Simulations of peristaltic slip-flow of hydromagnetic bio-fluid in a curved channel , N Ali, K Javid , and M Sajid Citation: AIP Advances 6, 025111 (2016); doi: 10.1063/1.4942200 View online: http://dx.doi.org/10.1063/1.4942200 View Table of Contents: http://aip.scitation.org/toc/adv/6/2 Published by the American Institute of Physics AIP ADVANCES 6, 025111 (2016) Simulations of peristaltic slip-flow of hydromagnetic bio-fluid in a curved channel N Ali,1 K Javid,1,a and M Sajid2 Department of Mathematics and Statistics, International Islamic University, Islamabad 44000, Pakistan Theoretical Physics Division, PINSTECH, P.O Nilore, Islamabad 44000, Pakistan (Received 10 December 2015; accepted February 2016; published online 12 February 2016) The influence of slip and magnetic field on transport characteristics of a bio-fluid are analyzed in a curved channel The problem is modeled in curvilinear coordinate system under the assumption that the wavelength of the peristaltic wave is larger in magnitude compared to the width of the channel The resulting nonlinear boundary value problem (BVP) is solved using an implicit finite difference technique (FDT) The flow velocity, pressure rise per wavelength and stream function are illustrated through graphs for various values of rheological and geometrical parameters of the problem The study reveals that a thin boundary layer exists at the channel wall for strong magnetic field Moreover, small values of Weissenberg number counteract the curvature and make the velocity profile symmetric It is also observed that pressure rise per wavelength in pumping region increases (decreases) by increasing magnetic field, Weissenberg number and curvature of the channel (slip parameter) C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4942200] I INTRODUCTION Fluid transport in many biological systems is achieved by peristaltic mechanism in which progressive waves of contraction or expansion propagate along the wall of the tubular organs of the system In particular, peristaltic mechanism is found in the urine transport from kidney to the bladder, the swallowing of food through the esophagus, the chyme movement in the small intestine, the transport of bile in duodenum, the transport of spermatozoa in the ductus efferents of the male reproductive tract and in the vasomotion of small blood vessels, such as arterioles, venules, and capillaries Peristaltic transport is also involved in design of medical equipment like heart-lung machine In addition, peristalsis is used in industrial pumps to transport very viscous or non-Newtonian fluid through flexible deformable tubes The initial theoretical studies on peristalsis were carried out by Latham,1 Shapiro et al.,2 Fung and Yih,3 Lykoudis and Roos,4 where either it is assumed that the amplitude of the peristaltic waves is low or the wavelength of the peristaltic waves is large Later studies on the subject extended the previous ones to include inertial effects,5,6 high Reynolds number effects,7,8 creeping flow condition,9 effects of peripheral layers10 etc The non-Newtonian nature of biological and industrial fluids is also incorporated in the studies on peristaltic transport For instance, Raju and Devanathan11,12 discussed the peristaltic motion of non-Newtonian fluids obeying the constitutive equations of Power-law and second grade models The transport of linear viscoelastic fluid satisfying the integral constitutive equation is analyzed by Bohme and Friedrich.13 