The term peristalsis is related to the Greek word peristaltikos, which defines compressing and clasping. The peristaltic passage is a device of fluid flow, which takes place due to the circulation of wave trains along the walls of tube/channel. In physical situations the prohibition of pumping machinery makes this mechanism indispensable. In general, its occurrence depends on the pressure difference variations. Pumping of biofluids through this mechanism can be observed in many physiological systems, e.g., lymph movement in lymphatic vessels, fluid mechanics in perivascular space of brain, chyme movement in intestinal tract, urine transport through the ureter, swallowing of food through esophagus and sperm transport in male reproductive tract etc. These peristaltic pumps are also widely used in industry and medicine, for example, to transport aggressive or corrosive fluids and for dialysis etc. From above explanation of the peristaltic transport mechanism, it is clear that the contaminations taking place due to direct contact with the external environment can be avoided by utilization of this mechanism. This advantage was convenient enough to encourage engineers to design various industrial devices using this mechanism like, hose pumps, pumps utilized to transport of corrosive and sensitive fluids in nuclear industry, roller/finger pumps and heart-lung and dialysis machines etc. The relevance about the physiological applications of peristalsis was mainly given by Kill’s^[1] research work. After some time, Latham^[2] and Shapiro et al.^[3] presented earliest peristaltic transport models for fluid motion. After above mentioned pioneering work, many researchers tended to explore the behavior of peristalsis by considering different aspects.^[4–10]

Lately, a new branch of fluid mechanics namely “nanofluid dynamics” has emerged into spotlight which has diverse applications in the field of energy, medical science and process systems engineering. Nanofluids are engineered by suspending nanometer-sized particles in the base fluid such as water, propylene-glycol, emulsions, lubricants, coolants, biofluids, oil and silk fibroin to enhance thermal conductivity of the fluid. Although, the great Scottish theoretical physicist, James Clerk Maxwell proposed the concept of nanofluids initially in 19th century, however, the term “nanofluid” was introduced after more than a century by Choi.^[11] After, Choi^[11] laid first stone in the field of nanofluids many researchers intended to explore this area of research.^[12–16]

Recently, the peristaltic flow of nanofluids gained significant attention in biomedical engineering, most particularly in hyperthermia, drug delivery schemes and cryosurgery as a way to annihilate the undesired tissues in cancer therapy. The study of peristaltic flow of nanofluids by adopting Buongiorno model through a channel is carried by Tripathi and Bég^[17] under the approximations of long wavelength and low Reynolds number. Zeid^[18] studied the combine effects of radiation, viscous dissipation and peristaltic motion of non-Newtonian nanofluid with heat transfer in an asymmetric channel while using the long-wavelength assumption. Sreenadh et al.^[19] assumed the small reynolds number and elongated wavelength approximations along with the asymmetric channel to scrutinize the peristaltic transport of nanofluid while taking into account the slip effects. Reddy and Makinde^[20] explored the peristaltic flow of Jeffrey nanofluid through an asymmetric channel while considering the combined effects of buoyancy forces, magnetic field, thermophoresis and Brownian motion. Their research revealed that with an upsurge in the power of magnetic field velocity and nanoparticles concentration decrease while the temperature increases. Ghasemi^[21] adopted the differential transformation method to determine the highly accurate analytical solution for the peristaltic nanofluid flow in drug delivery systems in the presence of thermophoresis and Brownian motion effects. Sucharitha et al.^[22] obtained the exact and numerical solutions for the problem of peristaltic flow of nanofluid confined in a uniform or in a non-uniform porous channel in the presence of Joule heating and wall flexibility. Their key findings revealed that for uniform channel velocity is low as compared to the diverging channel but, opposite effects can be seen for the temperature and concentration fields. The exact solutions for the combined effects of wall properties and variable viscosity on the peristaltic flow of single wall carbon nanotubes are determined by Shahzadi et al.^[23]

