Non-Newtonian characteristics of peristaltic flow of blood in micro-vessels

Non-Newtonian c haracteristics of p eristaltic flo w of blo o d in micro-v essels S. Maiti 1 ∗ , J.C.Misra 2 † 1 Scho ol of M e dic al Scienc e and T e chnolo gy & Center for The or et i c a l Studies , Indian Institute of T ec hnolo gy , Kharag pur-721 302, India 2 Pr ofess or of Mathema t i c s, In s t itute of T e chn ic al Educ ati o n & R ese ar ch, Siksha ’ O’ Anusandhan Universit y , Bh uba nesw ar-75 1030 , India Abstract Of concern i n the paper is a generalize d theo retical stud y of the non-Newtonian c har- acteristics of p eristaltic flo w of blo o d through micro-v essels, e.g. arterioles. The v essel is considered to b e of v ariable cross-sectio n and blo o d to b e a Hersc hel-Bulkley t yp e of fluid. T he progressiv e wa ve fron t of the p eristaltic flow is supp osed s inusoidal/straigh t section dominated (SSD) (expansion/con traction typ e); Reynolds num b er is c onsidered to b e sm all with reference to blo o d flo w in the m icro-circulatory system. The equations that go ve rn the non-Newtonian p eristaltic flo w of blo o d are considered to b e n on-linear. The ob jectiv e of the stud y has b een to examine the effect of amp litude ratio, mean p ressure gradien t, yield stress and the p o wer la w ind ex on the vel o cit y distribu tion, w all sh ear stress, str eamline p attern and trapp ing. It is obs erv ed that the numerical estimates for the aforesaid quan tities in the case of p eristaltic transp ort of the blo o d in a channel are m uc h different from those for fl o w in an axisymmetric ve ssel of circular cross-section. The study fu r ther shows that p eristaltic pumpin g, flo w vel o cit y and w all shear stress are sig- nifican tly altered du e to the non -u niformit y of the cross-sectional radius of blo o d vessel s of the m icro-circulato ry system. Moreo v er, the magnitude of th e amplitude ratio and th e v alue of the fluid ind ex are imp ortant parameters th at affect the flo w b eha viour. No v el features of SSD wa v e pr opagatio n that affect the flo w b eha viour of blo o d ha v e also b een discussed. Keywor ds: Non-Newtonian Fluid; Flo w Rev ersal; W all Shear Stress; T rappin g; SSD W a v e. ∗ Presently at Department of Applied Mathematics, I IT (BHU), V arana si 221 005, India. Email address: maiti0000 000somnath@gmail. c om/somnathm@cts.iitkgp.ernet.in (S . Maiti) † Email a ddress: m isr ajc@gmail.c om/misr ajc@r e diffmail.c om (J. C. Misr a) 1 1 In tro duct i on P eristaltic transp ort of fluids through differen t v essels of h uman ph ysiological systems is known to ph ysiologists as a natural mec ha nism of pumping materials. The phenomenon of p eristalsis has imp orta n t applications [1 , 2, 3, 4] in the design and construction of many useful devices of biomedical enginee ring and tec hnolog y , artificial blo o d device s suc h as heart-lung mac hine, blo o d pump ma chine, dialysis mac hine etc. Apart fr om phys iological studies, man y of the essen tial fluid mec hanical c haracteristics of p eristalsis ha v e also b een elucidated in analyses of differen t engineering problems carried o ut b y sev eral researc hers. These c haracteristics b ecome more prominen t when the flo w is induced by progressiv e w av es of area con t raction/expansion along the length of the b oundar y of a fluid-filled distensible tub e. Benefits that can b e deriv ed from studies on p eristaltic mov emen t hav e b een elab orately discussed in our earlier comm unications [5, 6 , 7, 8, 9, 10, 11, 12, 13, 14]. F urther discussion on p eristalsis was made b y sev eral other researc hers (cf. Guyton and Hall [15], Jaffrin and Shapiro [16], Sriv asta v a and Sriv asta v a [31], V a jra v elu et al. [18], Ha y at et al. [19]). 2 Nomenclature R, θ , Z Cylindrical co-ordinates c Sp eed of t he tra v elling w av e d W a ve amplitude a 0 Radius of the micro-v essel at t he inlet R 0 Radius of the plug flo w regio n H Displacemen t of the wall in the radia l direction I Iden tit y matrix n Flo w index num b er k Recipro cal of n k 1 A para metric constant P Fluid pressure Q Flux at an y lo cation t Time Re Reynolds n um b er U, V , W V elo cit y comp onen ts in Z -, R- and θ - directions resp ectiv ely δ W a v e n umber ∆ P Pressure difference b et w een the ends of t he v essel segmen t ˙ γ Strain rate of the fluid λ W av e length of the trav elling w av e motion a t the w all λ c A p ortion of length at which SSD expansion/con traction w av es a re confined µ Blo o d viscosit y µ 0 A constant denoting the limiting viscosit y of blo o d ν Kinematic viscosit y of blo o d φ Amplitude ratio Π The second inv ariant of the strain-ra te tensor Π 0 Limiting v alue of Π defined by (1 1 ) ρ Densit y o f blo o d τ 0 Yield stress o f blo o d τ h W all shear stress A theoretical foundation of p eristaltic transp ort in inertia-free Newtonian flo ws driv en b y si- n usoidal transv erse w a v es of small amplitude w as suggested b y F ung and Yih [20]. Shapiro 3 et a l. [21] presen ted the closed form solution for an infinite train o f p eristaltic w av es for small Reynolds num b er flo w, where the w a v e length is larg e and wa v e amplitude is arbitrary . The said in ves tigations witnessed a v ariety o f imp ortan t applications that are v ery useful in explaining the functioning of v arious ph ysiological systems, suc h as flows in ureter, g a stro-in testinal tract, small blo o d ves sels of the micro-circulator y system a nd glandular ducts. Conditions for the the existence of ph ysiologically significan t phenomena of tra pping and reflux in p eristaltic tr a nsp ort w ere also presen ted in the aforesaid commu nication b y Shapiro et al. [21]. References to some of the earlier lit era t ures on p eristaltic flo w of phys iological fluids are av ailable in comm unications of Jaffr in and Shapir o [16] as w ell as Sriv astav a and Sriv atav a [17], while br ief reviews o f some recen t lit era t ur es hav e b een ma de b y Tsiklauri and Beresnev [22], Mishra and Rao [23], Y aniv et al. [24], Jimenez-Lozano et al. [25 ], Nadeem and Akbar [26], Hay