**3. Solution Procedure**

In order to solve the resulting nonlinear system of partial differential equations, we applied the fast converging analytical technique (OHPM). According to the scheme, the deformation equations for the current problem may be written as [33–37]:

$$q(1-q)\left(\mathfrak{I}\left[\overline{w}\right]-\mathfrak{I}\left[\overline{w}\_{0}\right]\right)+q\left[\mathfrak{I}\left[\overline{w}\right]+\frac{1}{r^{2}}\overline{w}\_{\ell\ell\ell}+\left(1+\lambda\_{1}\right)\left(Br\Omega+Gr\Theta-p\_{z}\right)\right]=0,\tag{30}$$

$$\begin{split} (1-q)\left(\mathfrak{I}\left[\Theta\right]-\mathfrak{I}\left[\overline{\Theta}\ 0\right]\right) &+ q\Big(\mathfrak{I}\left[\Theta\right]+\frac{1}{r^{2}}\Theta\_{\partial\!\!\!0}+\mathrm{Nb}\Big(\Theta\_{r}\Omega\_{r}+\frac{1}{r^{2}}\Theta\_{\partial}\Omega\_{\partial}\Big)\Big) \\ &+ \mathrm{Nt}\Big(\Theta\_{r}^{-2}+\frac{1}{r^{2}}\Theta\_{\partial}{}^{2}\Big)+\mathrm{Gr}\Big(\frac{1}{1+\lambda\_{1}}\Big(\overline{w\_{z}w\_{r}}+\frac{1}{r^{2}}\overline{w\_{\partial}}^{2}\Big)\Big)\Big) = 0, \end{split} \tag{31}$$

$$q\left(\left(1-q\right)\left(\Im\left[\Omega\right]-\Im\left[\overline{\sigma}\_{o}\right]\right)+q\left(\Im\left[\Omega\right]+\frac{1}{r^{2}}\Omega\_{\ell\ell}+\frac{Nt}{Nb}\bigg(\Theta\_{rr}+\frac{1}{r}\Theta\_{r}+\frac{1}{r^{2}}\Theta\_{\ell\ell}\bigg)\right)=0.\tag{32}$$

The linear operator is chosen as = <sup>1</sup> *<sup>r</sup>* ∂*r*(*r*∂*r*). The initial guesses for *w*, θ, σ are selected as

$$
\widehat{\boldsymbol{w}\_{0}} = \boldsymbol{V} \ln[\boldsymbol{r}/\boldsymbol{r}^{V}] \left(\ln[\boldsymbol{r}\_{1}/\boldsymbol{r}\_{2}]\right)^{-1}, \\
\widehat{\boldsymbol{\theta}\_{0}} = \boldsymbol{\widetilde{\sigma}\_{0}} = \ln[\boldsymbol{r}\_{1}/\boldsymbol{r}] \left(\ln[\boldsymbol{r}\_{1}/\boldsymbol{r}\_{2}]\right)^{-1}.\tag{33}
$$

Now we describe the following series for complete solutions.

$$w = \lim\_{q \to 1} \overline{w}(r, \theta, z, t, q) = \lim\_{q \to 1} \sum\_{n=0}^{\infty} q^n \overline{w\_{n\prime}} \tag{34}$$

$$\overline{\partial} = \lim\_{q \to 1} \Theta(r, \theta, z, t, q) = \lim\_{q \to 1} \sum\_{n=0}^{\infty} q^n \overline{\partial\_{n\prime}} \tag{35}$$

$$\sigma = \lim\_{q \to 1} \Omega(r, \theta, z, q) = \lim\_{q \to 1} \sum\_{n=0}^{\infty} q^n \overline{\sigma\_n}. \tag{36}$$

Making use of Equations (34)–(36) into Equations (30)–(32) and equating the coefficients of exponents of *q*, we gather the system of ordinary differential equations, which can be solved easily on mathematical software by built-in commands. The volume flow rate *Q* can be noted as [26]:

$$\overline{Q} = 2\pi \int\_{r\_1}^{r\_2} rrw dr.\tag{37}$$

The mean volume flow rate *Q* over one period can be written as [26,30]:

$$Q(z,t) = \frac{\overline{Q}}{\pi} - \frac{\varrho^2}{2} + 2\varrho\cos[2\pi(z-t)] + \varrho^2\cos^2[2\pi(z-t)].\tag{38}$$

