D M
0 s
uou
20
lim ( ) 1 ,4
s
Fu r r
22
0
2 200
( )lim ( , ) 1 .
( )
i Sct
o
iF J i i ru r t Re e
J i i
ε
* 2/ 4P a 0
( , ) ( , , ) ,p
pC t r z C t r z dz
2 1
0 0
2 1
0 0
( , )
( ) .p
p
d rC t r dr
C t
d rdr
Mathematical model on magneto-hydrodynamic dispersion in a porous medium under the influence of bulk chemical…
Korea-Australia Rheology J., 32(4), 2020 291
(29)
with
(30)
Using Eq. (28) in Eqs. (29) and (30), gives
(31)
and
(32)
The pth order central moment of the tracer concentration
is introduced as
(33)
where is the mean of the distribution. The
other central moments acquired from Eq. (33), as
(34)
Now we solved Eq. (29) with Eq. (30) using a finite dif-
ference method based on the Crank-Nicolson implicit
scheme. The detailed procedure is shown in the Appendix.
5. Results and Discussion
In the presence of a transversely applied magnetic field
and axial periodic pressure gradient, the present study will
address a problem of tracer dispersion through a tube
filled with a porous medium. To discuss the numerical
consequences of the proposed dispersion problem, we
have chosen the values of various parameters based on the
available literature, which are listed in Table 2.
Since the velocity is unsteady due to the presence of
periodic pressure pulsation, we divide the velocity into
two parts: steady part and the oscillatory part
. Due to the pulsations of pressure, the fluid veloc-
ity is periodic with period , and it is sufficient to
consider the different phases in the range .
Flow velocities and dispersion coefficients are two cru-
cial parameters associate with the study of fluid and spe-
cies transport in porous media. In the beginning, we look
into various effects on the velocity components.
In most of our analysis, we observe the dependency of
both the velocity components as well as the combined
velocity. It is interesting to note that the findings show
identical qualitative dependence, regardless of their veloc-
ity components. Figure 2 demonstrates the dependency of
Darcy number , Hartmann number ( ) and phase
angles on velocity, the figure reveals that velocity
increases with Darcy Number but decreases with Hart-
1 2
1Pe ( 1) ,
p pp D p p
C Cr upC R p p C C
t r r r
for 0,
for
1
(0, )
0
0 at 1,
0 at 0.
0,
p
p
p
C r
Cr
r
p
Cr
p
r
1 2Pe ( ) ( 1) ,
p
p D p p
d Cp u r C R p p C C
dt
1(0) for 0,
(0) 0 for 0.
p
p
C p
C p
1 2
0 0
1 2
0 0
( )
( ) ,
(0, , )
p
g
p
r z z Cdrd dz
t
rC r z drd dz
1 0/
gz C C
2 2
2
0
3 3
3 2
0
4 2 4
4 3 2
0
( ) ,
( ) 3
( ) 4 6 .
,
g
g g
g g g
Ct z
C
Ct z z
C
Ct z z z
C
( )s
u r
( , )ou r t
22 / Sc
2( )Sct [0, ]
( )a
D M2Sct
Table 2. Range of controlling parameter in the present study.
Parameter Range or values
Womersley number () (Debnath et al., 2017) 0, 0.5, 1, 1.5, 2
Poiseuille number (F) (Debnath et al., 2018) 1
Schmidt number (Sc) (Mazumder and Das, 1992; Mazumder and Paul, 2008; Roy et al., 2020) 1000
Amplitude factor () (Debnath et al., 2018) 0 (Steady flow), 1.5 (Unsteady flow)
Darcy number (Da) (Ponalagusamy and Priydharshini, 2017) 0.01, 0.1, 0.5, 1, 5, 10
Hartmann number (M) (Ponalagusamy and Priydharshini, 2017) 0, 0.5, 1, 1.5, 2
Pèclet number (Pe) (Wang and Chen, 2015) 100
Porosity () (Jiang and Chen, 2019) 0.6, 0.75, 0.9
Bulk flow reaction rate () (Roy et al., 2017) 0, 10, 20, 50, 100
Ashis Kumar Roy, Apu Kumar Saha, R. Ponalagusamy and Sudip Debnath
292 Korea-Australia Rheology J., 32(4), 2020
mann number, on the other hand, phase angles change the
direction of the flow along with its magnitude. Moreover,
we can note that the Darcy number dependency on veloc-
ity will be negligible for the strong Darcy number .
Figure 3 shows how velocity depends on F, F = 0 means
no flow arises, positive F helps to move fluid in forward-
ing direction, whereas negative F causes backflow. The
reason is quite natural as the sign of F decides the direc-
tion of the driving force.
