**1. Introduction**

Nanofluids attract current predilection because of its heat conduction attributes. Changing the flow geometry, boundary conditions, or thermal conductivity of liquids can improve convective heat transfer. Over the years, researchers have tried to increase the thermal conductivity of liquids. For this purpose, with the idea of Maxwell [1] solid metal particles are introduced into the base liquid. The large micro-sized particles are used to make suspensions because the conductivity of solids is greater than that of liquids, but these particles tend to produce greater resistance to the flow of base fluid. Modern nanotechnology tends to take a new direction in this field. In 1995, Choi [2] proposed a liquid with nano-sized particles suspended in a base liquid to eliminate the disadvantages of micro-sized particles. These liquids have efficient convective heat transfer compared to pure liquids. Recently, the idea of nanofluid in peristalsis has been studied by some researchers [3–10].

Peristalsis is characterized as the extension and the arrival of a substance into a liquid that improves the formative waves that broaden the length of the conduit, blending and shipping the liquid toward the wave spread. It is a mechanism that is available in numerous organs of the human body. In some specific instruments—for example heart–lung machines, implantation gadgets, and other pumping apparatus—such types of processes are utilized. It is of specific significance in many species and especially in human body that the transportation of many tissues of the body under various conditions, for example, the sucking of blood by leaches, the heap from the kidneys to the bladder through filtration, transport of the spermatozoa to the male genital tract, the development of the bosom in the Fallopian tubes, vasomotion of little veins, just as the blending and transport of gastrointestinal entry material.

The use of heat is of particular importance in the field due to its wide scope in engineering and biomechanics. In addition, the common relationship of heat stress and peristalsis can be observed during the oxygenation process with the patient. The assessment of magnetic resonance in biological tissues has aroused great interest among researchers regarding physical problems such as blood.

The assessment of heat transfer is related to the conditions of convection used in processes such as thermal conductivity, mechanical properties, chemical reactions, and so on. Aziz [11] presented a similarity solution to incorporate the convective walls conditions for thermal boundary layer on a smooth plate. In another article, Makinde and Aziz [12] developed the MHD mixed model on a flat surface in a concise way in terms of compatibility. Makinde [13] also discussed the flow of the MHD component with the temperature and the mechanical evaluation of a plate on a flat surface with extended conditions. Merkin and Pop [14] considered the analysis of heat transfer by dynamically simulating the flow of a uniform current on a flat surface with a horizontal displacement. According to them, the heat flux near the main edge is dominated by the surface heat flux.

After knowing the significance of the above discussed phenomena, authors are keen to develop a series solution of peristaltic flow of nanofluid with Williamson fluid model as a base liquid with convective boundary conditions travelling through asymmetric channel. At least we know that this study has not been yet explored in the literature. This study will be a good base for the engineers to utilize the results in procedures like thermal energy storage, gas turbines, nuclear workshops, etc. The problem is modeled under the induction of lubrication approach. The series solutions of stream function, temperature distribution, and nanoparticle concentration are achieved by using a well-known converging method the homotopy perturbation method. The important features are analyzed more specifically by sketching graphs to estimate the impact of pertinent constant physical factors.

#### **2. Mathematical Modeling**

The incompressible Williamson model is chosen as a base fluid for nanofluid in between an asymmetric channel experiencing heat convection at the peristaltic type surfaces. The width of the channel is taken as (*d*<sup>11</sup> + *d*12). Flow is initiated due to the propagation of curved waves travelling with uniform speed *c* towards the flow. The exchange of heat is recognized by imposing temperatures *T*<sup>0</sup> and *T*<sup>1</sup> at the lower and upper areas, correspondingly. To discuss nano particle phenomenon, we have taken the nanoparticle concentration *C*<sup>0</sup> and on the lower side and upper one, accordingly (see Figure 1).

