The peristaltic transport phenomena have immense theoretical and technical applications in physiology and industry,for example in biomedical engineering,pumping of blood in dial- ysis and in industries,and transport of toxic liquid in nuclear industry. It also occurs in the transport of urine from kidney to bladder,chyme motion in the gastrointestinal tract,blood transport in capillaries,intra-uterine fluid motion,movement of ovum in the female fallopian tube,etc. Latham^[1] and Jaffrin and Shapiro^[2] initially discussed the process of peristalsis in tube/channel. Keeping in mind the importance of processes of hemodialysis and oxygenation, several researches were conducted on the study of heat transfer in peristalsis. Radhakrishna- macharya and Murty^[3] investigated the characteristics of heat transfer in the peristaltic flow through a non-uniform channel. Vajravelu et al.^[4] considered the peristaltic flow in an annulus by using the approximation of long wavelength. Srinivas and Kothandapani^[5] examined the heat transfer in peristaltic flow in an asymmetric channel. Heat transfer characteristics in the hydromagnetic flow with peristalsis were explored by Mekheimer and Elmaboud^[6]. Srinivas et al.^[7] studied the features of convective heat and mass transfer in the peristaltic transport through an asymmetric channel. Influence of chemical reaction and space porosity on the hydromagnetic peristaltic transport in an asymmetric channel was reported by Srinivas and Muthraj^[8]. Nadeem and Akbar^[9] considered the magnetohydrodynamics (MHD) peristaltic motion with heat and mass transfer. Induced magnetic field influences on peristalsis were described by Elmaboud^[10]. Tripathi et al.^[11] analyzed the peristaltic motion in gerenalized Burgers’ fluid. Further,Tripathi^[12] examined the peristaltic transport of chyme movement in small intestine.

There is a subclass of differential type fluids known as third-grade fluid. This fluid model is capable of describing the effects of shear thinning/thickening. In the past,the peristaltic motion of third order/third grade fluid in the planar channel was also reported. For example, the study of peristalsis through the third order fluid flow was presented by Haroun^[13]. The influence of slip conditions on the third-order fluid flow with peristalsis was addressed by Hayat et al.^[14]. Wang et al.^[15] studied the peristaltic motion of third grade fluid with slip conditions. Combined heat and mass transfer effects in the third-order fluid flow with peristalsis were addressed by Nadeem et al.^[16]. Peristaltic motion of third grade fluid in an asymmetric channel was investigated by Ali et al.^[17]. Peristaltic flow with variable fluid properties was described by Hayat and Abbasi^[18]. Hayat and Mehmood^[19] examined the peristaltic motion of third grade fluid in a planar channel with slip conditions.

In spite of the fact that most of the physiological systems such as arteries and glandular ducts are curved,it is noticed that not many studies have been carried out on peristaltic mechanism in a curved channel. Sato et al.^[20] discussed the peristaltic flow due to the transverse deflections of the walls of the curved channel. Ali et al.^[21] studied the peristaltic flow analysis in a curved channel with long wavelength approximation. Ali et al.^{[22, 23]} extended the analysis of Ref. ^[21] for a third grade fluid and heat transfer characteristics. Hayat et al.^[24] discussed the peristaltic transport of Newtonian fluid in a curved channel with compliant walls. Influence of wall properties on peristaltic transport of third grade fluid in a curved channel was discussed by Hina et al.^[25]. Hayat et al.^[26] extended the analysis of Ref. ^[25] for heat and mass transfer effects. The present work extends the flow analysis of Ref. ^[25] from no-slip to slip effects. Low Reynolds number and long wavelength approximation are used for the development of mathematical model. The numerical solutions are computed via the shooting method through computational software Mathematica. The analytic solutions are also obtained by a regular perturbation method. Graphs for the embedded flow quantities are portrayed and discussed.

Consider a two-dimensional laminar flow in a curved channel of thickness 2d₁ coiled in a circle with centre O and radius R* (see Fig. 1). x and r are the coordinates that denote the axial and radial directions respectively,whereas axial and radial components of velocity are denoted by u and v,respectively.

The wave shapes along the wall are presented as

Click Browse formula

Fig. 1 Model of problem

Figure options

Click Browse formula

The subjected slip boundary conditions are of the form

Click Browse formula

Click Browse formula

Click Browse formula

After non-dimensionalization,the stress components and the governing equations of the problem are written as^[25]

Click Browse formula

In the above equations,€ (= a/d₁) represents the amplitude ratio,ν is the kinematic viscos- ity,δ (= d₁/λ) is the wave number,k is the dimensionless curvature parameter,Re (= cd₁/υ) is the Reynolds number,and E₁,E₂,and E₃ are the non-dimensional elasticity parameters expressed by

Click Browse formula

The velocity components u and v in terms of the stream function ψ(x,y,t) are defined as

Click Browse formula

Click Browse formula

Click Browse formula

Click Browse formula

Click Browse formula

Click Browse formula

We have solved Eq. (16) subject to the boundary conditions (13) and (14) numerically using the computational software Mathematica. This software uses the shooting method as a default method for the numerical solution of nonlinear boundary value problem. In addition,the perturbation solution for small values of β₃ is also obtained in the form ψ = ψ₀ + β₃ψ₁ + · · ·. Here,our intention is to discuss the influence of embedding parameters on the axial velocity u and the stream function ψ. In particular,the behaviors of compliant wall parameters,i.e., the elastic tension in the membrane E₁,the mass per unit area E₂,the coefficient of viscous damping E₃,the slip parameter β,the curvature parameter k,and the third grade parameter β₃ have been predicted.