Subsequent attempts in this direction were made by Srivastava and Srivastava,14 Siddiqui and Schwarz,15 Siddiqui et al.,16 El-Shahaway and Mekheimer,17 Mekheimer et al.,18 Hayat et al.19 and Wang et al.20 More recent attempt encompassing the effect of tube mechanics and systematic exploration of the effects of the wave profile a Corresponding author Tel.: +92 051 9019756 E-mail address: khurram_javid1985@yahoo.com (K Javid) 2158-3226/2016/6(2)/025111/14 6, 025111-1 © Author(s) 2016 025111-2 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) are due to Takagi and Balmforth21 and Walker and Shelly.22 In addition, Tripathi and Beg23–26 and Tripathi et al.27 recently analyzed analytically the model of food transport through esophagus, chyme dynamics through the intestine and viscoelastic fluid propulsion It has been noticed in the literature that most of the studies on peristaltic transport were carried out for tube or channel geometry Nevertheless, the geometry of physiological ducts demands an inclusion of the curvature of the duct This motivated some researcher to include curvature effects on the peristaltic transport of biological fluids The study performed by Sato et al 28 is pioneering in this direction The results presented by Sato et al 28 were generalized by Ali et al.,29–31 Hayat et al.,32,33 Hina et al.,34–36 Ramanamurthy et al.37 and Narla et al.38 In this connection also the paper of Kalantari et al.39 is worth mentioning It is related to those situation where the curvature of the channel, applied magnetic field and non-Newtonian effects are equally important The non-Newtonian model chosen in Ref 39 is Phan-Thien-Tanner (PTT) model In the present paper we investigate the effects of fluid slippage at the channel walls, applied magnetic field and non-Newtonian rheology on peristaltic flow in a curved channel In this study, we opted for Williamson model to represent the rheology of the fluid inside the channel The Williamson model corresponds to fluids exhibiting strong shear-thinning and relaxation effects The analysis of present theoretical model is also driven by a recent study performed by Asghar40 where the boundary layer character of shear-thinning fluid subject to strong magnetic field is highlighted The paper is organized as fellows: The flow problem is formulated by stating the underline assumptions and deriving the governing equation in Section II Solution technique is described in Section III A discussion about the obtained results is presented in Section IV The paper is concluded in Section V II DESCRIPTION OF THE PROBLEM Let us consider a curved channel of radius R0 and width 2b coiled in a circle with centre O The channel is assumed to be filled with an incompressible Williamson fluid The motion in the fluid is generated by sinusoidal waves of amplitude a and speed c travelling along the walls of the channel A diagram of the curved channel is shown in Fig We choose a curvilinear coordinate system ¯ X) ¯ to analyze the flow in which R¯ is along the radial direction and X¯ is along the axial direction ( R, The mathematical expressions describing the wall geometry are: ( ) ¯ t¯ = b + a Sin 2π X¯ − ct¯ , Upper wall (1) W X, λ1 ( ) 2π ¯ ¯ ¯ ¯ − W X, t = −b − a Sin X − ct , Lower wall (2) λ1 where λ is the wavelength and t¯ is the time FIG A diagram of curved channel 025111-3 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) It is instructive to define velocity field for the flow under consideration as ¯ i = U¯ i X, ¯ R, ¯ t¯ , U i = 1, 2, (3) where U¯ and U¯ are the velocity components in radial and axial directions, respectively The balances of mass and linear momentum characterizing the present flow are U¯,ii = 0, ¯ ρa¯ i = T¯,ijj + J¯ × B, (Contiunity Equation) (4) (Momentum Equation) (5) where T¯ i j is the Cauchy stress tensor, ρ is the fluid density, J¯ is the current density, B¯ is an applied magnetic field in the radial direction and a¯ i = ∂U¯ i + U¯ r U¯,ri , ∂t (6) are the components of the accelerating vector and “,” denotes the covariant differentiation The constitutive law for Williamson fluid model is T¯ i j = − p¯ δ i j + S¯ i j , i, j = 1, where p¯ is the pressure, δ i j is the identity tensor and the extra stress tensor S¯ i j satisfies41 ¯ −1 γ¯˙ i j , S¯ i j = µ∞ + (µ0 + µ∞) − ΓΠ (7) (8) In Eq (8), µ∞ is the infinite shear rate viscosity, µ0 is the zero shear rate viscosity, Γ is the time constant, Π is the second invariant of first Rivlin-Ericksen tensor defined as ( i j ji) Π= trac γ˙¯ γ˙¯ , (9) rm where γ¯˙ = U¯ r, m + U¯ m, r , r, m = 1, We consider constitutive relation (8), for the case when µ∞ = The extra stress tensor, therefore, can be written as ¯ S¯ i j = µ0 − ΓΠ −1 i j ¯ γ˙ (10) Since peristaltic flow in the channel is subjected to magnetic field of strength B¯ in radial direction, therefore from Maxwell equations, we obtain ( ∗ ) ¯B = B R0 eR¯ , (11) R¯ + R0 where B∗ is the characteristic magnetic induction in the limit R0 → ∞, and eR¯ is the unit vector in the radial direction It is pointed out here that the magnetic field given by Eq (11) is solenoidal Using Eq (3), the term J¯ × B¯ in Eq (5) is given by39 ∗2 ¯ = − σB U2 R0 eX¯ , J¯ × B R¯ + R0 (12) where eX¯ is the unit vector in the azimuthal direction In view of Eq (7), we can write Eq (5) as ¯ ρa¯ i = − pδ ¯ ,i jj + S¯,ijj + J¯ × B (13) In view of the definition of velocity field given by (3), Eqs (4) and (13) takes the form ∂ ∂ R¯ ∂U¯ R¯ + R0 U¯ + R0 = 0, ∂ X¯ (14) 025111-4 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) ¯ ∂U1 ¯ ∂U¯ ∂ p¯ ∂ ¯ R0U¯ ∂U¯ U¯2 ¯ ¯ ρ + U1 =− R + R0 S¯ R R + − + ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ∂ t R + R0 ∂ X R + R0 ∂R ∂ R R + R0 ∂ R (15) ¯ ¯ R0 S¯ X X ∂ ¯ R¯ X¯ + S − , R¯ + R0 ∂ X¯ R¯ + R0 ( ) R0U¯ ∂U¯ U¯ 2U¯ R0 ∂ p¯ ∂ ¯ ∂U¯ ¯ ∂U¯ 2 R¯ X¯ ¯ + + =− + R + R S ρ + U1 ¯ ¯ ∂¯t ¯ + R0 ∂ R¯ ∂ R¯ R¯ + R0 ∂ X R¯ + R0 R¯ + R0 ∂ X R (16) ∗2 ¯ R0 ∂ ¯ X¯ X¯ σ β U2R0 + S − ¯ + R0 ∂ X ¯ ¯ +R R R ¯ X¯ with scale factor given by h1 = and h2 = We remark here that curvilinear coordinates R, ¯ ( R + R0)/R0 are used in the derivation of above equations ¯ X¯ can be treated as steady in a frame which is moving The unsteady flow in the fixed frame R, with speed of waves on the channel wall This frame denoted by (r, ¯ x) ¯ is called wave frame The coordinates, velocities and pressures in both frame are related according to the transformations ¯ u¯1 = U¯ 1, u¯2 = U¯ − c, p¯ = p x¯ = X¯ − ct¯, r¯ = R, ¯ (17) With the help of Eq (17), Eqs (14)–(16) can be put in the form ∂ u¯2 ∂ {(r¯ + R0) u¯1} + R0 = 0, (18) ∂ r¯ ∂ x¯ ∂ ∂ u¯1 R0 (u¯2 + c) ∂ u¯1 (u¯2 + c)2 ∂ p¯ ∂ u¯1 (r¯ + R0) S¯ r¯ r¯ ρ −c + u¯1 + − =− + ∂ x¯ ∂ r¯ r¯ + R0 ∂ x¯ r¯ + R0 ∂ r¯ r¯ + R0 ∂ r¯ (19) S¯ x¯ x¯ R0 ∂ ¯ r¯ x¯ S − , + r¯ + R0 ∂ x¯ r¯ + R0 ( ) ∂ u¯ ∂ u¯ R0 (¯u2 + c) ∂ u¯ (¯u2 + c) u¯1 R0 ∂ p¯ ∂ (¯r + R0)2 S¯ r¯ x¯ ρ −c =− + u¯1 + + + ∂ x¯ ∂¯r r¯ + R0 ∂ x¯ r¯ + R0 r¯ + R0 ∂ x¯ (¯r + R0)2 ∂¯r (20) R0 ∂ ¯ x¯ x¯ σ β ∗2R02 (¯u2 + c) S − + r¯ + R0 ∂ x¯ (¯r + R0)2 The above equation can be made dimensionless by scaling the variables and parameters as 2π r¯ u¯ u¯ ρcb 2πb2 2πb i j b ¯i j x, ¯ η = , u1 = , u2 = , Re = ,p= p, ¯δ= ,S = S , λ1 b c c µ λ µc λ1 µc R0 Γc σ k= ,W e = , Ha = β ∗ b b µ x= (21) In view of (21), Eqs (18)–(20) become ∂ ∂u {(η + k) u1} + δk = 0, (22) ∂η ∂x ( ∂u ∂u k (u2 + 1) ∂u1 (u2 + 1)2 ∂p ∂ (η + k) Sηη δRe −δ + u1 + δ − = − + δ ∂x ∂η η + k ∂x η + k ∂η η + k ∂η (23) ) k ∂ ηx Sxx S − +δ , η + k ∂x η+k ( ) ∂u k ∂p ∂ ∂u k (u2 + 1) ∂u2 (u2 + 1) u1 (η + k)2 Sη x Re −δ + u1 + δ + =− + ∂x ∂η η + k ∂x η+k η + k ∂ x (η + k) ∂η (24) k ∂ xx Ha2 k (u2 + 1) +δ S − η + k ∂x (η + k)2 025111-5 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) In components form, Eq (10) gives ∂u1 , (a) S x x = 2[1 − W eΠ]−1 (∂η ) (1 ) ∂u k ∂u + u 2 −1 xη S = [1 − W eΠ] + − , (b) ∂η η + k ∂η η + k ( ) k ∂u u −1 ηη S = 2[1 − W eΠ] + , (c) η + k ∂x η+k ( )2 ( )2 ( )2 ∂u1 ∂u2 k ∂u1 (1 + u2) 2k ∂u2 2u1 where Π = +2 + − + + ∂η ∂η η + k ∂η η+k η + k ∂x η+k (25) (26) In above equations Re is the Reynolds number, δ the wave number, k the dimensionless radius of curvature, We is Weissenberg number and Ha the Hartmann number In order to adopt stream function formulation, we define u1 = δ k ∂ψ ∂ψ , u2 = − k + η ∂x ∂η (27) The above definition of stream function along with the application of long wavelength approximation reduce Eqs (22)–(26) in the following form ∂p = 0, ∂η ( ) ) ∂ ( Ha2 k ∂ψ ∂p ηx (k + η) S + − 1− = 0, − ∂ x k(k + η) ∂η (k + η) ∂η S x x = 0, ( ( ) ) −1 ( ( )) ∂ ψ 1 ∂ψ ∂ ψ ∂ψ Sη x = − W e − − 1− − 2− 1− , η+k ∂η η+k ∂η ∂η ∂η Sηη = 0, ( ( ))2 ∂ ψ ∂ψ Π = − 2− 1− η+k ∂η ∂η (28) (29) (a) (b) (c) (d) (30) Inserting the expression of Sη x from Eq (30)(b) into Eq (29) yields ( ( ( ) ) −1 ( ( ) )) ∂p ∂ ∂ ψ ∂ψ ∂ψ ∂ 2ψ (k + η)2 + W e + + − − − − ∂x k(k + η) ∂η ∂η k+η ∂η ∂η η + k ∂η (31) ( ) ∂ψ Ha k 1− = 0, − (k + η) ∂η − Elimination of pressure between Eqs (29) and (31) results in the following compatibility equation ( ( )) −1 ( ( ) )) ∂ ∂ ψ 1 ∂ψ ∂ψ ∂ 2ψ (k + η)2 + W e + − − + (k + η) ∂η ∂η k+η ∂η ∂η η + k ∂η (32) ( )) Ha2 k ∂ψ − 1− = 0, (k + η) ∂η ∂ ∂η Eq (32) is subject to slip conditions at the walls i.e ∂ψ = βSη x + 1, at η = −w = −1 − Φ sin x, ∂η ∂ψ = − βSη x + 1, at η = w = + Φ sin x, ∂η (a) (b) (33) where Φ = a/b is the amplitude ratio and β is the dimensionless slip parameter It is interesting to note that due to slip condition at the walls, the dimensionless radius of curvature k comes in the boundary condition and thus makes the velocity at the wall a function of curvature of the channel and rheological parameters of the fluid 025111-6 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) It is important to mention that, Eq (32) is fourth order differential equation and two boundary conditions given by Eq (33) are not sufficient for the unique solution Therefore, two additional boundary conditions must be provided These additional conditions may be imposed by prescribing the dimensionless flow rate in the wave frame Following29 the additional boundary conditions on ψ are: ψ=∓ f∗ , at η = ±w (34) The dimensionless pressure rise over one wavelength is defined by29–31 2π ∆p = dp dx dx (35) III NUMERICAL METHOD An analytical solution of Eq (32) subject to boundary condition (33) and (34) is difficult to find Therefore, we intent to find a numerical solution Due to non-linear nature of Eq (32), direct application of finite difference method will not yield appropriate convergent