Inter national Journal of Pure and Applied Mathematics Volume 113 No. 11 2017, 65 74 ISSN: 1311-8080 printed version; ISSN: 1314-3395 on-line version url: http://www.ijpam.eu ijpam.eu Effects of Magnetic Field and Slip on a Two-Fluid Model for Couple Stress Fluid Flow through a Porous Medium Santhosh Nallapu 1, G. Radhakrishnamacharya 2 and G. Ravi Kiran 3 1 Department of Mathematics, SRM University, Kattankulathur, Kancheepuram, Chennai-603203 2 Department of Mathematics, National Institute of Technology, Warangal, Telangana-506004 3 Department of Mathematics,GITAM University, Bengaluru, Karnataka-562163 princenallapu@gmail.com, grk.nitw@yahoo.com, ravikiran.wgl@gmail.com February 10, 2017 Abstract A two-fluid mathematical model for a couple stress fluid flow in a porous medium has been studied in the presence of a magnetic field through a narrow channel under the influence of slip condition. It is assumed that core region contains couple stress fluid and Newtonian fluid in the peripheral region. It is found that the effective viscosity decreases with couple stress parameter and Darcy number but increases with core magnetic parameter slip at the wall and hematocrit. It is noticed that the effective viscosity increases with channel height. Further, the flow exhibits the anomalous Fahraeus-Lindqvist effect. AMS Subject Classification: 76Z05, 76A05, 92C10, 76S05 Key Words: Two-layered model, Non-Newtonian fluid, Effective viscosity, Fahraeus- Lindqvist effect, Darcy number, Slip. 1 Introduction Microcirculation is the study of flow in small blood vessels, particularly in the capillaries, which range in diameter from 20 µ micron to 500 µ in different species. In physiology, the most important functions of the circulation of blood through ijpam.eu 65 2017
capillaries are to supply nutrients to every living cell of the organism and also to remove various waste products from every cell. Further, the flow of blood through smaller diameter blood vessels is accompanied by anomalous effects. One such effect is a Fahraeus-Lindqvist effect, where the apparent viscosity of blood decreases with tube diameter. This effect has been confirmed by several investigators Fahraeus and Lindqvist[1]; Seshadri and Jaffrin[2]. To explain the Fahraeus-Lindqvist effect, Haynes[3] has considered a two-phase model, where both regions are occupied by Newtonian fluids with different viscosities. Cokelet[4] have experimentally shown that for blood flowing through small vessels, there is a peripheral layer which contains Newtonian fluid and core region which contains a non-newtonian fluid. Chaturani and Upadhya[5] and Srivastava[6] have studied two-fluid model for blood flow through small diameter tubes in which the core region consists of Non-newtonian fluids and Newtonian fluid in the peripheral region. Couple-stress fluid is a special type of a non-newtonian fluid, whose particle size is taken into account. The theory of couple-stress fluids was first introduced by Stokes[7] and represents the simplest generalization of classical theory which allows for polar effects such as the presence of couple-stresses and body couples. Pal et al.[8] developed a couple-stress fluid model of blood flow in microcirculation. Tripathi[9] investigated a theoretical study of the peristaltic hemodynamic flow of couple-stress fluids through a porous medium under the influence of wall slip condition. Further, several other authors studied couple-stress fluid flows under different conditions Mekheimer[10], Sankad and Radhakrishnamacharya[11]. Recently, Santhosh et al.[12] have studied a two-fluid model for the flow of Jeffrey fluid in tubes of small diameters. However, the flow through narrow channel with slip boundary condition has not been studied. In this paper, a mathematical model is presented to study the effects of slip boundary condition on a two-fluid model of couple stress fluid flow through a porous medium in narrow channel in the presence of a magnetic field. Following the analysis of Chaturani and Upadhya[5], the linearised equations of motion have been solved and analytical solution has been obtained. The analytical expressions for velocity, flow rate and effective viscosity have been derived and the effects of relevant parameters on these flow variables have been studied. 2 Formulation of The Problem Let us consider the steady, laminar flow of an incompressible and electrically conducting couple stress fluid through a porous medium in uniform channel of height 2a under slip boundary condition. It is assumed that flow is represented by a twofluid model in which core region of height 2b contains couple stress fluid surrounded by a peripheral layer of thickness ɛa b = ɛ which is occupied by Newtonian fluid Fig. 1. Cartesian coordinate system x, y is chosen so that the x-axis coincides with the center line of the channel and the y-axis normal to it. Further, it is assumed ijpam.eu 66 2017
