The Casson Dusty Nanofluid: Significance of Darcy–Forchheimer Law, Magnetic Field, and Non-Fourier Heat Flux Model Subject to Stretch Surface

Abstract

This work aims to offer a mathematical model for two-phase flow that investigates the interaction of Casson nanofluid and dust particles across a stretching surface. MHD Darcy–Forchheimer porous medium and Fourier’s law through Cattaneo–Christove thermal flux are also considered. The governing equations for the two phases model are partial differential equations later transmuted into ordinary ones via similarity transforms. The Runge–Kutta method with the shooting tool is utilized numerically to solve the boundary layer equations computed in MATLAB to obtain numerical results for various pertinent parameters. The numerical outcomes of momentum, temperature, and concentration distribution are visible for both phases. The results of the skin friction, heat transfer coefficients, and the Sherwood number are also visible in the graphs. Furthermore, by comparing the current findings to the existing literature, the validity of the results is confirmed and found to be in good agreement. The fluid velocity is reduced against increasing strength of Casson fluid parameter, enhanced the fluid phase and dust phase fluid temperature. The temperature declines against the growing values of the relaxation time parameter in both phases. Dusty fluids are used in various engineering and manufacturing sectors, including petroleum transportation, car smoke emissions, power plant pipes, and caustic granules in mining.

1. Introduction

Due to their two-phase structure, dusty fluid model flows have special researchers’ interest in recent studies. This effect occurs in fluid (gas or liquid) flows with the dispersion of solid particles. For illustration, the chemical reaction by which droplets are produced with the coalescence of relatively dust particles and the velocity of dusty air in fluidization difficulties. The fundamental antecedent for planetary systems is created by combining gas and dust particles called cosmic dust. Comet 238 produces tails due to dust particles, and ionized gas being ejected from the comet’s core. Many investigators used the dusty phase model to experiment with different flow configurations and boundary constrictions, managing to keep their intrigue in two-phase flows alive. As a result, the solutions they provide are approximate and numerical approaches. Saffman [1] is the one who started the research on dusty fluid laminar flow. The movement of dusty gas in the boundary layer sector was studied by Chakrabarti [2]. Datta and Mishra [3] explored the movement of a dusty liquid across a semi-infinite surface. Wei et al. [4] analyzed the significance of dust particles across a stretching surface.

Newton’s viscosity law states that the viscosity should stay constant regardless of stress in a non-Newtonian fluid. Viscosity changes when force is applied to non-Newtonian fluids; either it becomes less viscous or more. Many salt solutions and molten polymers, shampoo, melted butter, blood, toothpaste, and paints are non-Newtonian liquids. Many researchers work on several kinds of non-Newtonian fluids. Anwar et al. [5] deliberated the induced magnetohydrodynamic stream of the non-Newtonian nano-liquid past across a nonlinear stretchable sheet under the impact of thermic conductivity. Kumar et al. [6] analyzed the flow characteristics of non-Newtonian fluid passing across a rotating elliptical cylinder. Khashi et al. [7] analyzed viscous dissipation influence and Magnetohydrodynamic on radiating heat transportation of the non-Newtonian fluid flowing through a nonlinear extending sheet. Chaudhuri et al. [8] deliberated the behavior of MHD in a third-grade non-Newtonian liquid moving over vertical oriented similar plates. Khan et al. [9] presented the bioconvective stream of non-Newtonian nanoparticles with a partial slip under the impacts of unsteady viscosity and Arrhenius activation energy. Across thin peristaltic walls, Riaz et al. [10] studied solid particles’ influences on the fluid phase stream of non-newtonian fluid. By imposing slip conditions on the sheet’s surface, Ijaz et al. [11] presented a numerical study of the effect of non-linear mixed convection of non-newtonian liquid with entropy generation with thermophoretic effect and joule heating, mass and heat transference in various non-Newtonian fluids towards a stretched surface analyzed by Reddy et al. [12]. Fatunmbi et al. [13] analyzed the nonlinear mixed convection transport of hydromagnetic Casson nanofluid over a nonlinear stretching sheet. Fatunmbi et al. [14] deliberated the boundary layer flow and heat transfer analysis in an electromagnetohydrodynamic dissipative micropolar-Casson fluid over a two-dimensional stretching sheet in the porous medium.

