**1. Introduction**

In the present era, nanofluid is getting importance from the researchers due to its diverse application in the industrial field. For instance, they are available in polymer manufacturing, gas turbines, power generators, glass fabric, paper production, wire drawing, and many more. Nanofluid is a sort of heat transport medium containing nanoparticles under 100 nm, which are reliably and consistently scattered in a base fluid like water, oil, and ethylene glycol. These scattered nanoparticles, for the most part, a metal or metal oxide massively improve the thermal conductivity of the nanofluid, upgrades conduction, and convection coefficients, mulling over more heat transport. Enhancing the thermal specifications of liquids leads to a greater level of connective flow in thermal units. For heat transfer enhancement, adding additives to the operant liquids for modifying their thermal features are very attractive method. For this, a way has been represented by enhancement in nanotechnology. The meaning of 'nanofluid' has been expressed by Choi [1] in 1995 for increasing heat transfer specifications of convectional fluids. A total report on the productivity of temperature transformation

in the sunlight-based authority with the nanoliquid was done by Chen et al. [2]. They originate that the presentation of photo thermal change in the gatherer expanded by 96.93% and 52% at 30 and 75 ◦C separately, an entirely reasonable liquid for sun powered authorities which have the shortest fascination in low-temperatures. Oudina [3,4] examined nanomaterial conduct esoteric an annulus with different designs of temperature foundations. They used an arithmetical methodology and introduced soundness investigation. Mesoscopic line for investigating nanomaterial course through permeable area was introduced by Sheikholeslami [5]. They utilized Lorentz force to switch the stream style inside an opening. Chougule et al. [6] initiate that because of the low association of nano-powders, a pressure drop is critical in the curved cylinder when they utilized carbon nanotubes (CNTs) instead of unadulterated water. Besides, a curved cylinder with loop supplements improves better Nusselt number as a result of the rate of energy altercation increment because of this reality nano-powders have arbitrary and unpredictable movement in the liquids. Numerically, the plan assessment of a whirling stream microchannel for incredible warmth transition uses had been examined by Hartmann-Priesnitz et al. [7] who displayed the operant liquids as Cu-H2O nanomaterial in the laminar stream. Ding et al. [8] portrayed nanofluids that depend on CNTs, which depend upon the heat of the base liquid, and set up that when the liquid heat is 25 ◦C, the warm conductivity can be expanded by up to 30%, but can be increased by 79% observed at 40 ◦C. Pop and Watanabe [9] carried out a theoretical analysis with the main aim to discuss the influence of injection/suction on fluid flowing over a cone with free convection and heat flux. The authors used different differential methods to solve the existing equation describing the flow. Xu [10] recently studied time-dependent hybrid nanofluid with mixed convection in rotating disks multiple kinds of nanoparticles are taken here. A numerical approach is used for the solution. Flaccid devices are used by investigators in an earlier investigation to improve the convective coefficient [11]. Many researchers used a combination of both concepts to enhance convective coefficient by use of insert with nanofluids. Heat transfer and friction element characteristics on warped tape with Al2O3/water nanofluid are analyzed by Sharma et al. [12]. Zhang [13] has investigated adapted computational approaches in which a 2-D effective heat capacity model is used for forecasting the fleeting heat transmission procedure of the building envelopes equipped with phase change materials (PCM). The deviations against test data made by the principal procedures were viewed as especially bigger than the changed methodologies adjusted techniques. Sun et al. [14] have experimentally deliberated the heat transmission rate augmentation produced by natural convection of PCMs experiencing melting. The thermophysical properties of nanofluids have been discussed by Phuoc et al. [15]. The increase in the transfer of heat of SWCNTs-glycol-based nanofluids was examined by Harish et al. [16]. It was found that while 0.2% by volume of SNTs was added to ethylene glycol, the thermal conductivity increased by 14.8%. The magnetohydrodynamic (MHD) 3-D Maxwell nanofluid boundary layer flow with convective boundary conditions on a biaxially stretched sheet was explored by Hayat et al. [17].

Nowadays the main concerns of scientists are to make the strategies that control the ingesting of skilled vitality. In the field of thermal structuring, the key objective is to achieve the best viability of contraptions and with the base loss of warmth, scouring, and spread during the mechanical procedures. All the heat gadgets take a shot at the guideline of thermodynamics and produce. Thermodynamic second laws used to look at the irreversibility in terms of the entropy age rate. Entropy growth is abused to elucidate the exhibition of various settings in present-day and structure solicitations. Entropy is imitative from the Greek word Entropia, which suggests "moving toward" or "alteration". Entropy figuring is basic as it orders the factors for energy forfeiture. Bejan [18] offered the clue of an entropy generation problem. Ellahi et al. [19] studied the influence of entropy optimization on natural convective nanoliquid stream. A multiple turbulator has been engaged by Sheikholeslami et al. [20] to enlarge the involvement of nanomaterial inside a tube. They verified that thermal irreversibility improves with the upsurge of subordinate flow. The related research work in a similar filed can be seen in [21,22].