**Figure 1.** Geometry of the channel.

The magnetic field *B*<sup>0</sup> is exerted orthogonally. The wall surfaces are taken as

$$Y = H\_1 = d\_{11} + a\_{11} \cos[2\pi\lambda X'] \tag{1}$$

$$Y = H\_2 = -d\_{12} - b\_{11} \cos[2\pi\lambda X' + \overline{\varphi}], \text{ where } X' = X - ct \tag{2}$$

In upper defined equations, *a*<sup>11</sup> and *b*<sup>11</sup> represent the wave amplitudes, λ gives the wavelength, *t* suggests the time, *X* depicts the wave's direction, and *Y* is placed normally to *X*. The range of phase variance ϕ alters as 0 ≤ ϕ ≤ π. If ϕ = 0, we meant that a symmetric dimensional channel is having waves located out of the phase and ϕ = π, suggest the waves within the phase. Moreover *a*11, *b*11, *d*11, *d*<sup>12</sup> and ϕ overcome the following relation

$$a\_{11}^2 + b\_{11}^2 + 2a\_{11}b\_{11}\cos\overline{\varphi} \le (d\_{11} + d\_{12})^2\tag{3}$$

The mathematical models of the considered problem given as

$$
\nabla \cdot \widehat{V} = 0 \tag{4}
$$

$$
\rho \left( \frac{\partial \widehat{V}}{\partial t} + \widehat{V} \cdot \nabla \widehat{V} \right) = -\nabla \overline{P} + \nabla \cdot \mathbf{S} + \rho\_f \mathbf{g} \, a\_f \left( \overline{T} - T\_0 \right) + \rho\_f \mathbf{g} \, a\_f \left( \overline{\mathbf{C}} - \mathbf{C}\_0 \right) + \mathbf{J} \times \mathbf{B} \tag{5}
$$

$$\mathbf{E}\left(\rho c\right)\_f \left(\frac{\partial \overline{T}}{\partial t} + \widehat{\boldsymbol{V}} \cdot \nabla \overline{T}\right) = \nabla \cdot \boldsymbol{K} \nabla \overline{T} + \mathbf{S} \cdot \boldsymbol{\nabla} \overline{\boldsymbol{V}} + (\rho c)\_p \left(D\_\mathcal{B} \Big(\nabla \overline{\mathbf{C}} \cdot \nabla \overline{T}\Big) + \frac{D\_\mathcal{T}}{T\_m} (\nabla \overline{T} \cdot \nabla \overline{T})\right) \tag{6}$$

$$
\frac{\partial \widetilde{\mathbf{C}}}{\partial t} + \hat{\mathbf{V}} \cdot \nabla \widetilde{\mathbf{C}} = \nabla \cdot \left( D\_B \nabla \widetilde{\mathbf{C}} + D\_T \frac{\nabla \widetilde{T}}{T\_o} \right) \tag{7}
$$

where **g** is the gravitational body force and α*<sup>f</sup>* represents the volumetric volume distension nanofluid's coefficient. In above relations, (ρ*c*)*<sup>f</sup>* denotes the fluid's heat capacity, (ρ*c*)*<sup>p</sup>* accounts for effective nanoparticles heat capacity, **J** = σ( *<sup>V</sup>* <sup>×</sup> **<sup>B</sup>**) reveals the current density, *B* <sup>=</sup> (0, *<sup>B</sup>*0) notifies the external magnetic field, and **S** is placed for the Cauchy stress tensor for Williamson fluid and is determined as

$$\mathbf{r} = \left(\mu\_{\diamond} + (\mu\_0 + \mu\_{\diamond}) \left(1 - \Gamma \dot{\mathbf{y}}\right)^{-1}\right) \dot{\overline{\mathbf{y}}} \tag{8}$$

where · γ comprises the subsequent value

$$\dot{\overline{\mathbf{y}}} = \sqrt{\frac{1}{2} \sum\_{i} \sum\_{j} \dot{\mathbf{y}}\_{ij} \dot{\mathbf{y}}\_{ji}} = \sqrt{\frac{1}{2} \overline{\Pi}} \tag{9}$$