5.1 Concentration DecayFor p = 0, the moment Eq. (29) with boundary condition
Eq. (30) becomes:
(35)
(36)
. (37)
Using Eq. (37) and applying the cross-sectional average
of Eqs. (35) and (36), we obtain as
(38)
(39)
The solution of Eq. (38) with initial condition (39) is
(40)
which represents the total mass of the tracer material,
which is a function of , and t. For a fixed reaction
rate , the tracer material is depleted over time. When
there is no bulk reaction i.e., , then the mass of
tracer material , in the whole tube is con-
stant with respect to time. As expected, dimensionless
mass decays with the bulk reaction
rate and dimensionless time as illustrated in Fig. 4. This
figure also conveys the fact that, over time, the residuals
of the species goes to zero, moreover, with the increase of
reaction rate, the species mass degradation occurs rapidly.
( 5)a
D
0 0
0
10,
C Cr C
t r r r
0
1(0, ) ,C r
0 0 at 0,1C
rt
0
00,
d CC
dt
0 0
1| .t
C
0
1( , ) ,tC t e
0
0( ,0) 1 /C t
0 0( , ) / ( ,0)C t C t
Fig. 2. Velocity profile (a) for different values of Darcy number (Da) when F = 1, M = 1, and ; (b) for different
values of Hartmann number (M) when F = 1, Da = 0.5, = 0.5, and ; (c) for different phase angles when
F = 1, Da = 0.5, = 0.5, and M = 1.
1.5ε2
/ 2Sct
1.5ε2
/ 2Sct 2( )Sct
1.5ε
Fig. 3. Velocity profile for different Poiseuille number (F) when Da = 0.5, , = 0.5, and M = 1. (a) For steady
flow; (b) for Periodic component; (c) for combined flow (steady + periodic).
1.5ε2
/ 2Sct
Fig. 4. Solute residual with time due to bulk-flow reaction.
Mathematical model on magneto-hydrodynamic dispersion in a porous medium under the influence of bulk chemical…
Korea-Australia Rheology J., 32(4), 2020 293
5.2 Effective dispersion coefficientFollowing the method of moments of Aris (1956), the
apparent dispersion coefficient, is defined in regards
to the variance of the concentration distribution, as
. (41)
We have studied both the short and the large time behav-
ior of the dispersion coefficient (from which the ratio of
axial to radial diffusion is deduced); by the large time, we
mean that the time required to achieve a steady limit of
dispersion coefficients. Invariably all the figures of dis-
persion coefficient, with time show that the dis-
persion coefficient increases with time initially and reaches
its steady value over time. Figure 5 displays
against time due to various components of velocity dis-
tribution and Hartmann number. Figure 5 reveals the fact
that in all cases, the Hartmann number reduces the dis-
persion coefficient. This may be due to the decreasing
radial velocity gradient with the Hartmann number. To
check the dependency of the Hartmann number on the dis-
persion coefficient, Fig. 7 is plotted. The figure shows that
this decreasing tendency is valid only for the initial range
of Hartmann number and small Darcy number (< 1); how-
ever, after attained the minimum value of dispersion coef-
ficient, it is increased with Hartmann number and
ultimately reaches its steady value. One of the remarkable
results is that for high Darcy number (> 1) dispersion
curve has a local maximum followed by local minimum.
In all cases, the steady limit of the dispersion coefficient
is same. The maximum and minimum value of the dis-
persion coefficient is tabulated in Table 3, the table shows
that for all cases, the steady limit of the dispersion coef-
ficient is same. Also, the maximum and minimum disper-
sion coefficient is same irrespective of the value of Darcy
number.
appD
21
2app
dD
dt
app DD R
app DD R
Fig. 5. Dispersion coefficient Dapp against small (column 1) and large time (column 2) due to a various component of velocity dis-
tribution and Hartmann number (M) when F = 1, = 0.5, , Sc = 1000, = 0.75, Pe = 100, R = 20, Da = 0.5 and RD = 1; (a, b)
for a steady component of velocity; (c, d) for an oscillatory component of velocity; (e, f) for combined velocity.
1.5ε
Ashis Kumar Roy, Apu Kumar Saha, R. Ponalagusamy and Sudip Debnath
294 Korea-Australia Rheology J., 32(4), 2020
The influence of Darcy number in the Taylor-Aris dis-
persion process is illustrated through Fig. 6, figures show
that the dispersion coefficient increase with the Darcy
number. However, for small Darcy number dis-
persion coefficient decreases. Also, it can be seen from the
figure that the dispersion coefficient goes to negative for
small Darcy number, i.e., materiel move backed in the
flow. For detailed observation, we call Fig. 8, where the
dispersion coefficient is displayed with the Darcy number.