Now we can evaluate pressure gradient *pz* by solving Equations (37) and (38). The pressure rise Δ*p* in non-dimensional form contains the expression:

$$
\Delta p = \int\_0^1 (p\_z) dz.\tag{39}
$$

The two tables (Tables 1 and 2) are prepared through the numerical data of pressure rise Δ*p* against flow rate *Q* and temperature profile θ from Equation (26), and imposing values to defined parameters on the mathematical software "Mathematica".

**Table 1.** Data of Δ*p* for *Q* against β<sup>1</sup> and β<sup>2</sup> when *t* = 0.05, ε = 0.1,ϕ = 0.1, θ = 0.8, *V* = 0.1, λ<sup>1</sup> = 5, *Br* = 0.1.



**Table 1.** *Cont*.

**Table 2.** Error variation of temperature solution θ when other parameters are fixed.


#### **4. Entropy Generation**

Entropy evaluates the anarchy of the process. Due to this most important aspect of heat and mass transfer analysis, pivot concentrations are made to analyze the entropy effects and to minimize the entropy generation. The volumetric rate of entropy generation for a Jeffrey nanofluid in three-dimensional asymmetric annuli is defined as:

$$\begin{split} S'\_{\mathcal{S}\text{ren}} &= \frac{K}{T\_o^2} \Big( T\_r^2 + \frac{1}{r} T\_\theta^\* + T\_z^2 \Big) + \frac{D\_\mathcal{B}}{C\_o} \Big( \mathcal{C}\_r^2 + \frac{1}{r} \mathcal{C}\_\theta^\* + \mathcal{C}\_z^2 \Big) + \frac{D\_\mathcal{B}}{T\_o} \Big( \mathcal{C}\_r T\_r + \frac{1}{r} \mathcal{C}\_\theta T\_\theta + \mathcal{C}\_z T\_z \Big) \\ &+ \frac{1}{T\_o} \Big( \mathcal{S}\_{11} u\_r + \frac{1}{r} \mathcal{S}\_{12} u\_\theta + \mathcal{S}\_{13} (u\_z + w\_r) + \frac{1}{r} \mathcal{S}\_{32} w\_\theta + \mathcal{S}\_{33} w\_z + \frac{\mu}{r} \mathcal{S}\_{22} \Big). \end{split} \tag{40}$$

From the above expression, we can assume that the entropy generation is composed of four terms: The entropy generation for heat transfer irreversibility, the entropy generation because of nanoparticles irreversibility, the entropy due to irreversibility of the combined effects of heat transfer and nanoparticles, and the entropy in the presence of irreversibility of viscous dissipation of Jeffrey fluid, orderly. The non-dimensional parameters used in the above equation are defined as follows:

$$\text{Ns} = \frac{S'\_{\text{gen}}}{S\_{\text{G}}}, S\_{\text{G}} = \frac{K(T\_1 - T\_o)^2}{a^2 T\_o^2}, \Gamma = \frac{D\_B T\_o (\mathbb{C}\_1 - \mathbb{C}\_o)}{K(T\_1 - T\_o)}, \Lambda = \frac{(T\_1 - T\_o)}{T\_o}, \Omega = \frac{(\mathbb{C}\_1 - \mathbb{C}\_o)}{\mathbb{C}\_o} \Big|\_{\text{G}}.$$

where *Ns* is the entropy generation number, Λ gives the temperature difference parameter, Ω represents the concentration difference parameter, Γ suggests the ratio of temperature to concentration parameters. By transforming Equation (40) into a dimensionless form without primes, we receive:

