# Particulate suspension flow induced by peristaltic waves in a non-uniform channel.

Introduction

The transport of fluids by means of a progressive wave of area contraction or expansion along the walls of a distensible duct containing liquid or mixture in both the mechanical and physiological situations, has been the subject or engineering and scientific research for over four decades since the first investigation of Latham (1966). Physiologists term the phenomenon of such transport as peristalsis. Besides, its practical applications involving biomechanical systems such as heart-lung machine, finger and roller pumps, peristaltic pumping has been found to be involved in many biological organs including the vasomotion of small blood vessels (Srivastava and Srivastava, 1984).

Jaffrin and Shapiro (1971) explained the basic principles of peristaltic pumping and brought out clearly the significance of the various parameters governing the flow. The literature on the topic is quite extensive by now and a summery of most of the experimental and theoretical investigations reported up to the year 1983; arranged according to the geometry, the fluid, the Reynolds number, the wave number, the amplitude ratio and the wave shape; has been presented in an excellent article by Srivastava and Srivastava (1984). The important contributions to the subject between the years 1984 and 1994 are well referenced in Srivastava and Saxena (1995). The recent year's investigations include the works of Srivastava and Srivastava (1997), mekheimer et al. (1998). Muthu et al. (2001), Srivastava (2002), Misra and Pandey (2002), Hayat et al. (2002, 2003). Mekheimer (2002, 2003), Hayat et. al. (2004), Misra and Rao (2004), Hayat et al. (2005), Hayat and Ali (2006a,b), Srivastava (2007), Medhavi and Coworkers (2008a,b; 2009a,b), Hayat and coworkers (2008 a,b), Ali and Hayat (2008), etc.

The theoretical study of particle fluid mixture is very useful in understanding a number of diverse physical problems concerned with powder technology, fluidization transport of solid particles by a liquid and liquid slurries in chemical and nuclear processing, and metalized liquid fuel slurries for rocketry. The sedimentation of particles in a liquid is of interest in many chemical engineering processes, in medicine where erythrocytes sedimentation has become a standard clinical test, and in oceanography as well as other fields. Recently, interest is developing in applying the particulate suspension theory to physiological flows as it provides improved understanding of the subjects such as diffusion of protein, the rheology of blood, the swimming of microorganism, the particle deposition on respiratory tract, etc. A good number of investigations on the topic have been referenced in Srivastava and Srivastava (1989).

Peristaltic transport of a particle-fluid mixture has been studied by Hung and Brown (1976), Srivastava and Srivastava (1989, 1997), Mekheimer et al. (1998), Srivastava (2002), Medhavi and Singh (2008b, 2009a), and a few others. Barring a few (Gupta and Sechadvi, 1976; Srivastava and coworkers 1982, 1983, 1985, 1988; Mekheimer, 2002; etc.), most of the studies in the literature have been conducted in uniform geometry, whereas it is known that in most of the practical applications, the flow geometry is found to be non-uniform. With increasing interest in particulate suspension flow due to its direct applications to diverse physical problems, the present investigation is therefore devoted to study the flow of a particle-fluid mixture in a non-uniform channel induced by sinusoidal peristaltic waves. In view of the theoretical model for blood flow proposed in Srivastava and Srivastava (1983) and used extensively used in the literature for blood flow in small vessels, it is believed that the theoretical study presented here may be applied to study the peristaltic induced flow of blood in small vessels with varying cross-section.

Formulation of the Problem

Consider the flow of a particulate suspension through a two-dimensional channel of non-uniform thickness with a sinusoidal wave traveling down its wall. The geometry of the wall surface is described as (fig.1)

H(x,t) = d(x) + b sin 2[pi]/[lambda](x - ct), (1)

[FIGURE 1 OMITTED]

with d(x) = [d.sub.o] + kx, (2)

where d(x) is the half width of the channel at any axial distance x from inlet, [d.sub.o] is the half width of the channel at inlet, k(<<1) is a constant whose magnitude depends on the length of the channel and exit and inlet dimensions, b is the amplitude of the wave [lambda] is the wavelength, c is the wave propagation velocity and t is the time.

