Effects of MHD and heat transfer on an oscillatory flow of Jeffrey fluid in a tube.

Kavitha, K. ; Prasad, K. Ramakrishna

Introduction

The study of oscillatory flow of a viscous fluid in cylindrical tubes has received the attention of many researchers as they play a significant role in understanding the important physiological problem, namely the blood flow in arteriosclerotic blood vessel. Womersley [21] have investigated the oscillating flow of thin walled elastic tube. Detailed measurements of the oscillating velocity profiles were made by Linford and Ryan [12]. Unsteady and oscillatory flow of viscous fluids in locally constricted, rigid, axisymmetric tubes at low Reynolds number has been studied by Ramachandra Rao and Devanathan [14], Hall [8] and Schneck and Ostrach [17]. Haldar [7] have considered the oscillatory flow of a blood through an artery with a mild constriction. Several other workers, Misra and Singh [13], Ogulu and Alabraba [14], Tay and Ogulu [18] and Elshahed [6], to mention but a few, have in one way or the other modeled and studied the flow of blood through a rigid tube under the influence of pulsatile pressure gradient.

Many researchers have studied blood flow in the artery by considering blood as either Newtonian or non-Newtonian fluids, since blood is a suspension of red cells in plasma; it behaves as a non-Newtonian fluid at low shear rate. Barnes et al. [1] have studied the behavior of no-Newtonian fluid flow through a straight rigid tube of circular cross section under the action of sinusoidally oscillating pressure gradient about a non-zero mean. Chaturani and Upadhya [4] have developed a method for the study of the pulsatile flow of couple stress fluid through circular tubes. The Poiseuille flow of couple stress fluid has been critically examined by Chaturani and Rathod [5]. Moreover, the Jeffrey model is relatively simpler linear model using time derivatives instead of convected derivatives for example the Oldroyd-B model does, it represents rheology different from the Newtonian (Bird et al. [3]). None of these studies considered the effect of body temperature on the blood flow -prominent during deep heat muscle treatment.

The magnetohydrodynamic (MHD) flow between parallel plates is a classical problem that occurs in MHD power generators, MHD pumps, accelerators, aerodynamic heating, electrostatic precipitation, polymer technology, petroleum industry, purification of crude oil and fluid droplets and sprays. Especially the flow of non-Newtonian fluids in channels is encountered in various engineering applications. For example, injection molding of plastic parts involves the flow of polymers inside channels. During the last few years the industrial importance of non-Newtonian fluids is widely known. Such fluids in the presence of a magnetic field have applications in the electromagnetic propulsion, the flow of nuclear fuel slurries and the flows of liquid state metals and alloys. Sarparkaya [16] have presented the first study for MHD Bingham plastic and power law fluids. Effect of magnetic field on pulsatile flow of blood in a porous channel was investigated by Bhuyan and Hazarika [2]. Hayat et al. [9] have studied the Hall effects on the unsteady hydromagnetic oscillatory flow of a second grade fluid in a channel. Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field in a channel was investigated by Hayat et al. [10]. Hayat et al. [11] have studied the influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet. Vasudev et al. [19] have investigated the influence of magnetic field and heat transfer on peristaltic flow of Jeffrey fluid through a porous medium in an asymmetric channel. Vasudev et al. [20] have studied the MHD peristaltic flow of a Newtonian fluid through a porous medium in an asymmetric vertical channel with heat transfer.

In view of these, we studied the effects of magnetic field and heat transfer on oscillatory flow of Jeffrey fluid in a circular tube. The expressions for the velocity field and temperature field are obtained analytically. The effects of various pertinent parameters on the velocity field and temperature field studied in detail with the help of graphs.

Mathematical formulation

We consider an oscillatory flow of a Jeffrey fluid through in a heated uniform cylindrical tube of constant radiusR. A uniform magnetic field [B.sub.0] is applied in the transverse direction to the flow. The wall of the tube is maintained at a temperature [T.sub.w]. We choose the cylindrical coordinates (r,[theta,z)such that r = 0 is the axis of symmetry. The flow is considered as axially symmetric and fully developed. The geometry of the flow is shown in Fig. 1.