at et al. [19] as w ell as b y P andey a nd Chaub e [27]. Sev eral other studies dealing with analysis of differen t problems of p eristaltic transp ort of v ario us ph ysiological fluids w ere rep orted b y differen t in v estigato rs, e.g. T ak abatak e and Ayuk a w a [28], Jimenez-Lozano a nd Sen [29], Bohme a nd F riedric h [30], Sriv asta v a and Sriv asta v a [31], Pro vost and Sch w arz [32] and Chakrab o r t y [33 ]. P ast exp erimental o bserv at io ns indicate tha t the non-Newtonian b eha viour of whole blo o d mainly ow es its o rigin to the presenc e of erythro cytes. As early as in the sixth decade o f the last cen tury , differen t gr o ups o f scien tists, viz. Rand et al. [34], Bugliarello et al. [35] and Chien et al. [36] made an observ ation that the non-Newtonian character of blo o d is pro minen t as so on as t he hemato crit rises ab ov e 20%. Ho w ev er, it plays a dominant ro le, when hemato crit lev el lies b etw een 40 % and 70%. It is known that the nature of bloo d flow in sm all v essels (radius < 0.1 mm) at lo w shear rate ( < 2 0 sec − 1 ) can b e represen ted b y a p ow er law fluid (cf. Charm and Kurland [37, 38]). How ev er, Merrill et al. [39] observ ed that Casson mo del is somewhat satisfactor y for blo o d flo wing in tub es of 130-10 0 0 µ diameter. Later on Scott- Blair and Spanner [4 0] also rep orted tha t blo o d ob eys Casson mo del for mo derate shear rate flo ws. F urther, they observ ed that Hersc hel-Bulkley mo del is more appropriate than Casson mo del, more particularly for co w’s blo o d. Ho w ev er, t hey did not rep o r t an y difference b etw een Casson and Hersc hel-Bulkley plots ov er the range where Casson plot is v alid for blo o d. It is kno wn t ha t most v essels of ph ysiological system are of v arying cross-sectional diameter (cf. Wiedman [44], Wiede rhielm [45], Lee and F ung [46]). Some initial attempts to p erfo rm theoretical studies p ertaining to p eristaltic transp ort of ph ysiological fluids in v essels of v arying cross section w ere made b y sev eral researc hers (cf. Sriv astav a and Sriv asta v a [17], Wiederhielm [45], Lee and F ung [46], Gupta and Seshadri [47], Sriv asta v a and Sriv asta v a [48]). In all these 4 studies, the ph ysiological fluids w ere considered either as Newtonian fluids or non-Newtonian fluids describ ed by Casson/p ow er-law mo dels. Ho w eve r, different flow characteristics ha v e not b een a dequately explained in these studies. Hersc hel-Bulkley mo dels are, of course, more suit- able to repres en t some ph ysiological fluids like blo o d, because the fluids represe n ted b y this mo del describ e v ery w ell material flows with a non-linear constitutiv e relationship depicting the b ehaviour of shear-thinning/ shear-t hic ke ning fluids that are o f m uc h relev ance in the field of biomedical engineering ([49]). Moreo ver, among the v arious types of non- Newtonian mo dels used to represen t blo o d, Hersc hel-Bulkley fluid mo del is mor e general. Use of this mo del has the adv an tage that the corresponding r esults f o r fluids represen ted by Bingham plastic mo del, p o we r la w mo del and Newtonian fluid mo del can b e deriv ed from those of the Hersc hel-Bulkley fluid mo del, a s differen t particular cases. Also, Hersc hel-Bulkley fluid mo del yields more accurate results than man y other non-Newtonian mo dels. In view of all the ab o ve, a study concerning the analysis of a non- linear problem of p eri- staltic flow of blo o d in a v essel of the micro-circulator y system has b een undertak en here, by treating blo o d as a non-Newtonian fluid of Hersc hel-Bulkley type and the ves sel to b e of v arying cross-section. It is w orth while to men tion that although flo w through axisymmetric tub es is qualitativ ely almost similar to t he case of flow in channels , mag nitude of differen t ph ysiologi- cal quan tities a sso ciated with flo w and pressure differ a ppreciably . Since for blo o d flo w in the micro-circulatory system, the Reynolds n um b er is low and since the r a tio b etw een the radius of the tub e and t he w av e length is small, the theoretical analysis has b een p erformed b y using the lubrication theory [21]. Based up on the ana lytical study , extensiv e n umerical calculations hav e b een made. Keeping a sp ecific situation of micro-circulation in view, the pumping p erformance has been in v estigated n umerically . The computational res ults are presen ted f o r the v elo city distribution of blo o d, the w all shear stress distribution as w ell as the streamline pattern and trapping. Influence of SSD w av e fron t on v arious flo w v ariables concerned with p eristaltic trans- p ort has b een discuss ed. The plots g iv e a clear view o f the qualitativ e v ariat io n of v ario us fluid mec hanical par ameters. The results presen ted for shear thinning and shear thic k ening fluids are quite relev ant in the context of v ario us studies of blo o d rheolog y . It is imp ortant to men tion here that normal blo o d usually b ehav es like a shear thinning fluid for whic h with the increase in shear ra te, the viscosit y decreases. As mentioned in [41, 4 2, 43], in the case of hardened red blo o d cell susp ensions, the fluid b ehaviour is similar to that of a shear thick ening fluid for whic h the fluid viscosit y is enhanced due to increase in shear rate. 5 The study pro vides some no vel information b y whic h w e can hav e a b etter insigh t o f blo o d flo ws taking place in the micro- circulatory sy stem. More part icularly , it has an imp ortant b earing on the clinical pro cedure o f extra-corp oreal circulation of blo o d through the use o f the heart-lung mac hine that ma y result in damage of erythro cytes o wing to high v ar ia tion of the w all shear stress. In addition, the results should find useful a pplication in the dev elopment o f roller pumps and arthro-pumps b y whic h fluids are transp orted in living org ans. 2 F orm ulatio n A non-linear problem concerning the p eristaltic motion o f blo o d in a micro-ve ssel will b e stud- ied here , b y conside ring blo o d a s an incompre ssible viscous