The equations governing the mass and the linear momentum for both the fluid and particle phases using a continuum approach are expressed as (Drew, 1979, Srivastava and Srivastava, 1989; Medhavi and Singh, 2008 b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

[partial derivative]/[partial derivative]x (C[u.sub.p]) + [partial derivative]/[partial derivative]y (C[v.sub.p]) = 0, (8)

where [[nabla].sup.2] [[partial derivative].sup.2]/[partial derivative][x.sup.2] + [[partial derivative].sup.2]/[partial derivative][y.sup.2] is two-dimensional Laplacian operator, (x,y) are Cartesian coordinates with x measured in the direction of the wave and y measured in the direction normal to the mean position of the channel walls; ([u.sub.f], [v.sub,f]) denotes the fluid phase and ([u.sub.p], [v.sub.p]) particle phase velocity components along (x,y) directions respectively; [[rho].sub.f], [[rho].sub.p] be the actual densities of the material constituting fluid and particulate phases respectively; (1-C) [[rho].sub.f] is the fluid phase density, C [[rho].sub.f] the particulate phase density; p denotes the pressure and C denotes the volume fraction of the particles; [[mu].sub.s] (C) [[mu].sub.s] is the suspension viscosity and S being the drag coefficient of interaction for the force exerted by one phase on the other. The concentration of the particles is considered small enough so as to neglect particle-particle impacts due to the Brownian motion. The volume fraction density, C of the particle is chosen as constant which is a good approximation for the low concentration of small particles (Batchelor; 1974, 1976).

The expressions for the drag coefficient of interaction, S and the empirical relation for the viscosity of suspension, [[mu].sub.s] for the present study are selected as (Tam, 1969; Charm and Kurland, 1974; Srivastava and Srivastava, 1989)

S = 9/2 [[mu].sub.o]/[a.sub.o.sup.2] [lambda]'(C),

[lambda]'(C) = 4+3[[8C-3[C.sup.2]].sup.1/2+3C/[(2-3).sup.2] (9)

[[mu].sub.s](C) = [[mu].sub.o]/(1-mC),

m = 0.070 exp [2.49C+(1107/T) exp (-1.69C)], (10)

where [[mu].sub.o] is the fluid viscosity (suspending medium) [a.sub.o] is the radius of a particle and T is measured in absolute temperature ([sup.o]K). The viscosity of the suspension expressed by the formula (10) is found to be reasonably accurate up to C=0.6 (i.e., 60% particle concentration).

An introduction of following non-dimensional variables

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

into equations (3)-(8), after dropping primes, yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

[partial derivative]/[partial derivative]x[C[u.sub.p]]+[partial derivative]/[partial derivative]y[C[v.sub.p] = 0 (16)

The Reynolds number, [R.sub.e] is quite small when the wavelength is large, and therefore, the inertial convective acceleration terms may be neglected in comparison to viscous terms (Shapiro, et al. 1969). Under long wavelength approximation of Shapiro et al. (1969), the equations (11)-(16) governing the flow in its non-dimensional form in the laboratory frame of reference reduce to

(1-C) Dp/dx = (1-C) [mu] [d.sup.2][mu,sub,f]/d[y.sup.2] + C S ([u.sub.p] - [u.sub.f]), (17)

C dp/dx + C S ([u.sub.f] - [u.sub.p], (18)

The non-dimensional boundary conditions are

[u.sub.f] = 0, at y = h = H/[d.sub.o] = 1 + k [lambda]x/[d.sub.o] [phi] sin 2[pi] (x-t), (19)

[partial derivative][u.sub.f]/[partial derivative]y = [partial derivative] [u.sub.p]/[partial derivative]y = 0 at y = 0 (20)

with [phi] = b/[d.sub.o]

Analysis

An integration of equations (17) and (18) subject to the boundary conditions (1) and (20), yields the expressions for the velocity profiles, [u.sub.f] and [u.sub.p] as

[u.sub.f] = -1/2(1-C)[mu] dp/dx ([h.sup.2] - [y.sup.2]), (21)

[u.sub/p] = -1/2(1-C) [mu] dp/dx {[h.sup.2] - [y.sup.2] + 2(1- C) [mu]/S}, (22)

The instantaneous volume flow rate q (x, t) is thus calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

or

- dp/dx = 3(1-C)[mu]q(x,t)/[h.sup.3] + [beta]h (24)

with [beta] = 3C (1-C) [mu]/S, a non-dimensional suspension parameter.

