Peristaltic Pumping of a Non-Newtonian Fluid

Available at ttp://pvamu.edu/aam Appl. Appl. Mat. ISSN: 93-9466 Vol. 3, Issue (June 8), pp. 37 48 (Previously, Vol. 3, No. ) Applications and Applied Matematics: An International Journal (AAM) Peristaltic Pumping of a Non-Newtonian Fluid Amit Medavi Department of Mecanical Engineering Sri Ramswaroop Memorial College of Engineering and Management Tiwariganj, Faizabad Road, Lucknow-75, India amitmedavi@yaoo.co.in Received August 4, 7; accepted April 9, 8 Abstract Te flow induced by sinusoidal peristaltic motion of te tube wall of a non-newtonian fluid obeying Herscel-Bulkley equation (a general reological equation tat represents a powerlaw, Bingam and Newtonian fluid for particular coice of parameters) under long wavelengt and low Reynolds number approximation is investigated. Te results obtained for flow rate, pressure drop and friction force are discussed bot qualitatively and quantitatively and compared wit oter related studies. It is found tat te pressure drop increases wit te flow rate and yield stress but decreases wit te increasing amplitude ratio. Te flow beaviour index sows significant impact on te magnitude of te pressure drop. Te pressure-flow rate relationsips in Bingam and Newtonian fluid models are found to be linear wereas te same are non-linear in power- law and Herscel-Bulkley models. Te friction force possesses te caracter similar to te pressure drop (an opposite caracter to te pressure rise) wit respect to any parameter. Keyword: Herscel-Bulkley, Peristaltic Wave, Pressure Drop, Flow Rate, Friction Force, Reynolds Number AMS Subject Classification Numbers: 76Z5. Introduction Pumping of fluids troug flexible tubes by means of te peristaltic wave motion of te tube wall as been te subject of engineering and scientific researc for over four decades. Engineers and pysiologists term te penomenon of suc flow as peristalsis. It is a form of fluid transport induced by a progressive wave of area contraction or expansion along te 37

38 Amit Medavi walls of a distensible duct containing a liquid or mixture. Besides its various engineering applications (e.g., eart-lung macines, finger and roller pumps, etc.), it is known to be significant mecanism responsible for fluid transport in many biological organs including in swallowing food troug esopagus, urine transport from kidney to bladder troug te ureter, movement of cyme in gastrointestinal tract, transport of spermatozoa in te ductus efferentes of te male reproductive tracts, and, in cervical canal, in movement of ovum in te female fallopian tubes, transport of lymp in lympatic vessels, and in te vasomotion of small blood vessels suc as arterioles, venules and capillaries. Latam (966) was probably te first to study te mecanism of peristaltic pumping in is M.S. Tesis. Sapiro et al. (969) and Jaffrin and Sapiro (97) explained te basic principles and brougt out clearly te significance of te various parameters governing te flow. Te literature on te topic is quite extensive by now and a review of muc of te literature up to te year 983, arranged according to te geometry, te fluid, te Reynolds number, te wave number, te amplitude ratio and te wave sape was presented in an excellent article by Srivastava and Srivastava (984). Te significant contributions to te subject between te years 984 and 994 were well referenced by Srivastava and Saxena (995). Te important studies of te recent years include te investigations of Srivastava and Srivastava (997), Mekeimer et al. (998), El- Seawey and Husseny (), El-Hakeem and El-Misery (), Mutu et al. (), Srivastava (), Misra and Pandey (), Hayat et al. (,3), Mekeimer (3), Hayat et al. (4), Siddiqui et al. (4), Misra and Rao (4), Rao and Misra (4), Hayat et al. (5), Hayat and Ali (6a,b), Srivastava (7), Medavi (8), Medavi and Sing (8) and a few oters. In bot te mecanical and pysiological situations, most of te studies conducted in te literature considered te transported fluid to be a Newtonian fluid. It is well accepted tat a large number of fluids of practical importance beave like a non-newtonian fluid. A survey of te topic indicates tat a few studies (Raju and Devenatan, 97, 974; Becker, 98; Sukla and Gupta, 98; Bome and Friedric, 983; Srivastava and coworkers, 984, 985, 995; Misra and Pandey,, Vijravelua et al., 5, etc.) ave used non-newtonian fluids in teir works. It may be mentioned tat Srivastava and coworkers (984, 995) and Misra and Pandey () made teir studies a particular reference to peristaltic pumping of blood in small vessels. Te present article deals wit te flow of a special type of a non-newtonian fluid obeying te Herscel-Bulkley equation. It is to note tat Herscel-Bulkley equation can be reduced to matematical models wic describe te beavior of Bingam, power-law and Newtonian fluids for particular coice of parameters involved. Since te Herscel-Bulkley fluid is a general fluid representing power-law, Bingam and Newtonian fluids, it is strongly believed tat te study covers up a wide range of applications in engineering as well as in pysiology.. Formulation of te Problem and Analysis Consider te asymmetric flow of a non-newtonian (Herscel-Bulkley) fluid in a uniform circular tube wit a sinusoidal wave traveling down its wall. Te geometry of te wall surface is described (Fig. ) as