Here <sup>Π</sup> is the strain tensor. The velocity profile for the given problem is considered as *V* = (*U*- , *V*- ). Introducing a wavy frame we introduce the following transformations

$$\mathbf{x} = \mathbf{X}', \ y = \mathbf{Y}, \ u = \mathsf{U}' - \mathsf{c}, \ v = \mathsf{V}', \ p(\mathbf{x}) = \overline{\mathsf{P}}(\mathbf{X}, \ t) \tag{10}$$

We suggest the following dimensionless parameters to be used in the above expressions

*x* = <sup>2</sup>π*<sup>x</sup>* <sup>λ</sup> , *<sup>y</sup>* <sup>=</sup> *<sup>y</sup> <sup>d</sup>*<sup>11</sup> , *<sup>u</sup>* <sup>=</sup> *<sup>u</sup> <sup>c</sup>* , *<sup>v</sup>* <sup>=</sup> *<sup>v</sup> <sup>c</sup>*<sup>δ</sup> , <sup>δ</sup> <sup>=</sup> *<sup>d</sup>*<sup>11</sup> <sup>λ</sup> , *<sup>d</sup>* <sup>=</sup> *<sup>d</sup>*<sup>12</sup> *<sup>d</sup>*<sup>11</sup> , *<sup>p</sup>* <sup>=</sup> *<sup>d</sup>*<sup>2</sup> 11*p* <sup>μ</sup>*c*<sup>λ</sup> , *<sup>h</sup>*<sup>11</sup> <sup>=</sup> *<sup>H</sup>*<sup>11</sup> *<sup>d</sup>*<sup>11</sup> , *<sup>h</sup>*<sup>12</sup> <sup>=</sup> *<sup>H</sup>*<sup>12</sup> *<sup>d</sup>*<sup>12</sup> , *<sup>a</sup>*<sup>12</sup> = *<sup>a</sup>*<sup>11</sup> *<sup>d</sup>*<sup>11</sup> , *Br* <sup>=</sup> *Ec*Pr, *<sup>b</sup>* <sup>=</sup> *<sup>b</sup>*<sup>11</sup> *<sup>d</sup>*<sup>11</sup> , Re <sup>=</sup> <sup>ρ</sup>*cd*<sup>11</sup> <sup>μ</sup> , <sup>ψ</sup> <sup>=</sup> <sup>ψ</sup> *cd*<sup>11</sup> , <sup>θ</sup> <sup>=</sup> <sup>1</sup>*T*−*T*<sup>0</sup> *T*1−*T*<sup>0</sup> , *Ec* = *<sup>c</sup>*<sup>2</sup> *cp*(*T*1−*T*0), Pr = ρν*<sup>c</sup> <sup>K</sup>* , *<sup>S</sup>* <sup>=</sup> *Sd*<sup>11</sup> <sup>μ</sup>*<sup>c</sup>* , *We* <sup>=</sup> <sup>Γ</sup>*<sup>c</sup> <sup>d</sup>*<sup>11</sup> , <sup>ϕ</sup> <sup>=</sup> *<sup>C</sup>* 1−*C*<sup>0</sup> *C*1−*C*<sup>0</sup> , *Gr* <sup>=</sup> <sup>ρ</sup>*<sup>f</sup> <sup>g</sup>*α*<sup>f</sup> <sup>d</sup>*2(*T*1−*T*0) *<sup>c</sup>*<sup>μ</sup> , *Gc* <sup>=</sup> <sup>ρ</sup>*<sup>f</sup> <sup>g</sup>*α*<sup>f</sup> <sup>d</sup>*2(*C*1−*C*0) *<sup>c</sup>*<sup>μ</sup> , *Nb* <sup>=</sup> <sup>τ</sup>*DB*(*C*1−*C*0) <sup>ν</sup> , *Nt* <sup>=</sup> <sup>τ</sup>*DT*(*T*1−*T*0) <sup>τ</sup>*Tm* , *M* = σ <sup>μ</sup> *<sup>B</sup>*0*d*11, *We* <sup>=</sup> <sup>Γ</sup>*<sup>c</sup> d*<sup>11</sup> (11)

