**2. Mathematical Analysis**

Let us analyze the entropy generation in three-dimensional flow of nanofluid with the Jeffrey model by considering the passage in a space between two eccentric annuli, with flexible outer surface along with inner rigid cylinder, going with the fluid with constant speed *V*. The walls of the outer annulus produce peristaltic waves along its length, which helps in pushing the fluid forward. A concentration *C*<sup>0</sup> of nanoparticles is assumed at the inner boundary, while the outer is maintained at *C*1. The temperature distributions are described as *T*<sup>0</sup> and *T*<sup>1</sup> on the considered inner and lower walls accordingly (see Figure 1).

**Figure 1.** Diagram of flow mechanism and annuli.

The physical behavior of inner and outer layers of the annuli is manipulated mathematically as follows:

χ*<sup>i</sup>* = χ*<sup>j</sup>* + χ*<sup>k</sup>* cos γ, for *i* = 1, 2 ... , where χ1, χ*<sup>j</sup>* = δ, χ*<sup>k</sup>* = ε, and γ = θ are representing the inner cylinder walls. Similarly, χ2, χ*<sup>j</sup>* = *a*, χ*<sup>k</sup>* = *b*, and γ = <sup>2</sup><sup>π</sup> <sup>λ</sup> (*z* − *ct*) are suggesting the same for outer annuli. Above appearing δ, *a*, *b*, λ, and *c* are denoting the radii of inner and outer cylinders, the amplitude of the wave, the wavelength and the wave speed, orderly.

According to the considered geometry, the velocity components are suggested as [*w*1(*r*, θ, *z*), 0, *w*2(*r*, θ, *z*)]. The mathematical structure of the given problem can be entertained by the following expressions based on physical laws:

$$\frac{\partial w\_1}{\partial r} + \frac{\partial w\_2}{\partial z} + \frac{w\_1}{r} = 0,\tag{1}$$

$$
\rho\_f \left( \frac{\partial w\_1}{\partial t} + w\_1 \frac{\partial w\_1}{\partial r} + w\_2 \frac{\partial w\_1}{\partial z} \right) = -\frac{\partial p}{\partial r} + \frac{\partial \Upsilon\_{11}}{\partial r} + \frac{1}{r} \frac{\partial \Upsilon\_{12}}{\partial \theta} + \frac{\partial \Upsilon\_{13}}{\partial z} + \frac{\Upsilon\_{11}}{r} - \frac{\Upsilon\_{22}}{r}, \tag{2}
$$

$$0 = -\frac{1}{r}\frac{\partial p}{\partial \theta} + \frac{\partial \Upsilon\_{12}}{\partial r} + \frac{1}{r}\frac{\partial \Upsilon\_{22}}{\partial \theta} + \frac{\partial \Upsilon\_{23}}{\partial z} + \frac{\Upsilon\_{21}}{r} + \frac{\Upsilon\_{12}}{r},\tag{3}$$

$$\begin{split} \rho\_f \Big( \frac{\partial w\_2}{\partial t} + w\_1 \frac{\partial w\_2}{\partial r} + w\_2 \frac{\partial w\_2}{\partial z} \Big) &= -\frac{\partial p}{\partial z} + \frac{\partial \Upsilon\_{31}}{\partial r} + \frac{1}{r} \frac{\partial \Upsilon\_{32}}{\partial 0} + \frac{\partial \Upsilon\_{33}}{\partial z} + \frac{\Upsilon\_{31}}{r} \\ &+ \rho\_f \text{g} a (T - T\_o) + \rho\_f \text{g} a (\text{\"C} - \text{C}\_o), \end{split} \tag{4}$$