Magnetohydrodynamic (MHD) (hydromagnetics or magnetofluid dynamics) is the study of electrical conducting fluid’s behavior and magnetic properties. Liquid metals, plasmas, electrolytes, or saltwater are examples of such magneto liquids. In 1970, Hannes Alfven was awarded the Nobel Prize in Physics for their groundbreaking investigation of MHD. Magnetohydrodynamics has many uses in physics and astronomy, space plasma physics, solar physics, blood pump machines, cancer tumor therapy, and laboratory plasma studies. Sandhya et al. [15] studied the MHD flow under the impacts of mass and heat transference over an inclined plane in the existence of chemical reactions. Kumar et al. [16] studied the time-dependent Eyring–Powell MHD nanofluid past across an angled elastic surface by the Joule heating. Hussain et al. [17] analyzed the thermal transference of nanofluid and MHD stream across a vertical permeable surface. Rehman et al. [18] computed the impacts of radiation on micropolar MHD nanofluid stream over a shrinking surface. MHD nanofluid stream moving across a nonlinear elastic surface associated with convective conditions, examined by Hayat et al. [19]. Kardri et al. [20] discussed the MHD nanofluid stream across a nonlinear shrinking or stretchable cylinder. The general solution of hydromagnetic free convection flow over a moving infinite vertical plate with Newtonian heating, mass diffusion and chemical reaction in the presence of a heat source is completely solved by Fetecau et al. [21].

Khan et al. [22] studied the rotational flow of hybrid MHD nanofluid with entropy generation existence. Shoaib et al. [23] deliberated the numerical solutions of hybrid MHD nanofluid for rotatory motion along with thermic radiation past over an elastic sheet. Hussain et al. [24] analyzed the transfer of heat and 3D time-dependent hydromagnetic flow with the suspension of both nanoparticles and microorganisms. Krishna et al. [25] analyzed the convection behavior of the time-dependent MHD rotating flow under the impacts of thermic radiation past over a moving filled surface numerically. Utilizing the fractional Burgers model and by considering the heat and mass transference, the impacts of MHD, radiation, and chemical reaction are discussed by Jiang et al. [26]. Hossain et al. [27] analyzed to influence on MHD mixed convective carbon nanotubes nano-fluid utilizing FEM technique numerically. Ghasemi et al. [28] deliberated the impacts of solar radiation on 2D MHD flow of nano-liquid across an extending sheet using the DQM technique. Mandal et al. [29] signified the nanofluid flow in a porous media in the presence of the magnetic field.

A fluid that consists of nanometer-sized small particles forms a colloidal solution of nanoparticles due to suspension in a base fluid. Carbides, metals oxides, and carbon nanotubes are frequently utilized as nanoparticles in nanofluids, with ethylene glycol, oil, and water as base fluids. Choi et al. [30] originated the concept of “Nanofluids” as the small amount of suspension of nanoparticles overdraws the thermal conductivity in a much more reasonable way. Khan et al. [31] analyzed the resistive force impact on Williamson nanofluid flowing across a nonlinear elastic sheet. Saeed et al. [32] studied the convective flow over an angled elastic surface of the thin film nanofluid. Yahya et al. [33] investigated the thermic properties for the hybrid Williamson nanofluid flowing across a stretching sheet with engine oil. Gopal et al. [34] discussed the magnetic and electric strength as well as viscous dissipation for chemical reactions on MHD nanofluid towards a porous extending sheet. Gouran et al. [35] analyzed thermal radiation on magnetohydrodynamics nanofluid stream utilizing two effective computational algorithms. By utilizing the ISPH method, Rashidi et al. [36] studied an updated comprehensive study of the thermophysical properties of hybrid nanofluids and the proposed models. Aly et al. [37] deliberated the impact of magnetic field on nanofluid in the presence of solid particles. Wang et al. [38] analyzed the heat and mass transfer of MHD bio nanofluid across a permeable medium. Sheikholeslami et al. [39] deliberated the recent progress on flat plate solar collectors and photovoltaic systems in the presence of nanofluid.

By studying the all above literature survey, however, we concluded that the research work on the significance of magnetic field, Darcy–Forchheimer law, Bronian motion, and non-Fourier heat flux model on the dynamics of Casson dusty fluid has not been done yet. Motivated by the aforementioned wide scope of non-Newtonian fluid and nanofluid applications, therefore, we decided to evaluate the current elaborated fluid problem. Recently, study on non-Newtonian fluids has fascinated young researchers to extend due to large scale applications in different industries. By using similarity modifications, the relevant nonlinear PDEs are converted into a system of ODEs. The graphical outcomes of velocity, temperature, concentration of nanoparticles, skin friction factor, and Nusselt number are conducted for different inputs of physical parameters. This numerical investigation has many industrials and engineering applications such as including petroleum transportation, power plants, and heat exchangers, etc.

