**2. Statement**

Here, steady, laminar Darcy–Forchheimer 3D flow of viscous nanoliquid because of a rotating disk with binary chemical reaction and Arrhenius activation energy is examined. The disk at *z* = 0 rotates with constant angular velocity Ω (see Figure 1). Effects of thermophoresis and Brownian dissemination are additionally accounted for. Convection factors for warmth and mass exchange are employed. It is additionally accepted that the surface is warmed by hot liquid with concentration *Cf* and temperature *Tf* that give mass and warmth exchange coefficients *km* and *hf* respectively. Velocities are (*<sup>u</sup>*, *v*, *w*) in directions of (*r*, *ϕ*, *z*) separately. Ensuing boundary layer articulations are [22,38,44]:

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

$$
\mu \frac{\partial u}{\partial r} - \frac{v^2}{r} + w \frac{\partial u}{\partial z} = \nu \left( \frac{\partial^2 u}{\partial z^2} + \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} - \frac{u}{r^2} \right) - \frac{\nu}{k^\*} u - F u^2,\tag{2}
$$

$$
\mu \frac{\partial v}{\partial r} + \frac{\mu v}{r} + w \frac{\partial v}{\partial z} = \nu \left( \frac{\partial^2 v}{\partial z^2} + \frac{\partial^2 v}{\partial r^2} + \frac{1}{r} \frac{\partial v}{\partial r} - \frac{v}{r^2} \right) - \frac{\nu}{k^\*} v - Fv^2,\tag{3}
$$

$$
\mu \frac{\partial w}{\partial r} + w \frac{\partial w}{\partial z} = \nu \left( \frac{\partial^2 w}{\partial z^2} + \frac{\partial^2 w}{\partial r^2} + \frac{1}{r} \frac{\partial w}{\partial r} \right) - \frac{\nu}{k^\*} w - Fw^2,\tag{4}
$$

*Mathematics* **2019**, *7*, 921

$$\begin{split} u\frac{\partial T}{\partial r} + w\frac{\partial T}{\partial z} &= a^\* \left( \frac{\partial^2 T}{\partial z^2} + \frac{\partial^2 T}{\partial r^2} + \frac{1}{r}\frac{\partial T}{\partial r} \right) \\ &+ \frac{(\rho c)\_p}{(\rho c)\_f} \left( D\_B \left( \frac{\partial T}{\partial r} \frac{\partial \mathbb{C}}{\partial r} + \frac{\partial T}{\partial z} \frac{\partial \mathbb{C}}{\partial z} \right) + \frac{D\_T}{T\_\infty} \left( \left( \frac{\partial T}{\partial z} \right)^2 + \left( \frac{\partial T}{\partial r} \right)^2 \right) \right), \end{split} \tag{5}$$

$$\begin{split} u\frac{\partial\mathbb{C}}{\partial r} + w\frac{\partial\mathbb{C}}{\partial z} &= D\_{\mathbb{B}}\left(\frac{\partial^{2}\mathbb{C}}{\partial z^{2}} + \frac{\partial^{2}\mathbb{C}}{\partial r^{2}} + \frac{1}{r}\frac{\partial\mathbb{C}}{\partial r}\right) \\ &+ \frac{D\_{T}}{T\_{\infty}}\left(\frac{\partial^{2}T}{\partial z^{2}} + \frac{\partial^{2}T}{\partial r^{2}} + \frac{1}{r}\frac{\partial T}{\partial r}\right) - k\_{r}^{2}\left(\mathbb{C} - \mathbb{C}\_{\infty}\right)\left(\frac{T}{T\_{\infty}}\right)^{n} \exp\left(-\frac{E\_{d}}{\kappa T}\right). \end{split} \tag{6}$$

Subjected boundary conditions are

$$u = 0, \ v = r\Omega, \ w = 0, \ -k\frac{\partial T}{\partial z} = h\_f \left(T\_f - T\right), \ -D\_B\frac{\partial C}{\partial z} = k\_{m^\*}\left(\mathbb{C}\_f - \mathbb{C}\right) \text{ at } z = 0,\tag{7}$$

$$u \to 0, \quad v \to 0, \quad T \to T\_{\infty \star} \quad \mathbb{C} \to \mathbb{C}\_{\infty} \quad \text{as } z \to \infty. \tag{8}$$