The pressure rise [DELTA] [p.sub.L](t) and the friction force at the wall, [F.sub.L](t) in the channel of length L in their non-dimensional form are obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (26)

The use of equation (24) into equations (25) and (26), yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (28)

Setting k = 0 in equations (27) and (28), one obtains the expressions for pressure rise and friction force for a particle-fluid suspension in a uniform channel. In the absence of the particulate phase (i.e., C = 0), the expressions given in equations (27) and (28) reduce to the results obtained in Gupta and Seshadri (1976) for a Newtonian fluid in non-uniform channel.

Also, with k = 0 and C = 0 in equations (27) and (28) one derives the results obtained in Jaffrin and Shapiro (1971) in the laboratory frame of reference.

Numerical Results and Discussion

To discuss the results obtained in the study quantitatively, computer codes were developed for the numerical evaluations of the analytical results obtained in equations (20) and (21) for various values of the parameters at the temperature of 25.5[degrees]C. We assume the form of instantaneous flow rate q (x,t), periodic in (x-t) as (Gupta and Seshadri, 1976; Srivastava and Srivastava, 1988; Mekheimer, 2002)

q(x, t) = Q + [phi] sin 2[pi] (x-t), (22)

where Q is the time average of the flow over one period of the wave. This form of q(x, t) has been assumed in view of the fact that the constant value of q(x, t) gives [DELTA][p.sub.L] (t) always negative and hence there will be no pumping action. Using the form of q(x, t) given in equation (29), we compute the dimensionless pressure rise [DELTA] [p.sub.L] (t) and friction for [F.sub.L] (t) over the channel length for various values of the dimensionless flow average Q, amplitude ratio [phi] and particle concentration, C. The average rise [DELTA][p.sub.L] and the friction force [F.sub.L] are then evaluated by averaging [DELTA][p.sub.L] (t) and [F.sub.L] (t), respectively over one period of wave. Using the parameter values (Srivastava and Srivastava, 1984; Mekheimer, 2002)

[d.sub.o] = 0.01 cm, L = [lambda] = 10 cm, k = 0.5[d.sub.o]/L = 0.0005

the integrals involved in equations (27) and (28) are evaluated numerically. Some of the critical results are displayed graphically in Figs. 2-11.

The pressure rise [DELTA][p.sub.L] (t) increases with particle concentration, C for any given flow rate Q but for a given particle concentration, C the pressure rise decreases with increasing flow rate, Q (Fig. 2). A comparison of Figs. 2 and 3 reveals that the flow characteristic, [DELTA][p.sub.L] (t) assumes much smaller magnitude in non-uniform channel than its corresponding magnitude in the uniform channel. The average pressure rise [DELTA][p.sub.L] versus time average flow rate, Q has been plotted in Fig. 4 in a non-uniform channel which indicates a linear relationship between [DELTA][p.sub.L] and Q and thus the maximum flow rate is achieved at zero pressure rise and maximum pressure occurs at zero flow rate. The flow characteristic, [DELTA][p.sub.L] is found to be indefinitely increasing with increasing amplitude ratio, [phi] for any given flow rate, Q and the particle concentration C (Fig. 5). The magnitude of [DELTA][p.sub.L] assumes a

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

very high asymptotic value at about [phi] = 0.6 in both the uniform and non-uniform channels (Figs. 5 and 6). However, as expected [DELTA][p.sub.L] assumes reasonably smaller value in non-uniform channel than its corresponding magnitude in uniform channel.

The average pressure rise, [DELTA][p.sub.L] versus C has been displayed in Figs. 7 and 8 for non-uniform and uniform channels, respectively. The flow characteristic,

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[DELTA][p.sub.L] increases with C at zero flow rate for any given amplitude ratio, [phi] however, the nature in the variation of [DELTA][p.sub.L] with C is highly influenced with decreasing values of the amplitude ratio, [phi] for any given non-zero value of the average flow rate, Q. It is noticed that for any given flow rate, Q, the flow characteristic [DELTA][p.sub.L] increases with amplitude ratio, [phi] (Figs. 7 and 8).