AAM: Intern, J., Vol. 3, Issue (June 8) [Previously, Vol. 3, No. ] 39 H (x, t) = a + b Sin (x ct), () were a is te mean radius of te tube, b is te amplitude of te wave, is te wavelengt, c is te wave propagation speed and t is te time. Te appropriate equation in te wave frame of reference (moving wit speed c) under long wavelengt approximation and neglecting te inertia terms (Sapiro et al., 969), is written as r r d p ( r r x), () d x were rx for Herscel-Bulkley fluid is given (Caturani and Narasimman, 988) u f ( ) ( ), r k n - rx rx, (3) u f ( ), rx r, (4) n n were p (dyne cm ) is te pressure, k [(cp) s ] and n (>) are consistency and flow beavior indexes, respectively and representing te non-newtonian effects, (dyne cm ), is te yield stress, (x, r) are (axial, radial) coordinates, and u (cm s ) is te axial velocity of te fluid. Relation (4) corresponds to te vanising of te velocity gradient in te region in wic r x < and implies a plug flow. Wen sear stress in te fluid is very ig (i.e., r x ), te power-law beavior is indicated. It is wort mentioning tat above Herscel-Bulkley fluid model reduces to Bingam fluid wen n =, k= (Newtonian

4 Amit Medavi viscosity); to power-law fluid wen = and to Newtonian fluid wen n =, k = and =. It is important to note tat te plug core radius increases wit yield stress, and also wit te flow beavior index, n (Caturani and Narasimman, 988). An introduction of te following non-dimensional variables r' = a r, x ' = x, t' = ct, u = c u, ' (a/ck) /n, p = (a n+ /ck n ) /n p, (5) into equations () (4), after dropping primes, yields / n u dp r, rx, (6) r r r dx u r, rx, (7) were rx and are dimensionless searing and yield stresses, respectively. Te non-dimensional boundary conditions are were u = at r = H/a = = + sin x, rx is finite at r =, (8) b / a. Te expression for velocity, u obtained as te solution of equation (6) under te boundary conditions (8), is given as n n n ( ) rdp dp u. (9) n dp/ dx dx dx One derives te expression for te fluid velocity in te plug flow region by substituting te radius of te plug flow region for te radial variable r in equation (9) wic obviously satisfies equation (7) in te plug flow region. Now considering tat te radius of te plug flow area to be small as compared to te non-plug flow area (Srivastava and Saxena, 995), te instantaneous volumetric flow rate, q (= q ' / a c, q' being te flux in moving system wic is same as in stationary system) is calculated as

AAM: Intern, J., Vol. 3, Issue (June 8) [Previously, Vol. 3, No. ] 4 q = rudr = 3 n + ( n 3) n ( n ) ( )( ) n n n 3, () ( n)( n) were = ( /) dp/dx is te sear stress at te wall. Under te condition, / <<, equation () yields dp = dx 3 / n n n [ q n / ( 3) ( ) ( 3) ]. () ( n ) Following te analysis of Sapiro et al. (969), one obtains te mean volume flow rate, Q over a period as Q = q + +. () Since te pressure drop, p = p() p(), across one wavelengt is same weter measured in moving or stationary coordinate system, it is terefore calculated using equation () into () as p dp dx dx / n ( n 3) / n ( Q / ) = ( n 3) dx 3 / n ( n ). (3) n n n / n Te non-dimensional friction force, F (= ( a / ck ) F', F' is te friction force at te wall in te stationary coordinate system wic is same as in moving system) is tus obtained as F = dp ( ) dx dx / n ( n 3) / n ( Q / ) = ( n 3) dx. 3 / n ( n ) (4) Te closed from analytical evaluation of te complicated integrals involved in equations (3) and (4) seems to be a formidable task and terefore tey will be evaluated numerically. Furter, to compare te results of te analysis wit oter teoretical studies analytically and to observe te qualitative effects of non-newtonian beavior, we return to equations (3) and

