Peristaltic Induced Flow of a Two-Layered Suspension in Non-Uniform Channel

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1 Aailable at htt://am.ed/aam Al. Al. Math. ISSN: Vl. 6, Isse (December ), Alicatins and Alied Mathematics: An Internatinal Jrnal (AAM) Peristaltic Indced Flw a Tw-aered Ssensin in Nn-Unirm Channel Amit Medhai Deartment Mechanical Engineering Kamla Nehr Institte Technlg Sltanr-8 8, India amitmedhai@ah.c.in Dharmendra Singh and Aja S. Yada Deartment Mathematics S.M.S. Institte Technlg cknw, India dr.dsingh9@gmail.cm; ajasinghd@gmail.cm Ramesh S. Gatam Deartment Mathematics Kanr Institte Technlg Kanr, India rsg.kanr@gmail.cm Receied: Nember 9, ; Acceted: Agst 8, Abstract Peristaltic transrt a tw-laered articlate ssensin in a nn-nirm channel has been inestigated. The cled dierential eqatins r bth the lid and the article hases in the central as well as in the eriheral laers hae been sled and the eressin r the lw rate, the ressre rise and the rictin rce has been deried. The reslts btained are discssed bth qalitatiel and qantitatiel in brie. The signiicance the article cncentratin as well as the eriheral laer has been well elained. Kewrds: Particle cncentratin, eriheral laer, lw rate, ressre rise, rictin rce. MSC () N.: 76Z5 46

2 AAM: Intern. J., Vl. 6, Isse (December ) 463. Intrdctin Fr abt r and hal decades, the lw indced b eristaltic waes had been the sbject scientiic and engineering research. atham was rbabl the irst t intrdce the mechanism eristaltic transrt in his M. S. thesis in the ear 966. Peristalsis, as termed b the hsilgists, is a rm lid transrt that ccrs when a rgressie wae area cntractin r eansin ragates alng the length a distensible dct cntaining liqid r mitre. Inclding the ascmtin small bld essels, it has been nd t be inled in man bilgical rgans (Sriastaa and Sriastaa, 984).Certain bimechanical sstems sch heart-lng machine, inger and rller ms hae been abricated sing the rinciles eristalsis. Shair et al. (969) and Jarin and Shair (97) elained the basic rinciles eristaltic ming and brght t clearl the signiicance the aris arameters gerning the lw. The literatre n the sbject is qite etensie b nw. A reiew mst the earl theretical and eerimental inestigatins rerted t the ear 994 ma be nd in Sriastaa and Cwrkers (984, 995). The literatre bend this and recent ears inclde the inestigatins Sriastaa and Sriastaa (997), Mekheimer et al. (998), Hakeem et al. (), Sriastaa (), Misra and Pande (), Haat and cwrkers (, 3, 4, 5, 6a,b; 8a,b), Mekheimer (3), Misra and Ra (4), Sriastaa (7a), Ali and Haat (8), Medhai and cwrker (8, ), and a ew thers. The std the ther articlate ssensin is er sel in man areas technical imrtance (wder technlg, lidizatin, sedimentatin, cmbstin, aersl iltratin, atmsheric allt, lnar ash lws, enirnmental lltin, etc.). Recentl, interest is increasingl deeling in aling the articlate ssensin ther t hsilgical lws as it rides an imred nderstanding tics sch as disin rtein, the rhelg bld, the swimming micrrganism, the article desitin n resiratr tract, etc. Peristaltic ming mlti-hase lid has been addressed b Sriastaa and Sriastaa (989, 997), Mekheimer et al. (998), Sriastaa (), Medhai and cwrker (8, 9) and a ew thers. It is knwn that the eriheral laer las a signiicant rle wheneer it is reqired t reent the transrted lid rm cming in cntact with the mechanical arts the ms. Flws in man the bilgical rgans sch as chme in the small intestines and bld thrgh small essels are mainl tw-laered. Peristaltic transrt tw-laered mdels hae been addressed b a ew inestigatrs inclding Shkla et al. (98), Sriastaa and Sriastaa (98, 984), Brasser et al. (987), Sriastaa and Saena (995), Ra and Usha (995), Misra and Pande (), etc. It is t nte here that the interace shae deends n the iscsit rati the lids in the tw (central and eriheral) laers and nt n the rati the radii the ter (eriheral) and the central laers, in general (Brasser et al.,987; Ra and Usha,995). Hweer, the shae the interace is nt signiicantl aected when the iscsit the lid in ne the laers is aried with resect t the iscsit the lid in the ther laer (Misra and Pande, ).

