Jeffrey Fluid Flow through Porous Medium in the Presence of Magnetic Field in Narrow Tubes

Abstract

Jeffrey fluid flow in the presence of magnetic field through porous medium in tubes of small diameters is studied. It is assumed that the core region consists of a Jeffrey fluid and the peripheral region of a Newtonian fluid. Making the assumptions as in the work of Chaturani and Upadhya, the linearised equations of motion have been solved and analytical solution has been obtained. The influence of various pertinent parameters on the flow characteristics such as effective viscosity, core hematocrit, and mean hematocrit has been studied and discussed through graphs. It is found that the effective viscosity and mean hematocrit decrease with Jeffrey parameter and Darcy number but increase with tube hematocrit and tube radius. Also, the core hematocrit decreases with Jeffrey parameter, Darcy number, tube hematocrit, and tube radius. Further, it is noticed that the flow exhibits the anomalous Fahraeus-Lindquist effect.

1. Introduction

Microcirculation is a part of human circulatory system, which consists of a complex network of blood vessels, whose diameter ranges from approximately 20 μm (microns) to 500 μm. There are several types of vessels in microcirculation such as arterioles, capillaries, and venules. Its main functions are to supply oxygen and nutrients to every part of the human body. Several anomalous effects have been observed in microcirculation. In particular, the apparent viscosity of the blood increases with tube diameter and this is referred to as the Fahraeus-Lindquist effect. This effect has been confirmed by several investigators.

To explain the observed Fahraeus-Lindquist effect, Haynes [1] considered a two-fluid model with both fluids as Newtonian fluids with different viscosities. Bugliarello and Sevilla [2] have considered a two-fluid model where in the core region as well as peripheral region fluids are both Newtonian with different viscosities or both fluids are Casson’s fluid with different yield coefficients and viscosities. Sharan and Popel [3] and Srivastava [4] have reported that, for blood flowing through narrow tubes, there is a peripheral layer of plasma and a core region of suspension of all erythrocytes. Following the theoretical study of Haynes [1] and experimentally tested model of Bugliarello and Sevilla [2], two-fluid modeling of blood flow has been discussed and used by a good number of researchers. Several non-Newtonian fluid models have been considered for blood flow in small diameter tubes. Chaturani and Upadhya [5, 6] analyzed two-fluid models assuming Newtonian fluid in peripheral region and micropolar and couple stress fluids in the core region. Bali and Awasthi [7] presented a mathematical model for blood flow in a small blood vessel in the presence of a magnetic field. Kumar et al. [8] investigated a computational technique for flow in blood vessels with porous effects. Gupta [9] investigated computational study of blood flow through stenosed artery with magnetic effects.

A porous medium is a material volume consisting of solid matrix with an interconnected void. Flows through a porous medium have several practical applications present in nature: flow in sand beds, in petroleum reservoir rocks, slurries, sedimentation, and so forth. Examples of natural porous media are beach sand, sandstone, limestone, the human lung, bile duct, and gall bladder with stones in small blood vessels. Flow through porous medium has been studied by a number of workers employing Darcy’s law. Furthermore, the study of magnetohydrodynamics (MHD) flow problems has gained considerable interest because of its extensive engineering and medical applications. The MHD deals with the dynamics of electrically conducting fluids. Some investigators have considered the MHD studies of Newtonian and non-Newtonian fluids in different flow geometries. Misra et al. [10] have investigated a mathematical modeling of blood flow in porous vessel having double stenosis in the presence of an external magnetic field. Eldabe et al. [11] studied the peristaltic motion of non-Newtonian fluid with heat and mass transfer through a porous medium in the channel under the effect of magnetic field.

