Research Article Slip Effects on Peristaltic Transport of a Particle-Fluid Suspension in a Planar Channel

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1 Applied Bionics and Biomechanics Volume 5, Article ID 73574, 4 pages Research Article Slip Effects on Peristaltic Transport of a Particle-Fluid Suspension in a Planar Channel Mohammed H. Kamel, Islam M. Eldesok, Bilal M. Maher, and Ramz M. Abumandour Department of Engineering Mathematics and Phsics, Facult of Engineering, Cairo Universit, Giza, Egpt Department of Basic Engineering Sciences, Facult of Engineering, Menoufia Universit, Shebin El-Kom, Egpt Correspondence should be addressed to Islam M. Eldesok; eldesoki@ahoo.com Received August 4; Revised 6 March 5; Accepted 6 April 5 Academic Editor: Agnès Drochon Copright 5 Mohammed H. Kamel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in an medium, provided the original work is properl cited. Peristaltic pumping induced b a sinusoidal traveling wave in the walls of a two-dimensional channel filled with a viscous incompressible fluid mixed with rigid spherical particles is investigated theoreticall taking the slip effect on the wall into account. A perturbation solution is obtained which satisfies the momentum equations for the case in which amplitude ratio (wave amplitude/channel half width) is small. The analsis has been carried out b dul accounting for the nonlinear convective acceleration terms and the slip condition for the fluid part on the wav wall. The governing equations are developed up to the second order of the amplitude ratio. The zeroth-order terms ield the Poiseuille flow and the first-order terms give the Orr-Sommerfeld equation. The results show that the slip conditions have significant effect within certain range of concentration. The phenomenon of reflux (the mean flow reversal) is discussed under slip conditions. It is found that the critical reflux pressure is lower for the particle-fluid suspension than for the particle-free fluid and is affected b slip condition. A motivation of the present analsis has been the hope that such theor of two-phase flow process under slip condition is ver useful in understanding the role of peristaltic muscular contraction in transporting biofluid behaving like a particle-fluid mixture. Also the theor is important to the engineering applications of pumping solid-fluid mixture b peristalsis.. Introduction Peristalsis is a form of a fluid transport induced b a progressive wave of area contraction or expansion along the walls of a distensible duct containing liquid. In phsiolog, peristaltic mechanism is involved in man biological organs such as ureter, gastrointestinal tract, ducts afferents of the male reproductive tracts, cervical canal, female fallopian tube, lmphatic vessels, and small blood vessels. In addition, peristaltic pumping occurring in man practical applications involving biomechanical sstems such as roller, finger pumps, and heart-lung machine have been fabricated. Since the first investigation of Latham [], several theoretical and experimental studies have been conducted in the past to understand peristaltic action in different situations. The literature on peristalsis is b now quite extensive, and a summar of most of the investigations has been presented in detail b Rath [], L. M. Srivastava and V. P. Srivastava [3 5], Srivastava and Saxena [6], and V. P. Srivastava and L. M. Srivastava [7]. The theoretical stud of the theor of particle-fluid mixture is ver useful in understanding a number of diverse phsical problems concerned with powder technolog, fluidization, transportation of solid particles b a liquid, transportation liquid slurries in chemical and nuclear processing, and metalized liquid fuel slurries for rocketr. The sedimentation of particles in a liquid is of interest in much chemical engineering process, in medicine, where erthrocte sedimentation has become a standard clinical test, and in oceanograph as well as other fields. The particulate theor of blood has recentl become the object of scientific research, Hill and Bedford [8], L. M. Srivastava and V. P. Srivastava [3 5, 9], Trowbridge [], and Oka []. A number of research works on the topic, with and without peristalsis, have been reviewed b L. M. Srivastava and V. P. Srivastava [5]. Applications of the theor of particle-fluid mixture to the microcirculation and erthrocte sedimentation included theworkofwangandskalak[], Bunga and Brenner [3], Skalak et al. [4], and Karino et al. [5].