2. Mathematical Formulation

Let us a steady MHD Darcy–Forchhiemer porous medium flow of dusty Casson fluid across a stretching surface. The flow is expected to be restricted to the $y>0$, and it is created by two equivalent and inverse forces acting along the x-axis and the y-axis, typical to the flow. The sheet is extended along the x-axis at $u_x$ velocity, while a magnetic field of strength $B_0$ is applied along y-axis as shown in Figure 1. The volume fraction of dust particles is ignored, and the number density is assumed to remain constant. The only way that the fluid and dust particles’ motions are linked is by drag and thermal transfer. Stoke’s linear drag theory is used to model the drag force. Other communication forces, for example, the virtual force, twist lift force, and shear lift force will be overlooked contrasted with the drag force. Dust particles have a spherical physique, a consistent size, and a constant density. The mathematical model for the flow of the dusty phase, coupled with the constraints, is as follows, based on these physical assumptions and boundary layer approximations [40,41].

(1)$\begin{array}{c}\frac{\partial \widehat{u}}{\partial x}+\frac{\partial \widehat{v}}{\partial y}=0,\end{array}$

(2)$\begin{array}{c}\widehat{u}\frac{\partial \widehat{u}}{\partial x}+\widehat{v}\frac{\partial \widehat{u}}{\partial y}=\nu \left(1+\frac{1}{\Gamma }\right)\frac{{\partial }^2\widehat{u}}{\partial y^2}-\frac{\sigma B_0^2}{\rho }\widehat{u}-\frac{\nu }{k^{*}}\widehat{u}-F\widehat{u^2}+\frac{KN}{\rho }(\widehat{u_p}-\widehat{u}),\\ \left(\widehat{u}\frac{\partial T}{\partial x}+\widehat{v}\frac{\partial T}{\partial y}+{\lambda }_1\left[\widehat{u}\frac{\partial \widehat{u}}{\partial x}.\frac{\partial \widehat{T}}{\partial x}+\widehat{v}\frac{\partial \widehat{v}}{\partial y}.\frac{\partial \widehat{T}}{\partial y}+\widehat{u}\frac{\partial \widehat{v}}{\partial x}.\frac{\partial \widehat{T}}{\partial y}+\widehat{v}\frac{\partial \widehat{u}}{\partial y}.\frac{\partial \widehat{T}}{\partial x}+2\widehat{u}\widehat{v}\frac{{\partial }^{2}T}{\partial x\partial y}+{\widehat{u}}^2\frac{{\partial }^{2}T}{\partial x^2}+{\widehat{v}}^2\frac{{\partial }^{2}T}{\partial y^2}\right]\right)\end{array}$

(3)$\begin{array}{c}=k\frac{{\partial }^{2}T}{\partial y^2}+\tau \left[D_B\frac{\partial C}{\partial y}.\frac{\partial T}{\partial y}+\frac{D_T}{T_{\infty }}{\left(\frac{\partial T}{\partial y}\right)}^2\right]+\frac{{\rho }_p{C}_p}{{\rho }_{cp}{\tau }_T}(T_p-T),\end{array}$

(4)$\begin{array}{c}\widehat{u}\frac{\partial C}{\partial x}+\widehat{v}\frac{\partial C}{\partial y}=D_B\frac{{\partial }^{2}C}{\partial y^2}+\frac{D_T}{T_{\infty }}\frac{{\partial }^{2}T}{\partial y^2}+\frac{{\rho }_p}{\rho {\tau }_C}(C_p-C).\end{array}$

For dusty particle flow

(5)$\begin{array}{c}\frac{\partial \widehat{u_p}}{\partial x}+\frac{\partial \widehat{v_p}}{\partial y}=0,\end{array}$

(6)$\begin{array}{c}\widehat{u_p}\frac{\partial \widehat{u_p}}{\partial x}+\widehat{v_p}\frac{\partial \widehat{u_p}}{\partial y}=\frac{KN}{\rho }(\widehat{u}-\widehat{u_p}),\end{array}$

