Slip Effects on Peristaltic Transport of Casson Fluid in an Inclined Elastic Tube with Porous Walls

Transcription

1 3, Issue (8) 67-8 Journal of Advanced Researc in Fluid Mecanics and Termal Sciences Journal omepage: ISSN: Slip Effects on Peristaltic Transport of Casson Fluid in an Inclined Elastic Tube wit Porous Walls Open Access Manjunata Gudekote,, Rajasekar Coudari Department of Matematics, Manipal Institute of Tecnology, Manipal Academy of Higer Education, Udupi Karkala Road, Manipal, Karnataka 576, India Karnataka, India ARTICLE INFO Article istory: Received 5 December 7 Received in revised form 7 February 8 Accepted February 8 Available online 9 Marc 8 Keywords: Darcy number, elastic parameters, pressure rise, velocity slip, yield stress ABSTRACT Te present study investigates te combined effects of slip and inclination on peristaltic transport of Casson fluid in an elastic tube wit porous walls. Te modeled governing equations are solved analytically by considering te long wavelengt and small Reynolds number approximations. A parametric analysis as been presented to study te effects of Darcy number, te angle of inclination, elastic parameters, velocity slip, yield stress, amplitude ratio, inlet and outlet radius on volumetric flow rate. Te study reveals tat an increase in te angle of inclination as a proportional increase in te pressure rise. Also, an increase in te porosity as a significant reduction in te pressure rise. Copyrigt 8 PENERBIT AKADEMIA BARU - All rigts reserved. Introduction Peristalsis is a mecanism induced by te progressive wave of area contraction and expansion wic travels along te walls of te distensible tube. Over te past decades, numerous researcers ave investigated te peristaltic transport due to its broad application in te field of biomedical engineering to design and construct many useful devices suc as blood pump macine and dialysis macine []. Moreover, it is a neuromuscular property of a biological framework in wic biofluids are transported along a tube by te propulsive movements of te tube wall. Many biological fluids in te uman body ave te peristaltic nature, suc as movement of te bolus in te esopagus region, cime movement in te cervical canal, te flow of blood in arteries, transportation of urine troug te ureter, etc. Te preliminary study on peristaltic flow was carried out by Latam [] to investigate te flow of urine troug te ureter. Later, Burns and Parkes [3] studied peristaltic transport by taking two cases, in te first instance tey considered peristaltic flow witout pressure gradient, and for te second case, tey considered te flow under pressure along a cannel or tube. Te initial studies on peristalsis were carried by taking te Newtonian fluid to understand te pysiological beavior of Corresponding autor. address: manjunata.g@manipal.edu (Manjunata Gudekote) 67

2 Volume 3, Issue (8) 67-8 biological systems. Te Newtonian approac may be sufficient to understand te flow of classical fluids troug microcannel under various assumptions [,5] and urine flow troug te ureter, but it fails to explain te complex reological action in te stomac, lympatic vessels and flow of blood in arteries. Tis empasizes to use te non-newtonian models to investigate te pysiological beavior in suc systems. Te initial attempt on peristaltic transport of non-newtonian fluid was carried out by Raju and Devanatan [6] by using Power law model. Later, numerous researcers ave investigated te peristaltic mecanism by using different non-newtonian models flowing troug different geometries [7-9]. Te examination of non-newtonian nature of bloodstream as been of most significance to scientists lately because of teir application in exploring te conduct of blood in small arteries. In suc circumstances, te presence of slip on te boundary due to te porosity of te walls as a fundamental impact in reviewing te non-newtonian nature of blood. Hence, slip implications are more verbalized for liquids traveling troug geometries wic ave elastic property, like blood vessels. Tis slip flow of fluids is used in polising of te internal cavities and artificial eart valves. Studies on te use of porous boundaries on peristaltic transport ave been initially investigated by Elseawey et al., []. Later, numerous researcers studied peristaltic flow using Maxwell fluid by taking porous walls into account [-3]. Most of te work done in te literature are carried out by taking Newtonian, Maxwell or Herscel-Bulkley model. Tis approac neglects to clarify te pysiological conduct of bloodstream in supply routes because of te presence of yield stress. Even toug, Herscel-Bulkley flow contains yield stress parameter, it is found tat at low sear rates Casson model fits te blood flow better tan tat of Herscel-Bulkley fluid []. In tis way, various researcers did te investigation of Casson and Herscel-Bulkley fluid under different pysiological situations in recent times [5 ]. Te nonlinearity in vascular beds of warm-blooded creatures is ascribed to te flexible idea of veins and teir immense distensibility. Tis elastic property of veins was first perceived by Young []. Afterwards, Rubinow and Keller [] exibited tat te scope of te tube could be controlled by te strain in te dividers and te transmural weigt contrast by accepting tat te Poiseuille law olds locally. Consequently, tere is a necessity for te subjective speculation of blood flow troug tubes wic are elastic in nature. Te stream designs acquired by te models wit rigid tube can't clarify te conduct of te stream of blood troug contracted supply routes completely. Hencefort, it becomes important to consider te elasticity in te present model. To te best of autors knowledge, no attempts ave been made in te literature to investigate te combined effects of slip and inclination on peristaltic transport of Casson fluid in an axisymmetric porous tube. Te present investigation is elpful in filling te gap in tis direction. Te obtained results are in good agreement wit tat of Vajravelu et al., [6] and Sumalata and Sreenad [3] in te absence of velocity slip and porous walls. Te resulting equations are solved analytically under te appropriate slip boundary conditions. Te influence of yield stress, amplitude ratio, Darcy number, slip parameter and elastic parameters on flux are represented grapically. Te outcomes of te present model elp in understanding te complex pysiological response of blood in te circumstances mentioned above, wic intern elps medical people to investigate te blood flow in arteries muc better way tan te earlier and elps in modeling te eart-lung, dialysis macines, etc.. Matematical Formulation Te flow of blood is modelled to be laminar, steady, incompressible, two-dimensional, fullydeveloped, axisymmetric, inclined at an angle β wit te orizontal and exibiting peristaltic 68