2 Applied Bionics and Biomechanics λ Wave speed C d o x a η(x, t) = acos π (x ct) λ Figure : Geometr of the problem. Peristaltic transport of solid particle with fluid was first attempted b Hung and Brown [6]. The initiated the stud of the peristaltic transport of solid particles, which included an experimental work on the particle transport in twodimensional vertical channels having various geometries. In this connection also another paper b Brown and Hung [7] and a stud b Takabatake and Aukawa [8] areworth mentioning. Both studies have emploed finite difference technique to solve two-dimensional nonlinear peristaltic flows problem. L. M. Srivastava and V. P. Srivastava [5] studied the peristaltic pumping of a particle-fluid mixture in a two-dimensional channel carried out mathematicall; a perturbation solution is obtained. Mekheimer et al. [9] studied the peristaltic pumping of a particle-fluid suspension in a planar channel. El Miser et al. [] studied the peristaltic motion of an incompressible generalized Newtonian fluid in aplanarchannel. No-slip boundar conditions are convenient idealization of the behavior of viscous fluids near walls. The inadequac of the no-slip condition is quite evident in polmer melts which often exhibit microscopic wall slip. The slip condition plas an important role in shear skin, spurt, and hsteresis effects. The boundar conditions relevant to flowing fluids are ver important in predicting fluid flows in man applications. The fluids that exhibit boundar slip have important technological applications such as in polishing valves of artificial heart and internal cavities []. The slip effects on the peristaltic flow of a non-newtonian Maxwellian fluid have been investigated b Eldesok []. The influence of slip condition on peristaltic transport of a compressible Maxwell fluid through porous medium in a tube has been studied b Chu and Fang [3]. Man recent researches have been made in the subject of slip boundar conditions [4 33]. From the previous studies, there is no an attempt to stud the effect of slip condition on the flow of a particlefluid suspension with peristalsis. The purpose of this paper is to stud the slip effects on the peristaltic pumping of a particle-fluid mixture in a two-dimensional channel. It is an application of the two-dimensional analsis of peristaltic motion of a particle-fluid mixture b L. M. Srivastava and V. P. Srivastava [5] and the two-dimensionalanalsisof peristaltic motion of single phase fluid b Fung and Yih [34] inthe presence of slip effect. The mathematical model considers a particle-fluid mixture between infinite parallel walls with slip condition on which a sinusoidal traveling wave is imposed. A perturbation solution is obtained which satisfies the momentum equations for the case in which amplitude ratio (wave amplitude/channel half width) is small. Finall, the phenomenon of the mean flow reversal is presented and its phsiological implication is discussed. Beside the engineering applications of pumping particle-fluid mixture b peristalsis, the present analsis of two-phase flow process is potentiall important in regard to biofluid transport b peristalsis muscular contractions in bod organs where fluids behave like particle-fluid mixtures, namel, chime in small intestine, spermatic fluid in cervical canal, urine (from a diseased kidne) in ureter, and blood suspension in arteriole.. Formulation of the Problem Consider a two-dimensional infinite channel of mean width d (see Figure ), filled with a mixture of small spherical rigid particles in an incompressible Newtonian viscous fluid. The walls of the channel are flexible, on which are imposed travelling, sinusoidal wave of small amplitude. The equations governing conservation of mass and linear momentum for both fluid and particle phase using a continuum approach are expressed as follows (Drew [35]; L. M. Srivastava and V. P. Srivastava [5, 9]). Fluid Phase ( C) ρ f [ u f t +u u f f x +υ u f f ] = ( C) p x + ( C) μ s (C) u f +CS(u p u f ), ()

