The influence of a micropolar fluid on peristaltic transport in an annulus: application of the clot model

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1 Applied Bionics and Biomechanics Vol. 5, No. 1, Mach 28, The influence of a micopola fluid on peistaltic tanspot in an annulus: application of the clot model Kh. S. Mekheime a and Y. Abd Elmaboud b a Mathematics Depatment, Faculty of Science, Al-Azha Univesity, Nas City, Caio, Egypt; b Mathematics Depatment, Faculty of Science, Al-Azha Univesity Assiut Banch, Assiut, Egypt Received 14 Febuay 28; final vesion eceived 8 June 28 A seious pathological condition is encounteed when some blood constituents deposited on the blood vessels get detached fom the wall, join the blood steam again and fom a clot. Study of the peistaltic tanspot of a micopola fluid in an annula egion is investigated unde low Reynolds numbe and long wavelength appoximations. We model a small atey as a tube having a sinusoidal wave tavelling down its wall and a clot model inside it. Closed fom solutions ae obtained fo the velocity and the micootation components, as well as the steam function, and they contain new additional paametes, namely, δ, the height of the clot, N, the coupling numbe and m, the micopola paamete. The pessue ise and fiction foce on the inne and the oute tubes have been discussed fo vaious values of the physical paametes of inteest. Keywods: peistaltic tanspot; micopola fluid; clot model; pessue ise 1. Intoduction Peistaltic pumping is a fom of fluid tanspot that occus when a pogessive wave of aea contaction o expansion popagates along the length of a distensible duct. Peistalsis is an inheent popety of many biological systems having smooth muscle tubes that tanspots biofluids by its populsive movement and is found in the tanspot of uine fom kidney to the bladde, the movement of chyme in the gastointestinal tact, inta-uteine fluid motion, vasomotion of the small blood vessels and in many othe glandula ducts. The mechanism of peistaltic tanspot has been exploited fo industial applications such as sanitay fluid tanspot, blood pumps in heat lung machine and tanspot of coosive fluids whee the contact of the fluid with the machiney pats is pohibited. The behaviou of most of the physiological fluids is known to be non-newtonian. Hence, in ecent yeas, the study of peistaltic tanspot of non-newtonian fluids in channels o pipes has gained much attention. Seveal investigatos, Povost and Schwaz 1994, Sivastava and Sivastava 1988, Vajavelu et al. 25, Rao and Misha 24 and Hayat et al. 25, have analysed the peistaltic tanspot in physiological situations of inteest. Most of these analytical studies use asymptotic expansions with small Reynolds numbe, wave numbe, and amplitude atio as the petubation paametes. The model of micopola fluid intoduced by Eingen 1966 epesents fluids consisting of igid, andomly oiented o spheical paticles suspended in a viscous medium whee the defomation of the paticles is ignoed. Micopola fluids exhibit some micoscopic effects aising fom the local stuctue and micomotion of the fluid elements. Futhemoe, they can sustain couple stesses. The micopola fluid is consideed to model the blood flow in small ateies and the calculation of theoetical velocity pofiles is obseved in good ageement with the expeimental data. A seious pathological condition is encounteed when some blood constituents deposited on the atey wall get detached fom the wall, join the blood steam again and fom a clot. This can lead to patial o even complete blockage of the blood vessels. Jayaaman and Saka 25 studied the mathematical model fomulated fo solving flow in a stenosed atey with a clot inside it. In the pesent study, the peistaltic tanspot of an incompessible micopola fluid in an annula egion between two coaxial tubes, the oute tube is unifom and has a sinusoidal wave tavelling down its wall and the inne one has a clot on its wall, is investigated. The analysis has been caied out in the wave fame of efeence with long wavelength and zeo Reynolds numbe assumptions. The elationship between pessue gadient and time mean flow ate fo vaious micopola paametes, coupling numbes and maximum height attained by the clot is obtained. A motivation of the pesent analysis is the hope that such a poblem will be applicable in many clinical applications. 2. Fomulation of the poblem Conside the flow of an incompessible micopola fluid in an annula egion between two coaxial tubes, the oute tube is unifom and has a sinusoidal wave tavelling down Coesponding autho. kh mekheime@yahoo.com ISSN: pint / online Copyight C 28 Taylo & Fancis DOI: 1.18/