results Usually in solving such non-linear equations, iterative methods are commonly used For current problem, we can now construct an iterative procedure in the following form: ( ) ∂g ∂ 2g H a 2k ( ) −g + (η + k) ∂η + (η + k)2 ∂η − η+k ∂ 4ψ n+1 ∂g ∂ 3ψ n+1 ∂ 2ψ n+1 (η + k) g (η + g + + k) + + (η + k) ∂η ∂η ∂η ∂η ( ) ( ) ∂g ∂ 2g ∂g ∂ 2g H a 2k H a 2k g − (η + k) ∂η − (η + k)2 ∂η −g + (η + k) ∂η + (η + k)2 ∂η n+1 + − (η+k)2 ∂ψ (η+k)2 + = 0, ∂η (η + k)2 (η + k)2 ) ( ∂ψ n+1 ∂ 2ψ n+1 βg βg q =− + 1, at η = −w, + − ψ n+1 = , βg η+k ∂η η+k ∂η ) ( n+1 n+1 q ∂ ψ ∂ψ βg βg n+1 ψ = − , − βg = + 1, at η = w, + 1+ 2 η+k ∂η η+k ∂η where ( ( n ( ))) −1 ∂ ψ ∂ψ n + , g = + We − k+η ∂η ∂η (36) (37) (38) where the index (n) denotes the iterative step It is now clear that above boundary value problem is linear in ψ n+1 Employing central difference formulae n+1 ψ n+1 − ψi−1 ∂ψ n+1 = i+1 , ∂η 2∆η n+1 n+1 − 2ψin+1 + ψi−1 ∂ 2ψ n+1 ψi+1 = , ∂η ∆η n+1 n+1 n+1 n+1 − 2ψi+1 + 2ψi−1 + ψi−2 ∂ 3ψ n+1 ψi+2 = , ∂η 2∆η n+1 n+1 n+1 n+1 − 4ψi+1 + 6ψin+1 − 4ψi−1 + ψi−2 ∂ 2ψ n+1 ψi+2 = , ∂η ∆η (39) Eqs (36) and (37) can be put in the form n+1 n+1 n+1 n+1 a 1i ψ i+2 + a 2i ψ i+1 + a 3i ψ in+1 + a 4i ψ i−1 + a 5i ψ i−2 = 0, i = 3, 4, , M + 1, (40) 025111-7 Ali, Javid, and Sajid q , ψ 3n+1 − 2ψ 2n+1 + ψ 1n+1 AIP Advances 6, 025111 (2016) ψ 2n+1 = ψ 3n+1 − ψ 1n+1 βg βg =− + 1, + 1− η2 + k 2∆η η2 + k ∆η q ψ M +2n+1 = − , ) ( n+1 n+1 n+1 ψ n+1 ψ n+1 βg βg M +3 − 2ψ M +2 + ψ M +1 M +3 − ψ M +1 −βg = + 1, + + η M +2 + k 2∆η η M +2 + k ∆η βg where ai1 = ( ) at η = η 2, at η = η M +2, (41) (η i +k)g , ∆η ( ) ∂g ai2 = g + (η i + k) ∂η , ∆η ( ) Ha2 k ∂g 2∂ g −g + (η i + k) = + (η i + k) − , ∂η (η i + k) ∆η ∂η η i + k ( ) Ha2 k ∂g 2∂ g (η (η ai4 = − + k) + g − + k) , i i ∂η ∂η (η i + k)2 (η i + k)2∆η ( ) Ha2 k ∂g 2∂ g − −g + (η i + k) + (η i + k) = ∂η ∂η (η i + k)2 (η i + k)2∆η (42) Now at a fixed cross-section and for M uniformly discrete points η i , (i = 2, 3, , M + 2) with a grid size ∆η, Eq (40) with boundary conditions (41) define a system of linear algebraic equation which is to be solved for each iterative (n + 1)th step In the present case, we have used Gaussian elimination method for solving such a system To start the iterative process we should require some suitable initial numerical values of ψ n at each cross-section are required For the present problem, a linear initial guess satisfying the Drichlet conditions at the wall is provided to start the simulations However, it is not necessary that a convergent solution is not always possible, particularly when initial numerical values of ψ are not given suitably In these conditions we often used the method of successive under-relaxation In this method the estimated value of ψ at (n + 1)th iterative step i.e ψ˜ n+1 is refined to get the convergent value of ψ at the same step This can achieved by the following formula ψ n+1 = ψ n + τ ψ˜ n+1 − ψ n , τ ∈ (0, 1] , (43) where τ is an under-relaxation parameter In order to obtain convergent iterationτ should be very small than a given error chosen to be 10−3 The above described method has been already used by many authors We refer the reader to articles19,42 (especially, the article of Wang and Hayat19) for further details about this technique IV RESULTS AND DISCUSSION In this section, the quantities of interest such as flow velocity, pressure rise per wavelength and trapping phenomena are examined numerically for various values of involved parameter Figure illustrates the