Study of Peristaltic Flow of Blood in Artery

Transcription

1 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) ISSN Study of Peristaltic Flow of Blood i Artery Dr. Smita Dey a*, Aamol Kumar al b a Uiversity departmet of Mathematics, Morabadi Campus, Basic Sciece Buildig,Rachi Uiversity, Rachi 848, Jharkhad, Idia b Departmet of Mathematics, Marwari College, ake Road, Rachi 84, Jharkhad, Idia a smitadey@yahoo.com b myid.amol@gmail.com Abstract I the fuctioig of ureters, itesties, esophagus ad i medical istrumets such as the heart-lug machie, fluid is trasported by a uique process, arteries of cotractios of the wall propagates alog the legth of the tube causig fluid to be trasported i the directio of the wave. This is called peristaltic pumpig. The problem of peristaltic trasport i two dimesioal chaels ad i axisymmetric tubes has received cosiderable attetio. I the paper, we have studied the effect of asymmetric Peristalsis o blood flow through a artery with steosis. We kow that blood ca be take as a Newtoia fluid as well as a No-Newtoia fluid. Here, we have cosidered the blood as a Newtoia ad No Newtoia fluid. Here, the basic equatio is take as Navier-stokes equatio ad we have trasformed the statioary co-ordiates to movig Coordiates. The frictioal force ad the expressio for pressure rise have bee obtaied. Keywords:Peristalsis; Newtoia fluid; No-Newtoia fluid Nomeclature C = Wave Propagatio velocity. h = Mea radius of the chael. = Wavelegth. q = Dimesioless volume flow. p = Pressure. b = Amplitude of the Peristaltic wave o upper wall. b = Amplitude of the Peristaltic wave o lower wall. ε, ε = Dimesioless Amplitude. * Dr. Smita Dey. Tel.: Address : smitadey@yahoo.com

2 .Itroductio Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) () Volume, No, pp -8 Pumpig of fluids through flexible tubes by meas of the peristaltic wave motio of the tube wall has bee the subject of egieerig ad scietific research for over four decades. Egieers ad physiologists term the pheomeo of such flow as peristalsis. It is a form of iduced by a progressive wave of area cotractio or expasio alog the walls of a distesible duct cotaiig a liquid or mixture. Besides its various egieerig applicatios (e.g., heart-lug machies, figer ad roller pumps, etc.), it is kow to be sigificat mechaism resposible for fluid trasport i may biological orgas icludig i swallowig food through esophagus, urie trasport from kidey to bladder through the ureter, movemet of chyme i gastroitestial tract, trasport of spermatozoa i the ductusefferetes of the male reproductive tracts, ad, i cervical caal, i movemet of ovum i the female fallopia tubes, trasport of lymph i lymphatic vessels, ad i the vasomotio of small blood vessels such as arterioles, veules ad capillaries. Recetly Tripathi & Sriivas [,] studied asymmetric Peristalsis for heat trasfer. atham [] was probably the first to study the mechaism of peristaltic pumpig i his M.S. Thesis. Shapiro et al. [4] ad Jaffri ad Shapiro [5] explaied the basic priciples ad brought out clearly the sigificace of the various parameters goverig the flow. The literature o the topic is quite extesive by ow ad a review of much of the literature up to the year 98, arraged accordig to the geometry, the fluid, the Reyolds umber, the wave umber, the amplitude ratio ad the wave shape was preseted i a excellet article by Srivastava ad Srivastava [6]. The sigificat cotributios to the subject betwee the years 984 ad 994 were well refereced by Srivastava ad Saxea [7]. The importat studies of the recet years iclude the ivestigatios of Srivastava ad Srivastava [6], Mekheimer ad his colleagues [8], El-Shehawey ad Hussey[], El-Hakeem ad El-Misery [], Muthuet ad his colleagues [9], Mishra ad Rao [], Medhavi[],Medhavi ad Sigh () ad a few others. I both the mechaical ad physiological situatios, most of the studies coducted i the literature cosidered the trasported fluid to be a Newtoia fluid. It is well accepted that a large umber of fluids of practical importace behave like a o- Newtoia fluid also. The preset article deals with the flow of a Newtoia ad No- Newtoia fluid.. Basicequatios ad formulatio of the problem (for Newtoia Flow): et the Equatio of the tube surface (Fig. ) be give by:- π h( z, t) = a + cos ( z ct) Where a = udisturbed radius ad = the amplitude ratio. Blood is take as a Newtoia fluid with the assumptios of ifiite wave legth ad iertia-free flow, the Navier-stokes equatio reduces to: dp u = µ () dz y The boudary coditios are: u= cat π y = h + b. Cos. z ; u= c at π y = h b. Cos. z () Where c is the wave propagatio velocity, h is the mea radius of the chael, b ad b are the amplitudes of the peristaltic wave o the upper ad lower walls respectively. is the wavelegth. et us trasform the statioary Co-ordiates to movig Co-ordiates as z = Z - c.t, y = Y Ad u(z,y) = U(Z - c.t, Y) c for velocity compoet i the laboratory frame U

