Mathematical model on magneto-hydrodynamic dispersion in a porous medium under the influence of bulk chemical reaction

DOI: 10.1007/s13367-020-0027-0

Mathematical model on magneto-hydrodynamic dispersion in a porous medium under the influence of bulk chemical reaction

Ashis Kumar Roy^1,*, Apu Kumar Saha², R. Ponalagusamy³ and Sudip Debnath⁴

1Department of Science & Humanities, Tripura Institute of Technology, Agartala, Tripura-799009, India

2Department of Mathematics, National Institute of Technology, Agartala, 799046, Tripura, India

3Department of Mathematics, National Institute of Technology, Tiruchirappalli, 620015, Tamil Nadu, India

4Center 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 into the flow is experienced a first-order chemical reaction with flowing liquid. The dispersion coefficient is numerically determined using Aris’s moment equation of solute concentration. The numerical technique employed 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 of these 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- 0.5( / )a D2

*Corresponding author; E-mail: [email protected]

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 s^1 (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 

B0

r z

( ) ( )

1 p P* 1 Re e( ^{i t}) , z





  

    ε 

P*

ε 

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 assum ptions are m ade 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

 

( ) p 2 ,

t K

          

u u u u u J B

1 ( ) ( ) .

C k C k C C

t  

 

         

 u D

 

^.



  

J E u B

, , ,k

  D

B B1

B0

(J B ) B u₀²

2

1 0 ,

u p r u u B u

t 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, , ) m z , 0 .

C r z r a

a



   

2 eff

eff 2

( , ) _z ^r

C u r t C D C D r C C

t z z r r r  

         

eff ( / )

z z

D k D  D_r^eff k(D_r/) Dz

Dr

Fig. 1. Flow Geometry.

Table 1. List of variable and parameters.

Symbols Name Unit

Time s

Superficial velocity m s^1

Permeability m²

Porosity Dimensionless

Current density Cm^2

Total magnetic field T (Tesla) Superficial pressure including gravity Nm^2 Solute concentration Kg m^3 Concentration diffusivity m² s^2

Tortuosity Dimensionless

Concentration dispersivity tensor m² s^2

Bulk reaction rate s^1

t u K

 J Bp 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 p^th 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 p^th 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 a

 r 



0 at 0.

C r

 r 



( , , ) 0.

C t r  

eff 3

2 , , , , ,

D tr r z C a u

t r z C u

a a m U

a

     

 

²

2

1 1 1

1 ,

i Sct

a

u F Re e r u

Sc t r r r

D M u

        

 

  

 

ε

/ 2

DaK a M B a   ₀ /

/ a

  

( * 2/ ) F P a U

Fε Sc  /D_reff

0 at 0 ,

0 at 1,

u r

ru r

   

  

2 2

Pe ( , ) 1 _D ,

C u r t C r C R C C

t z r r r z 

        

(0, , ) ( )z , (0 1),

C r z  r

   

0 at 0,

C r

 r 



0 at 1,

C r

 r 



( , , ) 0, C t r  

2/ _reff

a D

  R_DD^eff_z /D_r^eff

Pe (Ua/D_reff)

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   

2 0 0

( )

( ) 1 ,

s F J i r( )

u r J i

  

    

2 2

1 2 2 0 2 2

0

( )

( ) 1 ,

( )

F J i i r

u r i J i i



 

   

 

 

 

     

2 (D_a^1 M2)

  

0

  u_s u_o

 

²

lim ( )0 1 ,

s F4

u r r

  

2 2

2 0 2

0 0

( )

lim ( , ) 1 .

( )

i Sct

o iF J i i r

u r t Re e

J i i

 

 



   

   

 

 

   

 

ε

* 2/ 4

P a  ^{ 0}

( , ) ^p ( , , ) , C t rp ^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 p^th 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

1 Pe ( 1) ,

p p

p D p p

C C

r upC R p p C C

t r r r _{ } 

       

for 0,

for 1 (0, )

0

0 at 1,

0 at 0.

p 0,

p p

C r

C r

r

p

C r

p

r



 

 

  

   

 

 













1 2

Pe ( ) ( 1) ,

p p D p p

d C p u r C R p p C C

dt  _   _ 

(0) 1 for 0, (0) 0 for 0.

p

p

C p

C p



  



  

1 2 0 01 2

0 0

( )

( ) ,

(0, , )

g p p

r z z Cdrd dz t

rC r z drd dz





















  

  

1 / 0

zg 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

t C z

C

t C z z

C

t C z z z

C



 

  

 

  

   

s( ) u r ( , )

u r to

2 / 2Sc

(2Sct) [0, ]

( )D_a M

2Sct



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 Decay

For 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.

(D _a 5)

0 0

1 0 0,

C r C C

t r r r 

     

0(0, ) 1,

C r



0 0 at 0,1

C r

t

  



0 0 0,

d C C

dt  

0 |_t 0 1. C ^ 

0( , ) 1 ^t,

C 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.

1.5



ε ²Sct/ 2 1.5



ε ²Sct/ 2 (²Sct)

1.5

 ε

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).ε1.5 ²Sct/ 2

Fig. 4. Solute residual with time due to bulk-flow reaction.

5.2 Effective dispersion coefficient

Following 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.

Dapp

1 2 app 2d

D dt

 

app D

D R

app D

D 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.ε1.5

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.ε1.5

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.ε1.5

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 D

D 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, ε1.5, Sc = 1000,  = 0.75, Pe = 100,  = 20,  = 20 and RD= 1.

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.ε1.5

ever, this nature can be reversed with suitable Hartmann number (see Fig. 10).

5.3 Mean concentration

The axial mean concentration distribution is approxi-

mated from the expansion of the following series.

2 4 (42)

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.

1.5

 ε

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 smal l er 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, (z z_g) / 2

  

0 2 0 3

0 1 2 3 4

2

1 , 0, 0, 2 , .

24 96

2

a a

b b b b  b 

     

Hi

1 1

0

( ) 2 ( ) 2 ( ), 0,1,2, ( ) 1.

i i i

H H iH i

H

   



     

 

(Z Z _g) /Pe 0.5

t 

( , ) C x t Pem  (Z Z _g) /Pe

RD

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.

1.5

 ε

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. ε1.5

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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1,2,3 p  (0 r 1)

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