The present study deals with simultaneous effects of Joule heating and slip on peristaltic flow of a Bingham fluid in an inclined tapered porous channel with elastic walls. The closed form solutions for the stream function, the velocity and the temperature fields are obtained. The effects of the physical parameters on the flow characteristics are presented through graphs for both slip and no-slip cases. In addition, the performance of the temperature is studied with and without Joule heating effects. Moreover, the trapping phenomenon is analysed. The size of the trapped bolus increases with increasing values of the slip parameter and decreasing values of the magnetic, the permeability and the yield stress parameters. The present results are compared with the available results in the literature and our results agree well with the available results for some special cases.
Peristaltic transport is a mechanism of a fluid flow produced by propagation of wave trains along the channel walls. This phenomenon has wide range of practical applications in physiology and biomedical engineering such as swallowing of foodstuff, blood movement in blood vessels, lymph drive in lymphatic vessels, urine transport through ureter, chyme movement in intestinal tract, ovum transport, bile flow in bile duct, etc. Initially Latham  and Shapiro et al.  investigated the mechanism of peristalsis. Later many investigators have tried to understand the behaviour of peristalsis by considering different fluids and geometries. Elshehawey and Mekhemier  studied the peristaltic flow of couple-stress fluids. Usha and Ramachandra Rao  investigated the impact of the peristalsis on two-layered power-law fluid flows. Misra and Pandey  discussed the peristaltic flow of a non-Newtonian fluid by considering a peripheral layer.
The study of non-Newtonian fluid flow problems with heat transfer has many applications in chemical engineering and related industries; also, due to its tremendous applications in biomedical engineering, researchers have paid considerable attention on peristaltic flow of convective non-Newtonian fluids. The effects of heat transfer on peristaltic transport of a Jeffrey fluid in a vertical channel with porous medium was analyzed by Vajravelu et al. . Tripathi  presented a mathematical model for explaining the impact of heat transfer on swallowing of food bolus through the esophagus. Vajravelu et al.  made a theoretical model to study the peristaltic flow of a MHD phan-thien-tanner fluid in an asymmetric channel with heat transfer. Akram et al.  and Nadeem et al.  studied the impact of peristaltic flow of non-Newtonian fluids. Very recently the authors [12, 13, 14, 15, 16, 17, 18] have studied the peristalsis by considering different fluids and geometries.
Not much work has been reported on peristaltic flow by considering simultaneous effects of slip and magnetic field with heat transfer. Furthermore, the consideration of wall properties is essential to understand the behavior of physiological fluids. Srinivas et al.  examined the impact of slip conditions and heat transfer on MHD peristaltic transport. The effects of slip and wall properties on the peristaltic transport of a MHD Bingham fluid with heat transfer, was presented by Lakshminarayana et al. . Satyanarayana et al.  presented a model to explain the effects of magnetohydrodynamics and heat transfer on peristaltic slip flow of Bingham fluid in porous channel with flexible walls. Moreover, the impact of wall properties is discussed by Srinivas and Kothandapani , Hayat et al. , Riaz et al.  and Sucharitha et al. [33, 34].
Present paper describes a mathematical model to investigate the impact of Joule heating and slip on magnetohydrodynamic peristaltic flow of a Bingham fluid in an inclined non-uniform porous channel with flexible walls. The expressions for the stream function, the velocity and the temperature fields are obtained. The effects of the physical parameters on the flow quantities are discussed in detail. The present study reveals many interesting results which could facilitate the further investigation in convective non-Newtonian fluid flow phenomenon.
2 Mathematical formulation
Consider the two dimensional flow of a MHD Bingham fluid in an inclined non-uniform channel with porous medium. Flow is due to sinusoidal wave trains propagating along the elastic walls of the channel with a constant speed c (see Fig. 1). The channel wall deformation is assumed as
where d(x) = d+
Using long wavelength and small Reynolds number assumptions (see [28, 30, 34, 35] for details), the simplified non-dimensional governing equations and corresponding boundary conditions for the present study can be written as
Momentum and energy equations are
where N2 = M2+σ2
(Flexible boundary condition)
The non-dimensional parameters and quantities used in the above governing equations are
where u, v, p and ψ are the velocity components, pressure and stream functions respectively, ρ is the density, μ is the viscosity, d is the mean width of the channel, a is the amplitude, λ is the wavelength, c is the wave speed, ξ is the specific heat, ν is the kinematic viscosity, k0 is the thermal conductivity, k is the permeability, σ0 is the electrical conductivity, B0 is the magnetic field, g is the acceleration due to gravity, T is the temperature, τ0 is the yield stress, Ec is the Eckert number, E1, E2 and E3 are the elasticity parameters, m is the non-uniform parameter, σ is the permeability parameter, α is the inclination angle, β is the slip parameter, Br is the Brinkman number, M is the magnetic parameter, δ is the wave number,ε is the amplitude ratio, Pr is the Prandtl number and Re is the Reynolds number.