Here *u*, *v* and *w* represent velocities in directions of *r*, *φ* and *z* while *ρf* , *ν* = *<sup>μ</sup>*/*ρf* and *μ* show density, kinematic and dynamic viscosities respectively, (*ρc*)*p* effective heat capacity of nanoparticles, *Ea* the activation energy, (*ρc*)*f* heat capacity of liquid, *k*∗ the permeability of porous space, *C* the concentration, *n* the fitted rate constant, *C*∞ the ambient concentration, *F* = *Cb*/*rk*∗1/2 the non-uniform inertia factor, *DT* the thermophoretic factor, *Cb* the drag factor, *hf* the uniform heat transfer factor, *α*<sup>∗</sup> = *<sup>k</sup>*/(*ρc*)*f* and *k* the thermal diffusivity and thermal conductivity respectively, *T* the fluid temperature, *kr* the reaction rate, *DB* the Brownian factor, *κ* the Boltzmann constant, *km*∗ the uniform mass transfer factor and *T*∞ the ambient temperature. Selecting

$$\begin{aligned} u &= r\Omega f'(\zeta), \; w = -(2\Omega v)^{1/2} f(\zeta), \; v = r\Omega g(\zeta), \\ \phi(\zeta) &= \frac{\mathbb{C} - \mathbb{C}\_{\infty}}{\mathbb{C}\_{f} - \mathbb{C}\_{\infty}}, \; \zeta = \left(\frac{2\Omega}{\mathbb{V}}\right)^{1/2} z, \; \theta(\zeta) = \frac{T - T\_{\infty}}{T\_{f} - T\_{\infty}}. \end{aligned} \tag{9}$$

Continuity expression (1) is verified while Equations (2)–(8) yield

$$2f^{\prime\prime\prime} + 2ff^{\prime\prime} - f^{\prime} + \text{g}^2 - \lambda f^{\prime} - Frf^{\prime} = 0,\tag{10}$$

$$2\,\mathrm{g}''' + 2f\,\mathrm{g}' - 2f'\,\mathrm{g} - \lambda\,\mathrm{g} - Fr\,\mathrm{g}^2 = 0,\tag{11}$$

$$\frac{1}{\text{Pr}}\boldsymbol{\theta}^{\prime\prime} + f\boldsymbol{\theta}^{\prime} + \text{N}\_{b}\boldsymbol{\theta}^{\prime}\boldsymbol{\phi}^{\prime} + \text{N}\_{l}\boldsymbol{\theta}^{\prime^2} = \boldsymbol{0},\tag{12}$$

$$\frac{1}{Sc}\phi'' + f\phi' + \frac{1}{Sc}\frac{N\_l}{N\_b}\theta'' - \sigma \left(1 + \delta\theta\right)^n \phi \exp\left(-\frac{E}{1 + \delta\theta}\right) = 0,\tag{13}$$

$$f(0) = 0, \ f'(0) = 0, \ g(0) = 1, \ \theta'(0) = -\gamma\_1 \left(1 - \theta \left(0\right)\right), \ \phi'(0) = -\gamma\_2 \left(1 - \phi \left(0\right)\right), \tag{14}$$

$$f'(\infty) \to 0, \ g(\infty) \to 0, \; \theta(\infty) \to 0, \; \phi(\infty) \to 0. \tag{15}$$

Here *Fr* stands for Forchheimer number, *γ*2 for concentration Biot number, *λ* for porosity parameter, *γ*1 for thermal Biot number, *Nt* thermophoresis parameter, Pr Prandtl number, *σ* for chemical reaction parameter, *Nb* for Brownian motion, *δ* for temperature difference parameter, *Sc* Schmidt number, and *E* for nondimensional activation energy. Nondimensional variables are defined by