Some new dynamic features come into view when nano/micro scaled fluidic devices are concerned due to direct contact between the solid surfaces of the fluid container and the fluid constituent. Investigations suggest that for the nano/micro scale fluid flow no-slip condition near walls is not true, however, a certain degree of tangential slip must be considered. Applications regarding no-slip conditions include the polishing of the optical coatings, artificial heart valves and internal cavities, refrigeration equipment, inexpensive lubricating and many industrial processes. After pioneering work done by Navier,^[24] Choi et al.^[25] and Matthews and Hill^[26–27] many researchers are seemed to be paying more attention to this phenomenon.^[28–30] Khan et al.^[31] discussed numerical study of partial slip on the MHD flow of an oldroyd 8-constant fluid. They concluded the result that the increase of the partial slip parameter, reduced the boundary resistance that leads to the increase of the velocity over the whole cross section. Derek et al.^[32] explained experimentally that the fluid shows non-continuum effects such as slip flow when the molecular mean free path length of the fluid is comparable to the distance between the plates as in nano channels or micro channels. Renksizbulut et al.^[33] explained the slip-flow and heat transfer in rectangular microchannels with constant wall temperature. They explained that the knudsen number is a measure of the degree of rarefaction, when lies in the range 10⁻³ < Kn < 10⁻¹, deviations from continuum behavior arise close to the walls where, in a thin layer named the Knudsen layer, molecular collisions with the walls lead over intermolecular collisions. They also explained that for this range of Knudsen numbers (slip-flow regime), the typical Navier-Stokes and energy equations can still be adopted with modifications to the boundary conditions allowing for velocity-slip and temperature-jump at the walls. Zhang^[34] discussed fluid-solid interfacial shear strength for some ordinary fluids. In an another paper Zhang,^[35] explained review of hydrodynamic lubrication with interfacial slippage and concluded that the limiting shear strength model was more sensible and suggested as an appropriate model for reviewing the interfacial slippage. Hayat et al.^[36] studied impact of wall properties and slip on peristaltic motion of nanofluid with MHD and Joule heating. Abbasi et al.^[37] discussed slip effects on peristaltic flow of copper-water nanofluid in an inclined channel with mixed convection.

The aforementioned work motivates us to venture the untouched regime of mixed convective peristaltic transport of Cu-blood nanofluid confined in non-uniform tube with slip and wall properties under large wavelength and low Reynolds number approximations. Due to slip effects non-continuum flow occurred at the peristaltic wall, also we are assuming flow motion for peristaltic problem so channel length is considered large. The exact solutions of the governing coupled differential equations will be obtained for nanoparticle temperature, nanoparticle velocity and stream function. These results will be then plotted and discussed at length.

We have assumed the peristaltic flow occurrence for the two-dimensional incompressible viscous fluid in an annulus in the presence of nanoparticles, velocity slip and wall properties effects are also occupied. The equations for mass conservation, momentum and energy equations along with nanoparticles are defined as

The suitable boundary conditions for flow geometry (see Fig. 1) are defined as^[14,36–37]

Fig. 1

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	Fig. 1 Flow geometry.

The leading equation of motion of the elastic wall may be specified as^[22–23]

Introduce the following dimensionless variables

Equation (10) specifies that p is independent of r and the dimensionless boundary conditions can be inscribed as

The exact solution of the Eqs. (11) and (12) along with boundary conditions (14) is straight forward written as

Table 1

Table 1

Table 1 Thermo-physical properties of fluid and nanoparticles. . Physical Properties Blood Cu cp/(J/kgK) 3594 385 ρ/(kg/m3) 1063 8933 κ/(W/mK) 0.492 400 β × 10−5/(1/K) 0.18 1.67

Table 1 Thermo-physical properties of fluid and nanoparticles. .

In order to perceive the assessable effects of numerous parameters convoluted in the study, the velocity, temperature and stream functions are designed for numerous values of the physical parameters.