3 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) () Volume, No, pp -8 Itegratig equatio (i). We get.. c. µ. c dz h h u h dp y y = Cos πz Cos π z ( Cos πz) () Where b b z =, =, Z =, <, < h h The rate of volume flow through each sectio q is give by y q = udy y H dp q = ch µ dz π H h b b. Cos z where = + ( + ) q = dimesioless volume flow = The istataeous volume flow rate q hc = H H dp h 4µ c dz (4) y i.e. (, ) (5) Q z t = u + c dy = q + ch y The time-mea volume flow at each cross sectio is Q Q = T Qdt T = + + ( + ) ( ) T C π q hc b b. Cos z ct. dt = q + T hc The pressure chage per wavelegth is P Where dp =. dz dz Q θ = hc h = + + µ + 8 p. 5. θ c ( + ) 4 + (6) The dimesioless pressure rise ( ( P P ) θ ) = θ = for Zero time -mea flow ad the dimesioless time mea flow θ for the zero pressure rise are give by h + ( p ) = µ c θ = (7)

4 ad Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) () Volume, No, pp -8 ( + ) θ = ( + ) + + θ = ( ) (8) Frictioal force at the wall a cross oe wave legth i.e. ' H dp 6µ c q ' H dp 6µ c q F = F =.. dx. =. dz = + µ c, + cf F dx dz4 ( + 4 ) ( + ) ' µ 6q = = + µc 4 + Dimesioless frictioal force i.e. P P Resistace to flow R = = θ P ( P ) 4 θ ( + ) µ c θ h θ ( + ) (9) Whe there is o. Peristaltic wave, the µ c R = h () Dimesioless Resistace to flow is + = =. + R 4 8 R 5 R θ ( + ) 4 + (). For No-Newtoia Flow The geometry of the wall surface {figure (vi) } is described as ' ' ' π m ' ' H( x, t ) = a( x ) + b. Si ( x ct ) () ' ' With ax = a + Kx () ' Where, ax is the radius of the tube at ay axial distace from ilet, a is the radius of the ilet, K is a costat whose magitude depeds o the legth of the tube, exit ad ilet dimesio, ad b is the amplitude of the wave. Assumig that Wave umber α ad Re are very small; appropriate Navier-Stokes equatios describig the flow are p u = r. x r r r (4) ad p = r (5)

5 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) () Volume, No, pp -8 Whe a o slip coditio is imposed as a boudary coditio for the U velocity the particle is assumed to move oly i the radial directio alog the boudary. u = r at r = (6) u = at kx r = h = + + φsi πm( x t) (7) a Where H b h = ; φ = < a a Solvig equatiosafter applyig the boudary coditio, The expressio for velocity profileis + + dp u( xrt,, ) = h r + dx (8) The istataeous volume flow rate Qxt (,) is obtaied as h π dp Q(,) x t = π rudr = h + dx + (9) Q dp = dx + (,) xt π + h () The Pressure rise P () t i the tube of legth i the o-dimesioal form is give by / = P () t = dp dx dx + / Q(,) xt. dx + π kx + + φsi πm( x t) a Qxt (,) / π P () t = dx + + kx + + φ. Si πm( x t) a The above equatio gives the Pressure flow rate relatioship. The frictioal force () i their No dimesioal forms are give by: () F t at the wall i the chael of legth 4

6 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) () Volume, No, pp -8 π / Q xt F () t (,) =+ + kx + + φsi πm( x t) a dx (). Results ad Discussio. For Newtoia Flow I orderto see the effects of various parameters o frictio, resistace to flow etc, the followig values of the parameters are take Table It has bee observed that ) O icreasig the value of qq from.5 to.55, there is a icrease i the value of F with ε [(fig ()]. ) O icreasig the value ofθ from.5 to, there is icrease i the value of RR with ε [fig ()]. ) O decreasig the value of ε + ε from to., F also decreases with qq [fig (4)]. 4) O icreasig the value of ε from. to., the value of θ icreases with εε ( P = )[fig (5)].. For No-Newtoia Flow ε ε q.5.55 θ.5. I order to see the effect of various parameters o frictio, pressure rise etc, the followig values of the parameters are take: Table t Q a. cm m = cm K a / =.8 Equatio () gives the formulas for pressure chage ad Equatio () gives the formulas for frictioal force chage ad Here; for umerical itegratio, Gaussia quadrature formula has bee used. Here, it has bee assumed that Q (,) x t = ( Q) + φsi πm( x t) whereq is the time average of the flow over oe period of the wave. This form of Qxt (,) has bee assumed due to the fact that costat value of Qxt (,) gives P () t always egative ad hece, there will be o pumpig actio. It has bee observed that 5