The attractive properties of carbon nanotubes (CNTs) include mechanical and chemical stability, excellent thermal and electrical conductivity, lightweight and physicochemical reliability, making them a desirable material in the manufacture of electrochemical devices. Considering this exciting feature of carbon nanotubes, in this research work, our goal was to scrutinize the case where water-based nanofluids having single-wall and multi-wall CNTs flow through a vertical cone. The body package is layered in convective heat and diluted permeable medium. The effects of Joule heating, rotary microorganisms, and heat generation/absorption, chemical reactions, and heat radiation increase the novelty of the established model. By using a local similarity transformation technique, the PDE is changed into a coupled differential equation. By using the Homotopy analysis method to get the solution of the conservation equations and their relevant Boundary conditions. The parameters appearing in the distribution analysis of the alliance are scrutinized, and the consequences are depicted graphically. It can be perceived that on account of the two nanotubes, the velocity of fluid decreases as the magnetic is increased. Moreover, the thickness of moving microorganisms is decreased compared to more estimation of biological convection constants.

#### **2. Mathematical Analysis**

We suppose the flow of magnetohydrodynamic (MHD) mixed convective viscous water-based micropolar nanofluidic of CNTs on a vertical cone in a penetrable medium. The coordinate system for the flow phenomena is chosen is such a way that x-coordinate is parallel with the direction of the fluid. Energy expression with thermal radiation dissipation, thermal flux, and Joule heating is measured. Irreversibility investigation with chemical reaction is investigated. Entropy rate is determined. Slip impact is likewise talked about. A magnetic field of constant strength β<sup>0</sup> is applied vertically. The temperature (*T*) and concentration ponder (*C*). Speeding up because gravity acts downwards. Problem geometry is featured in Figure 1.

**Figure 1.** Schematic diagram for the flow direction.

From the above suppositions, the resulting modeled equations are:

$$(ru)\_x + (rv)\_y = 0\tag{1}$$

$$\begin{aligned} \mu u\_{\rm x} + \upsilon u\_{\rm x} + \lambda \left( u^2 u\_{\rm xx} + \upsilon^2 u\_{\rm yy} + 2\mu \upsilon u\_{\rm xy} \right) &= \frac{\mu\_{nf}}{\rho\_{nf}} u\_{\rm yy} - \frac{\mu\_{nf}}{\rho\_{nf}} \frac{u}{\mathbb{X}} + \\ \log \left[ \beta (T - T\_{\rm oo}) - \beta \* (\mathbb{C} - \mathbb{C}\_{\rm oo}) - \beta \* \gamma (n - n\_{\rm oo}) \right] \cos \gamma\_1 - \frac{\sigma\_{nf} \beta\_0^2}{\rho\_{nf}} u + \text{KN}\_y \end{aligned} \tag{2}$$

$$uT\_X + \upsilon T\_Y = \alpha\_{nf} T\_{yy} - \frac{1}{\left(\rho\_{cp}\right)\_{nf}} (q\_f)\_y + \frac{Q\_0}{\left(\rho\_{cp}\right)\_{nf}} (T - T\_{so}) + \frac{\sigma \beta\_0^2}{\left(\rho\_{cp}\right)\_{nf}} u^2 \tag{3}$$

*Coatings* **2020**, *10*, 998

$$uN\_x + vN\_y = \frac{\mathcal{V}\_{nf}^\*}{\varrho\_{nf}} N\_{yy} - \frac{k}{\varrho\_{nf}} (2N + u\_y) \tag{4}$$

$$
u \mathbb{C}\_x + v \mathbb{C}\_y = D\_m \mathbb{C}\_{yy} - k\_r (\mathbb{C} - \mathbb{C}\_{\infty}) \tag{5}$$

$$
tau\_x + vn\_y + \frac{b\mathcal{W}}{\left(\mathbb{C}\_w - \mathbb{C}\_0\right)} \Big(n\mathbb{C}\_y\Big)\_y = D\_n n\_{yy} \tag{6}
$$