Tables 2 and 3 are considered to see the impact of velocity profile for diverse values of nano particle volume fraction for both cases (velocity slip and no slip). It is portrayed from Tables 2 and 3 that velocity profile gets a decreasing function when we insert nanoparticles into our base fluid for both cases e.g. velocity slip and no slip.

Table 2

Table 2

Table 2 Velocity profile for no slip flow for diverse values of ϕ with other fixed values of b2 = 0.24, λ = 0.02, ϵ = 0.05, z = 0.22, r1 = 0.11, Gr = 2.5, E3 = 1.5, t = 0.12, k = 0.16, E2 = 1.3, B = 0.8, E1 = 1.5. . No slip flow L = 0 r ϕ = 0 ϕ = 0.01 ϕ = 0.02 ϕ = 0.03 0.11 = r1 0 0 0 0 0.21 2.083 45 2.064 32 2.044 06 2.022 74 0.31 3.083 49 3.052 21 3.019 39 2.985 14 0.41 3.533 39 3.494 15 3.453 32 3.411 01 0.51 3.609 45 3.565 92 3.520 94 3.474 63 0.61 3.392 38 3.348 25 3.302 92 3.2565 0.71 2.926 12 2.885 27 2.843 56 2.801 04 0.81 2.237 37 2.204 01 2.170 11 2.135 72 0.91 1.343 69 1.322 38 1.300 82 1.279 03 1.01 0.257 332 0.253 006 0.248 645 0.244 257 1.03146 = r2 0 0 0 0

Table 2 Velocity profile for no slip flow for diverse values of ϕ with other fixed values of b2 = 0.24, λ = 0.02, ϵ = 0.05, z = 0.22, r1 = 0.11, Gr = 2.5, E3 = 1.5, t = 0.12, k = 0.16, E2 = 1.3, B = 0.8, E1 = 1.5. .

Table 3

Table 3

Table 3 Velocity profile for slip flow for diverse values of ϕ with other fixed values of b2 = 0.34, λ = 0.02, ϵ = 0.05, z = 0.22, r1 = 0.11, Gr = 2.5, E3 = 1.5, t = 0.12, k = 0.16, E2 = 1.3, B = 0.8, E1 = 1.5. . Slip flow L = 0.1 r ϕ = 0 ϕ = 0.01 ϕ = 0.02 ϕ = 0.03 0.11 0 0 0 0 0.21 3.334 59 3.263 54 3.193 17 3.1235 0.31 5.0882 4.973 71 4.8606 4.748 89 0.41 6.079 06 5.934 17 5.791 38 5.650 71 0.51 6.577 41 6.4107 6.246 86 6.085 86 0.61 6.706 77 6.525 08 6.347 02 6.172 53 0.71 6.534 24 6.343 65 6.157 42 5.975 49 0.81 6.100 44 5.906 76 5.718 14 5.534 48 0.91 5.432 5.241 02 5.055 72 4.875 96 1.01 4.547 38 4.365 4.188 83 4.018 67

Table 3 Velocity profile for slip flow for diverse values of ϕ with other fixed values of b2 = 0.34, λ = 0.02, ϵ = 0.05, z = 0.22, r1 = 0.11, Gr = 2.5, E3 = 1.5, t = 0.12, k = 0.16, E2 = 1.3, B = 0.8, E1 = 1.5. .

Tables 4 and 5 show the velocity profile for swelling values of L (slip parameter) for both cases pure blood as well as Cu-blood case. It is depicted that with the rise of L (partial slip parameter), the velocity gradient for both pure blood as well Cu-blood falls, and hence the velocity profiles become flatter due to the reducing shear force from the slip boundary. Moreover, by the increase of slip parameter L, the main result appears to be that the reduced boundary resistance leads to the increase of the velocity profile over the whole cross section (that agrees with Ref. [31]). Figures 2(a) and 2(b) are designed to see the impact of heat absorption parameter and volume fraction of the nanoparticle on the temperature profile. It is depicted from Fig. 2(a) that by increasing values of heat source parameter, temperature profile gets an increasing behavior (Metabolic process is one of the best heat source tool in the human body). Figure 2(b) is plotted to see the impact of nano particle volume fraction on the temperature profile. Temperature profile diminutions when we increase nanoparticle into our base fluid. This is owing to the fact that high thermal conductivity of the nanoparticles shows a significant role for rapid dissipation this leads to the results that the use of the nanoparticles in different type as a coolant.