no n-Newtonian fluid. The non- Newtonian viscous b ehaviour will b e represen ted by Hersc hel-Bulkley fluid mo del. The v essel will b e considered to b e of v arying cross-sectional radius. Ho w eve r, it will b e considered as an axisymmetric v esse l and the flo w that ta k es pla ce through it as a lso axisymmetric. W e take (R, θ ,Z) as the cylindrical co ordinates of the lo cation of an y fluid particle, Z b eing measured in the direction o f w a v e propagation. R = H (cf. Fig. 1) represen ts the wall of the micro-v esse l and the flo w is supp osed to b e induced by either a progressiv e sinus oidal w av e or an SSD expansion/con traction wa v e train propaga ting with a constan t sp eed c trav elling down the w all, such that H = a ( Z ) + d sin  2 π λ ( Z − ct )  , (1) H =    a ( Z ) + d sin  π λ c ( Z − ct )  : if 0 ≤ Z − ct ≤ λ c a ( Z ) : if λ c ≤ Z − ct ≤ λ, (2) H =    a ( Z ) − d sin  π λ c ( Z − ct )  : if 0 ≤ Z − ct ≤ λ c a ( Z ) : if λ c ≤ Z − ct ≤ λ ; (3) The SSD expansion/con traction wa v es defined by equations (2) and (3) a re confined to a po rtion of length λ c . Let us consider a(Z)= a 0 + k 1 Z, where a ( Z ) represen ts t he radius o f the ve ssel at an y axial distance Z f rom the inlet, a 0 is the radius at the inlet and k 1 ( < 1) is a constan t whose magnitude dep ends on the length of the ve ssel as w ell as on the dimensions of the inlet and the exit; b is the w av e amplitude, t is t he time v aria ble and λ denotes the wa v e length. 6 R=R 0 c R=H R a 0 λ R=0 Z d c R=R 0 R Z d λ c a 0 λ R=H R=0 ( a ) ( b ) λ λ c R c d a 0 R=R R=H R=0 0 Z (c) Figure 1: A phys ical sk etc h of the pro blem for a tap ered ve ssel in the case of (a) Sinus oidal w av e, (b) SSD expansion w av e and ( c) SSD contraction wa v e 7 3 Analysis Using the fixed frame of reference w e shall p erform the ana lysis of the non-linear problem. F or the mo del f o rm ula ted in the preceding section, flow of blo o d in the micro-ves sel can b e a ssumed to b e g ov erned by the partia l differential equations ∇ · V = 0 (4) and ρ d V dt = ∇ σ + ρ f , (5) where V is the velocity , f the b o dy fo rce p er unit mass, ρ the fluid densit y and d dt the material time-deriv ativ e. σ represen ts the Cauc h y stress tensor defined by σ = − P I + T , in whic h T = 2 µ E ij + η IS, and S = ∇ · V , E ij b eing the symmetric part of the v elo cit y gradient, defined by E ij = 1 2 [ L + L T ], where L = ∇ V . − P I represen ts the indeterminate part of the stress due to the constraint of incompressibilit y , while µ and η denote viscosit y parameters. As mentioned in Sec. 1, bloo d in micro-v essels is considere d in the pres en t study as a Hersc hel-Bulkley fluid [50, 51]. It ma y b e mentioned that Hersc hel-Bulkley mo del is the repre- sen tative of the com bined effect of Bingham plastic and p ow er-law b ehav ior of t he fluid. When strain-rate ˙ γ is lo w suc h that ˙ γ < τ 0 µ 0 , the fluid b ehav es lik e a viscous fluid with constan t viscos- it y µ 0 . But as the strain rate increases and the yield stress threshold ( τ 0 ) is reac hed, t he fluid b eha vior is b etter describ ed by a p ow er law of the form µ = τ 0 + α n ˙ γ n −  τ 0 µ 0  n o ˙ γ , in whic h α and n denote respective ly the consistency factor and the p ow er la w index. n < 1 corresp onds to a shear thinning fluid, while for a shear thic k ening fluid n > 1. In t he case of a uniformly circular v essel, if t he length of the ve ssel is an in tegra l multiple of the w av elength a nd the pressure difference b et w een the ends of the v essel is a constan t, the flo w is steady in the w av e fra me. Since in the presen t study , the geometry of w all surface is considered non- uniform, the flow is inheren tly unsteady in the lab or a tory frame as w ell as in the 8 w av e frame of reference. Disregarding the b o dy forces (i.e. taking f=0) the Hersc hel-Bulkley equations that are b eing considered here as the gov erning equations of the incompressible fluid motion in the micro-v essel, in the fixed frame of reference ma y b e put in t he fo rm ρ ∂ U ∂ t + U ∂ U ∂ Z + V ∂ U ∂ R ! = − ∂ P ∂ Z + 1 R ∂ ( Rτ RZ ) ∂ R + ∂ τ Z Z ∂ Z (6) ρ ∂ V ∂ t + U ∂ V ∂ Z + V ∂ V ∂ R ! = − ∂ P ∂ R + 1 R ∂ ( Rτ RR ) ∂ R + ∂ τ RZ ∂ Z (7) w ith τ ij = 2 µE ij = µ ∂ U i ∂ X j + ∂ U j ∂ X i ! , (8) µ =    µ 0 : f or Π ≤ Π 0 , α Π n − 1 + τ 0 Π − 1 : f or Π ≥ Π 0 (9) Π = q 2 E ij E ij (10) The limiting viscosit y µ 0 is considered suc h that µ 0 = α Π n − 1 0 + τ 0 Π − 1 0 . (11) The follow ing no n- dimensional v ariables will b e in tro duced in the analysis tha t fo llows: ¯ Z = Z λ , ¯ R = R a 0 , ¯ U = U c , ¯ V = V cδ , δ = a 0 λ , ¯ P = a n +1 0 P αc n λ , ¯ t = ct λ , h = H a 0 , φ = d a 0 , Re = ρa n 0 αc n − 2 , ¯ λ c = λ c λ , ¯ τ 0 = τ 0 α  c a 0  n , ¯ τ RZ = τ RZ α  c a 0  n , ¯ Ψ = Ψ a 0 c . (12) The equation gov erning the flow of the fluid can no w b e rewritten in the form (dropping the bars o v er the sym b ols) Reδ ∂ U ∂ t + U ∂ U ∂ Z + V ∂ U ∂ R ! = − ∂ P ∂ Z + 1 R ∂  Φ  R ∂ U ∂ R + Rδ 2 ∂ V ∂ Z  ∂ R + 2 δ 2 ∂  Φ ∂ U ∂ Z  ∂ Z (13) Reδ 3 ∂ V ∂ t + U ∂ V ∂ Z + V ∂ V ∂ R ! = − ∂ P ∂ R + δ 2 1 R ∂  R Φ ∂ V ∂ R  ∂ R + δ 2 ∂  Φ  ∂ U ∂ R + δ 2 ∂ V ∂ Z  ∂ Z (14) w her e Φ =        v u u u t 2 δ 2    ∂ V ∂ R ! 2 +  V R  2 + ∂ U ∂ Z ! 2    + ∂ U ∂ R + δ 2 ∂ V ∂ Z ! 2        n − 1 + τ 0        v u u u t 2 δ 2    ∂ V ∂ R ! 2 +  V R  2 + ∂ U ∂ Z ! 2    + ∂ U ∂ R + δ 2 ∂ V ∂ Z ! 