3 Applied Bionics and Biomechanics 3 ( C) ρ f [ V f t +u V f f x + V V f f ] = ( C) p + ( C) μ s (C) V f +CS(V p V f ), () x {( C) u f}+ {( C) V f}=. (3) Particulate Phase Cρ p [ u p t +u u p p x +υ u p p ] = C p x +CS(u f u p ), Cρ p [ V p t +u V p p x +υ V p p ] = C p +CS(V f V p ), (4) (5) x {Cu p}+ {CV p}=. (6) In () (6), x and are Cartesian coordinates with x measured in the direction of wave propagation and measured in the direction normal to the mean position of the channel walls, (u f, V f ) denotes fluid phase velocities, (u p, V p ) denotes particulate phase velocities, ρ f and ρ p are the actual densities of the materials constituting fluid and particulate phase, respectivel, ( C)ρ f isthefluidphasedensit,cρ p is the particulate phase densit, p denotes the pressure, C denotes the volume fraction densit of the particles, μ(c) is the particle-fluid mixture viscosit (also the effective viscosit of the suspension), and S the drag coefficient of interaction fortheforceexertedbonephaseontheother. The concentration of the particles is considered to be so small that the field interaction between particles ma be neglected. Thus, the diffusivit terms, which can model the effects of particle-particle impacts due to the Brownian motion, are neglected. It is worth mentioning here that the effect of Brownian motion was considered b others including Batchelor [36, 37]. The volume fraction densit, C, of the particles is chosen also constant. This is a good assumption for low concentration of small particles. The expression for the drag coefficient for the present problem is selected as S= 9 μ a λ (C), λ (C) = [8C 3C ] / + 3C [ 3C], where μ isthefluidviscositanda is the radius of the particle. Relation (7) represents the classical Stokes drag for (7) small particle Renolds number, modified to account for the finite particulate fractional volume through the function λ (C), obtained b Tam [38]. Man empirical relations have been suggested to express the viscosit of the suspension as a function of particle concentration and viscosit of the suspending medium. Einstein was the first to obtain theoreticall that the viscosit of the suspension μ S was related to that of the suspending medium μ for spheres in suspension b μ =μ S (.5C). However, the Einstein formula expresses the viscosit of the suspension onl when C is less than.5. As C increases from.5, the suspension viscosit varies from the Einstein equation. For the present problem, an empirical relation for the viscosit of the suspension is as follows: μ S (C) =μ qc, (8a) q=.7exp [.49C+ 7 exp (.69C)], (8b) T where T the absolute temperature ( K), suggested b Charm and Kurland (974) [39], is used. The viscosit of the suspension expressed b this formula is found to be reasonabl accurate up to C =.6. Charm and Kurland (974) [39]tested (8a) and (8b) with a cone and plate viscometer and found it to be in agreement within ten percentincaseofblood suspension. The boundar conditions that must be satisfied b the fluidonthewallsaretheslipandimpermeabilitconditions. The walls of the channel are assumed to be flexible but extensible with a travelling sinusoidal wave, and displacement in the channel walls is in transverse direction onl. Hence, boundar conditions are u f =Ψ f = A u f, V f = Ψ fx =± η t on =±d±η, V p = Ψ px =± η t, where Ψ,thestreamfunction,issuchthat u f = Ψ f, V f = Ψ f x, u p = Ψ p, V p = Ψ p x. (9a) (9b) The transverse displacement, η, of the wall is represented as (9c) () η (x, t) =acos π (x ct), () λ

4 4 Applied Bionics and Biomechanics where a is the amplitude, λ the wavelength, and c the wave speed. We now select the following set of nondimensional variables and parameters: x = x d, = d, Suspension parameter N= Sd ρ f ( C) ρ p μ S. (8) Thus, the sstems of () (6), and(9a) () now become, after dropping the primes, u f = u f c, V f = V f c, ( C) R[ t Ψ f +Ψ f Ψ fx Ψ fx Ψ f ] = 4 Ψ f +CM( Ψ p Ψ f ), (9) u p = u p c, V p = V p c, η = η d, Ψ f = Ψ f cd, Ψ p = Ψ p cd, t = ct d, p = p ρ f c, υ= μ ρ f. Suspension Renolds number Wave number Knudsen number Amplitude ratio R= Suspension parameter M= () cdρ f ( C) μ S. (3) α= πd λ. (4) Kn = A R. (5) ε= a d. (6) Sd ( C) μ S. (7) CR [ t Ψ p +Ψ p Ψ px Ψ px Ψ p ] =CN( Ψ f Ψ p ), () η=εcos α (x t), () Ψ f = KnΨ f, Ψ fx = αεsin α (x t), Ψ px = αεsin α (x t), where denotes the Laplacian operator. 3. Method of Solution on =±( +η), () Assuming the amplitude ratio e of the wave is small,we obtain the solution for the stream function as a power series in terms of ε, b expanding Ψ f, Ψ p,and p/ x in the form (Fung and Yih (968) [34]) Ψ f =Ψ f +εψ f +ε Ψ f +, (3) Ψ p =Ψ p +εψ p +ε Ψ p +, (4) p x = ( p x ) +ε( p x ) +ε ( p x ) + (5) In (5), the first term on the right-hand side corresponds to the imposed pressure gradient associated with the primar flow and the other terms correspond to the peristaltic motion or higher imposed pressure gradient. Substituting (3) and (4) in (9), (), and() and collecting terms of like powers of ε, weobtainthreesetsof coupled linear differential equations with their correspondingboundarconditionsinψ (f,p), Ψ (f,p),andψ (f,p),forthe first three powers of ε. The first set of differential equations in Ψ (f,p),subjectto the stead parallel flow and transverse smmetr assumption for a constant pressure gradient in the x-direction, ields