[FIGURE 1 OMITTED]

The constitute equation of S for Jeffrey fluid is

S = [mu] / 1 + [[lambda].sub.1] ([??] + [[lambda].sub.2] [??]) (2.1)

where [mu] is the dynamic viscosity, [[lambda].sub.1] is the ratio of relaxation to retardation times, [[lambda].sub.2] is the retardation time, [??] is the shear rate and dots over the quantities denote differentiation with time.

The equations governing the flow are given by

P = [partial derivative] / [partial derivative]t = - [partial derivative]p / [partial derivative]z + [[partial derivative]S.sub.rZ] / [partial derivative]r - [sigma] [B.sup.2.sub.0]w + pg[beta] (T - [T.sub.w]) (2.2)

[pc.sub.p] [partial derivative]T / [partial derivative]T = [k.sub.0] ([[partial derivative].sup.2]T / [[partial derivative].sup.2] + 1[partial derivative]T / r [partial derivative]r) (2.3)

where [rho] is the fluid density, [mu] is the fluid viscosity, p is the pressure, w is the velocity component in z - direction, g is the acceleration due to gravity, [sigma] is the electrical conductivity of the fluid, [beta] coefficient of thermal expansion, T is the temperature, [k.sub.0] is the thermal conductivity and [c.sub.p] is the specific heat at constant pressure.

The appropriate boundary conditions are

w = 0, T = [T.sub.w] at r = R [partial derivative]w / [partial derivative]r = 0, T = [T.sub.[infinity]] at r = 0 (2.4)

Introducing the following non-dimensional variables

[bar.r] = r / R, [bar.z] = z / R, [bar.t] = [bar.t] = [w.sub.0]t / R, [[alpha].sup.2] = [rho][R.sup.2] / [mu], [lambda] = R / [w.sub.0], [bar.w] = w / [w.sub.0]

[bar.p] = p - [p.sub.w] / [mu], [theta] = T - [T.sub.w] / [T.sub.w] - [T.sub.[infinity]] Pr = [[micro]c.sub.p] / [k.sub.0], Re = p[w.sub.0]R / [mu]

into the Eqs. (2.2) - (2.4), we get (after dropping bars)

Re [partial derivative]w / [partial derivative]t = -[lambda] dp / dz + 1 / 1 + [[lambda].sub.1] [[[partial derivative].sup.2]w / [[partial derivative].sup.2]r + 1 / r [partial derivative]w / [partial derivative]r] - [M.sup.2]w + Gr / Re [theta] (2.5)

PrRe [partial derivative][theta] / [partial derivative]t = [[partial derivative].sup.2][theta] / [[partial derivative]r.sup.2] + 1 [partial derivative][theta] / [partial derivative]r (2.6)

where Pr is the Prandtl number, M = [RB.sub.0] [square root of ([sigma] / [mu])] is the Hartmann number, Gr = [rho]g[beta][R.sup.2] ([T.sub.w] - [T.sub.[infinity]]) / [w.sub.0][mu] is the Grashoff number and Re is the Reynolds number.

The corresponding non-dimensional boundary conditions are

w = 0, T = 1 at r = 1

[partial derivative]w / [partial derivative]r = 0, T = 0 at r = 0 (2.7)

Solution

It is fairly unanimous that, the pumping action of the heart results in a pulsatile blood flow so that we can represent the pressure gradient (pressure in the left ventricle) as

- dp / dz = [p.sub.0][e.sup.i[omega]t] (3.1)

where and flow variables expresses as

[theta](y, t) = [[theta].sub.0] (r)[e.sup.i[omega]t] (3.2)

w(y, t) = [w.sub.0] (r)[e.sup.i[omega]t] (3.3)

Substituting Eqs. (3.1) - (3.2) into Eqs. (2.5) and (2.6) and solving the resultant equations subject to the boundary conditions in (2.7), we obtain

[[theta].sub.0] = [I.sub.0]([OMEGA]r) / [I.sub.0]([OMEGA]h) (3.4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

here [[OMEGA].sup.2] = i[omega] Pr Re and [[beta].sup.2.sub.1] = (1 / Da + i[omega]Re (1 + [[lambda].sub.1]).