2        − 1 (15) 9 W e now use the long w a v elength appro ximatio n ( δ ≪ 1) and the lubrication approac h [21, 23]. Then the gov erning equations stated earlier describing the flow in the lab oratory fra me o f reference in terms o f the dimensionless v ariables (12), assume the form 0 = − ∂ P ∂ Z + 1 R ∂ ( R ∂ U ∂ R | ∂ U ∂ R | n − 1 + τ 0 ) ∂ R (16) and 0 = − ∂ P ∂ R . (17) Also the form o f the b oundary conditions will no w b e Ψ = 0 , ∂ U ∂ R = ∂  1 R ∂ Ψ ∂ R  ∂ R = 0 , τ RZ = 0 at R = 0; (18) and U = 1 R ∂ Ψ ∂ R = 0 a t R = h . (19) The solution of equation (16) sub ject to the conditions (18) and (19) is found to b e giv en b y U ( R , Z, t ) = 1 ( k + 1) P 1 h ( P 1 h − τ 0 ) k +1 − ( P 1 R − τ 0 ) k +1 i , 0 ≤ R ≤ h (20) where P 1 = − 1 2 ∂ P ∂ Z and k = 1 n . If R 0 b e the ra dius of plug flo w region (where µ 0 = ∞ ), ∂ U ∂ R = 0 at R = R 0 . It follow s from (2 0) t ha t R 0 = τ 0 /P 1 . If τ RZ = τ h at R =h, w e find h = τ h /P . T hus R 0 h = τ 0 τ h = τ ( say ) , 0 < τ < 1 . (21) Then the expression of the plug velocity turns out to b e U P = ( P 1 h − τ 0 ) k +1 ( k + 1) P 1 . (22) In order to determine the stream function Ψ w e use the b oundary conditions Ψ P = 0 at R = 0 and Ψ = Ψ P at R = R 0 . In tegra ting (20) and (22), the stream function is found in the form Ψ = P k 1 k + 1 " R 2 2 ( h − R 0 ) k +1 − ( R − R 0 ) k +2 (( k + 2) R + R 0 ) ( k + 2)( k + 3) # , R 0 ≤ R ≤ h (23) 10 and Ψ P = P k 1 (h − R 0 ) k+1 R 2 2(k + 1) , 0 ≤ R ≤ R 0 (24) The instan taneous rate of volume flow through the micro- v essel, ¯ Q is giv en b y ¯ Q ( Z , t ) = 2 Z R 0 0 RU d R + 2 Z h R 0 RU d R = P k 1 ( h − R 0 ) k +1 ( h 2 ( k 2 + 3 k + 2) + 2 R 0 h ( k + 2) + 2 R 2 0 ) ( k + 1)( k + 2)( k + 3) , k = 1 n . (25) F rom ( 25), w e ha v e P 1 = − 1 2 ∂ P ∂ Z = " ¯ Q ( Z , t )( k + 1)( k + 2)( k + 3) ( h − R 0 ) k +1 ( h 2 ( k 2 + 3 k + 2) + 2 R 0 h ( k + 2) + 2 R 2 0 ) # n = " ¯ Q ( Z , t )( k + 1)( k + 2)( k + 3) h k +3 (1 − τ ) k +1 (( k 2 + 3 k + 2) + 2 τ ( k + 1) + 2 τ 2 ) # n . (26) The a v erag e pressure difference p er w av e length can no w b e calculated b y using the formula ∆ P = Z 1 0 Z 1 0 ∂ P ∂ Z ! d Z dt . (27) W e observ e that when n = 1 , τ = 0 and k 1 = 0, the equations (20), (23 ), (25) and (2 6 ) reduce to those obtained by Shapiro et al. [21] for a Newtonian fluid. Our results also mat ch with those o f La rdner and Shack [52], when the eccen tricit y of the elliptical motion of cilia tips is set equal to zero in their analysis fo r a Newtonian fluid flo wing on a uniform c hannel. Also, for n = 1 and R 0 = 0, equation (26) tallies with tha t o bta ined b y Gupta and Seshadri [47] for a Newtonian fluid of constan t viscosit y . T o obtain the quantitativ e results, the instan taneous rate of v olume flow ¯ Q ( Z , t ) has b een considered to b e p erio dic in (Z- t) [47, 48], so that ¯ Q app earing in equation (26) can b e repre- sen ted as ¯ Q n ( Z , t ) =          Q n + φ 2 sin 2 2 π ( Z − t ) + 2 φ (1+ k 1 λZ ) sin 2 π ( Z − t ) a 0 − φ 2 2 : f or sinusoidal w av e, Q n + h 2 − 1 − λ c φ 2 2 − 4 λ c φ π : f or S S D expansion w av e, Q n + h 2 − 1 − λ c φ 2 2 + 4 λ c φ π : f or S S D contr action w av e. (28) In (28), Q represen ts the time-av eraged flo w flux. Since t he right hand side of equation (27) cannot b e in tegrated in closed form, f o r non-uniform/uniform geometry , for further in ves tigation of the problem under consideration, w e had to resort to the use of softw are Mathematica. This help ed us calculate the n umerical estimate of the pressure difference, ∆ P giv en b y (27). 11 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 U R z=0.5, t=0.25, k 1 =0, φ =0.5, τ =0, n=1, ∆ P=0 Results of our study Results of Shapiro et al [21] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 U R z=0.5, t=0.25, k 1 =0, φ =0.5, τ =0, n=2/3, ∆ P=0 Results of our study Results of Srivastava and Srivastava [31] ( a ) ( b ) Figure 2: V ariatio n of axial v elo cit y in the radial direction at Z=0.5, (a) Newtonian fluid, (b) Shear-thinning fluid 4 Quan tit ati v e In v esti g ation Theoretical estimates of differen t ph ysical quantities tha t a r e of relev ance to the ph ysiological problem of blo o d flo w in the micro-circulatory system hav e b een obtained o n the basis of t he presen t study . F or this purp ose, the following data t ha t ar e v alid in the ph ysiological range [15, 17, 4 1, 53] ha v e b een used: a 0 = 10 to 60 µm , φ = 0.1 to 0.9, a 0 λ = 0 . 01 to 0 . 02, ∆ P = − 300 to 50; τ = 0 . 0 to 0.2; Q=0 to 2, n= 1 3 to 2. The v alue of k 1 for the non- uniform geometry of the micro-v essel, has b een chos en to matc h with ph ysiological system. Th us for arterioles, whic h are o f conv erging t yp e, the width of t he outlet o f one w a v e length has b een tak en to b e 2 5% less t ha n that of the inlet; in the case of div erging v essels (e.g. v en ules), the width of the outlet of one w a v e length has b een tak en 25% more than that of the inlet. 4.1 Distribution of V elo cit y F or differen t v alues o f the a mplitude ratio φ , flo w index nu m b er n, k 1 and τ , F ig s. 2- 6 presen t the distribution of axial ve lo cit y o f blo o d in the cases of free pumping, pumping and co-pumping zones. Fig . 2 sho ws that the results computed on the ba sis of o ur study for the particular case of a Newtonian fluid and shear-thinning fluid ( n = 2 / 3) tally w ell with the r esults rep orted b y Shapiro et al. [21] and Sriv asta v a and Sriv asta v a [3 1 ], resp ectiv ely when the amplitude ratio 12 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.5 0 0.5 1 1.5 U Z R U 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.5 0 0.5 1 1.5 U Z R U ( a ) ( b ) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.5 0 0.5 1 1.5 U Z R U 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.5 0 0.5 1 1.5 U Z R U ( c ) ( d ) Figure 3: Aerial view of the velocity distribution a t differen t instan ts of time, when n = 2 / 3 , k 1 = 0 , ∆ P = 0 , τ = 0 . 1 , φ = 0 . 5 fo r a non-Newtonian fluid of shear thinning type (a) t=0.0 (b) t=0.25 (c) t = 0.5 (d) t=0.75 13 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.25 0.5 0.75 R U k 1 =0, n=2/3, τ =0.1, ∆ P=0, φ =0.5 t=0.0 t=0.25 t=0.5 t=0.75 Figure 4: V elo city distribution a t differen t instants of time. φ = 0 . 