5 Applied Bionics and Biomechanics 5 the following classical Poiseuille flow for the fluid and the particulate phase: Ψ f =K[( 3 3 )+ Kn + Kn ( )], (6) Ψ p =K(+ M 3 3 ), (7) K= R ( p x ), (8) where K is the Poiseuille flow parameter. Thus, the effect of the particles on the fluid velocit profile is to cause an increase in the viscosit; that is, fluid viscosit μ is replaced b suspension viscosit (see (8a) and (8b)), and thus for a given pressure difference less fluid will flow through the channel. Further, the particles lead the fluid b a relative velocit proportional to /M( p/ x). The second and third sets of differential equations in Ψ (f,p) and Ψ (f,p) with their corresponding boundar conditions are satisfied b Ψ f (x,, t) = Φ f () e iα(x t) +Φ f () e iα(x t), Ψ p (x,, t) = Φ p () e iα(x t) +Φ p () e iα(x t), (9a) (9b) icαr [ K[( + M )+ Kn + Kn ( + M )]] ( d d α )Φ p ikcαrφ p =CN( d d α )(Φ p Φ f ), Φ KKn f (±) K = Kn [Φ f (±) K], + Kn (33a) Φ f (±) =±, Φ p (±) =±, (3) (33b) (33c) Φ ( C) αr f () = (Φ f Φ f Φ f Φ f ) (34) +CM(Φ f Φ p ), = icαr (Φ p Φ p Φ p Φ p ) +CN(Φ p Φ f ), (35) ( d d 4α )[ d d 4α + i ( C) αr] Φ f Ψ f (x,, t) = Φ f +Φ f () e iα(x t) +Φ f () e iα(x t), Ψ p (x,, t) = Φ p +Φ p () e iα(x t) +Φ p () e iα(x t). (3a) (3b) = i ( C) KαR [( )+ Kn ( )] + Kn ( d d 4α )Φ f + 4i ( C) KαRΦ f + i ( C) αr (Φ f Φ f Φ fφ f ) (36) Asubstitutionof(9a), (9b) and (3a), (3b) into the differential equations and their corresponding boundar conditions in Ψ (f,p) and Ψ (f,p) leads to the following set of differential equations: [ d d α +i( C) αr[ K[( )+ Kn ( )]]] + Kn ( d d α )Φ f ik ( C) αrφ f =CM( d d α )(Φ f Φ p ), (3) +CM( d d 4α )(Φ f Φ p ), icαr ( d d 4α )Φ p = ickαr ( d ) ( M d 4α )Φ p + 4iCKαRΦ p + icαr (Φ p Φ p Φ pφ p 4α )(Φ p Φ f ), d )+CN( d Φ f (±) ± [Φ f (±) +Φ f (±)] K = Kn [Φ f (±) ± [Φ f (37) (38a) (±) +Φ f (±)]],

6 6 Applied Bionics and Biomechanics Φ f (±) ± Φ f (±) K Φ f (±)], Φ f (±) ± 4 Φ f (±) =, (38c) Φ p (±) ± 4 Φ p (±) =. (38d) = Kn [Φ f (±) ± (38b) Thus, we obtained a set of differential equations together with the corresponding boundar conditions which are sufficient to determine the solution of the problem up to the second order in ε. Now, our main attention is to find out solution of differential equations for B f and B p.although (3) and (3) for B f and B p are coupled fourth-order ordinar differential equations with variable coefficients, it would, perhaps, be impossible to obtain solution of these differential equations for arbitrar values of R, α, andk. Thisisjustbecauseofthemovingboundarconsideredin the present problem. The condition of moving boundar has made the boundar condition nonhomogeneous and thus the problem is not an eigenvalue problem as in all problems of hdrodnamic stabilit for which solutions are available in the literature. However, we can restrict our investigation to the case of pumping of an initiall stagnant fluid, corresponding to no imposed pressure gradient. Thus, in this case ( p/ x) =, which means that constant K vanishesandwewouldbeabletoobtainasimpleclosedform analtical solution of this interesting case of free pumping. Phsicall, this assumption means that the fluid is stationar if there are no peristaltic waves. In fact, this assumption is not so restrictive because the maximum pressure gradient that small-amplitude waves can generate is of the order of ε and in the pumping range the zeroth-order mean pressure gradient must certainl vanish. Solutionsof (3) and (3)subject to the boundar condition ((33a), (33b), and(33c)), under the assumption, K=, ma be obtained as where Φ f () = A sinh (α) + B sinh (β), Φ p () = A sinh (α) + B sinh (β), (39) β =α iαr[ C+ A = CM N iαr ]. (βcosh (β) + Knβ sinh (β)) α cosh (α) sinh (β) β cosh (β) sinh (α) + Kn sinh (α) sinh (β) (α β ), B = α cosh (α) + Knα sinh (α) α cosh (α) sinh (β) β cosh (β) sinh (α) + Kn sinh (α) sinh (β) (α β ), (4) A = B sinh (β), sinh (α) B = B N N iαr. Next, in the expansion of Ψ (f,p), we need onl to concern ourselves with the terms Φ (f,p) () as our aim is to determine the mean flow onl. Thus, the solution of the coupled differential equations ((34), (35)) subject to the boundar conditions ((38a), (38b), (38c),and(38d)), under the assumption, K=, gives the expressions Φ f () = F () F () +D C ( + Kn), Φ p () = G () F () +D (4) +β 3 Kn cosh (β)) + B (β sinh (β ) +β 3 cosh (β ))], F()= ( C) α R 4 [γ (A B + CMA B ( C) N ) cosh (α + β) cosh (α β) [ (α + β) (α β) ] where C ( + M + Kn), +γ (A B + CMA B ( C) N ) [cosh (α + β ) (α + β ) D=Φ f (±) = [(A +A )(α sinh (α) +α 3 Kn cosh (α))+b (β sinh (β) cosh (α β ) (α β ) ]+(γ +γ ) (B B