As pointed in Fig. 6, here also we can see that the dis-
persion coefficient is increased with Darcy number fol-
lowed by an initial dramatic fall. The rate of increment is
gradually decreased with Hartmann number and goes to( 0.1)
Fig. 6. (Color online) Dispersion coefficient Dapp against small (column 1) and large time (column 2) due to various component of veloc-
ity distribution and Hartmann number (M) when F = 1, = 0.5, , Sc = 1000, = 0.75, Pe = 100, R = 20, Da = 0.5 and RD = 1;
(a, b) for steady component of velocity; (c, d) for oscillatory component of velocity; (e, f) for combined velocity.
1.5ε
Fig. 7. (Color online) Dispersion coefficient Dapp against Darcy
number at t = 0.5 for various Hartmann number when F = 1,
= 0.5, , Sc = 1000, = 0.75, Pe = 100, = 20, = 20
and RD = 1.
1.5ε
Table 3. Some observations on Fig. 7.
Darcy Num-
ber
Critical
point
Maximum
value
Minimum
Value Steady limit
0.5 2.18 - 0.04841
1 2.38 - 0.04841
5 0.84, 2.58 1.682 0.04842
10 0.84, 2.58 1.682 0.04842
1
1
1
1
Mathematical model on magneto-hydrodynamic dispersion in a porous medium under the influence of bulk chemical…
Korea-Australia Rheology J., 32(4), 2020 295
zero processes, Fig. 8(b, c) reflects this fact. This is
attributed to the fact that the existence of pores in the flow
passage reduces the flow resistance which, in turn, helps
to bring up the higher flow of blood. When a magnetic
field is applied to a moving and electrically conducting
blood, it does induce electric and magnetic fields. The
interaction between these fields produces a body force
known as the Lorentz force, which has a tendency to
oppose the movement of the fluid (blood) resulting decel-
erate the flow velocity of blood in the human arterial sys-
tem which results in reduces the dispersion coefficient.
Figure 9 presents a variety of with time for
various reaction rates, and it is noticed that the dispersion
coefficient decreases with the increase of the bulk reaction
rate. These facts can be inferred from the physical ground
that, with the increase of reaction rate, the number of
moles participating in the chemical reaction increases
resulting in a decrease in the dispersion coefficient. How-
app DD R
Fig. 8. (Color online) Dispersion coefficient Dapp against Darcy number (Da) at t = 0.5 for various Hartmann number (M) when F = 1,
= 0.5, , Sc = 1000, = 0.75, Pe = 100, = 20, = 20 and RD = 1.1.5ε
Fig. 9. (Color online) Dispersion coefficient Dapp against small (column 1) and large time (column 2) due to the various component
of velocity distribution and bulk reaction () when F = 1, = 0.5, , Sc = 1000, = 0.75, Pe = 100, M = 1, Da = 0.5 and RD = 1;
(a, b) for a steady component of velocity; (c, d) for an oscillatory component of velocity; (e, f) for combined velocity.
1.5ε
Ashis Kumar Roy, Apu Kumar Saha, R. Ponalagusamy and Sudip Debnath
296 Korea-Australia Rheology J., 32(4), 2020
ever, this nature can be reversed with suitable Hartmann
number (see Fig. 10).
5.3 Mean concentrationThe axial mean concentration distribution is approxi-
mated from the expansion of the following series.
(42)2
4
0
0
( , ) ( ) ( ) ( ) ,m n n
n
C z t C t e b t H
Fig. 10. (Color online) Dispersion coefficient Dapp with the reaction parameter at time t = 0.5 for (a, b, c, d) various Hartmann number,
and (e) Darcy number.
Fig. 11. (Color online) Mean concentration distribution due to combined flow at time t = 0.5 when F = 1, = 0.5, , Sc = 1000,
= 0.5, Pe = 100, R = 20, and RD = 1 (a) for different Hartmann Number and fixed Darcy number Da = 1; (b) for Darcy number and
fixed Hartmann number M = 0.5.
1.5ε
Mathematical model on magneto-hydrodynamic dispersion in a porous medium under the influence of bulk chemical…
Korea-Australia Rheology J., 32(4), 2020 297
where and the coefficient bi(i = 0, 1,
2, 3, 4) is estimated from the first four central moments of
the species concentration as
(43)
, the Hermite polynomials, satisfy the recurrence rela-
tion
(44)
Therefore, at any given location and time it is possible
to evaluate the axial mean concentration using statistical
parameters described in Eq. (34).