In Eqs. (3.4) and (3.5), [I.sub.0] (x) is the modified Bessel function of first kind of order zero.

Hence the temperature distribution and the axial velocity are given by

[theta] = [I.sub.0]([OMEGA]r) / [I.sub.0] ([OMEGA]H) [e.sup.i[omega]t] (3.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.7)

Discussion of the Results

Fig. 2 depicts the effects of material parameter [[lambda].sub.1] on w for Da = 0.1, p = 1, [omega] = 10, [lambda] = 0.5,Pr = 2, Gr = 1,Re = 1and t = 0.1. It is observed that, the axial velocity w increases at the axis of tube with increasing material parameter [[lambda].sub.1].

In order to see the effects of Hartmann number M on w for [[lambda].sub.1] = 0.3, p = 1, [omega] = 10, [lambda] = 0.5, Pr = 2, Gr = 1,Re = 1 and t = 0.1 we plotted Fig. 3. It is found that, the axial velocity w decreases with an increase in Hartmann number M .

Fig. 4 shows the effects of Prandtl number Pr on w for [[lambda].sub.1] = 0.3, p = 1, [omega] = 10, [lambda] = 0.5, Da = 0.1, Gr = 1,Re = 1and t = 0.1. It is noted that, an increase in the Prandtl number Pr decreases the axial velocity w.

Fig. 5 illustrates the effects of Grashof number Gr on w for [[lambda].sub.1] = 0.3, p = 1, [omega] = 10, [lambda]] = 0.5, Pr = 2, Da = 0.1,Re = 1 and t = 0.1. It is observed that, the axial velocity w increases with an increase in Grashof number Gr .

Fig. 6 shows the effects of Reynolds number Re on w for [[lambda].sub.1] = 0.3, p = 1, [omega] = 10, [lambda] = 0.5,Pr = 2,Da = 0.1,Re = 1and t = 0.1. It is found that, the axial velocity w decreases on increasing Reynolds number Re.

In order to see the effects of [lambda] on w for [[lambda].sub.1] = 0.3, Da = 0.1, Pr = 2, Gr = 1, p = 1, [omega] = 10, Re = 1 and t = 0.1 we plotted Fig. 7. It is observed that, the axial velocity increases with increasing [lambda] .

Fig. 8 shows the effects of [omega] on w for [[lambda].sub.1] = 0.3, Da = 0.1, Pr = 2, Gr = 1, p = 1, [lambda] = 0.5 , Re = 1 and t = 0.1 is shown in Fig. 7. It is observed that, the axial velocity decreases on increasing [omega].

Fig. 9 depicts the effects of Prandtl number Pr on [theta] for [omega] = 10,Re = 1and t = 0.1. It is found that, the temperature 0 decreases with increasing Prandtl number Pr .

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

Conclusions

In this paper, we studied the effects of magnetic field and heat transfer on oscillatory flow of Jeffrey fluid in a circular tube. The expressions for the velocity field and temperature field are obtained analytically. It is observed that, the axial velocity increases with increasing [[lambda].sub.1], Gr and [lambda], while it decreases with increasing M , Pr, Re and [lambda]. The temperature field decreases with increasing Pr .

References

[1] Barnes, H. A., Townsend, P. and Walters, K., 1971, On pulsatile flow of non-Newtonian liquids, Rheol. Acta, 10, pp. 517-527.

[2] Bhuyan, B. C. and Hazarika, G. C., 2001, Effect of magnetic field on pulsatile flow of blood in a porous channel, Bio-Sci Res. Bull. 17(2), pp. 105-112.