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

For any particle concentration, C the friction force, [F.sub.L] (t) increases with flow rate, Q but decreases with increasing particle concentration, C for a given flow rate, Q (Fig. 9). The flow characteristic, [F.sub.L] (t) assumes higher magnitude in uniform channel than its corresponding magnitude in non-uniform channel (Figs. 9 and 10). One notices that the average friction force [F.sub.L] increases with the flow rate, Q for any given particle concentration, C and the amplitude ratio, [phi] retaining the linear relationships between the average friction force [F.sub.L] and the flow rate, Q.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

(Fig.11). The inspection of Figs. 2 and 9 reveals that the flow characteristics, [F.sub.L] (t) possesses character opposite to that of the pressure rise, [DELTA][p.sub.L] (t) for any given set of the parameters. The similar conclusion may be drawn in the case of their averaged values, [F.sub.L] and [DELTA][p.sub.L] from Figs. 4 and 11.

[FIGURE 11 OMITTED]

References

[1] Ali, N. and Hayat, T. Effects of an endoscope on peristaltic flow of a micropolar fluid. Mathematical and Computer Modeling 48, 721-733, 2008.

[2] Batchelor, G. K. Diffusion of particles with homodynamic interaction. J. Fluid Mech. 74, 1-29, 1976.

[3] Batchelor, G. K. Transport properties of two-phase materials with random structure. Ann. Rev. Fluid Mech.6, 227-225, 1974.

[4] Charm, S.E. and Kurland, G.S. Blood Flow and Micro Circulation. John Wiley, Newyork, 1974.

[5] Drew, D. A. Stability of Stokes layer of a dusty gas. Phys. Fluids 19, 20812084, 1979.

[6] Gupta, B. B. and Seshadri, V. Peristaltic pumping in non-uniform tubes. J. Biomech. 9, 105-109, 1976.

[7] Hayat, T. and Ali, N. On mechanism of peristaltic flows for power-law fluid. Physica Acta 371, 188-194, 2006a.

[8] Hayat, T. and Ali, N. Peristaltic induced motion of a MHD third grade fluidin a deformable tube. Physica Acta 370, 225-239, 2006b.

[9] Hayat, T., Ahmed, N. and Ali, N. Effects of an endoscope and magnetic field on peristalsis involving Jeffery fluid. Communication in Nonlinear Science and Numerical Simulation 13, 1581-1591, 2008a.

[10] Hayat, T., and Ali, N. Slip effects on peristaltic transport of MHD fluid with variable viscosity. Physics Letters A 372, 1477-1489, 2008b.

[11] Hayat, T., Masood, S and Asghar, S. Peristaltic transport of magnetohydro dynamic Johnson- Segalman fluid. Nonlinear Dynamics 40, 375-385, 2005.

[12] Hayat, T., Wang, Y., Hutter, K., Asghar, S. and Siddiqui, A.M. Peristaltic transport of an Oldroyd-B fluid in a plannar channel. Math. Problems in Engng.4, 347-376, 2004.

[13] Hayat, T., Wang, Y., Siddiqui, A.M. and Hutter, K. Peristaltic motion of a third order fluid in a circular cylindrical tube. Mathematical Models and Methods in Applied Sciences 12, 1671-1706, 2002.

[14] Hayat, T., Wang, Y., Siddiqui, A.M. and Hutter, K. Peristaltic motion of a Segalman fluid in a plannar channel. Math. Problems in Engng.1, 1-23, 2003.

[15] Hung, T.K. and Brown, T. D. Solid particle motion in a two-dimensional peristaltic flows. J. Fluid Mech. 73, 77-96, 1976.

[16] Jaffrin, M. Y. and Shapiro, A. H. Peristaltic pumping. Ann. Rev. Fluid Mech. 3, 13-36, 1971.

[17] Latham, T. W. Fluid Motion in Peristaltic Pump. M.S. Thesis, M.I.T. Cambridge, 1966.

[18] Medhavi, A and Singh, U. K. Peristaltic pumping of a two-layered particulate suspension in a circular cylindrical tube. Int. J. Appl. Engng. Res. 4, 2291-2306, 2009a.

[19] Medhavi, A. and Singh, U. K. A two-layered suspension flow induced by peristaltic waves. Int. J. Fluid Mech. Res. 35, 258-272, 2008b.

[20] Medhavi, A. and Srivastava, R. Peristaltic pumping of a particulate suspension in a catheterized tube of circular cross-section. J. Appl. Fluid Mech., Submitted in Revised form, 2008b.

[21] Medhavi, A. peristaltic pumping of a non-Newtonian fluid. Applic. and Appl. Math. 3, 137-148, 2008a.

[22] Mekheimer, Kh. S. Nonlinear peristaltic transport through a porous medium in an inclined planner channel. Journal of Porous Media 6, 189-201, 2003.