2 14 K. S. Mekheime and Y. A. Elmaboud Figue 1. Geomety of the poblem. its wall and the inne one having a clot on its wall. The geometies of the wall sufaces ae see Figue 1. R 1 = R a + f Z,t, Z λ, 1 = R a othewise, 2π R 2 = R + b sin λ Z ct, 2 whee ar is the adius of the inne tube that keeps the clot model in position inside the tube with a 1, f Z,t the abitay shape along the axial diection that can be handled though suitable choice, R is the adius of the oute tube at any axial distance Z fom the inlet, b is the wave amplitude, λ is the wavelength, c is the popagation velocity and t is the time. Intoducing a wave fame,z moving with velocity c away fom the fixed fame R,Z by the tansfomation z = Z ct, = R, v z = V z c, v = V, 3 whee V,V z and v,v z ae adial and axial velocity components in the stationay and moving coodinate systems, espectively. Taking the axisymmetic flow, whee all the vaiables ae independent of θ, the velocity vecto is given by v = v,,v z and micootation vecto is w =,v θ,. Also, we intoduce the following non-dimensional vaiables: =,z= z R λ,v z = v z c,v = λ v cr,v θ = R c v θ, p = R2 λµc p,t= c λ t,j= j R 2, 1 = 1, 2 = 2 4 R R Afte using these tansfomation and non-dimensional vaiables, the non-dimensional equations govening the steady flow of an incompessible micopola fluid in the absence of body foce and body couple ae: v + v z z + v Re α 3 v v + v z =, 5 v = p z + α2 1 N N v θ v z Re α v + v z N v θ jreα1 N N + 2 N m 2 z + 2 v v z z + 2 v z 2 v v θ v v + 2 v 2 α2, 6 z 2 = p z N + 1 v z + 2 v z α2 z 2 = 2v θ + + v v θ z z 1 v θ + α 2 2 v θ z 2, 7 α 2 v z v z, 8 whee φ = b/r is the amplitude atio, α = R /λ, Re = ρcr /µ is the Reynolds numbe, N = k/µ + k is the coupling numbe N 1 Cowin 1968 and m 2 = R 2 k2µ + k/γ µ + k is the micopola paamete Einge Using the long wavelength appoximation α 1 and dopping tems of ode α and highe, it follows fom Equations 5 8 that the appopiate equations descibing the flow in the wave fame ae: v + v z z + v p N v θ =, 9 =, v z v z 2v θ + v z 2 N m 2 = 1 N p z, 11 1 v θ =. 12 The coesponding bounday conditions in the wave fame ae whee v z = 1, v θ = at = 1, = 2, 13 1 = a + f 1 z, z 1, 14 = a othewise, 2 = 1 + φ sin 2πz Solution of the poblem Fom Equations 1 and 11 and afte one integation, we get v z dp = 1 N 2 dz + A 1z Nv θ. 16

