On the exact solution for peristaltic flow of couple-stress fluid with wall properties

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1 Bulgarian Chemical Communications, Volume 7, Number (pp. 7) On the exact solution for peristaltic flow of couple-stress fluid with wall properties S. Hina, M. Mustafa, T. Haat, Department of Mathematical Sciences, Fatima Jinnah Women Universit, Rawalpindi, Pakistan School of Natural Sciences (SNS), National Universit of Sciences and Technolog (NUST), Islamabad, Pakistan Department of Mathematics, Quaid-I-Azam Universit, Islamabad, Pakistan Department of Mathematics, Facult of Science, King Abdulaziz Universit, Jedda, Saudi Arabia Received November 9, ; Revised Februar, This communication reports the peristaltic motion of an electricall conducting couple-stress fluid in a channel with complaint walls. Mathematical model subjected to long wavelength and low Renolds number approximations is presented. A closed form exact solution for the dimensionless stream function is derived. Behaviors of different phsical parameters including Hartman number ( M ), couple-stress fluid parameter ( ), elastic parameters (,,, ) and amplitude ratio ( ) are thoroughl examined through the graphical results for velocit, and stream function. The stud reveals that fluid velocit significantl increases with an increase in the couple-stress fluid parameter and decreases with an increase in the magnetic field strength. Kewords: Couple-stress fluid, compliant wall, peristalsis, magnetic field, analtic solution. INTRODUCTION Peristaltic motion is the form of fluid transport that occurs due to the waves travelling along the walls of a channel/tube. This mechanism appears in man industrial and phsiological processes including rollers and finger pumps to pump sanitar and corrosive fluids. The peristaltic activit is quite prevalent in the gastrointestinal tract for food bolus transportation. It also occurs in the urinar tract to transport urine from kidne to bladder, in small blood vessels to transport blood, spermatozoa transport in the ductus efferentes, etc. Haroun [] studied the peristaltic flow of third grade fluid in an asmmetric channel. Peristaltic flow of fourth grade fluid in an inclined channel is also discussed b Haroun []. Kothandapani and Srinivas [] presented the peristaltic flow of Newtonian fluid in an inclined asmmetric channel. Here, the fluid saturates the porous medium. Kothandapani and Srinivas [] discussed the effect of magnetic field on the peristaltic flow of Jeffre fluid in an asmmetric channel. baid [] studied the effects of magnetic field and wall slip conditions on the peristaltic transport of Newtonian fluid in an asmmetric channel. Muthuraj and Srinivas [6] presented the mixed convective heat and mass transfer effects on the * To whom all correspondence should be sent: -mail: address: quaidan8@ahoo.com peristaltic flow in a vertical wav channel with porous medium. Srinivas et al. [7] extended the work of Muthuraj and Srinivas [6] for an asmmetric channel. Srinivas and Muthuraj [8] discussed the effects of chemical reaction and space porosit on MHD mixed convective peristaltic flow in a vertical asmmetric channel. Heat transfer in the pulsatile flow through a vertical annulus was discussed b lmaboud and Mekheimer [9]. Peristaltic flow of an electricall conducting micropolar fluid was described b Mekheimer []. It is now recognized that fluids involved in various industrial and phsiological processes are non-newtonian. In view of flow diversit in nature, all non-newtonian fluids cannot be described b a single constitutive relationship between stress and shear rate. Due to this reason several constitutive equations describing the motions of such fluids have been proposed b the researchers. The associated equations are mathematicall more complex and higher-ordered than the governing equations for a Navier-Stokes fluid. Couple-stress fluid is one of the non-newtonian fluids that describe rheologicall complex fluids such as liquid crstals, polmeric suspensions, infected urine, animal and human blood and man lubricants. Recentl, man researchers discussed the peristaltic flow of couple- stress fluid. Mekheimer [] analzed the effect of induced magnetic field on the peristaltic flow of couple-stress fluid. Mekheimer and lmaboud [] studied the peristaltic flow of couple-stress fluid in an annulus. Nadeem and Akram [] extended the work of Bulgarian Academ of Sciences, Union of Chemists in Bulgaria