with the corresponding boundary conditions

$$\begin{aligned} \mu = 0, \upsilon = V\_1, \ N = 0, T\_y &= \frac{h\_f \{T\_f - T\}}{-k\_{nf}}, & \mathbb{C}\_w = \mathbb{C} &= d\mathbf{x} + \mathbb{C}\_{0, \prime} n = n\_{\mathrm{w}\prime} & \text{At} & \text{ } y = a & \text{or} \\ \mu \to 0, N \to 0, \mathbb{C} \to \mathbb{C}\_{\mathrm{oo}} &= \mathbb{C}\_0 + \text{ex}, T \to T\_{\mathrm{oo}}, n \to n\_{\mathrm{w}} & \text{At} & \text{ } y \to \infty \end{aligned} \tag{7}$$

where, (β, β\*), (μ*nf*, μ*f*), (ρ*CHT*, ρ*f*), β0, α*nf*, *V*0, *hf*, (*d*, *e*), Q0, ((ρ*cp*)*f*, (ρ*cp*)*nf*), (*kf*, *knf*, *k*), *Dn*, *kr*, *Wc*, *Dm*, *qr*, γ<sup>1</sup> indicate coefficients of thermal and solutal expansion, dynamic viscosities, densities, magnetic strength, thermal modified diffusivity, suction/injection parameter, convective parameter, dimensionless constants of concentration, heat generation/absorption parameter, heat capacities, thermal conductivity, diffusivity of microorganisms, chemical reaction rate, coefficient, extreme cell swimming motion, Brownian diffusion, radiation coefficient, and cone half-angle, respectively. In Equation (7) the term *V*<sup>1</sup> characterizes the mass transmission and defined as *V*<sup>1</sup> = − 3 <sup>4</sup> *ax*−1*R*(*ax*) 1 4 *V*0. In case of *V*<sup>1</sup> < 0, the mass transfer is for injection and *V*<sup>1</sup> > 0 shows suction.

<sup>α</sup>*n f* <sup>=</sup> *kn f* <sup>ρ</sup>*n f*(*cp*)*n f* , *kn f <sup>k</sup> <sup>f</sup>* <sup>=</sup> (1−φ)+2<sup>φ</sup> *kCNT kCNT*−*<sup>k</sup> <sup>f</sup> n kCNT*+*<sup>k</sup> <sup>f</sup>* 2*k f* (1−φ)+2<sup>φ</sup> *kn f kCNT*−*k f n kCNT*+*<sup>k</sup> <sup>f</sup>* 2*k f* <sup>ν</sup>*n f* <sup>=</sup> <sup>μ</sup>*n f* <sup>ρ</sup>*n f* , γ<sup>∗</sup> *n f* <sup>=</sup> μ*n f* + 0.5*k j*, ρ*n f* = (1 − φ)ρ*<sup>f</sup>* + φρ*CNT*, μ*n f* = μ(1 − φ) −2.5 σ*n f* <sup>σ</sup>*<sup>f</sup>* = <sup>1</sup> <sup>−</sup> 3 σ*s* σ *f* −1 φ σ*s* σ *f* −1 φ− σ*s* σ *f* +2 (8) ψ = α*Ra*1/4 *<sup>x</sup> <sup>f</sup>*(η), *<sup>g</sup>*(η) = *<sup>C</sup>*−*C*<sup>∞</sup> *Cw*−*C*<sup>0</sup> , η = *<sup>y</sup> xRa*1/4 *x* θ(η) = *<sup>T</sup>*−*T*<sup>∞</sup> *Tw*−*T*<sup>∞</sup> , h(η) <sup>=</sup> *<sup>n</sup>*−*n*<sup>∞</sup> *nw*−*n*<sup>∞</sup> , S(η) <sup>=</sup> <sup>ρ</sup>*<sup>f</sup> <sup>x</sup>*2*NRa*−3/4 *x* μ*f* (9)

