mathematical model on magneto-hydrodynamic dispersion in a

© 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: [email protected]

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



(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



(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



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.



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



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



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



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



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



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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