2 S. Hina et al.: On the exact solution for peristaltic flow of couple-stress fluid with wall properties Mekheimer [] for an asmmetric channel. Rao FORMULATION OF TH PROBLM and Rao [] discussed the influence of heat Consider the peristaltic flow of couple-stress fluid transfer on the peristaltic transport of couple-stress in a compliant walls channel. The channel width is fluid through a porous medium. ldabe et al. [] studied the ()MHD peristaltic flow of couplestress fluid with heat and mass transfer through a and perpendicular to the channel walls, respectivel. taken as d. The x and axes are along porous medium. Heat transfer analsis in the An incompressible and electricall conducting fluid peristaltic flow of couple-stress fluid through is considered. A uniform magnetic field of strength asmmetric channel is performed b lmaboud et B is applied in the direction. The flow is al. [6]. induced due to the sinusoidal waves on the compliant In all above mentioned studies, the effect of walls of the channel and the wave shapes are wall properties is not taken into account. The described as follows: consideration of wall properties of the channel in peristaltic motion is realistic. Due to this reason ( x, t) d asin ( x ct). () the peristaltic flow in a channel with wall properties has great importance in the natural In the above equation c is the wave speed, a processes that exist in industr and phsiolog. the wave amplitude and the wavelength. The Muthu et al. [7] discussed the effect of wall governing equations for the problem under properties on the peristaltic motion of micropolar consideration are fluid. Radhakrishnamachara and Srinivasulu [8] u v studied the influence of wall properties on the, x () peristaltic motion with heat transfer. Kumari and Radhakrishnamachara [9] analzed the effect of slip boundar condition on the peristaltic transport p in an inclined channel with wall properties. u v u u u B u, t x x Kothandapani and Srinivas [] discussed the influence of wall properties on the MHD () p peristaltic flow with heat transfer and porous u v v v v, medium. Srinivas and Kothandapani [] t x () presented the heat and mass transfer on MHD and the boundar conditions are peristaltic flow through a porous space with u compliant walls. Haat et al. [] extended the u, at, () work of Srinivas and Kothandapani [8] for the flow of second grade fluid. Haat et al. [] discussed the peristaltic flow of Maxwell fluid in m d B H an asmmetric channel with wall properties. x xt tx x x (6) Mustafa et al. [] analzed the effect of wall properties on the peristaltic transport of nanofluid. u u B u u v u at. Mustafa et al. [] extended their work [] b t x considering the slip boundar conditions. Hina et al. [6] studied the peristaltic flow of In the above equations u and v denote the pseudoplastic fluid in a curved channel with wall velocities in the x and directions, properties. respectivel, / x /, p The objective here is to examine the denotes the pressure, is the electrical magnetohdrodnamic (MHD) peristaltic flow of conductivit of the fluid, the elastic tension in couple-stress fluid with wall properties. To our knowledge, no such attempt is made et. Problem the membrane, m the mass per unit area, d the formulation is presented. Closed form exact coefficient of viscous damping, B the flexural solutions for velocit and stream function are rigidit of the plate, H the spring stiffness, the constructed. ffects of couple-stress parameter, couple-stress fluid parameter, the dnamic Hartman number and compliant wall parameters viscosit and the densit. are also discussed in detail. We define the following dimensionless variables

3 S. Hina et al.: On the exact solution for peristaltic flow of couple-stress fluid with wall properties u v x ct u, v, x,, t, c c d p, (7) d p d, p,. with boundar conditions c d The problem statement in the dimensionless, at, variables is given b p Re u v u u u M u, t x (7) x x xt tx x x (8) M at. (6) p Re u v v v v, t x quations () and () ield (9) 6 M. 6 (7) u u, at sin ( x ct), Solving equation (7) subject to the boundar conditions () and (6), we have () C sinh m C sinh m L, (8) u m x xt tx x x C cosh m mc cosh m L, (9) where u u M u M M () m, m, Re u v u at. m L m L t x C, C, In the above equations / x / a / d is, ( ) the amplitude ratio, ( d / ) the wave number, Re ( cd / ) the Renolds number, M B d / the Hartman number, the couple-stress fluid parameter and ( d / ), ( mcd / ), c ( dd / ) ( Hd / c, ) are the non-dimensional elasticit parameters. Denoting the stream function ( x,, t) b u, v, () x we can see that the equation of continuit () is automaticall satisfied. Introducing stream function in the flow problem and then appling long wavelength and low Renolds number approximations, we obtain p M, () x m L M m m m m m m cosh cosh m 8 cos x t sin x t. RSULTS AND DISCUSSION This section discusses the variations of different emerging parameters on the axial velocit u and stream function. The variation of Hartman, elastic tension in the membrane, mass per unit area, coefficient of viscous damping, flexural rigidit of the plate, spring stiffness number M, amplitude ratio and couple-stress parameter () () are examined. Fig. elucidates the behavior of different parameters involved in the axial velocit. Fig. a shows that velocit decreases with an increase in Hartman number M.