7 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) () Volume, No, pp -8 ) O icreasig the value of from to,there is a decrease i the value of P with icreasig value of time (as time icrease). (fig 7) ) O icreasig the value of from to,there is a icrease i the value of P with icreasig value of Q (as Q icrease). (fig 8) ) Whe value of amplitude ratio chages from (fig 9) 4) Whe (fig ) φ = to φ =.7 the at = ; F icreases with icreasig value ofq. =, the the frictioal forces F chages much more whe φ = compared to φ =.7. (as time icrease) 6

8 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) () Volume, No, pp -8 Fig. 6. Peristaltic trasport i a o-uiform Chael. (No-Newtoia) Fig. 7.Variatio of P with t for differet Fig. 8. Variatio of P with Q for differet Fig. 9. Variatio of F with Q for differet φ Fig.. Variatio of F with t for differet φ. Coclusios: 7

9 Iteratioal Joural of Scieces: Basic ad Applied Research (IJSBAR) () Volume, No, pp -8 A Newtoia ad No Newtoia fluid model have bee applied & study of the flow iduced by meas of siusoidal peristaltic waves usig log wavelegth approximatio has bee take ito accout.compariso ca be made for particular cases (Bigham, Power law, Newtoia fluids) both qualitatively ad quatitatively, pressure drop icreases with the flow rate ad yield stress. Frictioal force possesses the similar character with flow rate. It is strogly believed that the results of the aalysis may be applied to discuss the peristaltic flow of blood. The improvemet ca be doe by itroducig the elasticity parameter i the goverig equatio, it will be the more realistic work. Refereces: [] DharmedraTripathi. Numerical ad Aalytical Simulatio of Peristaltic flows of geeralized oldroyd B fluids, Iteratioal Joural for Numerical methods i fluidsvol 67 P.P [] S. Sriivas, R. Gayathri, M. Kothadapai. Mixed covective heat ad mass trasfer i a asymmetric chael with Peristalsis,Commuicatios i No-liear Silece ad Numerical SimulatioVol 6, P.P [] T. W. atham. Fluid Motio i a Peristaltic Pump.966.M.S. Thesis, MIT, Cambridge, Mass. [4] A. H. Shapiro, M.Y.Jaffri ad S.. Weiberg. Peristaltic Pumpig with ogwavelegth at ow Reyolds Number. 969.Joural of Fluid Mechaics, Vol. 7, pp [5] M. Y. JaffriadA. H. Shapiro. Peristaltic Pumpig, 97.Aual Rev. Fluid Mech., pp. -6. [6].M. Srivastava ad V.P. Srivastava. Peristaltic Trasport of Blood: Casso Model II. 984.Joural of Biomechaics, Vol. 7, pp M.Srivastava ad V.P. Srivastava Peristaltic Trasport of a No-NewtoiaFluid:Applicatios to the Vas Deferes ad Small Itestie, 985. Aals of BiomedicalEgieerig, Vol., pp [7] V.P.Srivastava ad M.Saxea. A two-fluid model of o-newtoia Blood FlowIduced by Peristaltic Waves, 995.RheologicaActa, Vol. 4, pp V.P. Srivastava Particle-Fluid Suspesio Flow Iduced by Peristaltic Waves icircular Cylidrical Tubes,.Bulleti of Calcutta Mathematical Society, Vol. 94, pp [8] S.Mekheimer, kh, E. F.El-Shehawey ad A. A.Elaw. Peristaltic Motio of Particle-Fluid Suspesio i a Plaar Chael, 998.Iteratioal Joural of Theoretical Physics, Vol. 7, pp S.Mekheimer, kh. Noliear Peristaltic Trasport Through a Porous Medium i a Iclied Plaar Chael,.Joural of Porous Media, Vol. 6, pp [9] P. Muthu, ad P. Chadra. Peristaltic Motio i Circular Cylidrical Tubes: Effects of Wall Properties,.Idia Joural of Pure ad Applied Mathematics, Vol. (9), pp [] A.El-Hakeem, A.El-Naby ad A.E.M. El-Misery. Peristaltic Trasport i the Presece of a Edoscope i Nouiform Tube for Newtoia Fluid,.Joural of Biomechaics, I Press. [] E. F. El-Shehawey ad Z.A. Hussey. Peristaltic Trasport of a Particle-fluid: Suspesio with Porous Boudaries,.Iteratioal Joural of Egieerig Sciece, I Press. [] M. Misra, ad A. R. Rao. Peristaltic Trasport of a Newtoia Fluid i a Asymmetric Chael, 4. ZAMP, Vol. 54, pp [] A. Medhavi. Peristaltic Pumpig of Blood: Suspesio Two-layered Model, 8. Joural of Mathematical Biology, Submitted. 8