3 On substituting Equation 16 into Equation 12, thus 2 v θ v θ m v 2 θ = m2 1 N 2 N and its geneal solution is Applied Bionics and Biomechanics 15 [ dp 2 dz + A ] 1z, 17 v θ = A 2 zi 1 m + A 3 zk 1 m [ 1 N dp 2 N 2 dz + A ] 1z, 18 whee I 1 and K 1 ae modified Bessel functions of the fist ode, fist and second kinds, espectively. Substituting Equation 18 into Equation 16 and integating, we obtain: v z = N m [A 3zK m A 2 zi m] + 1 N 2 N 2 2 dp dz + 2A 1zln + A 4 z, 19 + m [ a 35 ln ln a ln ]}. 22 The dimensionless flux q = q /πcr 2, q being the flux in the wave fame is given by 2 q = 2 v z d = 1 { η dp } 1 b 19 dz + ξ, 23 whee the constants in Equations 2 23 ae as given in Appendix 1. Thus, the pessue gadient is obtained fom Equation 11 as dp dz = 1 η {b 19q ξ}. 24 Following the analysis given by Shapio et al the mean volume flow, Q, ove a peiod is obtained as Q = 1 T T q dt = q + q2 q 1, 25 whee I and K ae modified Bessel functions of the zeoth ode. The expessions fo v z and v θ subject to the bounday conditions ae espectively given as v z = 1 { a 39 + AF dp a 24 dz a a } + a 41 K m + a 33 I m + 2a ln + a 34, v θ = A dp 2a 53 dz { F m 2 I 1 m 2 [ Fk m a16 1 K 1 m 1 K 1 m ln 2 1 ] 1 K 1 m K 1 m 1 + a 43 mi 1 m + m F [ a a 13 a 16 K 1 m + a ] + a a 54 K 1 m a 47 K 1 m 2 a 55 }. 21 The coesponding steam function v = 1 1 ψ is ψ z and v z = { ψ, z = b AF dp 4ma 24 dz m 2 2a 34 2b 13 + b 14 2 mb b a 33 I 1 m a 41 K 1 m whee q 2 = dz, q 1 = 1 3 into Equation 29, we get 2 1 dz. Substituting Equation dp dz = 1 η {Q q 2 + q 1 b 19 ξ}. 26 The pessue ise p and the fiction foce at the wall on the oute and inne tubes ae F o and F i, espectively, in thei non-dimensional foms Sivastava p = F i = dp dz dz, Fo = dp dz dp dz, dz dz. 27 In the absence of the inne tube the clot model i.e. 1, Equations ae v z = N dp 22 N dz [ N ] 2 I m 2 I m, 28 m I 1 m 2 v θ = 1 N [ ] dp 2 I 1 m 22 N dz I 1 m 2, 29 ψ, z = N [ dp 4 22 N dz N 2 mi 1 m 2 m 2 ] I m 2 2I 1 m. 3 2m

4 16 K. S. Mekheime and Y. A. Elmaboud These esults ae the same as those obtained by Sinivasachaya et al. 23. Also, if we take 1 and k i.e. N, the esults coincide with that given by Shapio et al Substituting Equation 31 into Equations 32 34, we get p = F o = F i = η {Q q 2 + q 1 b 19 ξ}dz, η {Q q 2 + q 1 b 19 ξ}dz, 2 1 η {Q q 2 + q 1 b 19 ξ}dz, 32 The integals involved in Equations 38 4 ae vey difficult to evaluate analytically and thus ae evaluated numeically. 4. Numeical esults and discussion The solutions ae obtained fo suitable choice of the clot model in the non-dimensional fom Jayaaman and Saka 25 as 1 = a + δe π 2 z z d.5 2, z 1 33 whee δ is the maximum height attained by the clot at z = z d +.5, a is adius atio of the inne tube that keeps the clot in position and z d epesents the axial displacement of the clot. With the help of the MATHEMATICA pogam, the numeical evaluations of the analytical esults ae obtained fo p, F o and F i fo diffeent paametes values Eingen 1966; Jayaaman and Saka 25; Sivastava 1986; Sinivaschaya et al. 23. The effect of the maximum height of the clot, δ, is analysed by vaying δ =.5 to.25, a =.1,.2, z d =,.125, N 1 and m = up to 1. In Figue 2 the vaiation of p with Q is shown fo diffeent values of N and δ by fixing the othe paametes, a =.2, φ =.4, z d = and m = 2. It is noticed that a linea elation between them and an incease in the flow ate educe the pessue ise and thus maximum flow ate is achieved at zeo pessue ise and maximum pessue occus at zeo flow ate. The pessue ise inceases as the coupling numbe N inceases, and N = coesponds to the case of Newtonian fluid. Also it is easy to obseve that the pessue ise inceases when the maximum height attained by the clot δ inceases. Figue 3 depicts the vaiation of p with Q, with N =.4, z d =, δ =.1 and a =.2 fo diffeent values of micopola paamete m and the amplitude atio φ. An inteesting obsevation hee is that the pessue ise deceases with inceases m and is geate than the case of a Newtonian fluid m =, but it inceases with inceases φ. In Figue 4 the vaiation of p with Q is shown fo diffeent values of z d and a, and fixing the othe paametes, it is clea that the effect of incease z d and a Figue 2. The vaiation of p with Q fo diffeent values of N and δ at a =.2, φ =.4, z d =, m = 2. leads to incease p. Also Figues 2 4 ae divided into sectos to illustate the pumping egions, peistaltic pumping Q > and p >, augmented pumping Q > and p < and etogade pumping Q < and p > and it is clea that the peistaltic pumping egion becomes wide with inceases in N, δ, amplitude atio φ and z d. The vaiation of p with amplitude atio φ fo diffeent values flow ate Q and the axial displacement of the clot z d fo othe given fixed set of paametes is pesented in Figue 5. It is obseved that the elation between p and φ is a non-linea elation: p inceases with inceases φ. It is also evident that an incease in the flow ate educes the pessue ise but an incease in the axial displacement of the clot z d inceases the pessue ise. Figue 6 depicts the vaiation of p with φ fo diffeent values of N and m fo othe given fixed set of Figue 3. The vaiation of p with Q fo diffeent values of m and φ at a =.2,N =.4, z d =, δ=.1.