$$\begin{pmatrix} \rho c \left( \rho c \right) \left( \frac{\partial T}{\partial t} + w\_1 \frac{\partial T}{\partial r} + w\_2 \frac{\partial T}{\partial z} \right) = k \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2} + \frac{\partial^2 T}{\partial z^2} + \frac{1}{r} \frac{\partial T}{\partial r} \right) + (\rho c)\_p \\ \frac{1}{r} \left[ Dg \left( \frac{\partial C}{\partial r} \frac{\partial T}{\partial r} + \frac{1}{r^2} \frac{\partial C}{\partial \theta} \frac{\partial T}{\partial \theta} + \frac{\partial C}{\partial z} \frac{\partial T}{\partial z} \right) + \frac{D\_T}{T\_o} \left( \left( \frac{\partial T}{\partial r} \right)^2 + \frac{1}{r^2} \left( \frac{\partial T}{\partial \theta} \right)^2 + \left( \frac{\partial T}{\partial z} \right)^2 \right) \right] \\ + \Upsilon\_{11} \frac{\partial w\_1}{\partial r} + \frac{1}{r} \Upsilon\_{12} \frac{\partial w\_1}{\partial \theta} + \Upsilon\_{13} \left( \frac{\partial w\_1}{\partial z} + \frac{\partial w\_2}{\partial r} \right) + \frac{1}{r} \Upsilon\_{32} \frac{\partial w\_2}{\partial \theta} + \Upsilon\_{33} \frac{\partial w\_2}{\partial z} + \frac{w\_1}{r} \Upsilon\_{22} \end{pmatrix} \tag{5}$$

$$\frac{\partial \mathbf{C}}{\partial t} + w\_1 \frac{\partial \mathbf{C}}{\partial r} + w\_2 \frac{\partial \mathbf{C}}{\partial z} = D\_\theta \left( \frac{\partial^2 \mathbf{C}}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 \mathbf{C}}{\partial \theta^2} + \frac{\partial^2 \mathbf{C}}{\partial z^2} + \frac{1}{r} \frac{\partial \mathbf{C}}{\partial r} \right) + \frac{D\_\Gamma}{T\_\theta} \left( \frac{\partial^2 T}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2} + \frac{\partial^2 T}{\partial z^2} + \frac{1}{r} \frac{\partial T}{\partial r} \right). \tag{6}$$

In this study, constitutive relation used for fluid is the Jeffrey mode [18], which has the following expression:

$$\Upsilon = \frac{\mu}{1 + \lambda\_1} (\dot{\mathbf{y}} + \lambda\_2 \bar{\mathbf{y}}). \tag{7}$$

To execute the collective effects of emerging parameters, we adopt the process of non-dimensionalization by introducing the following transformations [15,20–24]:

 $p' = \frac{\mu^2}{\mu c^2}$  $p$ ,  $w' = \frac{w\_2}{c}$ ,  $u' = \frac{h}{ac}w\_1$ ,  $V' = \frac{V}{c'}$ ,  $z' = \frac{T}{a}$ ,  $\theta' = \theta$ ,  $t' = \frac{\xi}{A}t$ ,  $\varphi = \frac{h}{a}$ ,  $\varepsilon' = \frac{\mu}{a}$ ,  $Re = \frac{\rho \alpha}{\mu}$ ,  $\delta' = \frac{\rho}{a}$ ,  $\overline{\theta} = \frac{T - T\_o}{T\_1 - T\_o}$ ,  $\delta\_o = \frac{\rho}{\mu}$ ,  $\overline{\theta} = \frac{T - T\_o}{T\_1 - T\_o}$ ,  $\delta\_o = \frac{\rho}{\mu c}$ ,  $\overline{\sigma} = \frac{\mu}{\rho \alpha}$ ,  $S\_c = \frac{\rho}{\rho D\_o}$ ,  $Br = \frac{\rho \rho \sin^2 \alpha}{\mu c}$ ,  $Tr = \frac{\rho \rho \sin^2 \alpha}{\mu c}$  ( $T\_1 - T\_o$ ),  $Nb = \frac{\tau D\_R}{a\_f}$  ( $C\_1 - C\_o$ ),  $Nb = \frac{\tau D\_R}{a\_f}$  ( $C\_1 - C\_o$ ),  $\alpha\_f = \frac{k}{(\rho c)\_f}$ ,  $\tau = \frac{(\rho c)\_p}{(\rho c)\_f}$ ,  $\tau = \frac{\rho c^2}{\kappa (T\_1 - T\_o)}$ ,  $S' = \frac{\mu c}{a}$ .