3 464 Amit Medhai et al. The stdies mentined abe hae cnsidered the cre lid t be either a single-hase Newtnian r nn-newtnian lid. With increasing interest in articlate ssensin lws, it is highl desired t address the tw-lid eristaltic transrt rblem in detail when the cre lid is reresented b a article-lid ssensin. In iew the abe discssin, an attemt has been made in the resent wrk t analze the lw a article-lid ssensin indced b eristaltic waes in the resence a eriheral laer in a nn-nirm channel. The theretical mdel cnsiders a tw-laered lw cnsisting a central laer as a article-lid mitre and a eriheral laer as article-ree Newtnian lid (same lid as ssending medim in the central laer).. Frmlatin the Prblem Cnsider the lw a tw-laered articlate ssensin thrgh a tw-dimensinal ininite channel nn-nirm width with a sinsidal wae traeling dwn its walls. The central laer (cre regin) cnsists a article-lid mitre and the eriheral laer article ree Newtnian iscs lid (same as the ssending medim in the cre regin). The gemetr the wall srace is described (Figre ) b with π H(,t) d() bsin ( ct), () λ d() d k, λ c b b d d () d H H d() Fig.. Tw laered lw gemetr eristaltic waes in a channel

4 AAM: Intern. J., Vl. 6, Isse (December ) 465 where d() is the hal width the channel at an aial distance rm inlet, d is the hal width the channel at the inlet, k (<<) is a cnstant whse magnitde deends n the length the channel and eist and inlet dimensins, b is the amlitde the wae, λ is the waelength, c is the wae ragatin elcit and t is the time. The eqatins gerning the linear mmentm and the cnseratin mass r bth the lid and article hases in the eriheral and the central regins are eressed (Sriastaa and Cwrker, 989, 7b) ρ t = - μ, H H, () ρ t = - μ, H H, (3), (4) (-C ) ρ t = - (- C ) ( C) μs (C) + CS ( ), H, (5) (-C ) ρ ρ ρ t ( C) ( C) t t = - (-C) ( C) μs (C) + CS ( ), H, (6) =, (7) z r [C ] = - C = - C CS ( ), H, (8) CS ( ), H, (9) [C ] =, () where = / / is tw-dimensinal alacian eratr with as the ertical crdinate measred in the directin nrmal t the tbe ais, (, ) and (, ) dente elcit cmnents the lid and article hases in (, ) directins, resectiel in the cre regin, H ; (, ) dentes elcit cmnents the lid in the eriheral laer H H ; ρ and ρ be the actal densities the material cnstitting lid and articlate hases, resectiel, (-C) ρ is the lid hase densit, C ρ the articlate hase densit in the cre regin; dentes ressre and C be the lme ractin articlate hase; μ is the lid