5 Applied Bionics and Biomechanics 17 Figue 4. The vaiation of p with Q fo diffeent values of a and z d at m = 1, N =.4, φ =.4, δ =.2. paametes. We obseve that the pessue ise p inceases with inceased N and deceased m. Figues 7 9 descibe the esults obtained fo the oute fiction foce vesus the flow ate Q, Figues 1 and 11 descibe the esults obtained fo the oute fiction foce vesus φ, Figues descibe the esults fo inne fiction foce vesus the flow ate Q and Figues 15 and 16 descibe the esults obtained fo the inne fiction foce vesus φ. Futhemoe, the effect of impotant paametes such as N, m, δ, a and φ on the inne and oute fiction foces has been investigated. We notice fom these figues that the inne and oute fiction foces have the opposite behaviou compaed to the pessue ise. The inne fiction foce behaves simila to the oute fiction foce fo the same values of the paametes; moeove, the Figue 6. The vaiation of p with φ fo diffeent values of N and m at Q =.1, z d =, a=.1, δ=.5. oute fiction foce is geate than the inne fiction foce at the same values of the paametes. 5. Steamlines and fluid tapping The phenomenon of tapping, wheeby a bolus defined as a volume of fluid bounded by a closed steamline in the wave fame is tanspoted at the wave speed. Figues 17 2 illustates the steamline gaphs fo diffeent values of the coupling numbe N fo a given fixed set of the othe paametes. It is obseved that the size of the tapping bolus inceases as N inceases nea the bounday wall oute tube. The effects of micopola paamete m on the tapping with fixed values of the paametes ae shown in Figues It is evident that the tapped bolus size inceases with Figue 5. The vaiation of p with φ fo diffeent values of Q and z d at m = 2,N =.2,a =.2, δ=.1. Figue 7. The vaiation of F o with Q fo diffeent values of N and δ at a =.2, m = 2, z d =, φ =.4.

6 18 K. S. Mekheime and Y. A. Elmaboud Figue 8. The vaiation of F o with Q fo diffeent values of m and φ at a =.2, N =.4, z d =, δ =.1. Figue 11. The vaiation of F o with φ fo diffeent values of N and m at z d =, Q =.1, a =.1, δ =.5. Figue 9. The vaiation of F o with Q fo diffeent values of a and z d at m = 1, N =.4, φ =.4, δ =.2. Figue 12. The vaiation of F i with Q fo diffeent values of N and δ at a =.2, m = 2, z d =, φ =.4. Figue 1. The vaiation of F o with φ fo diffeent values of Q and z d at m = 2, N =.2, a =.2, δ =.1. Figue 13. The vaiation of F i with Q fo diffeent values of m and φ at a =.2, N =.4, z d =, δ =.1.