3 Volume 3, Issue (8) 67-8 motion in an elastic tube (Fig. ). Te flow of blood is modelled by te Casson fluid and facilitates te coice of te cylindrical coordinate system to study te problem. Te wall deformation, due to an infinite sinusoidal propagation of peristaltic waves, is represented by π ( r, z) = + ε sin ( z ct). λ were ε is te amplitude ratio, λ is te wavelengt, c is te wave speed, t is te time and z is te axial direction. () Fig.. Geometrical representation of Peristaltic waves in an elastic tube 3. Matematical Modelling Te equations of motion and energy in te wave frame of reference, moving wit speed c, under te lubrication approac (Nadeem and Akbar []) is as follows: p Re u + w w = + ( rτ ) + δ ( τ zz ), () r z z r r r 3 p δ Reδ u + w u = + ( rτ rr ) + δ ( τ ), r z z r r r (3) were u and w are te axial and radial velocities, R e is te Reynolds number, δ is wave number, r is radial coordinate, τ rris sear stress in radial coordinates τ zr is sear stress in axial and radial coordinates, τ zz is sear stress in axial coordinate and τ is te sear stress along radial and axial coordinates. Te following nondimensional variables are introduced. 69

4 Volume 3, Issue (8) 67-8 τ τ ρ δ r =, z =, t =, =, =, P =, R =, r z ct Pa ca τ τ e a λ λ c c λ cµ µ µ µ a a rp b u w τ a µ c rp =, ε =,, u =, w =, τ a =, F =. c a c c c ρga µ a () Under te assumption of long wavelengt δ << and small Reynolds number ( R = ), equations () and (3) takes te form as, e p sin β ( rτ ) = +, r r z F p =. r (5) (6) Te constitutive equation for Casson's fluid in te non-dimensional form is given by w = τ τ, τ τ, r w =, τ τ. r (7) (8) Te corresponding non-dimensional boundary conditions are w α w = at r = r Da (9) τ is finite at r =. () Te slip boundary condition (equation (9)) is given by Saffman [5]. Furter, Da represents te Darcy number (porous parameter) and α represents te slip parameter. Solving equations (5) and (6) subjected to te boundary conditions (9) and () we obtain te velocity w as, 3 3 ( ) ( ) P + f ( ) Da P + f w = rp r r rp ( r ) rp r p, α () were P P =. z r p Prp Using te condition τ = at r = rp, te upper limit of plug flow region is obtained as τ =. Also by using te condition τ = τ at r = (Bird et al., [6]), we obtain P τ P =. () 7