5 466 Amit Medhai et al. iscsit in the eriheral regin; s (C) ~ s is the ssensin iscsit in the central laer and S being the drag ceicient interactin r the rce eerted b ne hase n the ther. In iew the argment stated earlier (Misra and Pande, ), ne ma assme the rm interace (Shkla et al., 98; Sriastaa and Saena, 995) as: H = d () + b sin / (z-ct) with d () = d +k, b resectiel as the central laer radis and interace wae amlitde and d being the radis the central laer at the inlet. The limitatins the resent theretical mdel are well described in Sriastaa and Sriastaa (983) and Sriastaa (7b). It is wrth t mentin here that lid densit in the central and the eriheral regins has been assmed the same, ρ in iew the act that the lid in the eriheral laer is same as that the ssending medim in the central cre regin. Intrdcing the llwing dimensinless ariables ct/λ )/c, t',, ( ') ',,,(' /d ' λ, / ' c c, (', ' c ) = (, c )/cd,, Sd /μ S', c d ' c / in t the eqatins () and (), ater dring the rimes, ields t δ Re =- δ, h h, () t δ Re 3 = - δ δ, h h, (), (4) t -C) δ ( Re = - (- C ) δ μ C) ( + CS ( ), h, (5) t -C) δ ( Re 3 = - (-C) δ μδ C) ( + CS ) ( δ, h, (6) C) ( C) ( =, (7) t )δ /ρ (ρ Re = - C CS ( ), h, (8)

6 AAM: Intern. J., Vl. 6, Isse (December ) (ρ/ρ )δ Re t [C ] = - C CS δ ( ), h, (9) [C ] =, () where Re= ρ cd / and =d / are Renlds nmber and wae nmber, resectiel, (,, ) ( d,b, b ) / d, μ μ, (, ) k λ /μ, sin ( ). c, (h,h ) (H,H )/d α ( ) π t d Jairin and Shair (97) bsered that the Renlds nmber is qite small when the waelength is lng and in sch a case the inertial terms ma be neglected. Ths, nder the lng wae arimatin (i.e., << ), eqatins () () redce t d d, h h, () d C) μ( C) CS( ), h, () d ( d C CS( ), h, (3) d The nn-dimensinal bndar cnditins are kλ at h sin π( t), (4) d and kλ at h α sin π( t), (5) d / at =, (6) with τ ( C)μs / and τ μ / ; τ,τ are shearing stress the cre and eriheral regins, resectiel. The eressin r the drag ceicient S and the emirical relatin r the iscsit the ssensin are selected (Sriastaa, 7b; Sriastaa et al., ) as

7 468 Amit Medhai et al. S / 9 μ 4 3( 8C 3C ) 3C, (7) a ( 3C) μ s μ, ( mc) m. 7e [. 49C ( 7/T) e(. 69C)], (8) where T is measred in abslte temeratre ( K). The iscsit ssensin eressed b this rmla is nd t be reasnabl accrate t C =.6 (i.e., 6% article cncentratin). 3. Analsis The eressin r the elcit riles,, and btained as the sltin eqatins () (3) sbject t the bndar cnditins (4)-(6), are gien as d (h ), h, (9) d d {h μ( C)(h h )}, h, (3) ( C)μ d d ( C)μ h μ( C)(h h ), h, (3) ( C)μ d S The lw l (instantanes lme lw rate), q q'/ cd ) is calclated as ( q(, t) h h 3( C)μ d h [( C) d {( C)(h d 3 C ]d 3 h ) h 3 δh }, (3) where δ 3 C( C)μ)μ, a nn-dimensinal ssensin arameter.

8 AAM: Intern. J., Vl. 6, Isse (December ) 469 Using nw the act that the ttal sm is eqal t the sm les acrss the tw regins : h and h h, ne arries at the relatins (Medhai, 9; Shkla et al., 98), α and h αh. Sbstittin these relatins int eqatin (3), ields d 3μ( C) q(,t), (33) 3 d ηh δαh with η ( C)( α 3 )μ α 3. The ressre rise, Δ (t) and the rictin rce at the walls, F (t) in the channel length, in their nn-dimensinal rm are btained as Δ (t) /λ d d d /λ q(,t) 3 ( C)μ d, (34) 3 ηh δαh /λ /λ d q(,t) F(t) h d 3 ( C)μ d. (35) d ηh δα Setting k = in eqatins (34) and (35), ne deries the eressins r the ressre rise and rictin rce r a tw-laered articlate ssensin in a nirm channel with α in eqatins (34) and (35), the reslts r a single-laered article-lid ssensin is deried. It is interesting t nte that when C =, the cre mitre redces t the same lid as in the eriheral laer and cnseqentl the rle the interace atmaticall disaears and the reslts btained abe redce t the crresnding reslt single-hase Newtnian iscs lid in a nn-nirm channel. 4. Nmerical Reslts and Discssin In rder t discss the reslts qantitatiel, cmter cdes are nw deeled r the nmerical ealatins the analtical reslts btained in eqatins (34) and (35) r aris arameter ales at the temeratre 37 C in a channel hal width. cm. In iew the act that the eriheral laer thickness strngl deends n the cre mitre iscsit besides ther actrs (Bgliarell and Seilla, 97; Sriastaa, 7b), we chse a (diameter a ssended article)=8µm, the eriheral laer thickness ε(μm) ε(c) 6. 8, 4. 67, 3. 6, 3.,. 58, 8,. crresnding t the article cncentratins, C =