7 Applied Bionics and Biomechanics 19 Figue 14. The vaiation of F i with Q fo diffeent values of a and z d at m = 1, N =.4, φ =.4, δ =.2. Figue 16. The vaiation of F i with φ fo diffeent values of N and m at z d =, Q =.1, a =.1, δ =.5. micopola paamete inceases m up to a cetain value of m and deceases late. It is clea fom Figues 25 and 26 that when the flow ate Q is small, thee is no tapping bolus but by inceasing the flow ate the tapping bolus appeas and the numbe of the tapping bolus inceases. The effects of the maximum height of the clot δ on the tapping ae illustated in Figues It is evident that the size of tapping bolus deceases with inceases δ, with the othe paametes being fixed. 6. Concluding emaks We have pesented a theoetical appoach to study the annulus peistaltic flow whee the inne tube has a clot on Figue 17. Gaph of the steamlines fo N =, m = 2, δ =.25, a =.2, Q =.5, φ =.4. Figue 15. The vaiation of F i with φ fo diffeent values of Q and z d at m = 2, N =.2, a =.2, δ =.1. Figue 18. Gaph of the steamlines fo N =.3, m = 2, δ =.25, a =.2, Q =.5, φ =.4

8 2 K. S. Mekheime and Y. A. Elmaboud Figue 19. Gaph of the steamlines fo N =.6, m = 2, δ =.25, a =.2, Q =.5, φ =.4. Figue 22. Gaph of the steamlines fo m = 1, N =.8, δ =.25, a =.2, Q =.5, φ =.4. Figue 2. Gaph of the steamlines fo N =.9, m = 2, δ =.25, a =.2, Q =.5, φ =.4. Figue 23. Gaph of the steamlines fo m = 7, N =.8, δ =.25, a =.2, Q =.5, φ =.4. Figue 21. Gaph of the steamlines fo m =.1, N =.8, δ =.25, a =.2, Q =.5, φ =.4. Figue 24. Gaph of the steamlines fo m = 1, N =.8, δ =.25, a =.2, Q =.5, φ =.4.

9 Applied Bionics and Biomechanics 21 Figue 25. Gaph of the steamlines fo Q =.1, N =.8, m = 2,δ =.25, a =.2, φ =.4. Figue 27. Gaph of the steamlines fo δ =, N =.8, m = 2, Q =.5, a =.2, φ =.4. Figue 26. Gaph of the steamlines fo Q =.6, N =.8, m = 2, δ =.25, a =.2, φ =.4. Figue 28. Gaph of the steamlines fo δ =.1, N =.8, m = 2, Q =.5, a =.2, φ =.4. its wall. The pesent analysis can seve as a model that may help in undestanding the mechanism of physiological blood flows in an annulus fo a fluid behaving like a micopola fluid. The main findings can be summaized as follows: The pessue ise inceases when the maximum height attained by the clot inceases. The pessue ise is highe in the case of a micopola fluid model than that fo a Newtonian fluid model. Thee is a linea elation between pessue ise and flow ate; an incease in the flow ate educes the pessue ise and thus maximum flow ate is achieved at zeo pessue ise and maximum pessue occus at zeo flow ate. The peistaltic pumping egion becomes wide with the incease in the coupling numbe, the maximum Figue 29. Gaph of the steamlines fo δ =.25, N =.8, m = 2, Q =.5, a =.2, φ =.4.