To validate the present results,a comparison between the numerical solutions with the exact solution for Newtonian fluid case (β₃ = 0) and first-order perturbation solutions has been discussed in Fig. 2. A very good agreement is noticed for different values of slip parameter β. Figure 3 analyzes the effects of parameters appearing in the velocity u. Figures 3(a) and 3(b) show that the axial velocity decreases when the Deborah number increases. It is observed that the magnitude of velocity is smaller for third grade fluid as compared with the viscous fluid. Figure 3(a) also indicates that in a curved channel,the profiles across the centre are not symmetric. Further,the maxima of the profiles shifts towards the lower wall of the channel when the curvature is significantly large. The results for the planar channel are deduced for the sufficiently large curvature parameter k → ∞ (see Fig. 3(b)). Here,the symmetric nature of the axial velocity is observed about the centre of the channel. These figures also reveal that maxima in the velocity profiles for viscous fluid (β₃ = 0) lies above the maxima in the profiles for third grade fluid. Figures 3(c) and 3(d) represent the behavior of slip parameter β on the axial velocity u for curved and planar channels,respectively. It is depicted that the axial velocity is an increasing function of β. In other words,as the slip effect intensifies,this corresponds to the larger fluid velocity. It is noticed that the velocity profiles are tilted below,i.e.,towards the lower wall of the channel for smaller values of k. Figures 3(e) describes that by increasing the curvature parameter k,the axial velocity decreases near the lower wall of the channel and increases in the upper half of the channel. For large values of curvature parameter the channel becomes straight and the profiles become symmetric about the central line (as noticed earlier in Figs. 3(a)-3(d)). Clearly,the maxima in the velocity curves is decreased for small values of k. Figures 4(a)-4(e) plot the axial velocity profiles at different cross-section x = −0.2 for different values of parameters. It can be seen that the influence of all the parameters on the magnitude of axial velocity is quite similar to those accounted at x = 0.2. However,the flow reversal occurs for negative values of x. The compliant wall effects (E1,E₂,and E₃) on the velocity are sketched in Figs. 5(a)-5(b). It is seen that velocity is a decreasing function of E₃. Further,the axial velocity increases when there is an increase in E₁ and E₂. It is notified that the maxima in the profiles for large curvature parameter k is tilted towards the central line of the channel.

Fig. 2 Comparison of numerical and analytic solutions when E1 = 0.01,E2 = 0.02,E3 = 0.01, € = 0.2,t = 0.1,and x = 0.2 (lines: numerical solutions,points: analytic solutions)

Figure options

Fig. 3 Influence of different parameters on u when E1 = 0.01,E2 = 0.02,E3 = 0.01,€ = 0.2,t = 0.1, and x = 0.2

Figure options

Figures 6-9 illustrate the behaviors of parameters on the dimensionless stream function ψ. It is clear that increasing the Deborah number β₃ decreases the size and circulation of the trapped bolus. Thus,the bolus size in the viscous fluid is greater as compared with a third grade fluid. Figure 7 plots the stream lines for a third grade fluid for different values of k. The bolus size is different in the two halves of the channel for small values of the curvature parameter k = 3,i.e.,for larger curvature; whereas the bolus attains its symmetric shape in the straight channel (→ ∞). Interestingly the size and circulation of trapped bolus increases by increas- ing k for r > 0. However,the circulation of the bolus decreases upon decreasing k for r≤0. It is clear from Fig. 8 that the size of trapped bolus increases in the lower half of the channel whereas in the upper half of the channel the circulation of the trapped boluses increases by increasing slip parameter β. Figures 9(a)-9(d) plot stream lines for different values of complaint wall parameters. It is revealed that an increase in wall elastance (E₁ and E₂) rapidly increases the size and circulation of the boluses. However,an increase in the wall damping (E₃) corresponds to a decrease in the size and circulation of the boluses.

Fig. 4 Influence of different parameters on u when E1 = 0.01,E2 = 0.02,E3 = 0.01,€ = 0.2,t = 0.1, and x = −0.2

Figure options

Fig. 5 Influence of compliant wall parameters E1,E2,and E3 on u when β3 = 0.1,β = 0.01,€ = 0.2, t = 0.1,x = 0.2

Figure options

Fig. 6 Influence of β3 on ψ when E1 = 0.05,E2 = 0.05,E3 = 0.001,€ = 0.1,k = 3,t = 0,and β = 0.05

Figure options

Fig. 7 Influence of k on ψ when E1 = 0.05,E2 = 0.05,E3 = 0.001,€ = 0.1,β3 = 0.002,β = 0.05, and t = 0

Figure options

Fig. 8 Influence of β on ψ when E1 = 0.05,E2 = 0.05,E3 = 0.001,€ = 0.1,β3 = 0.01,t = 0,and k = 3

Figure options

Fig. 9 Influence of E1,E2,and E3 on ψ when β = 0.05,€ = 0.1,β3 = 0.002,t = 0,and k = 5

Figure options

In this study,peristaltic flow of a third grade fluid in a curved channel with slip effects is addressed. The dimensionless differential system is solved numerically for the stream function ψ. The numerical results agree well with the exact solution for viscous fluid and perturbation solutions for small Deborah number (β₃ << 1). It is observed that the magnitude of axial velocity in a third grade fluid is larger than that in the viscous fluid. Further,at some point the fluid attains the maximum velocity which decreases in magnitude in the curved channel. The size and the number of circulations of the trapped boluses increase as the slip effect intensifies. The current analysis for viscous fluid which is not yet reported in the literature can be obtained by choosing β₃ = 0. The study of peristaltic transport in curved channels is still infancy and complementary work,which is required to understand new and unique applications.