In light of the above manufactured relations, the Equations (1) to (6) become

$$
\frac{\partial u'}{\partial r'} + \frac{\partial w'}{\partial z'} + \frac{u'}{r'} = 0,\tag{8}
$$

$$\operatorname{Re}\delta\_0 \left(\frac{\partial \boldsymbol{u}'}{\partial t'} + \boldsymbol{u}' \frac{\partial \boldsymbol{w}'}{\partial r'} + \boldsymbol{w}' \frac{\partial \boldsymbol{u}'}{\partial z'}\right) = -\frac{\partial p'}{\partial r'} + \delta\_0 \frac{\partial S'\_{11}}{\partial r'} + \delta\_0 \frac{1}{r'} \frac{\partial S'\_{12}}{\partial \theta'} + \delta\_0^2 \frac{\partial S'\_{13}}{\partial z'} + \delta\_0 \frac{S'\_{11}}{r'} - \delta\_0 \frac{S'\_{22}}{r'}, \tag{9}$$

$$0 = -\frac{1}{r'}\frac{\partial p'}{\partial \theta'} + \delta\_\vartheta \frac{\partial S'\_{21}}{\partial r'} + \delta\_\vartheta \frac{1}{r'}\frac{\partial S'\_{22}}{\partial \theta'} + \delta\_\vartheta^2 \frac{\partial S'\_{23}}{\partial z'} + \delta\_\vartheta \frac{S'\_{21}}{r'} + \delta\_\vartheta \frac{S'\_{12}}{r'},\tag{10}$$

$$\operatorname{Re}\delta\_{\theta}\left(\frac{\partial w'}{\partial t'} + u'\frac{\partial w'}{\partial r'} + w'\frac{\partial w'}{\partial z'}\right) = -\frac{\partial p'}{\partial z'} + \frac{\partial S'\_{\,\,31}}{\partial r'} + \frac{1}{r'}\frac{\partial S'\_{\,\,32}}{\partial \theta'} + \delta\_{\theta}\frac{\partial S'\_{\,\,33}}{\partial z'} + \frac{S'\_{\,\,31}}{r'} + Gr\overline{\theta} + Br\sigma,\tag{11}$$

δ*o*Re*Pr* ∂θ ∂*T*- + *u* - ∂θ ∂*r*- + *w*- ∂θ ∂*z*- = ∂2θ ∂*r*-<sup>2</sup> + <sup>1</sup> *r*-2 ∂2θ ∂θ-<sup>2</sup> + δ<sup>2</sup> *o* ∂2θ ∂*z*-<sup>2</sup> + <sup>1</sup> *r*- ∂θ ∂*r*- +*Nb* ∂σ ∂*r*- ∂θ ∂*r*- + <sup>1</sup> *r*-2 ∂σ ∂θ- ∂θ ∂θ- + δ<sup>2</sup> *<sup>o</sup>* ∂σ ∂*z*- ∂θ ∂*z*- <sup>+</sup> *Nt* ∂θ ∂*r*- 2 + <sup>1</sup> *r*-2 ∂θ ∂θ- 2 +δ<sup>2</sup> *o* ∂θ ∂*z*- 2 + *Gc*- δ*oS*- <sup>11</sup> <sup>∂</sup>*u*- ∂*r*- + δ*<sup>o</sup>* <sup>1</sup> *r*- *S*- <sup>12</sup> <sup>∂</sup>*u*- ∂θ- + *S*- 13- δ2 *o* ∂*u*- ∂*z*- + <sup>∂</sup>*w*- ∂*r*- + <sup>1</sup> *r*- *S*- <sup>32</sup> <sup>∂</sup>*w*- ∂θ- + δ*oS*- <sup>33</sup> <sup>∂</sup>*w*- ∂*z*- + δ*<sup>o</sup> u*- *r*- *S*- 22 , ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ (12)