5. Since t he v elo cit y pro files along with the radius of the blo o d v essels change with time, w e hav e in v estigated the distribution of v elo cit y at a time interv al o f a quarter of a w av e p erio d. Fig. 3 give s us the aerial view of a few t ypical axial velocity distributions for a Newtonian fluid and a shear-thinning fluid (n=2/3) flow ing in uniform micro-v essels , while Fig. 4 presen ts t he v elo cit y distribution in a v essel a t differen t instants of t ime. Fig. 5 rev eals that at a n y instan t of time, there exists a retrog r a de flo w regio n. Ho w ev er, the fo rw ar d flo w regio n is predominan t in this case, since the time-a v erag ed flo w rate is p ositive . F or a shear-thinning fluid (n=2 /3), the presen t study indicates that there exist t w o stagnatio n p oin ts on the axis. F or example, at time t=0.25, one of the stagnation p oin ts lies b et w een Z=0.0 and Z=0.25, while the other lies b et w een Z=0.75 and Z=1.0. Similar o bserv atio ns w ere made nume rically by T ak abata ke and Ayuk a wa [28] for a Newtonian fluid. Figs. 5(a)-(b) depict that in b oth the regio ns, a s τ increases, the magnitude of v elo cit y decreases for b o th types of fluids mentioned ab o ve. It can b e observ ed fr o m Fig. 5 ( c-d) that for a Hersc hel-Bulkley fluid (with n=2/ 3) when ∆ P = 0, the v elo cit y in b oth the regions of bac kw ard and forw a r d flo ws is enhanced as the v alue of φ increases in the interv al 0 < φ < 0 . 6, but b ey ond φ = 0 . 6, the v elo city in the backw ard flo w region decreases with φ increasing. These observ ations (cf. Figs. 5(c,d)) are in con trast t o the case of t w o dimensional c hannel flo w. In Figs. 5(e)-(k), it is w orth while to observ e the significan t influence of t he rheological fluid index ’n’ on t he velocity distribution f or flo ws in unifo rm/non-uniform v essels. These figures 14 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.25 0.5 0.75 R U k 1 =0, n=2/3 φ =0.5, ∆ P=0, t=0.0, τ =0.0 τ =0.1 τ =0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.25 0.5 0.75 R U k 1 =0, n=4/3 φ =0.5, ∆ P=0, t=0.0, τ =0.0 τ =0.1 τ =0.2 ( a ) ( b ) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.25 0.5 0.75 R U t=0.0, k 1 =0, τ =0.1 n=2/3, ∆ P=0, φ =0.2 φ =0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 0.25 0.5 0.75 R U t=0.0, k 1 =0, τ =0.1 n=2/3, ∆ P=0, φ =0.7 φ =0.9 ( c ) ( d ) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.25 0.5 0.75 R U    k 1 =0, τ =0.1 φ =0.5, t=0.0, ∆ P=0, n=1/3 n=1/2 n=2/3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.25 0.5 0.75 R U k 1 =0, τ =0.1, φ =0.5 t=0.0, ∆ P=0, n=4/3 n=5/3 n=2 ( e ) ( f ) 15 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.25 0.5 0.75 R U k 1 =0.25a 0 / λ , τ =0.1,       φ =0.5 t=0.0, ∆ P=0, n=1/3 n=1/2 n=2/3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.25 0.5 0.75 R U k 1 =0.25a 0 / λ , τ =0.1,   φ =0.5 t=0.0, ∆ P=0, n=4/3 n=5/3 ( g ) ( h ) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.25 0.5 0.75 R U k 1 =0.25a 0 / λ , τ =0.1,   φ =0.5 t=0.0, ∆ P=0, n=2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.25 0.5 0.75 R U k 1 = -0.25a 0 / λ , φ =0.5 τ =0.1, t=0.0, ∆ P=0, n=1/3 n=1/2 n=2/3 ( i ) ( j ) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.5 R U t=0.0, k 1 = -0.25a 0 / λ , τ =0.1, φ =0.5 ∆ P=0, n=4/3 n=5/3 n=2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.25 0.5 0.75 R U t=0.0, n=2/3 φ =0.5, τ =0.1, k 1 =0, ∆ P= -0.5 k 1 =0, ∆ P= -1.0 k 1 =0.25a 0 / λ , ∆ P= -0.5 k 1 =0.25a 0 / λ , ∆ P= -1.0 ( k ) ( l ) 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.25 0.5 0.75 R U t=0.0, n=4/3 φ =0.5, τ =0.1, k 1 =0, ∆ P= -0.5 k 1 =0, ∆ P= -1.0 k 1 =0.25a 0 / λ , ∆ P= -0.5 k 1 =0.25a 0 / λ , ∆ P= -1.0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.25 0.5 0.75 R U φ =0.5, t=0.0, k 1 = -0.25a 0 / λ τ =0.1, n=2/3, ∆ P= -0.5 n=2/3, ∆ P= -1.0 n=4/3, ∆ P= -0.5 n=4/3, ∆ P= -1.0 ( m ) ( n ) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.25 0.5 0.75 R U ∆ P=0, φ =0.5 (for expansion   waves) k 1 =0, n=2/3, τ =0.1, t=0.0, λ c =0.2 λ c =0.4 λ c =0.6 λ c =0.8 0 0.2 0.4 0.6 0.8 1 0 0.25 0.5 0.75 R U n=2/3, τ =0.1, t=0.0, k 1 =0, ∆ P=0, φ =0.5 (for contraction waves) λ c =0.2 λ c =0.4 λ c =0.6 ( o ) ( p ) Figure 5 : D istribution o f velocity in v arious cases. The fig ures reveal that flo w r eve rsal o ccur due to the contraction of the w av e. 17 rev eal that the parab olic nature of the v elo cit y profiles is disturb ed due to the non-Newtonian effect. Magnitude of the v elo cit y decreases at the maxim um expansion region; while in the remaining regions, a rev erse trend is noticed, when there is an increase in the v alue of ’n’ (cf. Figs. 5 (e-g,j-k)). F or a con v erg ing v essel, the magnitude of the velocity is greater t ha n that of a unifo rm ves sel; how ev er, for a div erging v essel, our observ ation is altogether differen t. Fig s. 5(l)-(n) illustrate the influence o f pressure on velocity distribution for shear thinning /shear thic kenin g fluids. The results presen ted for shear-thick ening fluid hav e b een computed b y taking n = 4 / 3, while tho se for shear-thinning fluid corresp ond to n = 2 / 3. It may b e noted that in the case of a uniform/dive rging v essel, if ∆ P decreases, the flo w rev ersal tends to decrease; how ev er, for a con v erging t ub e, a lt hough there is reduction in the region o f flow r eve rsal, the c hange is not to o significan t for shear thic k ening fluid. It is imp o rtan t to men tion that flo w rev ersal take s place due to c hang e in sign o f the vorticit y or the shear stress alo ng the wa vy w all. The results corresp onding to SSD w av e propagation are presen ted in Figs. 5 ( o-p). It is seen that for SSD expansion w a v es, there is no bac kw ard flo w and that the mag nitude of the ve lo city is considerably less in this case as compared to the case of sinus oidal w a v e propagation (cf. Fig. 4). Moreo v er, as λ c increases, v elo cit y is seen to rise except around Z=0.25 where it decreases when λ c exceeds 0 .6 . F or SSD con traction w a v es, Fig. 5(n) shows that if λ c = 0 . 2 , there is no bac kward flo w within one w av e length. If λ c > 0 . 3 , bac kw ard flow is observ ed from Z=0.15 to Z=0.7. 