7 Applied Bionics and Biomechanics 7 with + CMB B ( C) N )[cosh (β + β ) (β + β ) cosh (β β ) (β β ) ]], G()=F() ( C) α R [γ A 4N B [cosh (α + β) cosh (α β)]+γ A B [cosh (α + β ) cosh (α β )]+(γ +γ ) B B [cosh (β + β ) cosh (β β )]], (4) γ CM = + ( C)(N iαr). (43) Thus, we see that one constant C remains arbitrar in thesolutionwhichisfoundtobeproportionaltothesecondorder time-averaged pressure gradient. If we time-average () for the solution given b (), (), (4), (5), (3), (4), (5), ((9a), (9b)), ((3a), (3b)), (39),and(4),wefindthat c =R( p x ). (44) The constant c, which is related to the second-order pressure gradient distribution, ma be obtained using ends conditions of the real phsical problem. The mean time average velocities ma now be written as u f = ε Φ ε f () = [F () F () +D R( p x ) ( + Kn)], (45) u p = ε Φ ε p () = [G () F () +D (46) R( p x ) ( + M + Kn)]. If no-slip, that is, Kn =,resultsofthepresentproblems reduce exactl to the same as that found b L. M. Srivastava and V. P. Srivastava [5]. Also if no-slip, that is, Kn =and the fluid is particles free, that is, C =, results of the present problems reduce exactl to the same as that found b Fung and Yih [34]. 4. Numerical Results and Discussion A close look at (45) reveals that the mean axial velocit of the fluid phase u f is dominated b the constant D and the parabolic distribution term R( p/ x) (l +Kn). The term F() F(), alwas a negative quantit, is negligible compared to D. The constant D, which initiall arose from the slip condition of the axial velocit on the wall, is due to the value of Φ f at the boundar and is related to the mean velocit at the boundaries of the channel (at = ±)b u = (ε /)(D R( p/ x) (Kn)). This shows that the slip boundar condition applies to the wav wall and not to the mean position of the wall. It ma be reminded here that the corresponding D does not appear in the particulate phase mean axial velocit as the particulate phase velocit at the walls was unspecified. For the sake of comparison we define mean-velocit perturbation function J() in accordance with Fung and Yih [34] and L. M. Srivastava and V. P. Srivastava [5]as J= α (F () F ()), (47) R which gives mean time axial velocit of the fluid phase as u f () = ε p (D R ( x ) ( + Kn) α R J()). (48) It has been observed that urine, bacteria, or other materialssometimepassfromthebladder to the kidne or from one kidne to the other in direction opposite to the urine flow. Phsiologists term these phenomena as ureteral reflux. Two different definitions of reflux exist in the literature; Shapiro et al. [4] call a flow reflux whenever there is a negative net displacementofaparticletrajector,whileyinandfung[4] define a flow reflux whenever there is a negative mean velocit in the flow field. In the present analsis the latter definition of reflux is adopted as L. M. Srivastava and V. P. Srivastava [5]. Since D is alwas a positive quantit, u=(ε /)D at = ± showsthatthemeanflowreversalwillneveroccuratthe boundaries. Further, from (48), it is clear that the reflux would occur when the mean pressure gradient ( p/ x) reaches a certain critical value. Thus, the critical reflux condition ma be defined as one for which the mean velocit u() is equal to zero on the center line =; (48) ields C R =( p x ) critical = R ( + Kn) (D α R J ()). (49) For ( p/ x) <( p/ x) criticalreflux, there is no reflux and if ( p/ x) >( p/ x) criticalreflux there will be reflux and a backward flow in the neighborhood of the center line occurs. The value of ( p/ x) critical reflux for various values of C, R, Kn,andα is displaed in Figure. Fromthefigure,the results reveal that the value of ( p/ x) critical reflux with for different values of C, for a fixed value of R, ( p/ x) critical reflux

8 8 Applied Bionics and Biomechanics R(px) critical reflux R = α C =., Kn =. C =., Kn =. C =., Kn =. C =., Kn =. C =.4, Kn =. C =.4, Kn =. C =.59, Kn =. C =.59, Kn =. Figure : Effect of the Knudsen number Kn, particle concentration C,andwavenumberα on the critical reflux pressure gradient at R=. decreases with increasing particle concentration C.However, C have significant influence over ( p/ x) critical reflux onl at higher values of α. In the presence of Knudsen number Kn, we observed that, at α.4, ( p/ x) critical reflux decreases with increasing wave number α. However,α >.4 and ( p/ x) critical reflux increases with increasing wave number α. We observed that the critical reflux pressure ( p/ x) critical reflux at a given α and R is lower for particlefluid suspension than for particle-free fluid. This means that presence of particle in the fluid favors reversal flow. Finall, in Figures 3 6, themean-velocitdistribution withreversalflowisdisplaed.effectsofc, R, Kn,and ( p/ x) on mean velocit and reversal flow are shown. Figure 7 studies the effect of Knudsen number Kn and the mean second-order pressure gradient ( p/ x) on the meanvelocit distribution and reversal flow for C =.4, α =., and R = ; we notice that the mean-velocit distribution increases with increasing Knudsen number Kn forward for ( p/ x) < ( p/ x) critical reflux while for ( p/ x) > ( p/ x) critical reflux the velocit increases the reflux flow. Also we notice that the mean-velocit distribution decrease with increasing the value of ( p/ x). Also we notice that the value of ( p/ x) critical reflux decreases b increasing the Knudsen number Kn. Figure 4 studies the effect of Knudsen number Kn and particle concentration C on the mean-velocit distribution mean and reversal flow for α =., ( p/ x) =.3, and R=, and the figures reveal that the reversal flow increases with increasing particle concentration C, butthepresence of Knudsen number Kn results in a decrease in the reversal flow. Also we notice that the mean-velocit distribution increases with increasing Kn. Interpreted phsiologicall, this meansthat,underthesameconditions,urineinwhichsolute particles are suspended (i.e., urine from a diseased kidne) is more susceptible to reversal flow in ureter, in comparison to pure urine without solute particles. Figure 5 studies the effect of Knudsen number Kn and the Renolds number R on the mean-velocit distribution and reversal flow for α =.5, ( p/ x) =.5, andc =.3, and the figures reveal that the reversal flow increases with increasing Renolds number R. Also we notice that from Figures 5(a) and 5(b), atr < 3, the presence of Knudsen number Kn results a decrease in the mean-velocit distribution, from Figure 5(c) at R=3, the effect of Knudsen number vanishes, with increasing Kn at R > 3, andwe observed that the presence of Knudsen number Kn results in increase in the mean-velocit distribution and the reversal flow. Figure 6 studies the effect of Knudsen number Kn and thewavenumberα on the mean-velocit distribution and reversal flow for R=, ( p/ x) =.5, andc =.3; the figures reveal that the reversal flow increases with increasing wave number α. AlsowenoticethatfromFigures6(a) and 6(b),atα <.6, the presence of Knudsen number Kn results in an increase in the reversal flow, from Figure 6(c) at α=.6, the effect of Knudsen number vanishes, with increasing α at α >.6, and we observed that the presence of Knudsen number Kn results in decrease in the reversal flow. Next, we return to the dimensional flow problem; the dimensional mean axial velocit u f is equal to the dimensionless mean net axial velocit u f as given b (48) multiplied b the factor c. The properties of the blood are given b ρ f = 66 kg/m3 and μ = 4 3 Nm s. The particle concentration C various accurate up to C =.6 [39]. The frequenc f ofthewaveisrelatedtothewavespeedc and the wavelength λ according to λ = c/f. According to the Knudsen number, the flow regimes can be divided into various regions. These are continuum, slip, transition, and free molecular flow regimes. If Kn <., so that molecular mean free path of the molecules is negligible in comparison to the geometrical dimensions, the fluid can be treated as a continuous medium. If. < Kn <.,itis found that the fluid loses grip on the boundaries and tends to slip along the walls of the domain. If. < Kn < 3.,

9 Applied Bionics and Biomechanics u (mean velocit) 6 Kn =. Kn =. Kn =.5 R =. C =.4 α =. (Px) =(Px) critical reflux (a) R =. C =.4 α =. (Px) =. (Px) < u (mean velocit) R =. C =.4 α =. (Px) = u (mean velocit) 6 Kn =. Kn =. Kn =.5 (b) R =..4 C =.4 α =..6 (Px) =.5.8 (Px) > (Px) critical reflux u (mean velocit) 6 Kn =. Kn =. Kn =.5 (c) Kn =. Kn =. Kn =.5 (d) Figure 3: (a) Mean velocit distribution at ( p/ x) =.. (b) Mean velocit distribution at ( p/ x) =.. (c) Mean velocit distribution at ( p/ x) =( p/ x) Criticalreflux. (d) Mean velocit distribution at ( p/ x) =.5. it is transition flow regimes. Finall, the flow enters the free molecular regime when Kn > 3., each requiring a particular tpe of analsis [, 3]. For example, for the left main coronar arter, the range of the diameter is. 5.5 mm (mean 4 mm) and wavelength range λ =.5.56 cm [39]. The dimensional mean axial velocit u f (m/s) is plotted versus (m), forλ =.56 cm, the half of mean width d =. mm, the wave has amplitude a = 4 mm, and C =.3, withvariousvalues of Knudsen number Kn, for Kn =., Kn =., Kn =.3, Kn=.6,andKn=..Weobservethat,fromFigure 7(a), for ( p/ x) < ( p/ x) criticalreflux (( p/ x) =.), the mean axial velocit u f increases with increasing Knudsen number Kn and there is no reflux flow forward for ( p/ x) < ( p/ x) criticalreflux