[23] Mekheimer, Kh. S. Peristaltic transport of a couple stress fluid in a uniform and non-uniform channels. Biorheol. 39, 755-765, 2002.

[24] MeKheimer, Kh. S., El-Shehawey, F.E. and Elaw, A.A. Peristaltic motion of a particle-fluid suspension in a planar channel. Int. J. Theo. Phys. 37, 2895-2920, 1998.

[25] Misra, J. C. and Pandey, S. K. Peristaltic transport of blood in small vessels: study of a mathematical model. Computers and Mathematics with Applications 43, 1183-1193, 2002.

[26] Misra, M. and Rao, A. R. Peristaltic transport of a Newtonian fluid in an asymmetric channel. ZAMP 54, 532-550, 2004.

[27] Muthu, P., Kumar, B. V. R. and Chandra, P. Peristaltic motion in a circular cylindrical tubes: effects of wall properties. Ind. J. Pure and Applied Mathematics 32, 1317-1328, 2001.

[28] Shapiro, A. H., Jaffrin, M. Y. and Weinberg, S.L. Peristaltic pumping with long wavelength at low Reynolds number. J. Fluid Mech. 37, 799-825, 1969.

[29] Srivastava, L.M. and Srivastava, V.P. on two phase model of pulsatile blood flow with entrance effects. Biorheol. 20, 761-777, 1983.

[30] Srivastava, L.M. and Srivastava, V.P. Peristaltic transport of a particle-fluid suspension. Trans. ASME J. Biomech. Engng. 111, 157-165, 1989.

[31] Srivastava, L.M. and Srivastava, V.P. Peristaltic transport of a two-layered model of physiological fluid. J. Biomech. 15, 257-265, 1982.

[32] Srivastava, L.M. and Srivastava, V.P. Peristaltic transport of a non-Newtonian fluid. Annals of Biomed. Engng. 13, 137-153, 1985.

[33] Srivastava, L.M. and Srivastava, V.P. Peristaltic transport of a power-law fluid. Rheol. Acta 27, 428-433, 1988.

[34] Srivastava, L.M. and Srivastava, V.P. Peristaltic transport of blood: Casson model II. J. Biomech. 17, 821-829, 1984.

[35] Srivastava, L.M., Srivastava, V.P. and Sinha, S.N. Peristaltic transport of a physiological fluid. Part I, II & III. Biorheol. 20, 153-185, 1983.

[36] Srivastava, V. P. Effects of an inserted endoscope on chyme movement in small intestine. Applications and Appl. Math., 2, 79-91, 2007.

[37] Srivastava, V.P. and Saxena, M. A Two-fluid model of non-Newtonian blood flow induced by peristaltic waves. Rheol. Acta 34, 406-414, 1995

[38] Srivastava, V.P. and Srivastava, L.M. Influence of wall elasticity and Poiseuille flow on peristaltic induced flow of a particle-fluid mixture. Int. J. Engng. Sci. 35, 1359-1361, 1997.

[39] Srivastava, V.P. Particle-fluid suspension flow induced by peristaltic waves in a circular cylindrical tube. Bull. Cal. Math. Soc. 94, 167-184, 2002.

[40] Tam, C.K. W. The drag on a cloud of spherical particles in low Reynolds number flow. J. Fluid Mech. 38, 537-546, 1969.

Amit Medhavi and U. K. Singh

Department of Mechanical Engineering, Kamla Nehru Institute of Technology, Sultanpur-228118, India

The transport of fluids by means of a progressive wave of area contraction or expansion along the walls of a distensible duct containing liquid or mixture in both the mechanical and physiological situations, has been the subject or engineering and scientific research for over four decades since the first investigation of Latham (1966). Physiologists term the phenomenon of such transport as peristalsis. Besides, its practical applications involving biomechanical systems such as heart-lung machine, finger and roller pumps, peristaltic pumping has been found to be involved in many biological organs including the vasomotion of small blood vessels (Srivastava and Srivastava, 1984).