4.2 Pump ing Beha viour The pumping characteristics can b e determined through the v ariation of time a v eraged flux with difference in pressure across one w av e length ( cf. [21]). It is kno wn that if the flo w is steady in the w a ve frame, the instan taneous pressure difference b et w een t w o stations one w a v e length apart is a constan t. Since the pressure g radien t is a p erio dic f unction of ( Z − t ), pressure rise p er w av e length ∆ P λ is equal to λ times the time-av eraged pressure gradien t . In addition, the relation b et we en the fixed frame a nd the w av e fra me flux rates turns out to b e linear, a s in the case of inertia-free p eristaltic flo w of a Newtonian fluid. F rom the said relation, it is p ossible to calculate the amount of flo w pump ed b y p eristaltic w a v es, ev en in the absence of mean pressure gradien t. The region in whic h ∆ P = 0 is regarded to as the free pumping zone, while the region where ∆ P > 0 is said to b e the pumping zone. The situation when ∆ P < 0 is fav ourable for the flow to tak e place and t he corresp onding region is called the co-pumping zone. Since the 18 -20 -10 0 10 20 30 40 0 0.5 1 1.5 2 ∆ P Q k 1 =0, τ =0.1, n=2/3, φ =0.2 φ =0.5 φ =0.6 φ =0.7 -1000 -800 -600 -400 -200 0 200 400 600 800 0 0.5 1 1.5 2 ∆ P Q k 1 =0, τ =0.1, n=4/3, φ =0.2 φ =0.5 φ =0.6 φ =0.7 ( a ) ( b ) -500 -400 -300 -200 -100 0 100 200 0 0.5 1 1.5 2 ∆ P Q k 1 =0, τ =0.1, φ =0.5, n=1/3 n=2/3 n=4/3 n=5/3 -200 -150 -100 -50 0 50 0 0.5 1 1.5 2 ∆ P Q k 1 =0.25a 0 / λ , τ =0.1, φ =0.5, n=1/3 n=2/3 n=4/3 n=5/3 ( c ) ( d ) -5000 -4000 -3000 -2000 -1000 0 1000 0 0.5 1 1.5 2 ∆ P Q k 1 = -0.25a 0 / λ , τ =0.05, φ =0.5, n=1/3 n=2/3 n=4/3 n=5/3 -15 -10 -5 0 5 10 0 0.5 1 1.5 2 ∆ P Q k 1 =0, n=2/3, φ =0.5, τ =0.0 τ =0.1 τ =0.2 ( e ) ( f ) -150 -100 -50 0 50 0 0.5 1 1.5 2 ∆ P Q k 1 =0, n=4/3, φ =0.5, τ =0.0 τ =0.1 τ =0.2 -10 -5 0 5 10 15 0 0.5 1 1.5 2 ∆ P Q (for expansion waves) k 1 =0, τ =0.1, n=2/3, φ =0.5, λ c =0.2 λ c =0.4 λ c =0.6 λ c =0.8 ( g ) ( h ) 19 -10 -8 -6 -4 -2 0 0 0.5 1 1.5 2 ∆ P Q (for contraction waves) k 1 =0, τ =0.1, n=1/2, φ =0.5, λ c =0.2 λ c =0.4 λ c =0.6 λ c =0.8 (i) Figure 6: Pressure difference v ersus flow rate. It ma y b e observ ed that the area of t he pumping region increases when the amplitude r a tio φ is enhanced. In additio n, the v ariation of the v essel radius in b oth con v erging and dive rging cases as w ell as SSD expansion/con tractio n w a ve mo des ha v e significant impact on the pressure difference (∆P) a s w ell as on the volumetric flow rate (Q). presen t study is concerned with p eristaltic tra nsp ort of a non-Newtonian fluid, the aforesaid relationship b etw een the pressure difference and the mean flow rate is nonlinear. Fig. 6 illustrates the v ariation of v olumetric flow rate o f the fluid by wa y of propagation of p eristaltic w a v es, for differen t v alues of the amplitude ratio φ , flow index num b er n, a s w ell as the parameters τ and k 1 . Shapiro et al. [21] used lubrication theory to sho w that in the case of a New tonian fluid, the flo w rate a ve raged o v er one w a v e le ngth v aries linearly with pressure difference. But for our study of t he p eristaltic transp ort of a no n- Newtonian fluid, the relationship b et w een the pressure difference and the mean flow rate is found to b e non- linear (cf. Figs 6). The plots presen ted in this figure sho w that for a non-Newtonian fluid, the mean flow rate Q increase s as ∆ P decreases. Figs. 6(a-b) indicate that area o f the pumping region increases with an increase in the amplitude ratio φ for b oth shear thinning and shear t hick ening fluids. Figs. 6 ( c- e) illustrate the influence o f the rheological parameter ‘n’ on the pumping p erformance in uniform/ diverging/con v erging ves sels. It ma y b e noted that the pumping region (∆ P > 0) significan tly increases as the v alue of ‘n’ increases, while in the co-pumping region (∆ P < 0) the pressure-difference decreases when Q exceeds a certain v alue. Figs. 6(f-g) sho w that Q is not significan tly a ff ected b y the v alue o f the parameter τ in the case of free pumping. W e further find that for b oth shear-thinning and shear thic k ening fluids, pumping regio n increases with 20 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 φ =0.2, τ =0.1, n=2/3 Q=0.4 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 -0.18 -0.2 -0.22 -0.24 -0.26 -0.28 -0.3 z r -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.1, n=2/3 Q=0.4 0 -0.03 -0.06 -0.09 -0.12 -0.15 -0.18 -0.21 -0.24 -0.27 -0.3 -0.33 -0.36 z r ( a ) ( b ) -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 φ =0.7, τ =0.1, n=2/3 Q=0.4 0.03 0 -0.03 -0.06 -0.09 -0.12 -0.15 -0.18 -0.21 -0.24 -0.27 -0.3 -0.33 -0.36 -0.39 -0.42 z r -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 φ =0.8, τ =0.1, n=2/3 Q=0.4 0.06 0.03 -5.55e-17 -0.03 -0.06 -0.09 -0.12 -0.15 -0.18 -0.21 -0.24 -0.27 -0.3 -0.33 -0.36 -0.39 -0.42 -0.45 z r ( c ) ( d ) Figure 7: Streamline pa t t erns for p eristaltic flo w of a shear-thinning fluid for differen t v alues of φ when n=2/3, Q=0.4, k 1 = 0, τ = 0 τ increasing. Moreo ver, when Q excee ds a certain critical limit, ∆ P = 0, pressure difference decreases with an increase in τ . The effect of SSD w av e propagation on pumping is rev ealed in Figs. 6(h- i) . Unlike sin usoidal w av e form, Q increases significantly in pumping, free-pumping as w ell as co-pumping regions for SSD expansion w a ve f r on ts a s λ c increases. F or SSD contraction wa v e fronts, o ur observ ation is altogether differen t from SSD expansion wa v e fron ts. Ho we v er, Q ≤ 0 when ∆ P < 0 for shear-thinning fluids in the case of con traction w av es. This observ ation is in contrast to the case of sin usoidal w av e propagation. 