while, from Figure 7(b), for( p/ x) > ( p/ x) criticalreflux (( p/ x) =.5) there will be reflux flow and a backward flow in the neighborhood of the center line occurring, and the mean axial velocit u f increases in the reversal flow with increasing Knudsen number Kn. 5. Conclusions There is not an attempt to stud the effect of slip conditions on the flow of a particle-fluid suspension with peristalsis. The purpose of this paper is to stud the slip effects on the peristaltic pumping of a particle-fluid mixture in a two-dimensional channel. It is an application of the twodimensional analsis of peristaltic motion of a particle-fluid mixture b L. M. Srivastava and V. P. Srivastava [5] andthe two-dimensional analsis of peristaltic motion of single phase fluid b Fung and Yih [34] in the presence of slip effects. The mathematical model considers a particle-fluid mixture between infinite parallel walls with slip condition on which a sinusoidal traveling wave is imposed. A perturbation solution is obtained which satisfies the momentum equations for the case in which amplitude ratio (wave amplitude/channel half width) is small. Finall, the phenomenon of the mean flow reversal is presented and its phsiological implication is discussed. Beside the engineering applications of pumping particle-fluid mixture b peristalsis, the present analsis of

10 Applied Bionics and Biomechanics R =. α =.. (Px) = C = u (mean velocit) 6.8 R =..6 α =..4 (Px) = C = u (mean velocit) 6 Kn =. Kn =. Kn =.5 (a).8 R =..6 α =..4 (Px) = C = u (mean velocit) 6 Kn =. Kn =. Kn =.5 (b).8 R =..6 α =..4 (Px) = C = u (mean velocit) 6 Kn =. Kn =. Kn =.5 (c) Kn =. Kn =. Kn =.5 (d) Figure 4: (a) Mean velocit distribution at C =., α =., ( p/ x) =.3, and R=. (b) Mean velocit distribution at C =., α =., ( p/ x) =.3, and R=. (c) Mean velocit distribution at C =.4, α =., ( p/ x) =.3, and R=. (d) Mean velocit distribution at C =.59, α =., ( p/ x) =.3, and R=. two-phase flow process is potentiall important in regard to biofluid transport b peristalsis muscular contractions in bod organs where fluids behave like particle-fluid mixtures, namel, chime in small intestine, spermatic fluid in cervical canal, urine (from a diseased kidne) in ureter, and blood suspension in arteriole. Some concluding remarks are as follows. (i) The reversal flow increases with increasing particle concentration C, but the presence of Knudsen number Kn results in a decrease in the reversal flow. The presence of Knudsen number Kn results in a decrease in the reversal flow. (ii) Also we notice that the mean-velocit distribution increases with increasing Kn. Interpreted phsiologicall, this means that, under some conditions, urine in which solute particles are suspended (i.e., urine from a diseased kidne) is more susceptible to reversal flow in ureter, in comparison to pure urine without solute particles. (iii) For example, for the left main coronar arter, the mean axial velocit u f increases with increasing Knudsen number Kn and there is no reflux flow forward for ( p/ x) <( p/ x) criticalreflux while for ( p/ x) > ( p/ x) criticalreflux there will be reflux and a backward flow in the neighborhood of the center line occurring, and the mean axial velocit u f increases in the reversal flow with increasing Knudsen number Kn. Comparing with other models for verifications of results, the present model gives the most general form of velocit expression from which the other mathematical models can easil be obtained b proper substitutions. It is of interest to note that the result of the present model includes results of different mathematical models such as the following. () The results of L. M. Srivastava and V. P. Srivastava [5] have been recovered b taking Knudsen number kn =. (no-slip condition).

11 Applied Bionics and Biomechanics C =.3 α =.5 (Px) =.5 R = u (mean velocit) 6 (a).8 C =.3.6 α =.5.4 (Px) = R = u (mean velocit) 6 (c).8.6 C =.3.4 α =.5. (Px) =.5 R = u (mean velocit) 6 (e) C =.3.4 α =.5.6 (Px) =.5.8 R = u (mean velocit) 6 C =.3 α =.5 (Px) =.5 R = u (mean velocit) 6 (b) C =.3.6 α =.5 (Px).8 =.5 R = u (mean velocit) 6 (d) C =.3. α =.5.4 (Px) = R = u (mean velocit) 6 (f) Kn =. Kn =. Kn =.5 (g) Figure 5: (a) Mean velocit distribution at R =., α =.5, ( p/ x) =.5, and C =.3. (b) Mean velocit distribution at R =., α =.5, ( p/ x) =.5,andC =.3. (c) Mean velocit distribution at R = 3., α =.5, ( p/ x) =.5,andC =.3. (d) Mean velocit distribution at R = 4., α =.5, ( p/ x) =.5,andC =.3. (e) Mean velocit distribution at R =., α =.5, ( p/ x) =.5,andC =.3.(f)Mean velocit distribution at R = 5., α =.5, ( p/ x) =.5,andC =.3.(g)MeanvelocitdistributionatR =., α =.5, ( p/ x) =.5, and C =.3.