Fig. 2

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	Fig. 2 (Color online) Nanoparticle temperature profile for diverse values of B and ϕ with other fixed values of b2 = 0.34, λ = 0.22, ϵ = 0.15, z = 0.22, r1 = 0.11, t = 0.12, k = 0.34.

Table 4

Table 4

Table 4 Velocity profile for no slip flow for diverse values of ϕ with other fixed values of b2 = 0.24, λ = 0.02, ϵ = 0.05, z = 0.22, r1 = 0.11, Gr = 2.5, E3 = 1.5, t = 0.12, k = 0.16, E2 = 1.3. . Pure blood ϕ = 0 r L = 0.01 L = 0.02 L = 0.03 L = 0.04 0.11 = r1 0 0 0 0 0.21 2.118 51 2.153 26 2.187 72 2.221 89 0.31 3.139 66 3.195 35 3.250 56 3.305 31 0.41 3.604 72 3.675 43 3.745 55 3.815 06 0.51 3.692 61 3.775 06 3.8568 3.937 84 0.61 3.485 25 3.577 32 3.6686 3.759 11 0.71 3.027 22 3.127 45 3.226 82 3.325 35 0.81 2.345 61 2.452 92 2.559 32 2.6648 0.91 1.458 24 1.571 81 1.684 41 1.796 05 1.01 0.377 538 0.496 711 0.614 865 0.732 013 1.03146 = r2 0 0 0 0

Table 4 Velocity profile for no slip flow for diverse values of ϕ with other fixed values of b2 = 0.24, λ = 0.02, ϵ = 0.05, z = 0.22, r1 = 0.11, Gr = 2.5, E3 = 1.5, t = 0.12, k = 0.16, E2 = 1.3. .

Table 5

Table 5

Table 5 Velocity profile for no slip flow for diverse values of ϕ with other fixed values of b2 = 0.24, λ = 0.02, ϵ = 0.05, z = 0.22, r1 = 0.11, Gr = 2.5, E3 = 1.5, t = 0.12, k = 0.16, E2 = 1.3. . Cu-blood ϕ = 0.01 r L = 0.01 L = 0.02 L = 0.03 L = 0.04 0.11 = r1 0 0 0 0 0.21 2.097 77 2.130 95 2.163 85 2.196 48 0.31 3.105 81 3.158 97 3.211 68 3.263 96 0.41 3.562 22 3.629 72 3.696 66 3.763 05 0.51 3.645 28 3.723 98 3.802 02 3.879 42 0.61 3.436 87 3.524 75 3.611 91 3.698 34 0.71 2.981 75 3.077 42 3.1723 3.2664 0.81 2.307 31 2.409 74 2.511 33 2.612 07 0.91 1.4317 1.5401 1.647 61 1.754 22 1.01 0.367 717 0.481 472 0.594 282 0.706 16 1.03146 = r2 0 0 0 0

Table 5 Velocity profile for no slip flow for diverse values of ϕ with other fixed values of b2 = 0.24, λ = 0.02, ϵ = 0.05, z = 0.22, r1 = 0.11, Gr = 2.5, E3 = 1.5, t = 0.12, k = 0.16, E2 = 1.3. .