21 4.3 Streamlines and T rapping It is known that one o f the imp ort a n t characteristics of p eristaltic tra nsp ort is the phenomenon of trapping. It o ccurs when streamlines on the cen tral line are split to enclose a b olus of fluid particles circulating alo ng closed streamlines in the w a v e frame of reference. The n the trapp ed b olus mov es with a sp eed equal to the wa v e propagation v elo cit y . This ph ysical phenomenon ma y b e resp onsible for the formation o f thr o m bus in blo o d. Let us consider the w a ve frame transformations (x,y) mo ving with a v elo cit y c a wa y from a fixed frame ( X,Y) suc h that x = X − ct, y = Y , u = U − c, v = V , p ( x, y ) = P ( X , Y , t ) , in whic h (u,v) and (U,V) are the v elo cit y comp onen ts, p and P stand for pressure in w a ve frame and fixed fra me of reference resp ectiv ely . Under the purview of the presen t study , Figs. 7- 10 give an insigh t into the c hanges in the pattern of streamlines and trapping that o ccurs due to c hanges in the v alues of different parameters that go v ern t he flo w of blo o d in the w a v e frame of reference. Figs. 7 prov ide the streamline patterns and trapping in the case o f a shear-thinning fluid for differen t v alues o f φ . With an increase in φ , the b olus is found to a pp ear in a distinct manner. Streamlines for differen t v alues of the fluid index ‘n’ a re depicted in Fig. 8 . This fig ure indicates tha t o ccurrence of tr a pping is strongly influenced b y the v alue of the fluid index. Fig. 9 sho ws that trapp ed b olus increases in size and also that it has a t endency to mo v e tow ards the b o undary as the flow rate increases. Here it is imp ortant to not e that t he b olus app earing for small v alues of τ decreases in size with an increase in τ (Fig s. 10). 4.4 Distribution of W all Shear Stress If the shear stress generated on the w a ll of a blo o d v essel exceeds a certain limit, the constituen ts of blo o d are lik ely to b e damaged. The magnitude of the wall shear stress has a vital r o le in the pro cess of molecular con vec tion at high Prandtl or Sc hmidt n umber [54]. In view of these observ a tions, it is imp ortant to study the shear stress that is dev elop ed during the hemo dy- namical flow of blo o d in small arteries. The w all shear stress distributions fo r the presen t study are plot t ed in Figs. 11 under v aried conditions. The distributions of wall shear stress at fo ur differen t time instants during one complete w a v e p erio d hav e b een presen ted in Fig. 1 1(a). It ma y b e observ ed from this figure that a t each of these instants of time, there exist tw o p eaks in the w all shear stress distribution, with a gradual ramp in b et w een; how ev er, negativ e p eak of w all shear stress τ min is not as large a s the maxim um w all shear stress τ max . The transition from τ min to τ max of wall shear tak es pla ce in some zone b etw een the minimum and maxim um radii 22 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.1, n=1/3 Q=0.4 -0.03 -0.06 -0.09 -0.12 -0.15 -0.18 -0.21 -0.24 -0.27 -0.3 -0.33 -0.36 z r -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.1, n=2/3 Q=0.4 0 -0.03 -0.06 -0.09 -0.12 -0.15 -0.18 -0.21 -0.24 -0.27 -0.3 -0.33 -0.36 z r ( a ) ( b ) -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.1, n=4/3 Q=0.4 0.03 0 -0.03 -0.06 -0.09 -0.12 -0.15 -0.18 -0.21 -0.24 -0.27 -0.3 -0.33 -0.36 z r -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.1, n=5/3 Q=0.4 0.06 0.03 0 -0.03 -0.06 -0.09 -0.12 -0.15 -0.18 -0.21 -0.24 -0.27 -0.3 -0.33 -0.36 z r ( c ) ( d ) Figure 8: Streamline patterns a nd trapping in the case of p eristaltic flow for differen t v alues of the ph ysiological fluid index ‘n’ when φ = 0 . 5 , τ = 0 . 1 , k 1 = 0 , Q = 0 . 4 . The figures sho w t ha t the o ccurrence of trapping is highly dep enden t on the v alue of the fluid index ‘n’. 23 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.1, n=2/3 Q=0.2 -0.02 -0.06 -0.1 -0.14 -0.18 -0.22 -0.26 -0.3 -0.34 -0.38 -0.42 -0.46 z r -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.1, n=2/3 Q=0.6 0.065 0.04 0.015 -0.01 -0.035 -0.06 -0.085 -0.11 -0.135 -0.16 -0.185 -0.21 -0.235 -0.26 z r ( a ) ( b ) -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.1, n=2/3 Q=1.0 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 z r -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.1, n=2/3 Q=1.4 0.34 0.31 0.28 0.25 0.22 0.19 0.16 0.13 0.1 0.07 0.04 0.01 z r ( c ) ( d ) Figure 9: Streamline pa tterns and trapping for the effect of Q , when n = 2 / 3 , φ = 0 . 5 , k 1 = 0 , τ = 0 . 1 24 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.0, n=2/3 Q=0.4 0.015 -0.01 -0.035 -0.06 -0.085 -0.11 -0.135 -0.16 -0.185 -0.21 -0.235 -0.26 -0.285 -0.31 -0.335 -0.36 z r -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.1, n=2/3 Q=0.4 0.015 -0.01 -0.035 -0.06 -0.085 -0.11 -0.135 -0.16 -0.185 -0.21 -0.235 -0.26 -0.285 -0.31 -0.335 -0.36 z r ( a ) ( b ) -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.2, n=2/3 Q=0.4 0.015 -0.01 -0.035 -0.06 -0.085 -0.11 -0.135 -0.16 -0.185 -0.21 -0.235 -0.26 -0.285 -0.31 -0.335 -0.36 z r -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 φ =0.5, τ =0.3, n=2/3 Q=0.4 -0.01 -0.035 -0.06 -0.085 -0.11 -0.135 -0.16 -0.185 -0.21 -0.235 -0.26 -0.285 -0.31 -0.335 -0.36 z r ( c ) ( d ) Figure 10: Streamline patterns for different v alues of τ in the case of p eristaltic flo w when n = 2 / 3 , φ = 0 . 5 , k 1 = 0 , Q = 0 . 4. It ma y b e observ ed that the size o f the b olus reduces with the increase in the v alue of τ . The b o lus to t a lly disapp ears for τ ≥ 0 . 3. 25 of the ves sel. At the lo cation where the maxim um o cclusion o ccurs, w all shear stress along with the pressure is maximu m. Since the pressure gradien t to the left o f this lo cation ta kes a p ositiv e v alue, the lo cal instantaneous flow will take place tow ards the left of τ max . This ma y lead t o some serious consequences . F or example, if the shear rate at the crest exceeds some limit, a dissolving wa vy w all will hav e a tendency to lev el out. Moreov er, some bio-c hemical reaction b et w een the wall ma t erial and the constituents of blo o d may set in. As a result of this, the pro ducts of the che mical reaction may b e dep osited on the endothelium and consequen tly w a ll amplitude ma y increase at a rapid rate. This is like ly to lead to clogging of the blo o d v essel. The p eaks of the wall shear stress distribution o n b oth sides of τ max are small and hence the lo cal instan taneous flow will o ccur in t he direction of propagation of the p eristaltic wa v es. F or a Hersc hel-Bulkley fluid, Figs. 11(b-c) sho w that in t he con tracting regio n where o cclusion t a k es place, there is a remark able increase in the w all shear stress due to an increase in the v alue of φ for b oth shear-thinning a nd shear-thick ening fluids. With an increase in φ , magnitude of τ min increases in the expanding region for shear-thinning/shear-thick