A non-Newtonian fluid model that has attracted many researchers is the Jeffrey fluid as this is found to be a better model for physiological fluids [12]. Jeffrey fluid model is a significant generalization of Newtonian fluid model as the latter one can be deduced as a special case of the former. Several researchers have studied Jeffrey fluid flows under different conditions. Ebaid et al. [13] have studied the peristaltic transport in an asymmetric channel through a porous medium. Vajravelu et al. [14] investigated the influence of heat transfer on peristaltic transport of a Jeffrey fluid. Jyothi et al. [15] have considered the pulsatile flow of a Jeffrey fluid in a circular tube lined internally with porous material. Ebaid [16] has analyzed a mathematical model to study the peristaltic transport of an incompressible viscous fluid in an asymmetric channel under the effect of transverse magnetic field with slip boundary conditions. Akbar et al. [17] have studied the Jeffrey fluid model for the peristaltic flow of chyme in the small intestine with magnetic field. Abd-Alla et al. [18] have investigated the peristaltic flow of a Jeffrey fluid in an asymmetric channel. Recently, Nallapu and Radhakrishnamacharya [19] studied a two-fluid model for the flow of Jeffrey fluid in tubes of small diameters.

In the present paper, a two-fluid model for the flow of Jeffrey fluid through a porous medium in tubes of small diameters with magnetic effect has been investigated. It is assumed that the core region consists of Jeffrey fluid and the peripheral region consists of Newtonian fluid. Making the assumptions as in the work of Chaturani et al. [6], the linearised equations of motion have been solved and analytical solution has been obtained. The influence of various pertinent parameters on the flow characteristics such as effective viscosity, core hematocrit, and mean hematocrit has been studied.

2. Formulation of the Problem

Consider an axisymmetric, laminar, steady flow of an electrically conducting Jeffrey fluid through a circular tube of radius “𝑎” filled with porous medium. It is assumed that a uniform magnetic field 𝐵₀ is applied transversely to the flow. It is assumed that the system consists of two immiscible layers of fluids; the inside layer is non-Newtonian fluid obeying Jeffery model and takes cylindrical shape of radius 𝑏, while the outside layer is Newtonian fluid and takes cylindrical shape of radius 𝑎. Further, the cylinders have a common axis. It is clear that the surface of discontinuity between them is imaginary due to the two fluids being immiscible, as shown in Figure 1. Let 𝜇_𝑝 and 𝜇_𝑐 be the viscosities of Newtonian fluid in the peripheral region and Jeffrey fluid in the core region, respectively.

Figure 1 

Flow geometry of blood in small vessels.

Here, 𝑞 is the velocity of the fluid, 𝑗 is the current density, 𝐵=(𝐵₀+𝐵₁) is the total magnetic field, 𝐵₁ is the induced magnetic field (𝐵₁≪𝐵₀), and 𝑗×𝐵 is Lorentz’s force which is the body force acting on the fluid. The Maxwell equations and Ohm’s law (on neglecting the displacement currents) are ∇⋅𝐵=0,∇×𝐵=𝜇_𝑚𝑗,∇×𝜕𝐸=−𝐵,𝜕𝑡𝑗=𝜎𝐸+𝑞×𝐸, (1) where 𝜎 is the electrical conductivity, 𝜇_𝑚 is the magnetic permeability, and 𝐸 is the electric field. The imposed and induced electric fields are assumed to be negligible. The magnetic Reynolds number is assumed to be small so that the induced magnetic field is neglected.

Hence, the force 𝑗×𝐵 simplifies to 𝑗×𝐵=−𝜎𝐵²₀𝑤. (2) Cylindrical polar coordinate system (𝑟,𝜃,𝑧) is chosen, where the 𝑧-axis is taken along the axis of the tube. The equations governing the steady two-dimensional flow of an incompressible Jeffrey fluid for the present problem are as follows.

Equation of Continuity. One has 𝜕𝑣_𝑟+𝑣𝜕𝑟𝑟^𝑟+𝜕𝑣_𝑧𝜕𝑧=0. (3)