(7)$\begin{array}{c}\widehat{u}\frac{\partial T_p}{\partial x}+\widehat{v}\frac{\partial T_p}{\partial y}=\frac{c_p}{c_m{\tau }_T}(T-T_p),\end{array}$

(8)$\begin{array}{c}\widehat{u}\frac{\partial C_p}{\partial x}+\widehat{v}\frac{\partial C_p}{\partial y}=\frac{mN}{\rho {\tau }_c}(C-C_p).\end{array}$

where $\nu$ is kinematic viscosity, $\rho$ demonstrates the fluid density, $k^{*}$ is the porous medium permeability, $F=\frac{C_b}{\sqrt{k^{*}}}$ is the coefficient of inertia of porous material, and $-\frac{\nu }{k^{*}}\widehat{u}-F\widehat{u^2}$ is due to non-Darcian porous medium, $K=6\pi \mu r$ is the stoke’s drag constant, N is the number density of the dust particle, ${\lambda }_1$ is relaxation time for heat flux, k is thermal conductivity, $D_T$ is thermophoresis diffusion, $D_B$ is Brownian diffusion, ${\tau }_T$ is thermal equilibrium time, $c_p$ is fluid specific heat capacity, $c_m$ is dust particle specific heat, and m is dust particle mass.

The dimensional initial and boundary constraints are as follows:

(9)$\begin{array}{c}\left.\begin{array}{c}\hfill \widehat{u}=u_w=cx,\widehat{v}=0,T=T_w,C=C_w,aty=0,\\ \hfill \widehat{u_p}=u_w=ax,\widehat{v_p}=v,T_p=T_w,C_p=C_w,aty=0,\\ \hfill \widehat{u}\to 0,\widehat{v}\to 0,\widehat{T}\to T_{\infty },\widehat{C}\to C_{\infty },aty\to \infty ,\\ \hfill \widehat{u_p}\to 0,\widehat{v_p}\to 0,\widehat{T_p}\to T_{\infty },\widehat{C_p}\to C_{\infty },aty\to \infty .\end{array}\right\}\end{array}$

The following transformations are introduced for converting the leading differential Equations [40]:

(10)$\begin{array}{c}\left.\begin{array}{c}\hfill \widehat{u}=cx{F}^{\prime }\left(\eta \right),\widehat{v}=-\sqrt{c\nu }F\left(\eta \right),\eta ={\left(\frac{a}{\nu }\right)}^{\frac{1}{2}}y,\\ \hfill \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},\varphi =\frac{C-C_{\infty }}{C_w-C_{\infty }},\\ \hfill \widehat{u_p}=ax{F}_p^{\prime }\left(\eta \right),\widehat{v_p}=-\sqrt{a\nu }{F}_p\left(\eta \right),\eta =\sqrt{\frac{a}{\nu }}y,\\ \hfill {\theta }_p\left(\eta \right)=\frac{T_p-T_{\infty }}{T_w-T_{\infty }},{\varphi }_p=\frac{C_p-C_{\infty }}{C_w-C_{\infty }},\end{array}\right\}\end{array}$

where $\eta$ represents the similarity variable. $\widehat{u}$ (x-axis), and $\widehat{v}$ (y-axis) are the velocity components. The temperature of fluid is represented by T and C stands for fluid concentration. $T_w$, and $C_w$ is the temperature and concentration at the surface. $T_{\infty }$, is ambient temperature, and $C_{\infty }$ is fluid ambient concentration. Similarly $u_p$, $v_p$ are components of velocity of dust particles. $T_p$ and $C_p$ are the temperature, and concentration of dust particles.

After employing similarity transformation, the continuity Equations (1) and (5) are satisfied. However, Equations (2)–(4) and (6)–(8) are transmuted into the dimensionless form:

(11)$\begin{array}{c}\left(1+\frac{1}{\Gamma }\right)F^{‴}+F{F}^{″}-(M+K_p)F^{\prime }-(1+Fr)F^{\prime 2}+{\Gamma }_v.{\beta }_v(F_p^{\prime }-F^{\prime })=0,\end{array}$

(12)$\begin{array}{c}\frac{1}{Pr}{\theta }^{″}+F{\theta }^{\prime }-{\lambda }_t(F{F}^{\prime }{\theta }^{\prime }+F^2{\theta }^{″})+Nb{\theta }^{\prime }{\varphi }^{\prime }+Nt{\theta }^{\prime 2}+{\gamma }_t.{\beta }_t({\theta }_p-\theta ),\end{array}$