12 Applied Bionics and Biomechanics.8.6 C =.3.4 R =. (Px) =.5. α = u (mean velocit) C =.3 R = (Px) =.5 α = u (mean velocit) C =.3 R = (Px) =.5 α =.6 Kn =. Kn =. (a) Kn =.5 Kn = u (mean velocit) Kn =. Kn =. (b) Kn =.5 Kn =. C =.3 R = (Px) =.5 α = u (mean velocit) 6 Kn =. Kn =. Kn =.5 Kn =. Kn =. Kn =. Kn =.5 Kn =. (c) (d) Figure 6: (a) Mean velocit distribution at α =., ( p/ x) =.5,andC =.3. (b) Mean velocit distribution at α =.5, ( p/ x) =.5, and C =.3. (c) Mean velocit distribution at α =.6, ( p/ x) =.5,andC =.3. (d) Mean velocit distribution at α =., ( p/ x) =.5, and C =.3. (m) 3 (Px) > (Px) critical reflux.5 (Px) = d=mm (right main coronar arter) u (mean axial velocit) (m/s) 8 (m) 3 (Px) > (Px) critical reflux.5 (Px) = d=mm (left main coronar arter) u (mean axial velocit) (m/s) 9 Kn =. Kn =. Kn =.3 Kn =.6 Kn =. Kn =. Kn =. Kn =.3 Kn =.6 Kn =. (a) (b) Figure 7: (a) Mean axial velocit (m/s) distribution at ( p/ x) < ( p/ x) criticalreflux (( p/ x) =.). (b) Mean axial velocit (m/s) distribution at ( p/ x) >( p/ x) criticalreflux (( p/ x) =.5).

13 Applied Bionics and Biomechanics 3 ()TheresultsofFungandYih[34] have been recovered b taking Knudsen number kn =. and the fluid is particles-free; that is, C=. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [] T. W. Latham, Fluid motion in a peristaltic pump [M.S. thesis], Massachusetts Institute of Technolog, Cambridge, Mass, USA, 966. [] H.J.Rath,Peristaltische Strömungen, Springer, Berlin, German, 98. [3]L.M.SrivastavaandV.P.Srivastava, Peristaltictransportof blood: casson model II, JournalofBiomechanics,vol.7,no., pp. 8 89, 984. [4] L. M. Srivastava and V. P. Srivastava, Interaction of peristaltic flow with pulsatile flow in a circular clindrical tube, Journal of Biomechanical Engineering,vol.8,no.4,pp.47 53,985. [5]L.M.SrivastavaandV.P.Srivastava, Peristaltictransportof a particle-fluid suspension, Journal of Biomechanical Engineering, vol., no., pp , 989. [6] V. P. Srivastava and M. 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14 4 Applied Bionics and Biomechanics [34] Y.C.FungandC.S.Yih, Peristaltictransport, ASME Journal of Applied Mechanics,vol.33,pp ,968. [35] D. A. Drew, Stabilit of a stokes laer of a dust gas, Phsics of Fluids,vol.,no.,pp.8 84,979. [36] G. K. Batchelor, Transport properties of two-phase materials with random structure, Annual Review of Fluid Mechanics, vol. 6, pp. 7 55, 974. [37] G. K. Batchelor, Brownian diffusion of particles with hdrodnamic interaction, Journal of Fluid Mechanics,vol.74, no.,pp. 9, 976. [38] C. K. W. Tam, The drag on a cloud of spherical particles in low Renolds number flow, Journal of Fluid Mechanics,vol.38,no. 3, pp , 969. [39] S. E. Charm and G. S. Kurland, Blood Flow and Micro- Circulation, John Wile, New York, NY, USA, 974. [4] A. H. Shapiro, M. Y. Jaffrin, and S. L. Weinberg, Peristaltic pumping with long wavelengths at low Renolds number, Journal of Fluid Mechanics,vol.37,no.4,pp ,969. [4] F. C. Yin and Y. C. Fung, Comparison of theor and experiment in peristaltic transport, Journal of Fluid Mechanics,vol.47,no., pp. 93, 97.

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