[3] Bird, R.B., Stewart, W. E. and Lightfoot, E. N., 1960, Transport phenomena, New York, Wiley.

[4] Chaturani, P. and Upadhya, V. S., 1978, Pulsatile flow of a couple stress fluid through circular tubes with application to blood flow, Biorheology, 15, pp. 193-201.

[5] Chaturani, P. and Rathod, V. P., 1981, A critical study of Poiseuille flow of couple stress fluid with applications to blood flow, Biorheology, 18, pp. 235-244.

[6] Elshahed, M., 2003, Pulsatile flow of blood through a stenosed porous medium under periodic body acceleration, Appl. Mathematics and Computation, 138, pp. 479-488.

[7] Haldar, K., 1987, Oscillatory flow of blood in a stenosed artery, Bulletin of Mathematical Biology, 49, pp. 279-287.

[8] Hall, P., 1974, Unsteady viscous flow in a pipe of slowly varying cross section, J. Fluid Mech. 64, pp. 209-226.

[9] Hayat, T., Wang, Y. and Hutter, K., 2004, Hall effects on the unsteady hydromagnetic oscillatory flow of a second grade fluid, Int. J. Non-Linear Mech., 39, pp. 1027-1037.

[10] Hayat, T., Masood Khan and Ayub, M., 2004, Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field, J. Math. Anal. Appl., 298, pp. 225-244.

[11] Hayat, T., Saif, S. and Abbas, Z., 2008, The influence of heat transfer in an MHD second grade fluid film over an unsteady stretching sheet, Physics Letters A, 372, pp. 5037-5045.

[12] Linford, R.G. and Ryan, N. W., 1965, Pulsatile flow in rigid tube, J. Applied Physiology, 20, pp. 1078-1082.

[13] Misra, J. C. and Singh, S. I., A study on the non-linear flow of blood through arteries, Bulletin of Mathematical Biology, 49(1987), 257-277.

[14] Ogulu, A. and Alabraba, M. A., 1992, Magnetic effect on low Reynolds number flow in heated tube of slowly varying section, Acta Physica Hungarica, 72, pp. 223-234.

[15] Ramachandra Rao, A. and Devanathan, R. Pulsatile flow in tubes of varying cross-sections, ZAMP, 24, pp. 203-213.

[16] Sarpakaya, T., 1961, Flow of non-Newtonian fluids in a magnetic field, AIChE J., 7, pp. 324-328.

[17] Schneck, D. J. and Ostrach, S., 1975, Pulsatile blood flow in a channel of small exponential divergence-I: The linear approximation for low Reynolds number, J. Fluids Engineering, 97, pp. 353-360.

[18] Tay, G. and Ogulu, A. Low Reynolds number flow in a constricted tube under a time-dependent pressure gradient, Journal Fizik Malaysia, 18, pp. 105-112.

[19] Vasudev, C., Rajeswara Rao, U., Subba Reddy, M. V. and Prabhakara Rao, G., 2010, Influence of magnetic field and heat transfer on peristaltic flow of Jeffrey fluid through a porous medium in an asymmetric channel, ARPN Journal of Engineering and Applied Sciences, 5(12), pp. 87-103.

[20] Vasudev, C., Rajeswara Rao, U., Subba Reddy, M. V. and Prabhakara Rao, G., 2011, MHD peristaltic flow of a Newtonian fluid through a porous medium in an asymmetric vertical channel with heat transfer, International Journal of Science and Advanced Technology, 1(3), pp. 1-11.

[21] Womersley, J.R., 1955, Oscillatory motion of a viscous fluid in a thin walled elastic tube-I: The linear approximation for long waves, Phil. Mag., 46, pp. 199-221.

K. Kavitha (1) * and K. Ramakrishna Prasad (2)

(1) Department of Mathematics, Tirumala Engineering College, Hyderabad, A.P., India.

(2) Department of Mathematics, S.V. University, Tirupati-517 502, A.P., India.

* Corresponding Author E-mail: kosurikavitha.m1@gmail.com