9 47 Amit Medhai et al..,.,.3.4,.5,.6, resectiel rm Sriastaa (7b). The ale α is btained rm the relatin: α ε/d. We rther assme the rm (t)... K=.5 =.7 Q=.3 Nmbres C Single-aered Dble-aered t Fig. Variatin ressre rise, (t) in a nn-nirm channel with time, t r dierent C K=.5 =.7 C=.4 Nmbres Q Single-aered... Dble-aered (t) t.6.8. Fig. 3 Variatin ressre rise, (t) in a nn-nirm channel with time, t r dierent Q.

10 AAM: Intern. J., Vl. 6, Isse (December ) 47 instantanes lw rate q(,t), eridic in (-t) as (Gta and Seshadri, 976; Sriastaa and Sriastaa, 988; Mekheimer, ) q(,t) Q sin π( t), (36) where Q is the time aerage the lw er ne erid the wae. The abe rm q(,t) has been assmed in iew the act that the cnstant ale gies ΔP (t) alwas negatie and hence there wld be n ming actin. The dimensinless ressre rise, Δ (t) and rictin rce, F (t) er the channels length, r aris ales the lw rate, Q amlitde rati, and the article cncentratin. C is cmted sing eqatin (36). The aerage ressre rise Δ and the rictin rce, F are then ealated b aeraging Δ(t) and F(t), resectiel er ne erid the wae. Other arameters are selected (Sriastaa and Sriastaa, 984; Mekheimer, ) as. 5d λ cm, k. 5. The integrals inled in eqatins (34) and (35) are ealated sing Simsn rle and sme the critical reslts are dislaed in Figs. 5 grahicall (t) K=.5 =.7 Q=.3 Nmbres C Single-aered... Dble-aered t.6.8. Fig. 4 Variatin ressre rise, (t) in a nirm channel with time, t r dierent C.

11 47 Amit Medhai et al. In bth the single and dble-laered analses the ressre rise Δ (t) increases with article cncentratin, C. Deending n the article cncentratin, C, the eak ale Δ (t) ccrs between. 3 t. 4. (Fig. ). The ressre rise Δ (t) decreases with the increasing lw rate r an gien set ther arameters (Fig. 3). The lw characteristic, Δ (t) ssesses similar characteristics with resect t an arameter in bth the nirm and nn-nirm channel. Hweer, Δ (t) assmes mch smaller magnitde in nn-nirm channel than its crresnding ale in the nirm channel (Figs. -5). It is rther t nte that r an gien ale t, Δ (t) assmes signiicantl lwer magnitde in tw-laered analsis than its crresnding ale in single-laered analsis (Figs. 3 and 5). The aerage ressre rise, Δ erss time aerage lw rate, Q has been ltted in Figs. 6 and 7 which indicate a linear relatinshi between Δ and Q and ths the maimm lw 3. (t) K=.5 =.7 C=.4 Nmbres Q Single-aered Dble-aered t Fig. 5 Variatin ressre rise, (t) in a nirm channel with time, t r dierent Q.