$$\begin{array}{l} \text{Ns} = \left(\overline{\theta}\_{r}^{2} + \frac{1}{r^{2}}\overline{\theta}\_{\theta}{}^{2} + \delta\_{o}^{2}\overline{\theta}\_{z}{}^{2}\right) + \frac{\Gamma\Lambda}{\Omega}\Big(\sigma\_{r}^{2} + \frac{1}{r^{2}}\sigma\_{\theta}{}^{2} + \delta\_{o}^{2}\sigma\_{z}{}^{2}\Big) + \Gamma\Big(\sigma\_{r}\overline{\theta}\_{r} + \frac{1}{r^{2}}\sigma\_{\theta}\overline{\theta}\_{\theta} + \delta\_{o}^{2}\sigma\_{z}\overline{\theta}\_{z}\Big) \\ + \frac{\zeta\omega}{\Omega}\Big(\delta\_{o}\mathcal{S}\_{11}u\_{r} + \delta\_{o}\frac{1}{r}\mathcal{S}\_{12}u\_{\theta} + \mathcal{S}\_{13}\Big(\delta\_{o}^{2}u\_{z} + w\_{r}\Big) + \frac{1}{r}\mathcal{S}\_{32}w\_{\theta} + \delta\_{\theta}\mathcal{S}\_{33}w\_{z} + \delta\_{\theta}\frac{w}{r}\mathcal{S}\_{22}\Big). \end{array} \tag{41}$$

Incorporating the lubrication approach, we achieve:

$$\text{Ns} = \left(\overline{\theta}\_r^{\,2} + \frac{1}{r^2}\overline{\theta}\_\theta{}^2\right) + \frac{\Gamma\Lambda}{\Omega} \left(\sigma\_r^{\,2} + \frac{1}{r^2}\sigma\_\theta{}^2\right) + \Gamma \left(\sigma\_r\overline{\theta}\_r + \frac{1}{r^2}\sigma\_\theta\overline{\theta}\_\theta\right) + \frac{\text{G}c}{\Omega} \left(\mathbf{S}\_{13}w\_r + \frac{1}{r}\mathbf{S}\_{32}w\_\theta\right). \tag{42}$$

Invoking the values of *S*<sup>13</sup> and *S*<sup>32</sup> from Equations (21) and (23) into the above Equation (42), it becomes:

$$\text{Ns} = \left(\overline{\theta}\_{\text{r}}^{2} + \frac{1}{r^{2}}\overline{\theta}\_{\text{\textdegree 0}}^{2}\right) + \frac{\Gamma\Lambda}{\Omega} \left(\sigma\_{\text{r}}^{2} + \frac{1}{r^{2}}\sigma\_{\text{\textdegree 0}}^{2}\right) + \Gamma \left(\sigma\_{\text{r}}\overline{\theta}\_{\text{\textdegree 1}} + \frac{1}{r^{2}}\sigma\_{\text{\textdegree 0}}\overline{\theta}\_{\text{\textdegree 0}}\right) + \frac{\text{G}c}{\Omega} \left(\frac{1}{1+\lambda\_{1}}\left(w\_{\text{\textdegree 1}}w\_{\text{\textdegree 2}} + \frac{1}{r^{2}}w\_{\text{\textdegree 2}}\right)\right). \tag{43}$$

Moreover, the Bejan number, *Be*, being the ratio of entropy generation against the heat transfer irreversibility to the total entropy generation is described mathematically as:

$$\mathcal{B}c = \left(\overline{\theta}\_r^2 + \frac{1}{r^2}\overline{\theta}\_\theta^2\right) \begin{pmatrix} \left(\overline{\theta}\_r^2 + \frac{1}{r^2}\overline{\theta}\_\theta\right)^2 + \frac{\Gamma\Delta}{\Gamma\Omega} \left(\sigma\_r^2 + \frac{1}{r^2}\sigma\_\theta\right)^2 + \Gamma \left(\sigma\_r\overline{\theta}\_r\right)^{-1} \\\ + \frac{1}{r^2}\sigma\_\theta\overline{\theta}\_\theta\right) + \frac{G\varepsilon}{\Gamma\Omega} \left(\frac{1}{1+\lambda\_1}\left(w\_r w\_z + \frac{1}{r^2}w\_\theta^2\right)\right) \end{pmatrix}^{-1}.\tag{44}$$

Bejan number, *Be*, carries the values from the interval [0,1]. If *Be* < 1, it can be observed that total entropy generation surpasses the heat transfer entropy, and for *Be* = 1, the total entropy generation approaches the entropy generation against the heat transfer irreversibility.