Figures 3(a) and 2(b) are prepared to inspect the effects of embedded parameters such as rigidity parameter, stiffness parameter, viscous damping force parameter and Grashof number on the velocity profile for both cases (pure blood and Cu blood). It is portrayed in Figs. 3(a) and 2(b) that by increasing values of rigidity and stiffness parameter, velocity profile gets an increasing function for both cases (pure blood and Cu blood). Basically, the wall properties provides less resistance to the flow, thus velocity increases. It is also depicted from these figures, velocity profile for pure blood case gets the larger curves as equated to Cu-blood case. Figure 4(a) are planned to perceive the impact of viscous damping force parameter on velocity profile. Velocity profile decreases due to increase in viscous damping force parameter for pure blood as well as Cu-blood case. Velocity profile for swelling values of Grashof number for pure blood as well as Cu-blood is displayed in Fig. 4(b). Buoyancy forces plays a dominant role as compare to the viscous damping force parameter, this contributes in enhancing velocity profile for pure blood as well as Cu-blood case. One depiction from these figures is that velocity profile closer to the endoscopic duct gets closer curves whereas, near the peristaltic wall more variation is seen. Trapping is stimulating occurrence, whereby a bolus is elated at the wave speed and this trapped bolus pushed onward along the peristaltic wave.

Fig. 3

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	Fig. 3 (Color online) Velocity profile (a) for E2 = 1.5, (b) for E1 = 1.5, b2 = 0.34, λ = 0.02, ϵ = 0.05, z = 0.22, r1 = 0.11, t = 0.12, k = 0.16, Gr = 2.5, B = 0.8, E3 = 1.55.

Fig. 4

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	Fig. 4 (Color online) Velocity profile (a) for Gr = 2.5, (b) for E3 = 1.5, b2 = 0.34, λ = 0.02, ϵ = 0.05, z = 0.22, r1 = 0.11, t = 0.12, k = 0.16, E2 = 1.3, B = 0.8, E1 = 1.5.

Figures 5–7 are drawn to see the impact of rigidity parameter, stiffness parameter and viscous damping force parameter on trapping phenomenon. It is obviously understood from Figs. 5(a)–5(b) that number of the trapped boluses as well as size of the trapped bolus remains same. It is evident from the Figs. 5(a)–5(b) quantity of the trapped bolus remains same but magnitude, of the trapped bolus slightly decreases. It is also depicted from the stream Figures 6(a)–6(b), no change is observed from these plots when we upsurge the value of viscous damping force parameter.

Fig. 5

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	Fig. 5 (Color online) Stream plots for diverse values of E1 with fixed values of λ = 0.02, b2 = 0.34, E2 = 1.5, ϵ = 0.05, z = 0.34, r1 = 0.11, t = 0.12, k = 0.16, Gr = 2.5, B = 0.8, E3 = 1.55. (a) E1 = 1.1, (b) E1 = 1.5.

Fig. 6

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	Fig. 6 (Color online) Stream plots for diverse values of E2 with other fixed values of b2 = 0.34, λ = 0.02, E1 = 1.5, ϵ = 0.05, z = 0.34, r1 = 0.11, t = 0.12, k = 0.16, Gr = 2.5, B = 0.8, E3 = 1.55. (a) E2 = 1.1, (b) E2 = 1.5.

Fig. 7

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	Fig. 7 (Color online) Stream plots for diverse values of E3 with other fixed values of b2 = 0.34, E2 = 1.5, λ = 0.02, ε = 0.05, z = 0.34, r1 = 0.11, t = 0.12, k = 0.16, Gr = 2.5, B = 0.8, E1 = 1.55. (a) E3 = 1.1, (b) E3 = 1.5.

In the present study, we are deliberating the physiological fluid flow analysis for viscous nanofluid due to the combined effects of velocity profile and slip effects. The main deduction can be briefly as follows

(i) Velocity profile drops for swelling values of volume fraction of the nanoparticle for both cases (velocity slip and no slip case).

(ii) Velocity profile increases due to less resistance in the fluid for increasing values of rigidity and stiffness parameter.

(iii) Temperature profile declines with an upsurge in ϕ this justifies that the use of the copper nanoparticle plays a key role for coolant.

(iv) Number of trapped bolus almost remains unchanged for rigidity parameter, stiffness parameter, viscous damping force parameter cases.