Equations of Motion. Consider 𝜌𝑣_𝑟𝜕𝜕𝑟+𝑣_𝑧𝜕𝑣𝜕𝑧^𝑟=−𝜕𝑝+1𝜕𝑟𝑟𝜕𝑟𝜕𝑟𝑆_𝑟𝑟+𝜕𝜕𝑧𝑆_𝑟𝑧−𝜇_𝑐𝑘₀𝑣_𝑟,𝜌𝑣_𝑟𝜕𝜕𝑟+𝑣_𝑧𝜕𝑣𝜕𝑧^𝑧=−𝜕𝑝+1𝜕𝑧𝑟𝜕𝑟𝜕𝑟𝑆_𝑧𝑟+𝜕𝜕𝑧𝑆_𝑧𝑧−𝜎𝐵²₀𝑣_𝑧−𝜇_𝑐𝑘₀𝑣_𝑧, (4) in which 𝑆_𝑟𝑟=2𝜇_𝑐1+𝜆₁1+𝜆₂𝑣_𝑟𝜕𝜕𝑟+𝑣_𝑧𝜕𝜕𝑧𝜕𝑣_𝑟,𝜕𝑟𝑆_𝑟𝑧=𝑆_𝑧𝑟=𝜇_𝑐1+𝜆₁1+𝜆₂𝑣_𝑟𝜕𝜕𝑟+𝑣_𝑧𝜕𝜕𝑧𝜕𝑣_𝑧+𝜕𝑟𝜕𝑣_𝑟,𝜕𝑧𝑆_𝑧𝑧=2𝜇_𝑐1+𝜆₁1+𝜆₂𝑣_𝑟𝜕𝜕𝑟+𝑣_𝑧𝜕𝜕𝑧𝜕𝑣_𝑧𝜕𝑧 (5) are the extra stress components.

Further, 𝑣_𝑟 and 𝑣_𝑧 are the velocity components in the 𝑟- and 𝑧-directions, respectively, 𝜆₁ is the ratio of relaxation to retardation times, 𝜆₂ is the retardation time, 𝑝 is the pressure, 𝜌 is the density, 𝑘₀ is the permeability of the porous medium, 𝜎 is the electrical conductivity of the fluid, Da(=𝑘₀/𝑎²) is the Darcy number, and 𝑀(=𝜎/𝜇𝐵₀𝑎) is the magnetic parameter.

It is assumed that the flow is in the 𝑧-direction only and hence the velocity component 𝑣_𝑟=0. Consequently, the equations governing the flow of fluid (Jeffrey fluid) in the core region (0≤𝑟≤𝑏) reduce to 𝜕𝑝𝜇𝜕𝑟=0,_𝑐1+𝜆₁1𝑟𝜕𝑟𝜕𝑟𝜕𝑣_𝑧𝜕𝑟−𝜎𝐵²₀𝑣_𝑧−𝜇_𝑐𝑘₀𝑣_𝑧−𝜕𝑝𝜕𝑧=0. (6) Let 𝑣_𝑧(𝑟)=𝑣₁(𝑟) be the velocity in the peripheral region and 𝑣₂(𝑟) in the core region. The equations governing the flow of fluid are as follows.

Peripheral Region (Newtonian Fluid). Consider 𝜇_𝑝1𝑟𝜕𝑟𝜕𝑟𝜕𝑣₁𝜕𝑟−𝜎𝐵²₀𝑣₁−𝜇_𝑝𝑘₀𝑣₁−𝜕𝑝𝜕𝑧=0for𝑏≤𝑟≤𝑎. (7)Core Region (Jeffrey Fluid). One has 𝜇_𝑐1+𝜆₁1𝑟𝜕𝑟𝜕𝑟𝜕𝑣₂𝜕𝑟−𝜎𝐵²₀𝑣₂−𝜇_𝑐𝑘₀𝑣₂−𝜕𝑝𝜕𝑧=0for0≤𝑟≤𝑏, (8) where 𝜕𝑝/𝜕𝑧 is the constant pressure gradient.

The boundary conditions for the problem are given by𝑣₁=0,at𝑟=𝑎,𝑣₁=𝑣₂,𝜏₁=𝜏₂,at𝑟=𝑏,𝑣₂isfiniteat𝑟=0. (9a)(9b)(9c)Condition (9a) is the classical no-slip boundary condition for the velocity; (9b) denotes the continuity of velocities and stresses at the interface and (9c) is the regularity condition.