Jaffrin and Shapiro (1971) explained the basic principles of peristaltic pumping and brought out clearly the significance of the various parameters governing the flow. The literature on the topic is quite extensive by now and a summery of most of the experimental and theoretical investigations reported up to the year 1983; arranged according to the geometry, the fluid, the Reynolds number, the wave number, the amplitude ratio and the wave shape; has been presented in an excellent article by Srivastava and Srivastava (1984). The important contributions to the subject between the years 1984 and 1994 are well referenced in Srivastava and Saxena (1995). The recent year's investigations include the works of Srivastava and Srivastava (1997), mekheimer et al. (1998). Muthu et al. (2001), Srivastava (2002), Misra and Pandey (2002), Hayat et al. (2002, 2003). Mekheimer (2002, 2003), Hayat et. al. (2004), Misra and Rao (2004), Hayat et al. (2005), Hayat and Ali (2006a,b), Srivastava (2007), Medhavi and Coworkers (2008a,b; 2009a,b), Hayat and coworkers (2008 a,b), Ali and Hayat (2008), etc.

The theoretical study of particle fluid mixture is very useful in understanding a number of diverse physical problems concerned with powder technology, fluidization transport of solid particles by a liquid and liquid slurries in chemical and nuclear processing, and metalized liquid fuel slurries for rocketry. The sedimentation of particles in a liquid is of interest in many chemical engineering processes, in medicine where erythrocytes sedimentation has become a standard clinical test, and in oceanography as well as other fields. Recently, interest is developing in applying the particulate suspension theory to physiological flows as it provides improved understanding of the subjects such as diffusion of protein, the rheology of blood, the swimming of microorganism, the particle deposition on respiratory tract, etc. A good number of investigations on the topic have been referenced in Srivastava and Srivastava (1989).

Peristaltic transport of a particle-fluid mixture has been studied by Hung and Brown (1976), Srivastava and Srivastava (1989, 1997), Mekheimer et al. (1998), Srivastava (2002), Medhavi and Singh (2008b, 2009a), and a few others. Barring a few (Gupta and Sechadvi, 1976; Srivastava and coworkers 1982, 1983, 1985, 1988; Mekheimer, 2002; etc.), most of the studies in the literature have been conducted in uniform geometry, whereas it is known that in most of the practical applications, the flow geometry is found to be non-uniform. With increasing interest in particulate suspension flow due to its direct applications to diverse physical problems, the present investigation is therefore devoted to study the flow of a particle-fluid mixture in a non-uniform channel induced by sinusoidal peristaltic waves. In view of the theoretical model for blood flow proposed in Srivastava and Srivastava (1983) and used extensively used in the literature for blood flow in small vessels, it is believed that the theoretical study presented here may be applied to study the peristaltic induced flow of blood in small vessels with varying cross-section.

Formulation of the Problem

Consider the flow of a particulate suspension through a two-dimensional channel of non-uniform thickness with a sinusoidal wave traveling down its wall. The geometry of the wall surface is described as (fig.1)

H(x,t) = d(x) + b sin 2[pi]/[lambda](x - ct), (1)

[FIGURE 1 OMITTED]

with d(x) = [d.sub.o] + kx, (2)

where d(x) is the half width of the channel at any axial distance x from inlet, [d.sub.o] is the half width of the channel at inlet, k(<<1) is a constant whose magnitude depends on the length of the channel and exit and inlet dimensions, b is the amplitude of the wave [lambda] is the wavelength, c is the wave propagation velocity and t is the time.

The equations governing the mass and the linear momentum for both the fluid and particle phases using a continuum approach are expressed as (Drew, 1979, Srivastava and Srivastava, 1989; Medhavi and Singh, 2008 b)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

[partial derivative]/[partial derivative]x (C[u.sub.p]) + [partial derivative]/[partial derivative]y (C[v.sub.p]) = 0, (8)

where [[nabla].sup.2] [[partial derivative].sup.2]/[partial derivative][x.sup.2] + [[partial derivative].sup.2]/[partial derivative][y.sup.2] is two-dimensional Laplacian operator, (x,y) are Cartesian coordinates with x measured in the direction of the wave and y measured in the direction normal to the mean position of the channel walls; ([u.sub.f], [v.sub,f]) denotes the fluid phase and ([u.sub.p], [v.sub.p]) particle phase velocity components along (x,y) directions respectively; [[rho].sub.f], [[rho].sub.p] be the actual densities of the material constituting fluid and particulate phases respectively; (1-C) [[rho].sub.f] is the fluid phase density, C [[rho].sub.f] the particulate phase density; p denotes the pressure and C denotes the volume fraction of the particles; [[mu].sub.s] (C) [[mu].sub.s] is the suspension viscosity and S being the drag coefficient of interaction for the force exerted by one phase on the other. The concentration of the particles is considered small enough so as to neglect particle-particle impacts due to the Brownian motion. The volume fraction density, C of the particle is chosen as constant which is a good approximation for the low concentration of small particles (Batchelor; 1974, 1976).