4 Amit Medavi (4). Wen n = te results obtained in equations (3) and (4) reduce to te results for Bingam fluid analysis as 8 3 3 p ( ) Q ( ) ( 6), (5) 7 / ( ) 3 4 8 3 / F Q ( ) ( ). (6) 3 / ( ) 3 Wen =, equations (3) and (4) yield te results for a power-law fluid analysis as / n ( Q / ) p ( n 3) 3 / n / n dx, (7) F / n ( Q / ) ( n 3) 3 / n / n dx, (8) wic correspond to te results obtained in Srivastava and Srivastava (985) for uniform diameter tube. Also, wen n = and =, equations (3) and (4) yield te results for a Newtonian fluid model as 8 3 p 8 3/ F Q ( ), 3/ ( ) Q( ) ( 6) 7/ ( ) 4, (9) () wic are te same results as obtained in Sapiro et al. (969). Te results of Sapiro et al. (969) are also derived eiter by setting = in equations (5) and (6) or n = in equations (7) and (8). Finally, in view of te complicated form of te expression for p given in equation (3), it is difficult to obtain te analytical expression for te flow rate, Q for zero pressure drop. However, te pressure drop for zero flow rate, pq, wic is of particular interest is derived as / n ( n 3) / n ( / ) = ( n 3) dx. 3 / ( n ) n pq () Wit n = and =, equation () reduces to

AAM: Intern, J., Vol. 3, Issue (June 8) [Previously, Vol. 3, No. ] 43 = ( 6), () 7 / ( ) pq wic is te same expression as obtained in Sapiro et al. (969) for pressure drop at zero flow rate. 3. Numerical Results and Discussion To observe te quantitative effects of various parameters involved in te analysis, computer codes are developed to evaluate te expressions for dimensionless pressure drop, p, and friction force, F (equations (3) and (4)) for different values of te parameters (Sapiro et al., 969; Caturani and Narasimman, 988; Srivastava and Saxena, 995),, n, and Q and some critical results are displaced grapically in Figs. -7. Te results for te parameter values; n = and, n and =, and n = and = correspond to te results for Bingam, power-law, and Newtonian fluid models, respectively. For = te results of te study reduce to tose obtained in no peristalsis case. 9 8 7 =. =. n= n=.363 p 6 5 4 3 n=.857 = =. =.6...4.6.8...4.6 - Fig. Pressure -flow rate relationsip for different, and n. Q

44 Amit Medavi 6 p 5 4 3 Q=.6.363 =..857 =. Numbers n Q=..857.363 Q=.857....3.4.5 -.363 - Fig.3 Variation of pressure drop, p wit for different, n and Q. Te pressure drop, p increases wit te flow rate, Q and its minimum magnitude is acieved at zero flow rate (Fig. ). A linear relationsip exists between te pressure drop and flow rate in Bingam and Newtonian fluids analyses wereas te relationsip between tese ( p and Q) are seen non-linear in power-law and Herscel-Bulkey fluid models (Fig. ). 6 5 4 p 3 (,.6) (,.4) (.,.4) (.,.6) =. =. Numbers (,Q) (,.) (.,.) (.4,.6)...4.6.8...4.6.8 3. n - - (.,) (,) -3 Fig.4 Variation of pressure drop, p wit n for different, and Q.

AAM: Intern, J., Vol. 3, Issue (June 8) [Previously, Vol. 3, No. ] 45 Pressure drop, (Fig. 3). Increasing flow rate significantly affect te magnitudes of pressure drop wit te parameter n (Fig. 4) and for any given set of oter parameters, pressure drop increases wit yield stress (Fig. 5). Altoug, te radius of te plug flow region is small as compared to te p is found to be decreasing indefinitely wit increasing amplitude ratio, non- plug flow region, te numerical results reveal tat even relatively small values of yield stress, significantly influence te magnitudes of te flow caracteristics. Friction force, F possesses a caracter similar to te pressure drop, p wit respect to any parameter (Fig. 6 and Fig. 7). Fig. - Fig. 7 clearly present a good comparison of te results among different fluid models (Herscel-Bulkley, power-law, Bingam and Newtonian). 3. (.4,) (.4,.857).5. (.,.857) (.,.363).5 p. (.,) =. =. Numbers (Q,n) (.,.857) (,.857) (.,.363).5 (.,) (,.363) (,)...4.8..6. Fig.5 Variation of pressure drop, p wit for different, Q and n. 7 6 =. =. n= n=.363 F 5 4 n=.857 3 = =. =.6...4.6.8.. Q - Fig.6 Friction force - flow rate relationsip for different, and n.