where *M*, *We*, *Br*, Pr, *Nb*, *Nt*, *Gr*, and *Gc* represent the Hartman number, Weissenberg number, Brinkman number, Prandtl number, Brownian motion parameter, thermophoresis parameter, local temperature Grashof number, and local nanoparticle Grashof number, accordingly. After incorporating the above structured parameters and applying the conditions of large wavelength along with small Reynolds number in a wavy frame coordinates we have the final form of Equations (4)–(7)

$$\begin{aligned} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} &= 0\\ \frac{dp}{dx} = \frac{\partial}{\partial y} \left[ \frac{\partial^2 \psi}{\partial y^2} - M^2 \psi + \mathcal{W} \mathcal{E} \left( \frac{\partial^2 \psi}{\partial y^2} \right)^2 \right] + \mathcal{G}\_\mathbf{r} \boldsymbol{\theta} + \mathcal{G}\_\mathbf{r} \boldsymbol{\phi} \end{aligned} \tag{12}$$

or

$$-\frac{\partial^2}{\partial y^2} \left[ \frac{\partial^2 \psi}{\partial y^2} + \mathcal{W} \epsilon \left( \frac{\partial^2 \psi}{\partial y^2} \right)^2 - M^2 \psi \right] + G\_r \frac{\partial \mathcal{O}}{\partial y} + G\_c \frac{\partial \psi}{\partial y} = 0 \tag{13}$$

$$\Pr\left[N\_b \frac{\partial \boldsymbol{\psi}}{\partial \boldsymbol{y}} \frac{\partial \boldsymbol{\partial}}{\partial \boldsymbol{y}} + N\_t \left(\frac{\partial \boldsymbol{\partial}}{\partial \boldsymbol{y}}\right)^2\right] + \frac{\partial^2 \boldsymbol{\partial}}{\partial \boldsymbol{y}^2} + Br\left[\left(\frac{\partial^2 \boldsymbol{\psi}}{\partial \boldsymbol{y}^2}\right)^2 + \mathcal{W}\boldsymbol{\epsilon}\left(\frac{\partial^2 \boldsymbol{\psi}}{\partial \boldsymbol{y}^2}\right)^3\right] = 0\tag{14}$$

$$\frac{\partial^2 \varrho}{\partial y^2} + \frac{N\_\text{f}}{N\_b} \frac{\partial^2 \theta}{\partial y^2} = 0 \tag{15}$$

where ψ is stream function satisfying the relations *u* = ∂δ/∂*y* and *v* = −δ∂ψ/∂*x*, The no-slip boundary conditions for velocity *u* and nanoparticles fraction ϕ and convective boundaries are taken into consideration for temperature θ which have the following dimensionless form in the wave frame [15]

$$\begin{aligned} \psi = \frac{\mathbb{F}}{2}, \quad \frac{\partial \psi}{\partial y} = -1, \text{ at } y = h\_{11}, \; \psi = -\frac{\mathbb{F}}{2}, \; \frac{\partial \psi}{\partial y} = -1, \; \text{at } y = h\_{12}, \\\ \frac{\partial \theta}{\partial y} - B\_i \theta = -B\_i \text{ at } y = h\_{11} \text{ and } \theta = 0 \text{ at } y = h\_{12} \end{aligned} \tag{16}$$
 
$$\begin{aligned} \varphi = 1 \qquad \text{at } y = h\_{11} \quad \text{and} \quad \varphi = 0 \text{ at } y = h\_{12}. \end{aligned} \tag{16}$$

where *h*<sup>11</sup> = 1 + *a*<sup>12</sup> cos *x* and *h*<sup>12</sup> = −*d* − *b* cos(*x* + ϕ). Also *Bi* = *hf d*11/*K* is the Biot number, *hf* stands for the coefficient of convective thermal transport. The mean flow rate in dimensionless format is elaborated as

$$Q = F + 1 + d \tag{17}$$