The expressions for the drag coefficient of interaction, S and the empirical relation for the viscosity of suspension, [[mu].sub.s] for the present study are selected as (Tam, 1969; Charm and Kurland, 1974; Srivastava and Srivastava, 1989)

S = 9/2 [[mu].sub.o]/[a.sub.o.sup.2] [lambda]'(C),

[lambda]'(C) = 4+3[[8C-3[C.sup.2]].sup.1/2+3C/[(2-3).sup.2] (9)

[[mu].sub.s](C) = [[mu].sub.o]/(1-mC),

m = 0.070 exp [2.49C+(1107/T) exp (-1.69C)], (10)

where [[mu].sub.o] is the fluid viscosity (suspending medium) [a.sub.o] is the radius of a particle and T is measured in absolute temperature ([sup.o]K). The viscosity of the suspension expressed by the formula (10) is found to be reasonably accurate up to C=0.6 (i.e., 60% particle concentration).

An introduction of following non-dimensional variables

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

into equations (3)-(8), after dropping primes, yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

[partial derivative]/[partial derivative]x[C[u.sub.p]]+[partial derivative]/[partial derivative]y[C[v.sub.p] = 0 (16)

The Reynolds number, [R.sub.e] is quite small when the wavelength is large, and therefore, the inertial convective acceleration terms may be neglected in comparison to viscous terms (Shapiro, et al. 1969). Under long wavelength approximation of Shapiro et al. (1969), the equations (11)-(16) governing the flow in its non-dimensional form in the laboratory frame of reference reduce to

(1-C) Dp/dx = (1-C) [mu] [d.sup.2][mu,sub,f]/d[y.sup.2] + C S ([u.sub.p] - [u.sub.f]), (17)

C dp/dx + C S ([u.sub.f] - [u.sub.p], (18)

The non-dimensional boundary conditions are

[u.sub.f] = 0, at y = h = H/[d.sub.o] = 1 + k [lambda]x/[d.sub.o] [phi] sin 2[pi] (x-t), (19)

[partial derivative][u.sub.f]/[partial derivative]y = [partial derivative] [u.sub.p]/[partial derivative]y = 0 at y = 0 (20)

with [phi] = b/[d.sub.o]

Analysis

An integration of equations (17) and (18) subject to the boundary conditions (1) and (20), yields the expressions for the velocity profiles, [u.sub.f] and [u.sub.p] as

[u.sub.f] = -1/2(1-C)[mu] dp/dx ([h.sup.2] - [y.sup.2]), (21)

[u.sub/p] = -1/2(1-C) [mu] dp/dx {[h.sup.2] - [y.sup.2] + 2(1- C) [mu]/S}, (22)

The instantaneous volume flow rate q (x, t) is thus calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)

or

- dp/dx = 3(1-C)[mu]q(x,t)/[h.sup.3] + [beta]h (24)

with [beta] = 3C (1-C) [mu]/S, a non-dimensional suspension parameter.

The pressure rise [DELTA] [p.sub.L](t) and the friction force at the wall, [F.sub.L](t) in the channel of length L in their non-dimensional form are obtained as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (26)

The use of equation (24) into equations (25) and (26), yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (28)

Setting k = 0 in equations (27) and (28), one obtains the expressions for pressure rise and friction force for a particle-fluid suspension in a uniform channel. In the absence of the particulate phase (i.e., C = 0), the expressions given in equations (27) and (28) reduce to the results obtained in Gupta and Seshadri (1976) for a Newtonian fluid in non-uniform channel.

Also, with k = 0 and C = 0 in equations (27) and (28) one derives the results obtained in Jaffrin and Shapiro (1971) in the laboratory frame of reference.