46 Amit Medavi 6 =. =. Q=.6 5 Numbers n.363.857 F 4 3 Q=..363.857 Q=.363.857....3.4.5.6 - - Fig.7 Variation of friction force, F wit for different, n and Q. 4. Conclusions A non-newtonian fluid model obeying Herscel-Bulkley equation as been applied to study te flow induced by means of sinusoidal peristaltic waves using long wavelengt approximation. Particular cases (Bingam, power-law and Newtonian fluids), derived from te present non-newtonian fluid model, ave been discussed and compared bot qualitatively and quantitatively. Pressure-flow rate relationsips in Bingam and Newtonian fluids ave been found linear wereas te same are non-linear in power-law and Herscel-Bulkley models. Pressure drop increases wit te flow rate and yield stress but decreases wit amplitude ratio. Friction force possesses an opposite caracter to pressure rise (negative of pressure drop). It is strongly believed tat te results of te analysis may be applied to discuss te peristaltic induced flow of blood and oter pysiological fluids (Scott Blair and Spanner, 974, Witmore, 968). Acknowledgements Te autor gratefully acknowledges te valuable comments and suggestions by Prof. A. M. Hagigi, Te Editor-in-Cief and te Reviewers of te journal. I express sincere tanks to te Executive Director, Er. Pankaj Agarwal, te Principal Prof. A. K. Merotra, SRMCEM, Lucknow for invaluable support and to Dr. U. K. Sing, KNIT, Sultanpur and Prof. S.C. Rastogi, SRMCEM, Lucknow, India for teir kind guidance. REFERENCES Siddiqui, A. M., Hayat, T. and Masood, K. (4). Magnetic Fluid Induced by Peristaltic Waves. Journal of Pysical Society of Japan, Vol. 73, pp. 4-47. Rao, A. R. and Misra, M. (4). Nonlinear Curvature Effects on Peristaltic Flow of a Viscous Fluid in Asymmetric Cannel, Acta Mecanica, Vol. 68, pp. 35-39. Becker, E. (98). Simple non-newtonian flows. In: Advances in applied mecanics (C.S. Yi Ed.), Academic Press, N. Y. and London.

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48 Amit Medavi Sapiro, A. H., Jaffrin M.Y. and Weinberg S.L., (969). Peristaltic Pumping wit Long Wavelengt at Low Reynolds Number. Journal of Fluid Mecanics, Vol.7, pp. 799-85. Sukla, J. B. and Gupta, S.P. (98). Peristaltic Transport of a Power-law Fluid wit Variable Consistency, Journal of Biomecanical Engineering. Vol. 4, pp. 8-86. Srivastava, L.M. and Srivastava, V.P. (984). Peristaltic Transport of Blood: Casson Model II. Journal of Biomecanics, Vol. 7, pp. 8-89. Srivastava, L.M. and Srivastava, V.P. (985). Peristaltic Transport of a Non-Newtonian Fluid: Applications to te Vas Deferens and Small Intestine, Annals of Biomedical Engineering, Vol.3, pp. 37-53. Srivastava, V.P. and Saxena, M. (995). A two-fluid model of non-newtonian Blood Flow Induced by Peristaltic Waves, Reologica Acta, Vol. 34, pp. 46-44. Srivastava, V.P. (). Particle-Fluid Suspension Flow Induced by Peristaltic Waves in Circular Cylindrical Tubes, Bulletin of Calcutta Matematical Society, Vol. 94, pp. 67-83. Srivastava, V.P. (7). Effects of an Inserted Endoscope on Cyme Movement in Small Intestine. Applications and Applied Matematics, Vol., pp. 79-9. Vajravelua, S., Sreenad S. and Rames Babu, V. (5). Peristaltic Transport of a Herscel- Bulkley Fluid in an Inclined Tube, International Journal of Nonlinear Mecanics, Vol. 4, pp. 83-9. Witmore, R. L. (968). Reology of te Circulation, Pergamon Press, N. Y.