4 u u u u u S. Hina et al.: On the exact solution for peristaltic flow of couple-stress fluid with wall properties M,, 8,......,,.,..... (a) (b) Fig. (a). Variation of M on u when ;. ;. ;. ;. ;. ;.; x. ; t.... Fig. (b). Variation of on u when ;. ;. ;. ;. ;. ; M ; x. ; t ,.,.7, x,.,.,.6,.8... (c) (d) Fig. (c). Variation of when ;. ;. ;. ;. ;. ; M ; x. t.. Fig. (d). Variation of x when ;. ;. ;. ;. ;. ;. ; M ; t ,,,,.7,.,.,.,.,,,,.,.,.,.,.,,,,.7,.,.,.,.,,,,.7,.,.6,.,.,,,,.7,.,.,.,.,,,,.7,.,.,., Fig. (e). Variation of compliant wall parameters when. ;. ; M ; x. ; t.. (e)

5 (a) Fig.. Streamlines for ;. ;. ;. ; ;. 8 ; t ;. ; (a): M ; (b): M.. (b) (a) Fig.. Streamlines for ;. ;. ;. ; ;. 8 ; t ; M ; (a):. ; (b):.. (b) (a) Fig.. Streamlines for ;. ;. ;. ; ;. ; t ; M ; (a):. ; (b):.8. (b)

6 (a) (b) (c) (d) (e) Fig.. Streamlines for. 8 ; M ;. ; t ; (a):. 7 ;. ;. ;. ; ; (b): ;. ;. ;. ; ; (c):. 7 ;. 7 ;. ;. ; ; (d):. 7 ;. ;. ;. ; ; (e):. 7 ;. ;. ;. ; ; (f):. 7 ;. ;. ;. ;.. (f)

7 S. Hina et al.: On the exact solution for peristaltic flow of couple-stress fluid with wall properties CONCLUSIONS Hartman number is the ratio of magnetic force to viscous force. Magnetic force being transverse to the flow direction, causes resistance to the flow and hence slows down the fluid motion. Fig. b depicts that velocit increases b increasing the couple-stress fluid parameter. Fig. c analzes the behavior of amplitude ratio on axial velocit. It is observed that b increasing the amplitude ratio the axial velocit increases. This is due to the fact that increasing values of the amplitude ratio correspond to an increase in the amplitude of the wave across the channel which thereb increases the fluid velocit within the channel. Fig. d is sketched to perceive the behavior of velocit distribution at different locations in the channel. It is seen that the axial velocit increases along the positive x direction. The influences of compliant wall parameters (,,, and ) on the velocit can be captured from Fig.e. It is observed that the velocit distribution increases with an increase in and, whereas it decreases when, and are increased. These observations ma be inferred to the fact that increase in the elasticit of the channel walls assists the flow whereas damping and stiffness of the wall resists the flow. Figs. - illustrate the behaviour of embedded parameters on the stream function Generall, the shape of streamlines is analogous to the wave travelling along the walls of the channel. Under certain conditions these streamlines split and enclose a bolus which moves along with the wave across the channel. The circulation and size of the trapped bolus are presented in these Figs. Fig. shows the impact of Hartman number or equivalentl the magnetic field on the streamlines. It is observed that the size of trapped bolus and the number of circulations decrease with an increase in M. Fig. shows that the size of bolus increases with an increase in couple-stress fluid parameter. Fig. analzes the effect of amplitude ratio on the stream function. It is clear that size of the trapped bolus and number of circulations are increased b increasing the amplitude ratio. Fig. is plotted to examine the behavior of the wall elasticit parameters. It is observed that the size of trapped bolus increases b increasing, and it decreases b increasing, and. In this article, MHD peristaltic motion of couplestress fluid is discussed in a compliant walls channel. A closed form exact solution of the problem is presented after appling long wavelength and low Renolds number assumptions. The effects of different parameters on the axial velocit and streamlines are thoroughl discussed. It is observed that the axial velocit and the size of trapped bolus decreases b increasing the Hartman number, wall damping parameters and wall stiffness parameters, whereas axial velocit and size of trapped bolus increases b increasing the couple-stress fluid parameter, amplitude ratio and wall elasticit parameters. The present stud finds an important application in describing blood flow in arteries since the rheological properties of blood can be adequatel described through the couple-stress fluid model. Moreover, the human arteries and veins possess the properties of elasticit, damping and stiffness which are also taken into consideration in this work. RFRNCS. M.H. Haroun, Comm. Nonlinear Sci. Numer. Simul.,, 6 (7).. M.H. Haroun, Comput. Material Sci., 9, (7).. M. Kothandapani and S. Srinivas, Phs. Lett. A, 7, 6 (8).. M. Kothandapani and S. Srinivas, Int. J. Non-Linear Mech.,, 9 (8).. A. baid, Phs. Lett. A, 7, 9 (8). 6. R. Muthuraj and S. Srinivas, Comp. Math. Applicat., 9, 6 (). 7. S. Srinivas, R. Gaathri and M. Kothandapani, Comm. Nonlinear Sci. Numer. Simul., 6, 8 (). 8. S. Srinivas and R. Muthuraj, Math. Comp. Model.,, (). 9. Y.A. lmaboud and Kh. S. Mekheimer, Z. Naturforsch., 67a, 8 ().. Kh. S. Mekheimer, Appl. Math., (8) Article ID 78 doi:./8/78.. Kh.S. Mekheimer, Phs. Lett. A, 7, 7 (8).. Kh.S. Mekheimer and Y. Abd elmaboud, Phsica A, 87, (8).. S. Nadeem and S. Akram, Arch. Appl. Mech., 8, 97 ().. T.R. Rao and D.R.V.P. Rao, Advances Appl. Sci. Research,, ().. N.T. ldabe, S.M. lshabour, A.A. Hasan and M.A. logail, Design and ngineering,, (). 6. Y.A. elmaboud, Kh. S. Mekheimer and A. I. Abdellateef, J. Heat Transf. ASM, (), doi:./ P. Muthu, B.V.R. Kumar and P. Chandra, ANZIAM J.,,, (). 8. G. Radhakrishnamachara and C. Srinivasulu, C. R. Mecanique,, 69 (7). 6