(13)$\begin{array}{c}{\varphi }^{″}+ScF{\varphi }^{\prime }+\frac{Nt}{Nb}{\theta }^{″}+{\gamma }_c.{\beta }_c({\varphi }_p-\varphi )=0.\end{array}$

For dusty phase

(14)$\begin{array}{c}{F}_p{F}_p^{″}-F_p^{\prime 2}+{\beta }_v(F_p^{\prime }-F^{\prime })=0,\end{array}$

(15)$\begin{array}{c}{F}_p{\theta }_p^{{}^{\prime }}+{\gamma }_t.{\beta }_t(\theta -{\theta }_p)=0,\end{array}$

(16)$\begin{array}{c}{F}_p{\varphi }_p^{{}^{\prime }}+{\gamma }_c.{\beta }_c(\varphi -{\varphi }_p)=0,\end{array}$

with the following dimensionless physical constraints:

(17)$\begin{array}{c}\left.\begin{array}{c}\hfill F\left(\eta \right)=0,F^{\prime }\left(\eta \right)=1,\theta \left(\eta \right)=1,\varphi \left(\eta \right)=1,at\eta =0,\\ \hfill F_p\left(\eta \right)=F\left(\eta \right),F_p^{{}^{\prime }}\left(\eta \right)=0,{\theta }_p\left(\eta \right)=1,{\varphi }_p\left(\eta \right)=1,at\eta =0,\\ \hfill F^{\prime }\left(\eta \right)\to 0,\theta \left(\eta \right)\to 0,\varphi \left(\eta \right)\to 0,at\eta \to \infty .\\ \hfill F_p\left(\eta \right)\to 0,{\theta }_p\left(\eta \right)\to 0,{\varphi }_p\left(\eta \right)\to 0,at\eta \to \infty .\end{array}\right\}\end{array}$

where $\Gamma$ is the Casson Parameter, $M=\frac{\sigma B_0^2}{\rho c}$ is the magnetic parameter, $Fr=\frac{C_b}{\sqrt{k^{*}}}$ is the inertia coefficient, ${\Gamma }_v=\frac{Nm}{\rho }$ is the mass concentration of dusty granules, ${\beta }_v=\frac{1}{{\tau }_{v}c}$ is the interaction parameter for the velocity of a fluid particle, ${\tau }_v=\frac{m}{K}$ is the dust particle relaxation time, ${\rho }_p=mN$ is the dust particle density, ${\gamma }_c=\frac{c_p}{c_m}$ is the specific heat ratio, $Nb=\frac{\tau D_B(C_w-C_{\infty })}{\nu }$ is Brownian motion, $Pr=\frac{{\mu }_{Cp}}{k}$ is the Prandtl number, ${\lambda }_t={\lambda }_{1}c$ is the relaxation time parameter, $Nt=\frac{\tau D_T(T_w-T_{\infty })}{\nu T_{\infty }}$ is the thermophoretic parameter, ${\beta }_t=\frac{1}{c{\tau }_T}$ is fluid interaction temperature parameter and $Sc=\frac{\nu }{D_B}$ is the Schmidt number.

3. Physical Quantities

The influence of the significant engineering parameters may be adequately investigated in this physical problem by calculating the localized magnitude of drag forces and the rate of thermal transport at the slender sheet. In terms of $C_{fx}$(skin friction), $N{u}_x$ (Nusselt number) and $S{h}_x$ (Sherwood number) are as follows:

(18)$\begin{array}{c}{C}_{fx}=\frac{-{\tau }_w}{\rho u_w^2},\end{array}$

(19)$\begin{array}{c}N{u}_x=\frac{x{q}_T}{k(T_w-T_{\infty })},\end{array}$

(20)$\begin{array}{c}S{h}_x=\frac{x{q}_w}{D_B(\widehat{C}-C_{\infty })},\end{array}$

where $q_T$, and $q_w$ are the heat flux, and mass flux, respectively,

(21)$\begin{array}{c}\hfill {\tau }_w=\mu (1+{\Gamma }^{-1}){\left(\frac{\partial \widehat{u}}{\partial y}\right)}_{y=0}\end{array}$

(22)$\begin{array}{c}\hfill q_T=-k{\left(\frac{\partial T}{\partial y}\right)}_{y=0},\end{array}$