Using Equations (8) and (9), Equation (1) is satisfied and Equations (2)–(6) are written as

$$\begin{cases} f''' + -k\_1 f' + \left(1 - \phi\right)^{2.5} \Big(1 - \phi + \phi \frac{\rho\_{\rm C\gamma T}}{\rho\_f} \Big) \left[\Theta - N\_{\rm r} g - R\_b h\right] - \left(1 - \phi\right)^{2.5} M f' + \phi \left(K(1 - \phi)^{-1} \right) \left(\phi - \phi\right)^{-2.5} \Big) \left(\phi - \phi\right)^{-2.5} \Big|\_{\mathcal{F}} \\\ K(1 - \phi)^{2.5} \Big(1 - \phi + \phi \frac{\rho\_{\rm C\gamma T}}{\rho\_f} \Big) S' + \frac{1}{2\text{Tr}} (1 - \phi)^{2.5} \Big(1 - \phi + \phi \frac{\rho\_{\rm C\gamma T}}{\rho\_f} \Big) \Big(3f'^3 + \frac{1}{4} f'' f^2 - \frac{5}{2} f'' f' f\Big) = 0 \end{cases} \tag{10}$$

$$\frac{k\_{nf}}{k\_f}(1+Rd)\Theta'' + \frac{3}{4}(1-\Phi+\Phi\frac{\rho\_{CNT}}{\rho\_f})f\Theta' + \gamma\theta + \text{Pr} EcMf'^2 = 0\tag{11}$$

$$(S'' - \gamma \* (2S + \alpha f'') + \frac{1}{4\text{Pr}} \frac{(1 - \phi)^{2.5} \Big(1 - \phi + \Phi \frac{\rho\_{\text{CNT}}}{\rho\_f} \Big)}{(1 + 0.5K)} (5Sf' + S'f) = 0\tag{12}$$

$$\text{g}''' + \frac{3}{4} \text{Sc} f \text{g}' - \text{Sc} nf' - \text{G}\_r \text{g} = 0 \tag{13}$$

$$h'' + \frac{3}{4} L\_b f h' - P\_c(h'g' + (h + \delta)g'') = 0\tag{14}$$

$$\begin{aligned} f'(0) = 0, f(0) = V\_0, \frac{k\_{\pi f}}{k\_f} \theta'(0) = -B\_1(1 - \theta(0)), \mathbb{S}(0) = 0, h(0) = 1, \mathbb{g}(0) = 1 - n, \\\ f'(\infty) \to 0, \mathbb{S}(\infty) \to 0, \theta(\infty) \to 0, h(\infty) \to 0, \mathbb{g}(\infty) \to 0 \end{aligned} \tag{15}$$

Non-dimensional form of parameters is specified and defined as below in Abbreviations.

#### **3. Entropy Generation Modeling**

To include the irreversibility sources, below equations can be used:

$$\begin{split} S^{\prime\prime}\_{\prime\prime} &= \frac{k\_{nf}}{k\_f} \Big( 1 + \frac{16T\_{\text{ov}}^3 \sigma^\*}{3k \cdot k\_{nf}} \Big) \Big( T\_{yy} \Big) + \frac{\mu\_{nf}}{T\_{\text{ov}}} \Big( u\_y \Big)^2 + \frac{\sigma}{T\_{\text{ov}}} \beta\_0^2 u^2 + \frac{\mu\_{nf}}{T\_{\text{ov}} k} u^2 + \\ &\stackrel{\text{BD}}{\text{C}\_{\text{ov}}} \Big( \mathbb{C}\_y \Big)^2 + \frac{\mu\_{\text{D}}}{T\_{\text{ov}}} \Big( T\_y \Big) \Big( \mathbb{C}\_y \Big) \end{split} \tag{16}$$

where

$$N\_G = \frac{S^{\prime\prime}\_{\!\!\!\!\!\!\!\!\!\!\/}\_{\!\!\!\!\!\!\!\!\!\/]\_0} \tag{17}$$

 *<sup>S</sup> gen* is irreversibility optimization rate and (*S*) the characteristic irreversibility optimization rate signified by:

 $N\_G = \frac{k\_{af}}{k\_f} (1 + R)R a\_x \theta'^2 + \frac{1}{(1 - \phi)^{2.5}} \frac{Br \text{Ra}\_5}{\alpha} \left(f'^2 + k\_1 f'^2\right) + \frac{Br \text{Ra}\_x \text{M}}{\alpha} f'^2 + \lambda \left(\frac{\xi}{\alpha}\right)$  $\text{Ra}\_x$  $g'^2 + \frac{\xi}{\alpha}$  $\text{Ra}\_x \lambda \theta' g'$ 

where *Br* <sup>=</sup> <sup>μ</sup>*<sup>f</sup> Uw k <sup>f</sup>* Δ*T* Brinkman number, α = <sup>Δ</sup>*<sup>T</sup> T*<sup>∞</sup> diffusion parameter, ξ = <sup>Δ</sup>*<sup>C</sup> C*<sup>∞</sup> concentration ratio parameter, and λ = *RDC*<sup>∞</sup> *k f* temperature difference parameter, respectively.

#### **4. Engineering Quantities**