that a uniform magnetic field is applied normal to the walls of the channel. Figure 1: Geometry of the problem The appropriate governing equations by neglecting body forces and body couples for the present problem are Sankad and Radhakrishnamacharya[11] Core region 0 y b couple-stress fluid Dq1 ρ Dt q 1 = 0 1 = p + µ c 2 q 1 η 4 q 1 + j B µ c k 0 q 1 2 Peripheral Region b y a Newtonian fluid Dq2 ρ Dt q 2 = 0 3 = p + µ p 2 q 2 + j B µ p k 0 q 2 4 where D is the material derivative, q Dt 1 = u 1, v 1, 0 and q 2 = u 2, v 2, 0 are the velocity vectors of the fluid in the core region and peripheral region respectively, µ p and µ c are the viscosities of the fluid in peripheral region and core region respectively, ρ is the density, p is the pressure, t is the time, η is the couple-stress fluid viscosity, k 0 is the permeability constant of the medium, j is the current density, B = B 0 + B 1 is the total magnetic field, B 1 is the induced magnetic field 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 m j, E = B, j = σe + q E t ijpam.eu 67 2017
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. Hence, the force j B simplifies to j B = σb 2 0u Using the above assumptions and considering the flow to be steady and onedimensional, the governing equations 1 4 get reduced to Core region 0 y b couple-stress fluid 2 u 1 µ c y 4 u 1 2 η y 4 σb2 0u 1 µ c u 1 p = 0 for 0 y b 5 k 0 x Peripheral Region b y a Newtonian fluid µ p 2 u 2 y 2 σb2 0u 2 µ p u 2 p = 0 for b y a 6 k 0 x The boundary and interface conditions for the problem are u 1 = 0 at y = 0 7 y Da u 2 u 2 = a at y = ±a 8 α y 2 u 1 = 0 at y = ±b 9 y 2 u 1 = u 2 and τ 1 = τ 2 at y = ±b 10 Here 7 is the regularity condition, 8 is Saffman s slip boundary condition Saffman[13], 9 indicates the vanishing of couple stress at the interface and 10 expresses the continuity of velocities and stresses at the interface. Further, Da is the Darcy number and α is the slip at the wall, τ 1 and τ 2 are the shear stresses in the core and peripheral layers, respectively. Solving equations 5 and 6, under conditions 7 10, the expressions for velocities u 1 and u 2 can be obtained as u 1 y = 1 dp 1 + T1 coshα M 1 dx 1y T2 coshα2y for 0 y b 11 u 2 y = 1 dp 1 cosh M 2 y for b y a 12 M 2 dx T3 ijpam.eu 68 2017
where α1 = µ c 1 + 1 4 η M 2η µ 2 1, M 1 = σb0+ 2 µ c, α2 = µ c 1 1 4 η M c k 0 2η µ 2 1, c T 2 = T 1 = cosh M 2 bα 2 2 T 3 coshα 1bα 1 2 α 2 2, M 2 = σb 2 0 + µ p cosh M 2 bα 1 2 T 3 coshα 2bα 1 2 α 2 2, T 3 = cosh M 2 a 1+a k 0 Da α tanh M 2 a M 2, The flow flux in the core region and peripheral region, denoted by Q c and Q p, are given by and and Q c = 2 Q p = 2 b 0 a b 13 u 1 ydy 14 u 2 ydy 15 Substituting for u 1 and u 2 from 11 and 12 into 14 and 15, we get α 1 = α 1a = α 2 = α 2a = Q c = 2a3 P d + T 1 sinhα 1 d T 2 sinhα 2 d µ c M1c 2 α 1 α 2 Q p = 2a3 P 1 d sinhm 2p + sinhm 2pd µ p M2p 2 T 3 M 2p T 3 M 2p m 2 2 16 17 1 + 1 4 m M 2 1c, M 1c = Mc 2 + 1 Da, M c 2 = σ B µ 0a 2 2, Da = k 0 c a, 2 m 2 1 2 1 4 m M 2 1c, m = µc η a, d = b a, P = dp dx, coshm 2p dα2 2 T 1 = T 3 coshα 1 dα1 2 α2, T coshm 2p dα1 2 2 2 = T 3 coshα 2 dα1 2 α2, 2 Da T 3 = coshm 2p 1 + α tanhm 2pM 2p, M 2p = Mp 2 + 1 σ Da, M p = B 0 a µ p where m is the couple-stress parameter and d is the non-dimensional core channel height. Thus, the total flow flux is given by ijpam.eu 69 2017
Using 16 and 17 in 18, we get Q = 2a3 P µ p M2p 2 where 1 d sinhm 2p T 3 M 2p Q = Q c + Q p 18 + sinhm 2pd +µ β d+ T 1 sinhα 1 d T 3 M 2p α 2 α 1 T 2 sinhα 2 d µ = µ p µ c, β = M 2 2p M 2 1c Comparing equation 19 with flow flux for Poiseuille s flow, we get the effective viscosity as 19 µ eff = 3 1 d sinhm 2p T 3 M 2p + sinhm 2pd T 3 M 2p µ p M 2 2p + µ β 20 d + T 1 sinhα 1 d α 1 T 2 sinhα 2 d α 2 3 RESULTS AND DISCUSSION In order to discuss the effects of various parameters, the effective viscosity µ eff and mean hematocrit H m have been numerically evaluated and the results are graphically presented in Figs. 2-6. The effects of various parameters on the effective viscosity µ eff are shown in