$$\begin{array}{l} \delta\_o \text{Res}\_{\mathcal{L}} \Big( \frac{\partial \mathcal{L}}{\partial t'} + \boldsymbol{\mu}' \frac{\partial \boldsymbol{\sigma}}{\partial r'} + \boldsymbol{w}' \frac{\partial \boldsymbol{\sigma}}{\partial z'} \Big) = \left( \frac{\partial^2 \mathcal{L}}{\partial r'^2} + \frac{1}{r'^2} \frac{\partial^2 \mathcal{L}}{\partial \theta'^2} + \delta\_o^2 \frac{\partial^2 \mathcal{L}}{\partial z'^2} + \frac{1}{r'} \frac{\partial \mathcal{L}}{\partial r'} \right) \\ + \frac{\mathcal{N}t}{\mathcal{N}\mathcal{b}} \Big( \frac{\partial^2 \overline{\mathcal{D}}}{\partial r'^2} + \frac{1}{r'^2} \frac{\partial^2 \overline{\mathcal{D}}}{\partial \theta'^2} + \delta\_o^2 \frac{\partial^2 \overline{\mathcal{D}}}{\partial z'^2} + \frac{1}{r'} \frac{\partial \overline{\mathcal{D}}}{\partial r'} \Big). \end{array} \tag{13}$$

Here, the quantities like *Re*, δ0, *Gr*, *Br*, *Pr*, *Nb*, *Nt*, *Gc* and *Sc* represent the Reynolds number, wave number, local temperature Grashof number, local nanoparticle Grashof number, Prandtl number, Brownian motion parameter, thermophoresis parameter, Brinkman number, and Schmidt number, consecutively. After incorporating the theory of lubrication in this problem and disregarding the prime symbols, Equations (8) to (13) can be viewed as:

$$
\frac{\partial \mu}{\partial r} + \frac{\partial w}{\partial z} + \frac{u}{r} = 0,
\tag{14}
$$

$$\frac{\partial p}{\partial r} = 0 = \frac{\partial p}{\partial \theta'}\tag{15}$$

$$0 = -\frac{\partial p}{\partial z} + \frac{\partial S\_{31}}{\partial r} + \frac{1}{r}\frac{\partial S\_{32}}{\partial \theta} + \frac{S\_{31}}{r} + Gr\overline{\theta} + Br\sigma\_{\prime} \tag{16}$$

$$\begin{cases} 0 = \left(\frac{\partial^2 \overline{Q}}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 \overline{Q}}{\partial \vartheta^2} + \frac{1}{r} \frac{\partial \overline{Q}}{\partial r}\right) + Nb \left(\frac{\partial x}{\partial r} \frac{\partial \overline{Q}}{\partial r} + \frac{1}{r^2} \frac{\partial x}{\partial \vartheta} \frac{\partial \overline{Q}}{\partial \vartheta}\right) \\ + Nt \Big(\left(\frac{\partial \overline{Q}}{\partial r}\right)^2 + \frac{1}{r^2} \left(\frac{\partial \overline{Q}}{\partial \vartheta}\right)^2\Big) + Gr \{S\_{13} \frac{\partial w}{\partial r} + \frac{1}{r} S\_{32} \frac{\partial w}{\partial \vartheta}\Big)\_{\prime} \end{cases} \tag{17}$$

$$0 = \left(\frac{\partial^2 \sigma}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 \sigma}{\partial \theta^2} + \frac{1}{r} \frac{\partial \sigma}{\partial r}\right) + \frac{Nt}{Nb} \left(\frac{\partial^2 \overline{\partial}}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2 \overline{\partial}}{\partial \theta^2} + \frac{1}{r} \frac{\partial \overline{\partial}}{\partial r}\right). \tag{18}$$