effects of Weissenberg number on flow velocity by keeping the other parameters fixed It is interesting to note that for We = (Newtonian fluid), the velocity profile is not symmetric FIG Variation of u (η) for different values of We with k = 3, H a = 0, β = 0, Θ = 1.5, and Φ = 0.4 025111-8 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) FIG Variation of u (η) for different values of Ha with k = 3, We = 0.4, β = 0, Θ = 0.5, and Φ = 0.4 about η = and maximum in it lies below η = However, the maximum shifts toward the upper wall of the channel and amplitude of the flow velocity is enhanced with increasing We The asymmetric profile of velocity for We = and k = indicates that the effect of curvature is dominant in Newtonian fluid Such effect of curvature diminishes and velocity profile become symmetric by slightly increasing We In such a case there is a balance between viscoelastic and curvature effects However, for large values of We the viscoelastic effects counteract the curvature effects resulting in asymmetric velocity profile with maxima lying near the upper wall The effects of Hartmann number on flow velocity can be observed through Fig Figure shows that the magnitude of flow velocity decreases by increasing Hartmann number For strong magnetic field a thin boundary layer exist at both walls of the channel Moreover, the peak in velocity profile, which was originally in the lower half of the channel for small values of Hartmann number, shifts in the upper half of the channel for large values of Hartmann number The boundary layer character of velocity for strong magnetic field in a straight channel for peristaltic motion of a shear-thinning fluid is already highlighted through singular perturbation technique by Asghar et al.40 But for a curved channel no such results are reported yet The results of Asghar et al.40 indicate that for strong magnetic field two thin sharp boundary layers are formed near both the wall The middle region outside both boundary layer moves with constant speed a though it were a plug flow Our results for a curved channel also indicate the formation of two thin boundary layer at the channel wall However, the middle most region outside the boundary layer does not move with a constant speed rather the speed of fluid in this region increases linearly from a maximum at the lower wall to a maximum at the upper wall Figure demonstrates the effects of curvature on flow velocity It is evident from this figure that for small values of k i.e., for large channel curvature, the velocity profile becomes asymmetric Moreover, the peak in velocity profile shifts in the lower half of the channel In the limiting case when k → ∞, the results of the straight channel are recovered The present theoretical model of peristalsis also bears the potential to investigate the effects of slip at the channel wall on various flow characteristics We have plotted Fig to illustrate such effects on velocity profile This figure reveals that velocity at both walls increases by increasing slip parameter Figure demonstrates the FIG Variation of u (η) for different values of k with We = 0.5, Ha = 0, β = 0, Θ = 0.5, and Φ = 0.4 025111-9 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) FIG Variation of u (η) for different values of β with k = 2, We = 0.5, Ha = 0, Θ = 1.5, and Φ = 0.4 FIG Variation of u (η) for different values of Φ with k = 3, We = 0.5, Ha = 0, Θ = 1.5, and β = FIG Variation of ∆p for different values of We with k = 2, Ha = 0, β = 0, and Φ = 0.4 FIG Variation of ∆p for different values of Ha with k = 2, We = 0.1, β = 0.01, and Φ = 0.4 effects of Φ on velocity profile It shows that not only the amplitude of the flow velocity decreases but also the contraction of the peristaltic wall is observed with increasing Φ An important feature of peristaltic motion is pumping against the pressure rise per wavelength Figures 7–11 are plotted to highlight the effects of Weissenberg number, Hartmann number, slip parameter, amplitude ratio and curvature of the channel on pressure rise per wavelength (∆p) 025111-10 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) FIG Variation of ∆p for different values of β with k = 2, Ha = 0.01, We = 0.01, and Φ = 0.4 FIG 10 Variation of ∆p for different values of k with Ha = 0.01, We = 2, β = 0, and Φ = 0.4 FIG 11 Variation of ∆p for different values of Φ with We = 0.4, β = 0, Ha = 0, and k = Figure shows that for a given value of flow rate, ∆p in the pumping region (∆p ≥ 0, Θ ≥ 0) increases by increasing Weissenberg number Thus shear-thinning fluid with greater relaxation time offers more resistance to peristaltic activity in comparison with Newtonian fluid The effect of Hartmann number on ∆p in pumping region is similar to the effect of Weissenberg number (Fig 8) However, an increase in slip parameter significantly reduces the pressure rise per wavelength in the pumping region (Fig 9) This observation may have certain implications in situations where it is desired to reduce the magnitude of resistance offered by pressure to the peristaltic motion A mechanism may be devised to produce fluid slippage at the walls of channel which could then be utilized to reduce the pressure rise wavelength Fig 10 illustrates the effects of curvature of the channel on ∆p It is clear from this figure that ∆p in pumping region for a curved channel is significantly lesser in magnitude than that for a straight channel Figures 11 shows that the effects of amplitude ratio (Φ) on ∆p in pumping region are similar to the effects of Weissenberg number and Hartmann number Figures 12–16 are plotted to see the influence of Weissenberg number, Hartmann number, slip parameter, amplitude ratio and curvature of the channel on trapping phenomenon Figure 12 reveals that a bolus of fluid concentrated in the upper half of the channel exists for W e = The size of this bolus reduces and it shrinks by increasing We There is no indication of splitting of this bolus 025111-11 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) FIG 12 (a) – (d): Streamlines for different values of Weissenberg number We (0, 0.125, 0.25, 1.5) with k = 2, Ha = 0, β = 0, Θ = 1.5, and Φ = 0.8 FIG 13 (a) – (d): Streamlines for different values of Hartmann number Ha (0, 0.4, 1.1, 2.5) with k = 2, We = 0.1, β = 0, Θ = 1.5, and Φ = 0.8 into two halves by increasing We However, the splitting of bolus into two halves is observed by increasing Hartmann number as evident from Fig 13 The panel (c) of Fig 13 demonstrates the splitting of bolus, which was originally concentrated in the upper half of the channel, into two halves A further increase in Hartmann number increases the size of bolus in lower half of the channel Moreover, large values of Hartmann number confirm the appearance of a squeezed bolus concentrated in the lower half of the channel The effects of slip parameter on trapping phenomenon are shown in Fig 14 It is observed that the size of trapped bolus reduces by increasing the