(23)$\begin{array}{c}\hfill q_w=-D_B{\left(\frac{\partial C}{\partial y}\right)}_{y=0}.\end{array}$

implementing the above equations, the following terms are obtained,

(24)$\begin{array}{c}R{e}_x^{\frac{1}{2}}{C}_{fx}=-(1+{\Gamma }^{-1})f^{″}\left(0\right),\end{array}$

(25)$\begin{array}{c}R{e}_x^{\frac{-1}{2}}N{u}_x=(-{\theta }^{\prime }\left(0\right)),\end{array}$

(26)$\begin{array}{c}R{e}_x^{\frac{-1}{2}}S{h}_x=(-{\varphi }^{\prime }\left(0\right)).\end{array}$

The local Reynolds number $R{e}_x=\frac{u_w^2}{c\nu }$.

4. Solution Procedure

A system of ordinary differential Eqs represents the flow model. Equations (11)–(16) subjected to boundary conditions (17) are solved using the Runge–Kutta approach along shooting technique using MATLAB software because this scheme is stable, easy to implement and self-starting. The numerical results of flow and heat transfer for the two-phase flow of dusty Casson fluid are determined in terms of velocity and temperature profiles with the influences of several physical parameters. The Runge–Kutta technique has been used to transform the system of ODEs (11)–(16) into first-order ODEs for a solution as follows [42,43]:

$f_1^{\prime }=f_2,$

$f_2^{\prime }=f_3,$

$f_3^{\prime }=\frac{(-1)}{(1+{\Gamma }^{-1})}[f_1{f}_3-(M+K_p)f_2-(1+Fr)f_2^2+{\Gamma }_v{\beta }_v(f_9-f_2)],$

$f_4^{\prime }=f_5,$

$f_5^{\prime }=\frac{(-1)}{(\frac{1}{Pr}+\lambda f_1^2)}[f_1{f}_5+{\Lambda }_t{f}_1{f}_2{f}_5+Nb{f}_5{f}_7+Nt{f}_5^2+{\gamma }_t{\beta }_t(f_{10}-f_4)],$

$f_6^{\prime }=f_7,$

$f_7^{\prime }=(-1)[Sc{f}_1{f}_7+\frac{Nt}{Nb}{f}_5^{\prime }+{\gamma }_c{\beta }_c(f_{12}-f_6)],$

$f_8^{\prime }=f_9$

$f_9^{\prime }=\frac{(-1)}{f_8}[{\beta }_v(f_9-f_2)-f_9^2],$

$f_{10}^{{}^{\prime }}=\frac{(-1)}{f_8}\left[{\gamma }_t{\beta }_t(f_4-f_{10})\right],$

$f_{11}^{{}^{\prime }}=\frac{(-1)}{f_8}\left[{\gamma }_c{\beta }_c(f_6-f_{11})\right],$ and the boundary conditions (17) are enumerated as

$f_1=0,f_2=1,f_4=0,f_6=1f_3,at\eta =0,$

$f_8=f_1,f_9=0,f_{10}=1,f_{11}=1,at\eta =0,$

$f_2\to 0,f_5\to 0,f_6\to 0,as\eta \to \infty .$

$f_8\to 0,f_{10}\to 0,f_{11}\to 0,as\eta \to \infty .$

5. Results and Discussion

This research aims to begin to investigate Casson fluid boundary layer flow and melting thermal transport across a stretched surface with fluid-particle suspension, melting effect, and Cattaneo–Christov heat flux. The Runge–Kutta approach is used to solve the governed highly nonlinear expressions with associated conditions and is coded in MATLAB. The numerical code validation is obtained by comparing the numerical outcomes of the current and past results of skin friction (Gireesha [44], Jalil [40]). Our findings are obtained to be in good agreement with them (see Table 1). The impact of parameters on momentum, thermal, and concentration layers in dual fluid and dusty phases. We take values of parameters as $$\begin{gathered} \Gamma =0.5,M=0.5,Sc=2.0,Pr=2.0,K_p=0.3,Nb=Nt=0.2,{\lambda }_t=0.1,{\gamma }_v={\beta }_v=1.0,{\gamma }_t={\beta }_t=0.2, \\ {\gamma }_c={\beta }_c=1.0. \end{gathered}$$