Figs. 2-6. It is observed from Fig. 2 and Fig. 3, that the effective viscosity µ eff decreases with increase in magnitude of couple stress parameter m and Darcy number Da. It is found that increasing core magnetic parameter M c Fig. 4 and hematocrit H 0 Fig. 5 has an increasing effect on effective viscosity µ eff. It is also noticed from Fig. 6 that effective viscosity µ eff increases with increasing slip at the wall α. That is, the more the fluid slips at the wall, the more its effective viscosity is affected. Quantitative descriptions of these effects and dependence of blood viscosity on hematocrit at different channel heights a are required for the development of hydrodynamic models of blood flow through microcirculation. The values of effective viscosity computed from the present model are in good agreement, within the acceptable range, with the corresponding values of the effective viscosity obtained in the theoretical models of Haynes[3], Chaturani and Upadhya[5]. Further, the effective viscosity µ eff increases with channel height afigs. 2-6, but the increase in effective viscosity µ eff with channel height a is not very significant for higher values of Darcy number Da Fig. 3. Further, It can be noticed that the present model exhibits the anomalous effect observed in microcirculation, namely, "Fahraeus-Lindqvist Effect". ijpam.eu 70 2017
Figure 2: Effect of couple stress parameter m on µ eff M p = 1, M c = 0.5, α = 0.2, Da = 0.8 and H 0 = 40% Figure 3: Effect of Darcy number Da on µ eff M p = 1, m = 5, α = 0.2, M c = 0.5 and H 0 = 40% Figure 4: Effect of core magnetic parameter M c on µ eff M p = 1, m = 5, α = 0.2, Da = 0.8 and H 0 = 40% Figure 5: Effect of hematocrit H 0 on µ eff M p = 1, m = 5, M c = 0.5, Da = 0.8 and α = 0.2 Figure 6: Effect of slip at the wall α on µ eff M p = 1, m = 5, M c = 0.5, Da = 0.8 and H 0 = 40% ijpam.eu 71 2017
4 Conclusion Effects of the induced magnetic field and slip condition on a couple stress fluid flow in a porous medium through a narrow channel is studied. The exact solutions of velocity, flow rate, effective viscosity, core hematocrit and mean hematocrit are obtained. Graphical results are presented for the various relevant parameters, effective viscosity and mean hematocrit. It is found that the mean hematocrit decreases with couple stress parameter, core magnetic parameter and slip at the wall. It is noticed that the effective viscosity decreases with couple stress parameter and Darcy number but increases with hematocrit and channel height. It is observed that the more the fluid slips at the wall, the more its effective viscosity is affected. References [1] R. Fahraeus and T. Lindqvist, Viscosity of Blood in Narrow Capillary Tubes, Am. J. Phys. 96 1931 562 568. [2] V. Seshadri and N. Y. Jaffrin, Anomalous Effects in Blood Flow through Narrow Tubes, Inserm-Euromech 92 71 1977 265 282. [3] Haynes, R. H., Physical Basis of the Dependence of Blood Viscosity on Tube Radius, Am. J. Physiol. 198, 1960 1193 1200. [4] G. R. Cokelet, The Rheology of Human Blood. in: Y.C. Fung Ed. Biomechancis. Prentice-Hall, Englewood Cliffs, NJ, 1972 63âĂŞ-103. [5] P. Chaturani and V.S. Upadhya, A Two-Fluid Model for Blood Flow through Small diameter Tubes, Biorheology 18 1981 245 253. [6] V.P. Srivastava, Two phase model of blood flow through stenosed tubes in the presence of a peripheral layer: applications J. Biomechanics 2910 1996 1377 1382. [7] Stokes, V. K., Couple Stresses in Fluids, Phys. Fluids. 9 1966 1709 1715. [8] D. Pal, N. Rudraiah and R. Devanathan, A Couple Stress Model of Blood Flow in the Microcirculation, Bull. Math. Biol 50 1988 329 349. [9] D. Tripathi, Peristaltic Hemodynamic Flow of Couple-Stress Fluids Through a Porous Medium with Slip Effect, Transp Porous Med. 92 2012 559âĂŞ572. [10] Kh. S. Mekheimer, Effect of the induced magnetic field on peristaltic flow of a couple stress fluid, Physics Letters A 372 2008 4271âĂŞ4278. [11] G. C. Sankad and G. Radhakrishnamacharya, Effect of Magnetic field on the Peristaltic Transport of Couple Stress Fluid in a Channel with Wall Properties, Int. J. Biomath. 4 2011 365 378. ijpam.eu 72 2017
[12] N. Santhosh, G. Radhakrishnamacharya and Ali. J. Chamkha, Flow of a Jeffrey Fluid Through a Porous Medium in Narrow Tubes, J. Porous Media 181 2015 71 78. [13] P. G. Saffman, On the Boundary Conditions at the Surface of a Porous Medium, Stud. Appl. Math. 1 2015 93 101. ijpam.eu 73 2017
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