$$\begin{array}{l} \lambda = \frac{\nu}{\mathbb{E} \cdot \mathbb{T} \mathsf{L}}, \ Pr = \frac{\mathsf{C}\_{\mathsf{b}}}{\mathsf{k}^{\*} \mathsf{T}^{\mathsf{T}} \mathsf{L}^{\prime}}, \ \gamma\_{1} = \frac{\mathsf{h}\_{\mathsf{f}}}{\mathsf{k}} \sqrt{\frac{\mathsf{V}}{\mathsf{2} \mathsf{T} \mathsf{L}^{\prime}}}, \ \gamma\_{2} = \frac{\mathsf{k}\_{\mathsf{u}} \mathsf{v}}{\mathsf{D}\_{\mathsf{b}}} \sqrt{\frac{\mathsf{V}}{\mathsf{2} \mathsf{T} \mathsf{L}^{\prime}}}, \ \mathrm{N}\_{\mathsf{b}} = \frac{(\mathsf{c} \mathsf{c})\_{\mathsf{p}} D\_{\mathsf{b}} (\mathsf{C}\_{\mathsf{f}} - \mathsf{C}\_{\mathsf{u}})}{(\mathsf{c} \mathsf{c})\_{\mathsf{f}} \mathsf{v}},\\ \ \mathrm{Pr} = \frac{\mathsf{v}}{\mathsf{d}^{\mathsf{u}}}, \ \mathrm{N}\_{\mathsf{l}} = \frac{(\mathsf{c} \mathsf{c})\_{\mathsf{p}} D\_{\mathsf{l}} \left(\mathsf{T}\_{\mathsf{f}} - \mathsf{T}\_{\mathsf{u}}\right)}{(\mathsf{c} \mathsf{c})\_{\mathsf{f}} \sqrt{\mathsf{T}\_{\mathsf{u}}}}, \ \mathrm{S} \mathsf{c} = \frac{\mathsf{v}}{\mathsf{D}\_{\mathsf{b}} \mathsf{v}}, \ \sigma = \frac{\mathsf{k}\_{\mathsf{f}}^{2}}{\mathsf{T}\_{\mathsf{u}}}, \ \delta = \frac{\mathsf{T}\_{\mathsf{f}} - \mathsf{T}\_{\mathsf{u}}}{\mathsf{T}\_{\mathsf{u}}}, \ E = \frac{\mathsf{E}\_{\mathsf{u}}}{\mathsf{k} \mathsf{T}\_{\mathsf{u}}}. \end{array} \tag{16}$$

The coefficients of skin-friction and Nusselt and Sherwood expressions are

$$\begin{array}{l} \text{Re}\_r^{1/2} \mathbb{C}\_f = f''(0), \ \text{Re}\_r^{1/2} \mathbb{C}\_\mathfrak{g} = \mathfrak{g}'(0),\\ \text{Re}\_r^{-1/2} \operatorname{Nu} = -\theta'(0), \ \text{Re}\_r^{-1/2} \operatorname{Sh} = -\phi'(0), \end{array} \tag{17}$$

where Re*r* = <sup>2</sup>(<sup>Ω</sup>*r*)*r*/*ν* represents local rotational Reynolds number.