slip parameter The splitting of bolus into two halves is also seen as a result of increasing slip parameter It is interesting to note that the trapped bolus disappears for large values of slip parameter It is therefore concluded that both Hartmann number and slip parameter have significant effects on the trapping phenomena In fact, the slip parameter reduces the circulation of the bolus while magnetic parameter shifts the concentration of the bolus from the upper half of the channel to the lower one Figure 15 depicts that the symmetry of the circulating bolus about the centerline owes to large value of the parameter k For small values of k i.e for greater channel curvature, the bolus becomes asymmetric and it circulates about a point 025111-12 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) FIG 14 (a) – (d): Streamlines for different values of slip parameter β (0, 0.5, 1.1, 3) with k = 2, Ha = 0, We = 0.1, Θ = 1.5, and Φ = 0.8 FIG 15 (a) – (d): Streamlines for different values of k (2, 3.5, 5, ∞) with We = 0.1, Ha = 0, β = 0, Θ = 1.5, and Φ = 0.8 FIG 16 (a) – (d): Streamlines for different values of Φ (0.2, 0.4, 0.6, 0.7) with We = 0.1, Ha = 0, β = 0, Θ = 1.5, and k = 025111-13 Ali, Javid, and Sajid AIP Advances 6, 025111 (2016) in the upper half of the channel Figure 16 shows the streamlines patterns for different values of Φ Panel (a) shows a circulating bolus of fluid concentrated in the upper part of the channel for Φ = 0.2 It is observed that both size and circulation of bolus increase with increasing Φ from 0.2 to 0.7 V CONCLUDING REMARKS A theoretical model capable of illustrating the effects of fluid rheology, applied magnetic field, amplitude ratio and slip at walls on peristaltic motion in a curved channel is presented The governing equation of the model is simulated by a suitable finite difference technique for various value of the involved parameters The main findings of the present study are • For large values of applied magnetic field, the boundary layer character exhibited by the velocity profile in a curved channel is found to be markedly different than that in a straight channel • The asymmetric velocity profile in a curved channel becomes symmetric by increasing We In fact for given value of k, a certain value of We could be found which neutralizes the effects of curvature and produces a symmetric velocity profile For large values of We, the viscoelastic effects dominate and velocity profile again become asymmetric with peak appearing in the upper half of the channel • The velocity at both walls of the channel increases by increasing slip parameter • The amplitude of the flow velocity is suppressed with increasing the amplitude of the peristaltic wave • The pressure rise per wavelength in the pumping region increases by increasing the Weissenberg number, Hartmann number, amplitude ratio and curvature of the channel However, it reduces significantly by increasing the slip parameter • A circulating bolus of fluid concentrated in upper half of the channel exists in a curved channel • The size of trapped bolus decreases (increases) by increasing Weissenberg number (amplitude ratio) • The bolus of fluid concentrated in the upper half of the channel shifts in the lower half for large value of the Hartmann number • The trapped bolus of fluid disappears for large value of the slip parameter ACKNOWLEDGEMENT The second author is grateful to the Higher Education 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