10 22 K. S. Mekheime and Y. A. Elmaboud height attained by the clot, amplitude atio and the axial displacement of the clot. Elevating the values of the amplitude atio poduces moe occlusion in the gap between the inne and oute tubes; this leads to incease in the pessue ise. The inne fiction foce behaves simila to the oute fiction foce fo the same values of the paametes; moeove, the oute fiction foce is geate than the inne fiction foce fo the same values of the paametes. The size of the tapping bolus inceases as the coupling numbe inceases nea the bounday wall oute tube, wheeas it inceases with inceasing micopola paamete micopola spin paamete effect up to a cetain value and then deceases fo lage values. The size of tapping bolus deceases with inceases in the height attained by the clot. In the absence of the inne tube the clot model, the esults ae the same as those obtained by Sinivasachaya et al. 23. Refeences Cowin SC Pola fluids. Phys Fluids 11:1919. Eingen AC Theoy of micopola fluids. J Math Mech. 16:1. Hayat T, Mahomed FM, Asgha S. 25. Peistaltic flow of a magnetohydodynamic JohnsonSegalman fluid. Nonlinea Dyn. 4:375. Jayaaman G, Saka A. 25. Nonlinea analysis of ateial blood flow-steady steaming effect. Nonlinea Anal. 63:88. Povost AM, Schwaz WH A theoetical study of viscous effects in peistaltic pumping. J Fluid Mech. 279:177. Rao AR, Misha M. 24. Peistaltic tanspot of a powe-law fluid in a poous tube. J Non-Newtonian Fluid Mech. 121:163. Shapio AH, Jaffin MY, Weinbeg SL Peistaltic pumping with long wavelengths at low Reynolds numbe. J Fluid Mech. 37:799. Sinivasachaya D, Misha M, Rao AR. 23. Peistaltic pumping of a micopola fluid in a tube. Acta Mech. 161:165. Sivastava LM Peistaltic tanspot of a couple stess fluid. Rheol Acta. 25:638. Sivastava LM, Sivastava VP Peistaltic tanspot of powelaw fluid: application to the ductus of effeuntus of the epoductive tact. Rheol Acta 27:428. Vajavelu K, Seenadh S, Ramesh Babu V. 25. Peistaltic tanspot of a Heschel-Bulkley fluid in an inclined tube. Int J Non-Linea Mech. 4:83. Appendix a 11 = 2 I 1 m 2 1 I 1 m 1, a 12 = I 1 m 2 K 1 m 1 I 1 m 1 K 1 m 2, a 13 = I m 1 I m 2 a 14 = 1 K 1 m 1 2 K 1 m 2, a 15 = K m 1 K m 2, a 16 = ,A= 1 N 2 N,F = N m a 17 = 1 I 1 m 2 2 I 1 m 1, a 18 = 1 K 1 m 2 2 K 1 m 1, a 19 = 2 K 1 m K 1 m 2, a 21 = I m 1 K m 2 I m 2 K m 1, a 22 = ln 1 I m 2 ln 2 I m 1, a 23 = a 11 a 15 a 13 a 14, 1 a 24 = 2Fa 23 4a ln, 2 a 25 = ln 1 K m 2 ln 2 K m 1, a 26 = a 14 I m 1 a 11 K m 1, a 27 = a 11 K m 2 a 14 I m 2, a 28 = 1 3 K 1m K 1m 2, a 29 = Fa 13 a 11,a 32 = 1 a 16 a 12, a 31 = 1 FA [a 17a 15 a 13 a 18 ], a 33 = Fa 16 a 15 a a a 18 ln a 35 = 1 F [2a ], 2 1, a 34 = Fa 16 a a 18 a 22 + a 31 + a 38, a 36 = a 16 a 29, 2 a 37 = a 17 ln, 1 a 38 = a 17 a 25, 1 a 39 = 2F a 23 + a 35 ln, 2 a 41 = a 36 + a 37, 1 a 43 = 2a ln a 16 Fa 15 + a 14, 2 a 44 = 2 I m 1 K 1 m 2, a 45 = 1 I m 2 K 1 m 1, a 46 = F 1 I 1 m 1 K m 2, 1 a 47 = I 1 m 1 ln, 2 a 48 = a 26 + a 35 ln 1, a 49 = a 27 + a 35 ln 2, a 51 = I 1 m 1 + I m 2, a 52 = I m 1 K 1 m 2 + I 1 m 2 K m 1, a 53 = 2F m I 1 m 1 a 51a 46 + Fa 52 2 I 1 m 1 + a 12 a 47,

11 Applied Bionics and Biomechanics 23 a 54 = 2 a a 16 I 1 m 1, a 55 = a I 1 m 1 K 1 m 2, b 11 = a 39, 2a 24 b 14 = a 26 + a 27, b 12 = 2a a 48 a 49 2a a , b 13 = a a a , b 15 = a 33 I 1 m 1 a 41 K 1 m 1 + a m ln 1, b 16 = 2 2 ln ln 1, b 17 = a 33 a 11 + a 41 a 14, b 18 = a 48 a 49, b 19 = 2ma 24 AF, ξ = 2ma 16a 39, AF η = 4b 17 + a 32 b 16 m 1 2 ma 42 2a 34 b 18 a 16

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