12 AAM: Intern. J., Vl. 6, Isse (December ) 473 C=.4 Nmbers..6 Single-aered... Dble-aered Q Fig. 6 Pressre-lw rate relatinshi in a nn-nirm channel r dierent =.4 Nmbers C Single-aered Dble-aered Q Fig. 7 Pressre-lw rate relatinshi in a nn-nirm channel r dierent C. rate is achieed at zer ressre rise and maimm ressre ccrs at zer lw rate. The lw characteristic, Δ is nd t be indeinitel increasing with the amlitde rati, r an

13 474 Amit Medhai et al. gien lw rate, Q and the article cncentratin, C in bth the single and dble-laered analses and assme a er =.4 Nmbers C Single-aered Dble-aered Fig. 8 Variatin aerage ressre rise, with in a nn-nirm channel. high asmttic magnitde as. 6 (Figs. 8 and 9). The aerage ressre rise, Δ seems t be steel increasing with the article cncentratin, C r small ales the lw..5 C=.4 Nmbers Q Single-aered... Dble-aered Fig. 9 Variatin aerage ressre rise, with in a nn-nirm channelr dierent Q.

14 AAM: Intern. J., Vl. 6, Isse (December ) =.4 Nmbers Q Single-aered... Dble-aered C Fig. Variatin aerage ressre rise, with C in a nn-nirm channel r dierent Q. rate Q(. ), hweer, the lw characteristic, Δ is nd t be decreasing with increasing lw rate, Q r higher ales Q (Fig.). Fr an gien amlitde rati, Q, Δ decreases with increasing lw rate, Q (Fig.) Q=.3 Nmbers Single-aered Dble-aered C Fig. Variatin aerage ressre rise, with C in a nn-nirm channel r dierent.

15 476 Amit Medhai et al F (t) K=.5 =.7 Q=.3 Nmbres C Single-aered Dble-aered t.6.8. Fig. Variatin rictin rce, F (t) in a nn-nirm channel with time, t r dierent C. The nn-dimensinal rictin, F (t) decreases with increasing article cncentratin, C in bth the single and dble-laered analses (Fig. ). F (t) is nd t be decreasing with decreasing the lw rate, Q. The lwest magnitde F (t) ccrs r. 3 t. 4. The lw characteristic, F (t) assmes mch lwer magnitde in nn-nirm channel than its crresnding ale in nirm channel (Figs. 4 and 5). Nmerical reslts rther reeals that there eists a linear relatinshi between the aerage rictin rce, F and the aerage lw rate, Q. An insectin the illstratins als reeal that the rictin rce F (t) and its aeraged ale, F ssesses characteristics similar t that Δ (t) and Δ, resectiel with resect t an gien arameter. Present inestigatin has been carried t nder aris arimatins and assmtins. Cmments needs t made here regarding the same and se the sme the arameters inled in the std. It is nted that the eriheral laer thickness, ε decreases with increasing article cncentratin, C in the cre regin, cnseqentl the arameter α increases with article cncentratin, C. The elanatin regarding the shae the interace needs t be gien here. It is knwn rm the blished wrks (Brasser et.al., 987; Ra and Usha, 995) that the interace shae deends n the iscsit rati the lids in the tw regins (central and eriheral) and it des nt bear a cnstant rati radii the central and eriheral laers dring the crse eristaltic actin. Hweer, the shae interace is nt signiicantl aected when the iscsit

16 AAM: Intern. J., Vl. 6, Isse (December ) K=.5 =.7 C=.4 Nmbres Q Single-aered... Dble-aered -.5 F (t) t Fig. 3 Variatin rictin rce, F (t) in a nn-nirm channel with time, t r dierent Q. the lid in ne the laers is aried with resect t the lid iscsit in the ther laer and cnseqentl the radii rati remains arimatel cnstant (Misra and Pande, ). The eriheral laer lid iscsit µ remains alwas cnstant and it is ssensin iscsit, µ s aries with article cncentratin, C in the central cre laer. The cnditin stated in Misra and Pande () is bisl satisied which allws the se cnstant ale r a gien article cncentratin, C. Other limitatins the std are well addressed b earlier inestigatrs inclding Shair et al. (969), Shkla et al (98), Sriastaa and Saena (995), Medhai (), etc.