8 S. Hina et al.: On the exact solution for peristaltic flow of couple-stress fluid with wall properties 9. A.V.R. Kumari and G. Radhakrishnamachara, Int.. T. Haat, S. Hina, S. Asghar and S. Obaidat, Int. J. J. Appl. Math. Mech., 7, (). Phsical Sci., 7, ().. M. Kothandapani and Srinivas, Phs. Lett. A, 7,. M. Mustafa, S. Hina, T. Haat and A. Alsaedi, Int. J. 86 (8). Heat Mass Transfer,, 87 ().. S. Srinivas and M. Kothandapani, Appl. Math.. M. Mustafa, S. Hina, T. Haat and A. Alseadi, J. Heat Comput.,, 97 (9). Transfer,, 7(-7) ().. T. Haat, S. Hina and A. A. Hendi, Heat Transfer-. 6. S. Hina, M. Mustafa, T. Haat and A. Asian Research,, 77 (). Alseadi, J. Appl. Mech., 8, (-7) (). ОТНОСНО ТОЧНОТО РЕШЕНИЕ ЗА ПЕРИСТАЛТИЧНО ТЕЧЕНИЕ НА ФЛУИД СЪС СПРЕГНАТИ НАПРЕЖЕНИЯ И ПРОМЕНЛИВИ СВОЙСТВА НА СТЕНАТА С. Хина, М. Мустафа, Т. Хаят, Департамент по математика, Дамски университет Фатима Джина, Равалпинди, Пакистан Колеж по природни науки, Национален университет по наука и технология, Исламабад, Пакистан Департамент по математика, Университет Куаид-и-Азам, Исламабад, Пакистан Департамент по математика, Научен факултет, Университет Крал Абдулазис, Джеда, Саудитска Арабия Получена на 9 ноември г.; коригирана на февруари г. (Резюме) В тази работа се разглежда перисталтичното движение на електропроводящ флуид със спрегнати напрежения (couple-stress fluid) в канал със стени с променлива твърдост. Въведен е математичен модел за ламинарно вълново течение при голяма дължина на вълната. Получено е точно решение за безизмерната токова функция. Подробно са изследвани различни параметри, като числото на Hartman number (M), параметъра на спрегнати напрежения (), параметрите на еластичността (; ; ; ; ) и отношението на амплитудите () чрез резултатите за скоростта на течението и токовата функция. Изследването показва, че скоростта на течението се повишава значително с нарастване на параметъра на спрегнатите напрежения и се понижава, когато се приложи магнитно поле. 7