Numerical Results and Discussion

To discuss the results obtained in the study quantitatively, computer codes were developed for the numerical evaluations of the analytical results obtained in equations (20) and (21) for various values of the parameters at the temperature of 25.5[degrees]C. We assume the form of instantaneous flow rate q (x,t), periodic in (x-t) as (Gupta and Seshadri, 1976; Srivastava and Srivastava, 1988; Mekheimer, 2002)

q(x, t) = Q + [phi] sin 2[pi] (x-t), (22)

where Q is the time average of the flow over one period of the wave. This form of q(x, t) has been assumed in view of the fact that the constant value of q(x, t) gives [DELTA][p.sub.L] (t) always negative and hence there will be no pumping action. Using the form of q(x, t) given in equation (29), we compute the dimensionless pressure rise [DELTA] [p.sub.L] (t) and friction for [F.sub.L] (t) over the channel length for various values of the dimensionless flow average Q, amplitude ratio [phi] and particle concentration, C. The average rise [DELTA][p.sub.L] and the friction force [F.sub.L] are then evaluated by averaging [DELTA][p.sub.L] (t) and [F.sub.L] (t), respectively over one period of wave. Using the parameter values (Srivastava and Srivastava, 1984; Mekheimer, 2002)

[d.sub.o] = 0.01 cm, L = [lambda] = 10 cm, k = 0.5[d.sub.o]/L = 0.0005

the integrals involved in equations (27) and (28) are evaluated numerically. Some of the critical results are displayed graphically in Figs. 2-11.

The pressure rise [DELTA][p.sub.L] (t) increases with particle concentration, C for any given flow rate Q but for a given particle concentration, C the pressure rise decreases with increasing flow rate, Q (Fig. 2). A comparison of Figs. 2 and 3 reveals that the flow characteristic, [DELTA][p.sub.L] (t) assumes much smaller magnitude in non-uniform channel than its corresponding magnitude in the uniform channel. The average pressure rise [DELTA][p.sub.L] versus time average flow rate, Q has been plotted in Fig. 4 in a non-uniform channel which indicates a linear relationship between [DELTA][p.sub.L] and Q and thus the maximum flow rate is achieved at zero pressure rise and maximum pressure occurs at zero flow rate. The flow characteristic, [DELTA][p.sub.L] is found to be indefinitely increasing with increasing amplitude ratio, [phi] for any given flow rate, Q and the particle concentration C (Fig. 5). The magnitude of [DELTA][p.sub.L] assumes a

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

very high asymptotic value at about [phi] = 0.6 in both the uniform and non-uniform channels (Figs. 5 and 6). However, as expected [DELTA][p.sub.L] assumes reasonably smaller value in non-uniform channel than its corresponding magnitude in uniform channel.

The average pressure rise, [DELTA][p.sub.L] versus C has been displayed in Figs. 7 and 8 for non-uniform and uniform channels, respectively. The flow characteristic,

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[DELTA][p.sub.L] increases with C at zero flow rate for any given amplitude ratio, [phi] however, the nature in the variation of [DELTA][p.sub.L] with C is highly influenced with decreasing values of the amplitude ratio, [phi] for any given non-zero value of the average flow rate, Q. It is noticed that for any given flow rate, Q, the flow characteristic [DELTA][p.sub.L] increases with amplitude ratio, [phi] (Figs. 7 and 8).

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

For any particle concentration, C the friction force, [F.sub.L] (t) increases with flow rate, Q but decreases with increasing particle concentration, C for a given flow rate, Q (Fig. 9). The flow characteristic, [F.sub.L] (t) assumes higher magnitude in uniform channel than its corresponding magnitude in non-uniform channel (Figs. 9 and 10). One notices that the average friction force [F.sub.L] increases with the flow rate, Q for any given particle concentration, C and the amplitude ratio, [phi] retaining the linear relationships between the average friction force [F.sub.L] and the flow rate, Q.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

(Fig.11). The inspection of Figs. 2 and 9 reveals that the flow characteristics, [F.sub.L] (t) possesses character opposite to that of the pressure rise, [DELTA][p.sub.L] (t) for any given set of the parameters. The similar conclusion may be drawn in the case of their averaged values, [F.sub.L] and [DELTA][p.sub.L] from Figs. 4 and 11.

[FIGURE 11 OMITTED]

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Amit Medhavi and U. K. Singh

Department of Mechanical Engineering, Kamla Nehru Institute of Technology, Sultanpur-228118, India

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Author: | Medhavi, Amit; Singh, U.K. |
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Publication: | International Journal of Applied Engineering Research |

Article Type: | Report |

Date: | Nov 1, 2009 |

Words: | 3366 |

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