The effect of magnetic parameter M on momentum boundary layer $F^{\prime }\left(\eta \right)$, dust phase velocity $F_p\left(\eta \right)$ is portrayed in Figure 2a,b, it is noticed that improving the values of M lowers the velocity profiles in both fluid $F^{\prime }\left(\eta \right)$, and dusty phase $F_p\left(\eta \right)$. The enhanced M causes a greater opposing (Lorentz force) force on the stream, slowing it down. On the other hand, the Casson parameter $\Gamma$ also diminishes for fluid and dust phase velocity. Improvement in $\Gamma$ leads to enhanced dynamic viscosity, induces deficiency in fluid flow, and diminishes velocity for fluid and dust. The drawing for fluid temperature $\theta \left(\eta \right)$, fluid concentration $\varphi \left(\eta \right)$ and dust phase temperature ${\theta }_p\left(\eta \right)$ and dust concentration ${\varphi }_p\left(\eta \right)$ as influenced by M are portraited in Figure 3a,b and Figure 4a,b. Increasing the value of M rises the curve for fluid and dust phase temperature and concentration profile. The increase in M causes the stronger the resistive force. Meanwhile, the fluid loss of kinetic energy is offset by a gain in heat energy, and then the temperature and concentration are enhanced. Casson parameter $\Gamma$ also enhanced fluid temperature, concentration, and dust phase. Improvement in $\Gamma$ enhanced temperature and fluid and dust concentration.

Figure 5a,b are portraits of the impacts of $K_p$ for both fluid and dusty particles. From these, it is observed that by increasing the values of $K_p$ depreciation in velocity and momentum boundary layer thickness, caused due to $K_p$$(K_p=\frac{\nu }{K^{*}a})$ is converse to the permeability $k^{*}$ of the porous medium. The higher inputs $K_p$ means lower $k^{*}$ and hence higher resistance is offered to the flow to the velocity and rise in temperature and concentration for both fluid and dusty phase portraits in Figure 6a,b and Figure 7a,b. Similarly, local inertia parameter $Fr$ on velocity for fluid and dust diminished for growing values of $Fr$, caused due to inertia coefficient is a direct relation to medium porosity and drag force. So when $Cb$ enhances, the liquid resistive force is stronger, velocity is lower for the fluid and dust phase and rise in temperature and concentration for fluid and dust particles. The influence of temperature distribution of Casson nanofluid and dust phase $\theta$ and ${\theta }_p$ depicted in Figure 8a,b. The larger values of $Nb$, the faster the nanoparticles migrate, increasing heat diffusion and then rising in temperature for both phases. The quick motion also impairs focus. Thermophoretic impact happens when particles go from hot to cold. The dependence of the relaxation time parameter $\Gamma$ on the temperature distribution for both Casson fluid and dusty phases is portrayed in Figure 9a,b. As can be seen in this diagram, as the relaxation time parameter increases, the temperature distribution $(\theta ,{\theta }_p)$ and thickness of the thermal boundary layer decline. Physically, it indicates that the relaxation time parameter’s progressive nature necessitates a longer time for thermal transport from densely packed fluid particles to low-energy fluid particles. As a result, both the fluid and dust phases decrease the temperature distribution.

Figure 9a,b are also depicted to understand the effect of specific heat ratio parameter ${\gamma }_t$ on the temperature profile. It is noticed that the thermal boundary layer thickness and temperature profile for both fluid and dust phases diminished by enhancing specific heat ratio. Brownian motion $Nb$ affects fluid concentration distribution and dust phase elucidated in Figure 10a,b. By growing $Nb$, the fluid concentration and dust phase are lower. The thermal energy is contributed by nanoparticle dispersion and random motion, and festally distributed particles have a lower fluid concentration and dust phases. The thermophoresis effect represents the movement of nanoparticles from an upper to a lower temperature. As a result, as shown in Figure 10a,b, concentration $\varphi ,{\varphi }_p$ rises in an indirect relationship with $Nt$. The variations in Nusselt $N{u}_x$ and Sherwood $S{h}_x$ are shown in Figure 11a,b. The parameters $Nt$, $Nb$, and M offer various values. With modest increases in similar values of $Nb$ and $Nt$, the amount Nusselt decreases noticeably, although Sheroowd achieves larger values in this scenario. The effect of skin friction $C_{fx}$ and Nusselt number $N{u}_x$ are visible in Figure 12a,b. The parameters M, $K_p$ and $\Gamma$ give different values. Increase in values of $K_P$ sand $\Gamma$ the skin friction and Nusselt number decrease.