5 Volume 3, Issue (8) 67-8 Hence, r p τ τ = = τ. (3) Using relation (3) and taking r = rpin equation (), we obtain te plug flow velocity p w as, P + f 3 rp Da( P + f ) wp = rp rp rp r p. 3 6 α () Te instantaneous flow rate Q across any cross section of te artery is defined as given below: r p Q = wp r dr + w r dr. r p (5) ( P + f ) F Q =, (6) 8 6 τ Da were, F = τ + τ + ( τ ). 7 3 α It is wort mentioning tat, te results of Vajravelu et al., [6] can be obtained as a special case by taking Da = and β = in equation (6).. Teoretical Determination of Flux wit an Application to Flow troug an Artery A teoretical calculation of te flux Q is carried out for an incompressible Casson fluid troug an elastic tube of radius ( z, t) = '( z, t) + ''( z). Te fluid is assumed to enter te tube wit a pressure Pand leave te tube wit pressure P, wile te pressure outside te tube is P. If z denotes te distance along te tube from te inlet end, ten te pressure P( z ) in te fluid at z diminises from P() = P to P( λ ) = P. Te tube may contract or expand due to te difference in pressure of te fluid P( z) P. Subsequently, te cross section of te tube may ave a deformation due to te elastic property of te walls. Tus, te difference in pressure influences te conductivity σ of te tube at z. We consider te conductivity σ = σ[ P( z) P ] to be a known function of te pressure difference ( P( z) P ). Tis conductivity is assumed to be te same as tat of a uniform tube aving an identical cross section at z. Te relation between Q and te pressure gradient is given by Q = σ ( P P )( P + f ). (7) Under te considerations of peristaltic motion and te elastic property of te tube wall, equation (7) can be written as, 7

6 Volume 3, Issue (8) 67-8 F σ( P P ) = ( ' + ''), (8) 8 were '' is te cange in radius of te tube due to elasticity and is a function of pressure P P at eac cross section due to te Poiseuille flow i.e., [ ''( P P )]. Equation (7) wit te inlet condition P() = P gives P P F σ( ') ' ( ' ''), 8 P( z) P (9) Q = P dp + f + were P ' = P( z) P. Tis equation gives P( z ) in terms of z and Q. Setting z = and P() = P in equation (9), we get Q as, P P P P F ( ') ' ( ' ''). 8 () Q = σ P dp + f + Now, using equation (8) in equation (), we ave F Q = + dp + f + 8 P P ( ' '') ' ( ' ''). () P P Equation () can be solved if we explicitly know te function ''( P P ). If '' is known as a function of te tension T( '') or stress, ten ''( P ') can be determined from te equilibrium condition (Rubinow and Keller []) given by T( '') = P P. '' Rubinow and Keller [] carried out experimental investigations by controlling static pressure volume connection of a -cm long piece of a uman iliac artery and gave an expression for tension in an elastic tube as: T t t 5 ( '') = ( '' ) + ( '' ). Using equation (3) wit t = 3 and t = 3, equation () takes te following form t 3 dp ' = + t '' 5 '' + '' + d ''. '' '' Using equation (), equation () can be written as F Q = + + t + + d + f + 8 '' '' P P t 3 ( ' '') '' 5 '' '' '' ( ' '') P P (5) () (3) () Letting P = P and P = P in equation () te solutions are obtained for Equation (5) can be rewritten as ( ) F Q = g( ) g( ) + f, 8 '' and '' respectively. (6) 7

7 Volume 3, Issue (8) 67-8 were, '' 3 ' '' '' g( ) = t + ' '' + 6 ' '' + ' log log '' + t + ( 6 ' 5) + 3 '' '' '' 3 ( ' 6 ' + ) + ( 6 ' 9 ' + 8 ' ) '' '' '' 3 3 ( ' 6 ' ' ') ( 5 ' 8 ' 6 ' ) 3 + ( ' ' + ' ) + ' ( ) 3 ' ' ' + 6 ' + ' log log ''. '' (7) 5. Results and Discussion Te present paper empasizes te flow beavior of peristaltic transport of blood in an elastic tube, modeled as a Casson fluid. Te present model is te extension of te work carried out by Rubinow and Keller [] and Vajravelu et al., [6] by considering te effects of slip and inclination in a porous tube. Te results are obtained for te effects of various pysiological parameters on te flow rate are analyzed grapically wit te elp of MATLAB and are presented in Figures to. Fig.. Q versus z for varying τ wit =., t = 3, t = 3, π ε =.6, β =,.,.,.3. Da = α = = 73