Solving (7) and (8) under conditions (9a), (9b), and (9c), we get 𝑣₁(𝜉)=𝑎²𝑃𝜇_𝑝𝛼𝐼1−⁰√𝛼⋅𝜉𝐼₀√𝛼for𝑑≤𝜉≤1,𝑣₂(𝜉)=𝑎²𝑃𝜇_𝑝𝛼𝐼1−⁰√𝛼⋅𝑑𝐼₀√𝛼+𝜇^′𝛽𝐼1−⁰(𝜔⋅𝜉)𝐼₀(𝜔𝑑)for0≤𝜉≤𝑑, (10)(11) where 𝑟𝜉=𝑎𝑏,𝑑=𝑎,𝑃=−𝜕𝑝,𝜇𝜕𝑧_′=𝜇_𝑝𝜇_𝑐,𝛼=𝑀²_𝑝+1,𝛼Da𝛽=𝑀²_𝑐+(),1/Da𝜔=1+𝜆₁𝑀²_𝑐+1.Da (12) The flow flux in the peripheral region, denoted by 𝑄_𝑝, is defined by 𝑄_𝑝=2𝜋𝑎²¹_𝑑𝑣₁(𝜉)𝜉𝑑𝜉. (13) Substituting for 𝑣₁(𝜉) from (10) in (13), we get 𝑄_𝑝=𝑎⁴𝑃𝜋𝜇_𝑝𝛼1−𝑑²+2𝑑𝐼₁√𝛼⋅𝑑−𝐼₁√𝛼√𝛼𝐼₀√𝛼. (14) Similarly, the flow flux in the core region is given by 𝑄_𝑐=2𝜋𝑎^2𝑑₀𝑣₂(𝜉)=𝑎𝜉𝑑𝜉₄𝑃𝜋𝜇_𝑝𝛼𝑑²−𝑑²𝐼₀√𝛼⋅𝑑𝐼₀√𝛼+𝜇^′𝛽𝑑²𝐼−2𝑑¹(𝜔𝑑)𝜔𝐼₀().𝜔𝑑 (15) Thus, the flow flux through the tube is given by 𝑄=𝑄_𝑝+𝑄_𝑐. (16) Using (14) and (15) in (16), we get 𝑎𝑄=⁴𝑃𝜋𝜇_𝑝𝛼1+2𝑑𝐼₁√𝛼⋅𝑑−𝐼₁√𝛼√𝛼𝐼₀√𝛼−𝑑²𝐼₀√𝛼⋅𝑑𝐼₀√𝛼+𝜇^′𝛽𝑑²𝐼−2𝑑¹(𝜔𝑑)𝜔𝐼₀().𝜔𝑑 (17) Comparing (17) with flow flux for Poiseuille flow, we get the effective viscosity as follows: 𝜇_eff=𝜇_𝑝𝛼×8Da1+2𝑑𝐼₁√𝛼⋅𝑑−𝐼₁√𝛼√𝛼𝐼₀√𝛼−𝑑²𝐼₀√𝛼⋅𝑑𝐼₀√𝛼+𝜇^′𝛽𝑑²𝐼−2𝑑¹(𝜔𝑑)𝜔𝐼₀(𝜔𝑑)⁻¹. (18)

2.1. Mean Hematocrit for Cell-Free Wall Layer

The percentage volume of red blood cells is called the hematocrit and is approximately 40–45% for human adults.

The core hematocrit 𝐻_𝑐 is related to the hematocrit 𝐻₀ of blood leaving or entering the tube by 𝐻₀𝑄=𝐻_𝑐𝑄_𝑐. (19) Substituting for 𝑄_𝑐 and 𝑄 from (15) and (17) in (19), we get (after simplification) 𝐻_𝑐=𝐻_𝑐𝐻₀+=11−𝑑²+2𝑑𝐼₁√𝛼⋅𝑑−𝐼₁√𝛼√𝛼𝐼₀√𝛼×𝑑²−𝑑²𝐼₀√𝛼⋅𝑑𝐼₀√𝛼+𝜇^′𝛽𝑑²𝐼−2𝑑¹(𝜔𝑑)𝜔𝐼₀(𝜔𝑑)⁻¹, (20) where 𝐻_𝑐 is the normalized core hematocrit.