The dimensionless components of the stress tensor for Jeffrey model in eccentric annuli, by using cylindrical coordinates, are given by the following relations [18–20] after ignoring the prime symbols:

$$S\_{11} = \frac{2\delta\_o}{1 + \lambda\_1} \Big( 1 + \lambda\_2 \delta\_o \frac{c}{a} (u\_{rt} + uu\_{rr} + wu\_{rz}) \Big),\tag{19}$$

$$S\_{12} = \frac{\delta\_o}{1 + \lambda\_1} \Big( 1 + \lambda\_2 \delta\_o \frac{c}{ar} (u\_{t0} + uu\_{r0} - r^{-1}u\_0 + wu\_{z0}) \Big),\tag{20}$$

$$S\_{13} = \frac{1}{1+\lambda\_1} (1 + \lambda\_2 \delta\_o \frac{c}{a} (\partial\_t + u \partial\_r + w \partial\_z)) \Big(\delta\_o^2 u\_z + w\_r\Big),\tag{21}$$

$$S\_{22} = \frac{2\delta\_0}{1 + \lambda\_1} \Big( 1 + \lambda\_2 \delta\_o \frac{c}{ar} (u\_l + uu\_r - r^{-1}u + vu\_z) \Big),\tag{22}$$

$$S\_{23} = \frac{1}{1 + \lambda\_1} \left( \frac{1}{r} w\_{\theta} + \lambda\_2 \delta\_o \frac{c}{ar} (w\_{\theta t} + \mu w\_{\theta r} - r^{-1} w\_{\theta} + w w\_{\theta z}) \right),\tag{23}$$

$$S\_{33} = \frac{2\delta\_0}{1 + \lambda\_1} \Big( 1 + \lambda\_2 \delta\_o \frac{c}{a} (w\_{zt} + \mu w\_{rz} + w w\_{zz}) \Big). \tag{24}$$

So by switching expressions of the above stresses (after applying the constraints of long wavelength and low Reynolds number) in Equations (16)–(18), we get:

$$\frac{1}{1+\lambda\_1}p\_z = w\_{rr} + \frac{1}{r}w\_{00} + \frac{1}{r}w\_r + \left(Gr\overline{\theta} + Br\sigma\right)(1+\lambda\_1),\tag{25}$$

$$\begin{cases} 0 = \left(\overline{\theta}\_{\mathcal{T}} + \frac{1}{r^2}\overline{\theta}\_{\partial\mathcal{O}} + \frac{1}{r}\overline{\theta}\_{\mathcal{I}}\right) + \mathrm{Nb}\Big(\sigma\_{r}\overline{\theta}\_{\mathcal{I}} + \frac{1}{r^2}\sigma\_{\partial}\overline{\theta}\_{\partial}\Big) + \mathrm{Nt}\Big(\overline{\theta}\_{r}^{\prime} + \frac{1}{r^2}\overline{\theta}\_{\partial}{}^{\prime}\Big) \\ + \mathrm{Cr}\Big(\frac{1}{1+\lambda\_1}\Big(w\_{r}w\_{z} + \frac{1}{r^2}w\_{\partial}{}^{\prime}\Big)\Big). \end{cases} \tag{26}$$

$$0 = \left(\sigma\_{rr} + \frac{1}{r^2}\sigma\_{\theta\theta} + \frac{1}{r}\sigma\_r\right) + \frac{Nt}{Nb}\left(\overline{\theta}\_{rr} + \frac{1}{r^2}\overline{\theta}\_{\theta\theta} + \frac{1}{r}\overline{\theta}\_r\right). \tag{27}$$

The subscripts of *u*, *w*, *p*, ∂, θ and σ denote the velocity components, pressure, partial differentiation, temperature, and concentration, respectively. The non-dimensional form of radii will take the following form [32]:

$$\begin{aligned} r\_1 &= \delta + \varepsilon \cos \theta, \\ r\_2 &= 1 + q \cos 2\pi (z - t). \end{aligned} \tag{28}$$

The respective boundary conditions may be put in the form [32]:

$$\begin{aligned} w &= V, \overline{\theta} = 0, \sigma = 0 \text{ at } r = r\_{1\prime} \\ w &= 0, \overline{\theta} = 1, \sigma = 1 \text{ at } r = r\_{2\prime} \end{aligned} \tag{29}$$