8 Volume 3, Issue (8) 67-8 Figure sows te variation of yield stress ( τ ) on flow rate ( Q ). It is noticed tat an increase in te values of τ decreases te flow rate in an inclined elastic tube. Tis beavior is expected due to te presence of yield stress in te model. Figure 3 compares te results of present and Vajravelu et al. [6] model for a fixed value of yield stress ( τ =.). Te results of te present model are plotted by taking Da = and β = to sow te comparison wit teir model. It is clear from Figure 3 tat te results of te present model exactly matc wit teir results, wic also validate our results. Furter, te sligt variation is observed due to te nature of sinusoidal wave considered in te models. Figures and 5 represent te beavior of amplitude ratio ( ε ) and porous parameter ( Da ) on Q respectively. It is observed tat an increase in te values of ε and Da increases te flow rate Q. Te effect of velocity slip parameter sows te opposite trend as tat of ε and Da (Figure 6). Te variation of angle of inclination ( β ) on flow rate Q is plotted in Figure 7. It is noticed tat Q increase wit an increase in te angle β. Te observation of β is in concurrence wit te results of Sumalata and Sreenad [3]. Figures 8 and 9 are plotted to examine te variation of elastic parameters t and t on flow rate Q respectively. Flux in an elastic tube increases wit an increase in te values of elastic parameter t (Figure 8). Te same trend olds good for te oter parameter, namely t (Figure 9). Te flux profiles wit inlet and outlet elastic radius variations are sown grapically in Figures and. For a fixed value of outlet radius, te effect of increasing values of inlet elastic radius makes te flux to decrease and ence flow rate decreases as te elastic radius increases (Figure ). However, te opposite beavior is observed wen we fix inlet elastic radius and vary outlet elastic radius (Figure ). Fig. 3. Q versus z wit =., t = 3, t = 3, π ε =.6, β =,.,.,.3. Da = α = = 7

9 Volume 3, Issue (8) 67-8 Fig.. Q versus z for varying ε wit =., t = 3, t = 3, π τ =.5, β =,.,.,.3. Da = α = = Fig. 5. Q versus z for varying Da wit =., t = 3, t = 3, π τ =.5, β =, ε =.5, α =., =.3. 75

10 Volume 3, Issue (8) 67-8 Fig. 6. Q versus z for varying α wit =., t = 3, t = 3, π τ =.5, β =, ε =.5, Da =., =.3. Fig. 7. Q versus z for varying β wit =., t = 3, t = 3, τ =.5, α =., ε =.5, Da =., =.3. 76

11 Volume 3, Issue (8) 67-8 π Fig. 8. Q versus z for varying t wit =., β =, t = 3, τ =.5, α =., ε =.5, Da =., =.3. π Fig. 9. Q versus z for varying t wit =., β =, t = 3, τ =.5, α =., ε =.5, Da =., =.3. 77

12 Volume 3, Issue (8) 67-8 π Fig.. Q versus z for varying wit =.6, β =, t = 3, τ =.5, α =., ε =.5, Da =., t = 3. π Fig.. Q versus z for varying wit =.3, β =, t = 3, τ =.5, α =., ε =.5, Da =., t = 3. 78