The mean hematocrit within the tube 𝐻_𝑚 is related to the core hematocrit 𝐻_𝑐 by 𝐻_𝑚𝜋𝑎²=𝐻_𝑐𝜋𝑏². (21) On simplification, we get 𝐻_𝑚=𝐻_𝑚𝐻₀=𝐻_𝑐𝐻₀𝑑²=𝐻_𝑐𝑑², (22) where 𝐻_𝑚 is the normalized mean hematocrit.

Substituting for 𝐻_𝑐 from (20) in (22), we get 𝐻_𝑚=𝐻_𝑚𝐻₀=𝐻_𝑐𝐻₀𝑑²=1+1−𝑑²+2𝑑𝐼₁√𝛼⋅𝑑−𝐼₁√𝛼√𝛼𝐼₀√𝛼×𝑑²−𝑑²𝐼₀√𝛼⋅𝑑𝐼₀√𝛼+𝜇^′𝛽𝑑²𝐼−2𝑑¹(𝜔𝑑)𝜔𝐼₀(𝜔𝑑)⁻¹𝑑². (23)

3. Results and Discussion

In order to discuss the effects of Jeffrey parameter (𝜆₁), core magnetic parameter (𝑀_𝑐), Darcy number (Da), tube hematocrit (𝐻₀), and tube radius (𝑎) on effective viscosity 𝜇_eff, core hematocrit 𝐻_𝑐, and mean hematocrit 𝐻_𝑚, they have been numerically computed and graphically presented in Figures 2–13. In the present analysis, the values of 𝜇_𝑝 are taken as 1.2 μm, 𝜇_𝑐 = 4.0 μm and the value of 𝑑 is calculated from the relation 𝑑=1−(𝜖/𝑎) in which 𝜖=3.12 μm for 40% hematocrit, 3.60 μm for 30%, and 4.67 μm for 20% [5].

Figure 2 

Effect of 𝜆1 on 𝜇eff (𝐻0=40%, 𝑀𝑝=1, 𝑀𝑐=0.5, and Da = 0.5).

Figure 3 

Effect of Da on 𝜇eff (𝐻0=40%, 𝜆1=2, 𝑀𝑝=1, and 𝑀𝑐=0.5).

Figure 4 

Effect of 𝑀𝑐 on 𝜇eff (𝐻0=40%, 𝜆1=2, 𝑀𝑝=1, and Da = 0.5).

Figure 5 

Effect of 𝐻0 on 𝜇eff (𝜆1=2, 𝑀𝑝=1, 𝑀𝑐=0.5, and Da = 0.5).

Figure 6 

Effect of 𝜆1 on 𝐻𝑐 (𝐻0=40%, 𝑀𝑝=1, 𝑀𝑐=0.5, and Da = 0.5).

Figure 7 

Effect of Da on 𝐻𝑐 (𝐻0=40%, 𝜆1=2, 𝑀𝑝=1, and 𝑀𝑐=0.5).

Figure 8 

Effect of 𝐻0 on 𝐻𝑐 (𝜆1=2, 𝑀𝑝=1, 𝑀𝑐=0.5, and Da = 0.5).

Figure 9 

Effect of 𝑀𝑐 on 𝐻𝑐 (𝐻0=40%, 𝑀𝑝=1, 𝜆1=2, and Da = 0.5).

Figure 10 

Effect of 𝜆1 on 𝐻𝑚 (𝐻0=40%, 𝑀𝑝=1, 𝑀𝑐=0.5, and Da = 0.5).

Figure 11 

Effect of Da on 𝐻𝑚 (𝐻0=40%, 𝜆1=2, 𝑀𝑝=1, and 𝑀𝑐=0.5).

Figure 12 

Effect of 𝐻0 on 𝐻𝑚 (𝜆1=2, 𝑀𝑝=1, 𝑀𝑐=0.5, and Da = 0.5).

Figure 13 

Effect of 𝑀𝑐 on 𝐻𝑚 (𝐻0=40%, 𝑀𝑝=1, 𝜆1=2, and Da = 0.5).