17 478 Amit Medhai et al F (t) =.7 Q=.3 Nmbres C Single-aered Dble-aered t.6.8. Fig. 4 Variatin rictin rce, F (t) in a nirm channel with time, t r dierent C F (t) =.7 C=.4 Nmbres Q Single-aered Dble-aered t.6.8. Fig. 5 Variatin rictin rce, F (t) in a nirm channel with time, t r dierent Q.

18 AAM: Intern. J., Vl. 6, Isse (December ) Cnclsins T inestigate the simltanes eects the eriheral laer and the article cncentratin n the eristaltic ming, the lw indced b eristaltic waes a tw-laered articlate ssensin has been stdied. It has been nd that the ressre rise increases with the article cncentratin in the cre regin and assmes a signiicantl lwer magnitde in tw-laered analsis than its crresnding ale in ne-lid mdel. A linear relatinshi between the ressre and the lw is ehibited. The rictin rce ssesses characteristics site t thse the ressre rise with resect t an gien arameter. As eident rm the blished literatre (Misra and Pande, ; Medhai and Singh, 9) eristalsis des la an imrtant rle in asmtin small bld essels in additin t the lsatile lw, the indings the resent theretical wrk (Sriastaa, 7b), ma be alied t elain the lw behair bld in small essels with aring crss-sectin. REFERENCES Ali, N. and Haat, T.(8). Eects an endsce n eristaltic lw a micrlar lid, Mathematical and Cmter Mdeling, Vl. 48, Brasser, J. G. Crrsin, S. and, N. Q. (987). The inlence a eriheral laer dierent iscsit n eristaltic ming with Newtnian lids, J. Flid Mech., Vl. 74, Bgliarell G. and Seilla, J.(97). Velcit distribtin and ther characteristics stead and lsatile bld lw in ine glass tbes, Birhel., V. 7, Drew, D. A.(979).Stabilit Stke s laer a dst gas, Phs. Flids, Vl. 9, Gta, B. B. and Seshadri, V. Peristaltic ming in nn-nirm tbes. J. Bimech. 9, 5-9, 976. Hakeem, A.E., Nab, A.E. and Miser, A.M.E. ().Eects an endsce and generalized Newtnian lid n eristaltic mtin, Al. Math. & Cmt., Vl. 8, Haat, T. and Ali, N. (6b). On mechanism eristaltic lws r wer-law lid, Phsica Acta, Vl. 37, Haat, T. and Ali, N. (6a). Peristaltic indced mtin a MHD third grade lid in a dermable tbe, Phsica Acta, Vl. 37, Haat, T., Ahmed, N. and Ali, N. (8a). Eects an endsce and magnetic ield n eristalsis inling Jeer lid, Cmmnicatin in Nnlinear Science and Nmerical Simlatin, Vl. 3, Haat, T., Ali, N., Asghar, S. and Siddiqi, A.M. (6c). Eact eristaltic lw in tbes with an endsce, Al. Math. & Cmt., Vl. 8, Haat, T., and Ali, N. (8b). Sli eects n eristaltic transrt MHD lid with ariable iscsit, Phsics etters A, Vl. 37, Haat, T., Masd, S. and Asghar, S. (5). Peristaltic transrt magnet hdrdnamic Jhnsn-Segalman lid, Nnlinear Dnamics, Vl. 4, Haat, T., Wang, Y., Htter, K., Asghar, S. and Siddiqi, A.M. (4). Peristaltic transrt an Oldrd-B lid in a lanar channel, Math. Prblems in Engng., Vl. 4,