3 Solution of the problem
By differentiating Eq. (2.2) with respect to y we obtain
The nusselt number at the wall is given by
4 Results and discussion
To find the impact of physical parameters, we have plotted the velocity, temperature and heat transfer coefficient profiles in figures 2 – 22 with the fixed values of x = 0.2, t = 0.1, ε = 0.24, m = 0.2, M = 1, σ = 1, τ0 = 0.4, η = 1, α = π/4, E1 = 0.2, E2 = 0.2, E3 = 0.1, Br = 0.2 (for temperature E1 = 0.005, E2 = 0.005, E3 = 0.001). It is observed that the velocity is higher in slip flow when compared with the nonslip flow whereas the temperature exhibits the opposite behavior. In fact, the growth in slip diminishes the friction between the wall and the fluid. This may be the cause to increase in the velocity and reduction in the internal heat production. The impacts of magnetic parameter M and permeability parameter σ on the velocity and the temperature fields are presented in figures 2 – 5. We found that an increase in M and σ reduces the velocity profiles. This is due to the influence of drag force (Lorentz force) which opposes the flow. We also noticed an enhancement in the temperature field. This may be due to the Joule heating impact. Similar behavior has been noticed in the study of Hayat et al. .
From figures 6–9, it is observed that an increase in the yield stress τ0 is to reduce the velocity field and to increase the temperature field. We noticed that the higher values of the non-uniform parameter m increase both the velocity and the temperature profiles. The effects of the wall flexibility parameters are described trough the figures 10–13. We notice that the velocity increases with increasing E1 and E2 (due to the wall tension and mass charectrization property); but the velocity drop is noticed for increasing values of E3 (due to the damping force). The influence of the elasticity parameters on the temperature has similar effects as in the case of the velocity. This agrees with the results of Hayat et al. .
From figures 14 and 15 we see that for increasing values of the inclination angle al pha both the velocity and temperature fields increase. The impact of the Brinkman number Br is shown in figure 16. An increase in Br enhances the temperature field.
Figures 17 and 18 are drawn to validate the present results with the published availableresults in the literature. It is perceived that for α = 0 and in the absence of Joule heating our results reduce to those of Satyanarayana et al. . Further, we notice an increase in the values of the velocity and temperature fields when compared with the results in , due to the presence of Joule heating. The deviations in heat transfer coefficient are shown in figures 19–22. It can be seen that the absolute value of the Nusselt number is higher in the slip flow compared to the nonslip flow. Figures 19 and 20 reveal that the heat transfer coefficient increases for large values of M (due to the existence of Joule heating) whereas it is a decreasing function of σ. From figures 21 and 22, we observe an increase in the Nusselt number with the increasing Br and decreasing α.
5 Trapping phenomenon
The process of the formation of fluid bolus by the closed streamlines in the fluid flow is called trapping phenomenon and the trapped bolus moves forward with the peristaltic wave. The behaviour of streamlines is presented in figures 23–26. The effect of the slip parameter β is shown in figure 23. An increase in β increases the size of the bolus. From figures 24–26, we notice that an increase in the magnetic parameter M, the permeability parameter sigma and the yield stress τ0 reduces the size of the trapped bolus.
The effects of the wall slip and the Joule heating on MHD peristaltic flow of a Bingham fluid in a porous channel are studied. Long wavelength and small Reynolds number approximations are used to obtain the exact solutions of the problem. The results have applications in biomedical engineering and oil industries. Some of the interesting results are summarised as follows:
- Velocity is an increasing function of the slip parameter β whereas it is a decreasing function of the magnetic parameter M, the yield stress τ0 and the permeability parameter σ.
- The effect of the Joule heating is to increase the temperature.
- Slip parameter β reduces the temperature while it is an increasing function of the magnetic parameter M and the permeability parameter σ.
- Inclination angle α and and the non-uniform parameter m increase both the velocity and the temperature fields.
- Increase in both the velocity and the temperature is identified due to increase in the wall parameters E1 and E2, while reduction is noticed in the case of other wall parameter E2.
- Nusselt number is an increasing function of the slip parameter β and the magnetic parameter M.
- Size of the trapped bolus is reduced for the increasing values of M and σ while it increases with increasing σ.
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