Figures 2–5 show the variation of effective viscosity (𝜇_eff) for different values of Jeffrey parameter 𝜆₁, core magnetic parameter (𝑀_𝑐), Darcy number (Da), tube hematocrit (𝐻₀), and tube radius (𝑎). It can be seen that the effective viscosity decreases with Jeffrey parameter (𝜆₁) (Figure 2) and Darcy number (Da) (Figure 3) but increases with core magnetic parameter (𝑀_𝑐) (Figure 4) and tube hematocrit 𝐻₀ (Figure 5). These results are in agreement with the results of Bugliarello and Sevilla [2], Srivastava [4], and Chaturani and Upadhya [5]. Further, for given values of all the parameters, the effective viscosity (𝜇_eff) increases with tube radius (𝑎); that is, the flow exhibits the Fahraeus-Lindquist effect. For higher values of Jeffrey parameter (𝜆₁) and Darcy number (Da), the increase in effective viscosity with tube radius is not very significant for values of tube radius larger than 75 μm (Figures 2 and 3).

The effects of Jeffrey parameter (𝜆₁), core magnetic parameter (𝑀_𝑐), Darcy number (Da), tube hematocrit (𝐻₀), and tube radius (𝑎) on core hematocrit (𝐻_𝑐) and mean hematocrit (𝐻_𝑚) are depicted in Figures 6–13. It can be seen that the core hematocrit (𝐻_𝑐) decreases with Jeffrey parameter (𝜆₁) (Figure 6), Darcy number (Da) (Figure 7), tube hematocrit (𝐻₀) (Figure 8), and tube radius “𝑎.” It can be observed that core hematocrit (𝐻_𝑐) decreases with tube radius (Figures 6–9). Figure 9 shows that the influence of magnetic parameter (𝑀_𝑐) on core hematocrit is very insignificant. Also, the mean hematocrit (𝐻_𝑚) decreases with Jeffrey parameter (𝜆₁) (Figure 10) and Darcy number (Da) (Figure 11) but increases with tube hematocrit (𝐻₀) (Figure 12) and tube radius “𝑎.” It can be observed that mean hematocrit (𝐻_𝑚) increases with tube radius (Figures 10–13). The influence of magnetic parameter (𝑀_𝑐) on mean hematocrit (𝐻_𝑚) is very insignificant (Figure 13).

4. Conclusions

A two-fluid model has been proposed to describe flow through porous medium in small diameter tubes with Jeffrey fluid in the core region and Newtonian fluid in the peripheral region in the presence of a magnetic field. It is found that the effective viscosity increases with Jeffrey parameter, core magnetic parameter, Darcy number, tube hematocrit, and tube radius. Further, the core hematocrit decreases with Jeffrey parameter, Darcy number, tube hematocrit, and tube radius and the mean hematocrit increases with Jeffrey parameter, Darcy number, tube hematocrit, and tube radius. The influence of magnetic parameter on core hematocrit and mean hematocrit is very insignificant.

Nomenclature

𝑎:	Tube radius
𝑏:	Radius of the core region
𝑑:	Dimensionless core radius = 1−(𝜖/𝑎)
𝜖:	Peripheral region thickness =𝑎−𝑏
𝜉:	Dimensionless radial direction 𝑟/𝑎
𝜇′:	Ratio of viscosity =𝜇𝑝/𝜇𝑐
𝜇𝑝:	Peripheral region viscosity
𝜆1:	Jeffrey parameter
Da:	Darcy number = 𝑘0/𝑎2
𝜎:	Electrical conductivity
𝑀𝑝:	Magnetic parameter in peripheral region = 𝜎/𝜇𝑝𝐵0𝑎
𝑀𝑐:	Magnetic parameter in core region = 𝜎/𝜇𝑐𝐵0𝑎
𝑟:	Radial direction
𝑧:	Axial direction
𝑣1:	Velocity in peripheral region
𝑣2:	Velocity in core region
𝑝:	Pressure
𝑃:	Constant pressure gradient = −(𝜕𝑝/𝜕𝑧)
𝜇𝑐:	Core region viscosity
𝜌:	Density
𝑘0:	Permeability of porous medium
𝐵0:	Uniform magnetic field.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank the reviewers for their constructive suggestions which led to definite improvement in the paper.