Figure 11 displays the mean concentration distribution
against axial distance , due to combined flow
at time instance for various Hartmann numbers and
Darcy numbers. It can be seen from the figure that the
increase of Hartmann's number increases the peak of mean
concentration. However, the increase of Darcy number
decreases the peak. The increase of Hartmann number
reduces the flow velocity, and thus the dispersion coeffi-
cient decreases; as a result, axial mean concentration
increases. The decrement of the peak of axial mean con-
centration with the Darcy number is based on the same
analogy.
The axial mean concentration for oscillatory flow also
reports the same phenomenon (see Fig. 11 and 12). It is
worth mentioning that for combined flow and the larger
Hartmann numbers and the smaller Darcy numbers, the
mean concentration is non-symmetric and consists of dou-
ble peaks. However, the breakthrough curve converges to
the Gaussian curve with the growth of Hartmann number
and Darcy number (see Fig. 11).
The axial mean concentration vs. axial dis-
tance for the different reaction parameters is
presented in Fig. 13a. The increase of the reaction param-
eter ensures that the reactive material is exhausted, and
thus the peak of the mean concentration distribution grad-
ually decreases. Figure 13b reflects that the peak of axial
mean concentration fall as increases, which means
that means concentration has contributed more to radial
diffusivity then axial diffusivity. This is consistent with
the implication reported in Wang et al. (2015).
6. Conclusion
The dispersion coefficient of oscillatory flow in porous
media with reactive solute in the presence of a trans-
2, ( ) / 2
gz z
0 2 0 30 1 2 3 4
2
1 2, 0, 0, , .
24 962
a ab b b b b
iH
1 1
0
( ) 2 ( ) 2 ( ), 0,1,2,
( ) 1.
i i iH H iH i
H
( ) /g
Z Z Pe
0.5t
( , )m
C x t Pe
( ) /g
Z Z Pe
DR
Fig. 12. (Color online) Mean concentration distribution due to purely periodic flow at time t = 0.5 when F = 1, = 0.5, ,
Sc = 1000, = 0.5, Pe = 100, R = 20, and RD = 1 (a) for different Hartmann Number and fixed Darcy number Da = 0.5; (b) for Darcy
number and fixed Hartmann number M = 0.5.
1.5ε
Fig. 13. (Color online) Mean concentration distribution due to purely periodic flow at time t = 0.5 when F = 1, = 0.5, ,
Sc = 1000, = 0.5, and Pe = 100 (a) for different reaction rates and RD = 1; (b) for different RD and R = 20.
1.5ε
Ashis Kumar Roy, Apu Kumar Saha, R. Ponalagusamy and Sudip Debnath
298 Korea-Australia Rheology J., 32(4), 2020
versely applied magnetic field is computed numerically.
An investigation has been done for different flow veloc-
ities-steady , periodic , and combined
. The conclusions drawn from the above
analysis are as follows.
(a) An increase in Hartmann number, decrease in the
dispersion coefficient in its initial range after that
dispersion coefficient increases and reaches to its
steady limit.
(b) For small Hartmann number, the dispersion coeffi-
cient increase with Darcy number followed by a
drastic fall.
(c) The peak of the mean concentration increase with
the increase of Hartmann number but decrease with
Darcy number.
(d) The dispersion coefficient is expected to decrease
with the bulk reaction rate.
(e) In all cases, the distribution curve of the mean con-
centration tends to flatten with the increase of the
bulk reaction rate.
Acknowledgment
We thank the anonymous reviewers for their helpful
suggestions.
Appendix
In order to solve Eq. (29) numerically for and
4, we have partitioned the space domain into
mesh ( ) of equal length , where
denotes the axis of the tube and denotes the surface
of the tube. Similarly, we have partitioned time uniformly
using the mesh . The step length for time
is , thus each node can be estimated from the relation
. Hence the value of the continuous variables
at each mesh point is address by , further,
the spatial derivative and time derivative in Eq. (29) is
approximated by
, (A1)
, (A2)
. (A3)
The resulting finite difference equation turns into a sys-
tem of linear algebraic equation with a tri-diagonal coef-
ficient matrix,
(A4)
the associate initial and boundary condition becomes
(A5)
, (A6)
. (A7)
The above tri-diagonal system is solved by the Thomas
algorithm. This finite-difference technique is known as
Crank–Nicolson method, and the scheme is always
numerically stable and convergent. However, to capture
the fine behavior of , the time step is considered to be
very small. For our study, we have taken and
.
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