13 Volume 3, Issue (8) Summary and Conclusion Te present investigation deals wit te study of peristaltic motion of blood in te uman circulatory system. Te blood flow is modeled by Casson fluid in an elastic tube wit porous walls. Te present investigation gives an agreeable result tat speaks to a portion of te natural penomena, mainly, te stream of blood in narrow arteries wic can be taken care of and prepared if tere sould be an occurrence of dysfunction. Furter, tere is a possibility of extending te present model under te influence of external magnetic field wic will elp te doctors to control te flow of blood during surgeries. Some of te interesting findings are Te effects of increasing values of yield stress and inlet elastic radius decreases te flux in an elastic tube, wereas, flux increases wit increasing values of amplitude ratio, elastic parameters and outlet elastic radius. Te flux in an elastic tube increases wit an increasing value of te Porous parameter (Darcy number) and decreases wit slip parameter. Te effect of angle of inclination as a significant role in increasing te flow rate in an elastic tube. Te effects of velocity slip, angle of inclination and porous walls ave a major role in determining te flux in an elastic tube. References [] Jaggy, C., M. Lacat, B. Leskosek, G. Zünd, and M. Turina. Affinity pump system: a new peristaltic blood pump for cardiopulmonary bypass. Perfusion 5, no. (): [] Latam, Tomas Walker. Fluid motions in a peristaltic pump. PD diss., Massacusetts Institute of Tecnology, 966. [3] Burns, J. C., and T. Parkes. Peristaltic motion. Journal of Fluid Mecanics 9, no. (967): [] Abubakar, S. B., and NA Ce Sidik. Numerical prediction of laminar nanofluid flow in rectangular microcannel eat sink. J. Adv. Res. Fluid Mec. Term. Sci. 7, no. (5): [5] Soo Weng Beng and Wan Mod. Arif Aziz Japar. Numerical analysis of eat and fluid flow in microcannel eat sink wit triangular cavities. 3 (7): -8. [6] Raju, K. K., and Ratna Devanatan. Peristaltic motion of a non-newtonian fluid. Reologica Acta, no. (97): [7] Usa, S., and A. Ramacandra Rao. Peristaltic transport of two-layered power-law fluids. Journal of biomecanical engineering 9, no. (997): [8] Misra, J. C., and S. K. Pandey. Peristaltic transport of blood in small vessels: study of a matematical model. Computers & Matematics wit Applications 3, no. 8-9 (): [9] Maiti, S., and J. C. Misra. Non-Newtonian caracteristics of peristaltic flow of blood in microvessels. Communications in Nonlinear Science and Numerical Simulation 8, no. 8 (3): [] El Seawey, E. F., K S. Mekeimer, S. F. Kaldas, and N. A. S. Afifi. Peristaltic transport troug a porous medium. J. Biomat, no. (999): -3. [] Hayat, T., N. Ali, and S. Asgar. Hall effects on peristaltic flow of a Maxwell fluid in a porous medium. Pysics Letters A 363, no. 5-6 (7): [] Nadeem, S., and Safia Akram. Peristaltic flow of a Maxwell model troug porous boundaries in a porous medium. Transport in Porous Media 86, no. 3 (): [3] Tripati, Darmendra, and O. Anwar Bég. A numerical study of oscillating peristaltic flow of generalized Maxwell viscoelastic fluids troug a porous medium. Transport in porous media 95, no. (): [] Blair, GW Scott. An equation for te flow of blood, plasma and serum troug glass capillaries. Nature 83, no. 66 (959): 63. [5] Nagarani, P. Peristaltic transport of a Casson fluid in an inclined cannel. Korea-Australia Reology Journal, no. (): 5-. [6] Vajravelu, K., S. Sreenad, P. Devaki, and K. V. Prasad. Peristaltic Pumping of a Casson Fluid in an Elastic Tube. Journal of Applied Fluid Mecanics 9, no. (6):

14 Volume 3, Issue (8) 67-8 [7] Manjunata, G., and K. S. Basavarajappa. Peristaltic Transport of Tree Layered Viscous Incompressible Fluid. Global Journal of Pure and Applied Matematics 9, no. (3): [8] Manjunata, G., Basavarajappa, K. S. and Katiyar, V. K. Matematical modelling on peristaltic transport of two layered viscous incompressible fluid wit varying amplitude. International Journal of Matematical Modelling and Pysical Sciences ():-9. [9] Vajravelu, K., S. Sreenad, P. Devaki, and K. V. Prasad. Peristaltic transport of a Herscel Bulkley fluid in an elastic tube. Heat Transfer Asian Researc, no. 7 (5): [] Caturani, P., and S. Narasimman. Teory for flow of Casson and Herscel-Bulkley fluids in cone-plate viscometers. Bioreology 5, no. - (988): [] Young, D. F. Effect of a time-dependent stenosis on flow troug a tube. Journal of Engineering for Industry 9, no. (968): 8-5. [] Rubinow, S. I., and Josep B. Keller. Flow of a viscous fluid troug an elastic tube wit applications to blood flow. Journal of teoretical biology 35, no. (97): [3] Sumalata, B. and Sreenad, S. Poiseuille flow of a Jeffery fluid in an inclined elastic tube. International Journal of Engineering Sciences and Researc Tecnology 7 (8): 5-6. [] Nadeem, S., and Noreen Ser Akbar. Influence of eat transfer on a peristaltic transport of Herscel Bulkley fluid in a non-uniform inclined tube. Communications in Nonlinear Science and Numerical Simulation, no. (9): -3. [5] Saffman, Pilip Geoffrey. On te boundary condition at te surface of a porous medium. Studies in applied matematics5, no. (97): 93-. [6] Bird, R. B., Stewart, W. E. and Ligtfoot, E. N. Transport Penomena, Wiley, New York, 976 8