19 48 Amit Medhai et al. Haat, T., Wang, Y., Siddiqi, A.M. and Htter, K. (3). Peristaltic mtin a Segalman lid in a lanar channel. Math. Prblems in Engng., -3. Haat, T., Wang, Y., Siddiqi, A.M., Htter, K. and Asghar, S. (). Peristaltic transrt a third rder lid in a circlar clindrical tbe, Math. Mdel and Methds in Al. Sci., Vl., Jarin, M. Y. and Shair, A. H. (97). Peristaltic ming, Ann. Re. Flid Mech., Vl. 3, atham, T.W. (966). Flid Mtin in a Peristaltic Pm. M.S. Thesis, M.I.T. Mass. Medhai, A. and Singh, U. K. (9). Peristaltic ming a tw-laered articlate ssensin in circlar clindrical tbe, Int. The. and Al. Mech., Vl. 4, Medhai, A. (8). Peristaltic ming a nn-newtnian lid, Alic. and Al. Math., Vl. 3, Medhai, Amit. Peristaltic ming a articlate ssensin in a catheterized tbe. E-Jrnal Sci. and Tech. 5, 77-93,. Mekheimer, Kh. S. (3). Nnlinear eristaltic transrt thrgh a rs medim in an inclined lanner channel, J. Prs Media, Vl. 6, Mekheimer, Kh. S. (). Peristaltic transrt a cle stress lid in a nirm and nnnirm channels, Birhel., Vl. 39, Mekheimer, kh. S., El-Shehawe, E. F. and Elaw, A. A. (998). Peristaltic mtin a articlelid ssensin in a lanar channel, Int. J. The. Phs., Vl. 37, Misra, J.C. and Pande, S.K. (). Peristaltic transrt bld in small essels: std a mathematical mdel, Cmt. and Math. with Alic., Vl. 43, Misra, M. and Ra, A.R. (4). Peristaltic transrt a Newtnian lid in an asmmetric channel, ZAMP, Vl. 54, Ra, A. R. and Usha, S. (995). Peristaltic transrt tw immicible lids with lng waelength at lw Renlds nmber, J. Flid Mech., 98, Rs, R. and kdis, P.S. (97). The lid mechanics the reter with an inserted catheter, J. Flid Mech., Vl. 46, Shair, A.H., Jarin, M. Y. and Weinberg, S.. (969). Peristaltic ming with lng waelength at lw Renlds nmber, J. Flid Mech., Vl. 37, Shkla, J. B., Parihar, R. S. and Ra, B. R. P. (98). Eects eriheral laer iscsit n eristaltic transrt a bi-lid, J. Flid Mech., Vl. 97, Sriastaa,. M. and Sriastaa, V. P. (98). Peristaltic transrt a tw-laered mdel hsilgical lid, J. Bimech., Vl. 5, Sriastaa,. M. and Sriastaa, V. P. (983). On tw-hase mdel lsatile bld lw with entrance eects, Birhel., Vl., Sriastaa,.M. and Sriastaa, V.P. (984). Peristaltic transrt bld: Cassn mdel II. J. Bimech., Vl. 7, Sriastaa,.M. and Sriastaa, V.P. (988). Peristaltic transrt a wer-law lid, Rhel. Acta, Vl. 7, Sriastaa,. M. and Sriastaa, V. P. (989). Peristaltic transrt a article-lid ssensin, Trans. ASME J. Bimech. Engng., Vl., Sriastaa, V.P. and Saena, M. (995). A tw-lid mdel nn-newtnian bld lw indced b eristaltic waes, Rhel. Acta, Vl. 34,

20 AAM: Intern. J., Vl. 6, Isse (December ) 48 Sriastaa, V.P. and Sriastaa,.M. (997). Inlence wall elasticit and Piseille lw n eristaltic indced lw a article-lid mitre. Int. J. Engng. Sci., Vl. 35, Sriastaa, V.P. (). Particle-lid ssensin lw indced b eristaltic waes in a circlar clindrical tbe, Bll. Cal. Math. Sc., Vl. 94, Sriastaa, V.P. (7a). Eects an inserted endsce n chme mement in small intestine, Alc. and Al. Math., Vl., Sriastaa, V.P. (7b). A Theretical mdel r Bld Flw in Small Vessels, Alicatins and Alied Mathematics, Vl.,