#### **3. Numerical Results and Discussion**

The present section outlines the commitment of various relevant parameters including Schmidt number *Sc*, porosity parameter *λ*, thermophoresis parameter *Nt*, Prandtl number Pr, Forchheimer number *Fr*, nondimensional activation energy *E*, thermal Biot *γ*1, chemical reaction parameter *σ*, concentration Biot *γ*2 and Brownian number *Nb* on velocities *f* (*ζ*) and *g*(*ζ*), concentration *φ*(*ζ*) and temperature *θ* (*ζ*) distributions. Figure 2 portrays how porosity parameter *λ* influences the speed appropriation *f* (*ζ*). It has been discovered that the speed profile *f* (*ζ*) and its related energy layer are devalued by upgrading porosity *λ*. The presence of permeable space improves the protection from liquid stream which relates to bringing down liquid speed and its related energy layer. Figure 3 delineates the impact of Forchheimer variable Fr on *f* (*ζ*). Higher estimations of Forchheimer variable Fr establish lower speed profile *f* (*ζ*). Figure 4 shows how the speed conveyance *g*(*ζ*) is influenced by porosity parameter *λ*. Here the speed dissemination is rotted by expanding *λ*. Figure 5 delineates a variety of speed circulation *g*(*ζ*) for unmistakable Fr. By expanding Fr, a decrease showed up in speed dissemination and related layer. Figure 6 shows warm Biot *γ*1 impact on temperature *θ* (*ζ*). More grounded convection is delivered by upgrading warm Biot number *γ*1. Thus, temperature and warm layer are raised by expanding warm Biot number *γ*1. Figure 7 presents a variety in temperature field *θ* (*ζ*) for Pr. Here, temperature is rotted for bigger Pr. The proportion of force diffusivity to warm diffusivity is termed as the Prandtl number. Higher estimations of Pr depict more fragile warm diffusivity, which compares to diminishing in the warm layer. Figure 8 is shown to investigate *Nt* impact on temperature field *θ* (*ζ*). Bigger thermophoresis parameter *Nt* establishes a higher temperature field and progressively warm layer thickness. The purpose of such contention is that augmentation in *Nt* yields high grounded thermophoresis power which further permits motion of the nanoparticles in liquid zone. Far from surface in this way shapes a more grounded temperature dispersion *θ* (*ζ*) and progressively warm layer. The effect of *Nb* on temperature profile *θ* (*ζ*) is depicted in Figure 9. From a physical perspective, an unpredictable movement of nanoparticles increments by improving Brownian movement parameter *Nb* causes a crash of particle. As a result, the active vitality is changed into warmth vitality which causes upgrade in *θ* (*ζ*) and associated warm layer. Figure 10 shows how concentration *φ*(*ζ*) is influenced by concentration Biot number *γ*2. Concentration is upgraded for higher estimations of *γ*2. From Figure 11 we can see that bigger Sc rots concentration *φ*(*ζ*). Schmidt number Sc is conversely relative to Brownian diffusivity. Higher Sc yields a more fragile Brownian diffusivity. Such Brownian diffusivity prompts low concentration *φ*(*ζ*). Figure 12 demonstrates how the thermophoresis parameter *Nt* influences the concentration *φ*(*ζ*). By improving thermophoresis parameter *Nt*, concentration *φ*(*ζ*) and related concentration layers are upgraded. Figure 13 depicts the Brownian movement *Nb* and minor departure from concentration *φ*(*ζ*). It can be seen that a more fragile concentration *φ*(*ζ*) is produced by using higher *Nb*. Figure 14 explains the impact of nondimensional initiation vitality E on concentration *φ*(*ζ*). An improvement in E rots altered Arrhenius work *TT*∞ *n* exp − *Ea κT* . Such inevitably builds up the generative synthetic response because of which concentration *φ*(*ζ*) upgrades. Figure 15 shows that an improvement in *σ* shows a rot in concentration *φ*(*ζ*) and its related layer. Highlights of *Nt* and *Nb* on *Nu*(Re*r*)−1/2 are revealed through Figures 16 and 17 respectively. True to form, *Nu*(Re*r*)−1/2 reduces for *Nt* and *Nb*. Effects of *Nt* and *Nb* on *Sh*(Re*r*)−1/2 have been portrayed in Figures 18 and 19 respectively. Here *Sh*(Re*r*)−1/2 is an expanding capacity of *Nt*, while the inverse pattern is seen for *Nb*. Table 1 is developed to validate the present results with the previously published results in a limiting case. Here, we demonstrate that the present numerical solution has good agreemen<sup>t</sup> with the previous solution by Naqvi et al. [48] in a limiting case.

**Figure 2.** Curves of *f* (*ζ*) for *λ*.

**Figure 4.** Curves of *g*(*ζ*) for *λ*.

**Figure 6.** Curves of *θ*(*ζ*) for *γ*1.

**Figure 8.** Curves of *θ*(*ζ*) for *Nt*.

**Figure 10.** Curves of *φ*(*ζ*) for *γ*2.

**Figure 12.** Curves of *φ*(*ζ*) for *Nt*.

**Figure 14.** Curves of *φ*(*ζ*) for *E*.

**Figure 16.** Curves of *Nu*(Re*r*)−1/2 for *Nt*.

**Figure 18.** Curves of *Sh*(Re*r*)−1/2 for *Nt*.

**Figure 19.** Curves of *Sh*(Re*r*)−1/2 for *Nb*.

**Table 1.** Comparative values of *f* (0) and *g*(0) for value of *Fr* when *λ* = 0.2.