ening fluids, although the effect is not very prominent. It may b e observ ed from Fig. 11(d) that τ max increases with increase in τ , while τ has little effect on τ min . The quantum of influence of the rheological fluid index ’n’ on the distribution of w a ll shear stress is sho wn in Figs. 11(e-g) for uniform/non-unif o rm blo o d ves sels. In all t yp es of v essels studied here, τ max increases with increase in ‘n’. It is v ery imp orta n t to men tion that the difference of shear stress b et w een the outlet and the inlet in the case of a conv erging ve ssel is exceedingly large in comparison to the case of a div erging v essel. As the time av eraged flo w rate increases, Figs. 11(h-j) indicate very clearly that the w all shear stress tends to decrease for a ll t yp es of v essels examined here. One can observ e from Figs. 11(k-l) that τ h c hang es its v alues within the r egion of SSD w a ve a ctiv atio n; b ey o nd this, it main tains a constan t v alue. 5 Summary and Conc l usion The pr esen t pap er deals with a study of the p eristaltic motion of blo o d in the micro- circulatory system, b y taking into account the non-Newtonian nature of blo o d and the non-uniform geom- etry of the micro-ves sels, e.g. arterioles and v en ules. The non-Newtonian viscosit y of blo o d is considered to b e of Hersc hel-Bulkley type. The effects of amplitude ratio, mean pres sure gradien t, yield stress and the rheolog ical fluid index n on the distribution of velocity and wall shear stress as w ell as on the pumping phenomena, formation of the streamline pattern and the 26 -3 -2 -1 0 1 2 3 4 5 6 0 0.5 1 τ h Z τ =0.1, n=2/3, Q=0.25, k 1 =0, φ =0.5, t=0.0 t=0.25 t=0.5 t=0.75 0 5 10 15 20 25 30 35 40 0 0.2 0.4 0.6 0.8 1 τ h Z t=0.25, τ =0.1, Q=0, k 1 =0, n=2/3, φ =0.2 φ =0.5 φ =0.6 φ =0.7 ( a ) ( b ) 0 200 400 600 800 0 0.2 0.4 0.6 0.8 1 τ h Z t=0.25, τ =0.1, Q=0, k 1 =0, n=4/3, φ =0.2 φ =0.5 φ =0.6 φ =0.7 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 τ h Z t=0.25, Q=0, k 1 =0, φ =0.5, n=2/3, τ =0.0 n=2/3, τ =0.1 n=2/3, τ =0.2 n=4/3, τ =0.0 n=4/3, τ =0.1 n=4/3, τ =0.2 ( c ) ( d ) 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1 τ h Z t=0.25, Q=0, k 1 =0, φ =0.5, τ =0.1, n=1/3 n=2/3 n=4/3 n=5/3 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 τ h Z t=0.25, Q=0, k 1 =0.25a 0 / λ , φ =0.5, τ =0.1, n=1/3 n=2/3 n=4/3 n=5/3 ( e ) ( f ) 27 0 1000 2000 3000 4000 5000 6000 7000 0 0.2 0.4 0.6 0.8 1 τ h Z t=0.25, Q=0, k 1 = -0.25a 0 / λ , φ =0.5, τ =0.1, n=1/3 n=2/3 n=4/3 n=5/3 0 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 τ h Z t=0.25, k 1 =0, φ =0.5, τ =0.1, n=2/3, Q=0.25 n=2/3, Q=0.5 n=4/3, Q=0.25 n=4/3, Q=0.5 ( g ) ( h ) 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 τ h Z t=0.25, k 1 =0.25a 0 / λ , φ =0.5, τ =0.1, n=2/3, Q=0.25 n=2/3, Q=0.5 n=4/3, Q=0.25 n=4/3, Q=0.5 0 100 200 300 400 500 600 700 800 0 0.2 0.4 0.6 0.8 1 τ h Z t=0.25, k 1 = -0.25a 0 / λ , φ =0.5, τ =0.1, n=2/3, Q=0.25 n=2/3, Q=0.5 n=4/3, Q=0.25 n=4/3, Q=0.5 ( i ) ( j ) -2 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 τ h Z (for expansion waves) k 1 =0, t=0.25, τ =0.1, n=2/3, Q=0.25, φ =0.5, λ c =0.2 λ c =0.4 λ c =0.6 λ c =0.2 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 τ h Z (for contraction waves) Q=0.25, τ =0.1, n=2/3, k 1 =0, t=0.25, φ =0.5, λ c =0.2 λ c =0.4 λ c =0.6 λ c =0.2 ( k ) ( l ) Figure 11: W all shear stress distribution in different situations. W a ll shear stress a ttains its maxim um/minimum at the lo cation where the radius of the v essel is smallest/largest. 28 o ccurrence of trapping are inv estigated under the purview of the lubrication theory . Exp erimen- tal observ ations hav e rev ealed that in t he case of roller pumps, the fluid elemen ts are prone to significan t damage. Moreov er, during the pro cess of transp o rtation of fluids in living structures executed by using arthro - pumps, the fluid particles are lik ely to b e appreciably damaged. Qua l- itativ e and quantitativ e studies of the presen t pro blem for the w all shear stress ha v e a significant b earing on extracorp or eal circulation. When the heart-lung machine is used for patien t care, the erythro cytes of blo o d a re like ly to b e damaged. Thu s the presen t study b ears the p oten tial of significan t application in biomedical engineering and tech nology . The study rev eals that at a n y instan t of time, there is a retrograde flow region for Hersc hel- Bulkley type of non-Newtonian fluids like blo o d when ∆ P ≤ 0. The regio ns of forward/retrograde flo w a dv ance a t a faster ra t e, if the v alues of n and φ are raised. In the case of a uniform/diverging tub e, the flow rev ersal decreases when ∆ P tends to b e negative; ho w ev er, for a con verging tub e suc h an observ ation is not v ery prominen t. In addition, this study shows that non-uniform geometry of the v essel affects quite significan tly the distributions of v elo cit y and the w all shear stress as we ll a s pumping and o ther flow c haracteristics. It is also observ ed that the ampli- tude ratio φ and the rheolog ical fluid index ‘n’ a re v ery sensitiv e parameters that c hange the p eristaltic pumping c haracteristics and the distribution of v elo cit y and w all shear tress. The parab olic nat ure of t he v elo cit y profiles is also significan tly disturb ed by the n umerical v alue of the rheological fluid index ‘n’. F rom t he presen t study w e ma y conclude that for the p eristaltic flo w of blo o d throug h propagation of SSD expansion w a v es, the pumping p erformance is b etter than that in the case of sin usoidal w av e propagation and that the bac kw ard flo w region is to tally absen t in SSD ex- pansion wa v e propag ation mo de. On the basis of this study one may also dra w the conclusion that bac kw ard flo w originates due to the contraction o f v essels. Ac kno wledgmen t: The authors wish to c onvey their thanks to al l the r eviewe rs (a n ony- mous) fo r their c omments and suggestion s b ase d up on which the pr esent version of the manuscript has b e en pr e p ar e d. One of the authors, S. 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