6. Conclusions

In the majority of the industrial, science and engineering processes convection is nonlinear. It offers a wide range of uses, including large surface area, electron mobility, and stability. It possesses exceptional electrical, optical, material, physical, and chemical characteristics. In this research, we have examined the role of modified Fourier law in the flow of an MHD Casson nanofluid flow with dust particles over a stretched surface. The combined effect of Brownian motion and thermophoresis in nanofluid modeling is retained. Firstly, the partial differential expressions are transmuted into ordinary differential equations by implementing a similarity approach. Afterward, the solutions of attained equations are solved by implementing the Runge–Kutta–Fehlberg–45 method. A comparison is also made with a published paper to ascertain the validity of the presented model. The key observations of this study are listed below:

• The impacts of porosity, Forchheimer and magnetic parameters on fluid flow show decreasing behavior for both fluid and dust phases.

• The temperature profile increases as porosity, Forchheimer and magnetic parameters are augmented, but the reverse behavior is viewed when the values of ${\lambda }_t$ increase for both dust and fluid phase.

• Brownian diffusion enhanced the temperature profile and reduced the concentration profile.

• The current results were compared with the available results in the existing literature for the special case, and there was good agreement between them showing the validation of the present study.

• The growing strength of magnetic field (M), porosity ($K_p$), and Casson fluid parameter ($\Gamma$) caused to decline in the skin friction and Nusselt number.

This analysis may be extended for the hybrid-based dusty nano liquid, Williamson dusty fluid, Oldroyd-B dusty fluid and other non-Newtonian dusty fluids.

Author Contributions

Data curation, N.A.S.; Formal analysis, J.D.C.; Investigation, L.A.; Methodology, N.F.; Software, B.A.; Validation, M.I. and J.D.C.; Writing—original draft, S.U.R.; Writing—review & editing, N.A.S. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not Applicable.

Informed Consent Statement

Not Applicable.

Data Availability Statement

Not Applicable.

Acknowledgments

This work was supported by the Technology Innovation Program (20018869, Development of Waste Heat and Waste Cold Recovery Bus Air-conditioning System to Reduce Heating and Cooling Load by $10\%$) funded By the Ministry of Trade, Industry & Energy(MOTIE, Korea).

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

$\widehat{u}$	Fluid velocity components along the x-axis	$\widehat{v}$	Fluid velocity components along the y-axis
${\rho }_f$	Density of fluid	${\rho }_p$	Density of dust particles
$\nu$	Kinematic viscosity of fluid	$\Gamma$	Casson parameter
$\sigma$	Electrical conductivity	$B_0$	Magnetic field strength
$k^{*}$	Permeability of porous medium	F	Co-efficient of inertia of porous material
K	Stoke’s drag constant	N	Dust particle number constant
m	Mass of dust particle	${\tau }_T$	Thermal equilibrium time
${\lambda }_1$	Relaxation time for heat flux	$c_m$	Specific heat of dust particle
T	non-dimensional temperature	$T_w$	Temperature at surface
C	non-dimensional nanoparticles concentration	$C_w$	Concentration at surface
$C_p$	Concentration of dust particles	$\mu$	Dynamic viscosity
$T_{\infty }$	temperature away from the surface, K	$c_p$	Specific heat capacity of the fluid
$C_{\infty }$	concentration away from the surface	$\Omega$	angular velocity
$T_p$	Temperature of the dust particle	$u_w$	velocity of stretching sheet
$C{f}_x$	skin friction at x-direction	$({\widehat{u}}_p,{\widehat{v}}_p)$	Velocity components of dust particles
$N{u}_x$	Nusselt number	$S{h}_x$	Sherwood number
k	Thermal conductivity	$Nb$	Brownian motion parameter
$Nt$	thermophoresis parameter	$Cp$	Specific heat and constant pressure

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Figures and Table

Figure 1. Flow diagram.

Figure 2. Variation of M to influence the fluid and dust velocity profile.

Figure 3. Variation of M to influence the fluid and dust temperature profile.

Figure 4. Variation of M to influence the fluid and dust concentration profile.

Figure 5. Variation of [Forumla omitted. See PDF.] to influence the fluid and dust velocity profile.

Figure 6. Variation of [Forumla omitted. See PDF.] to influence the fluid and dust temperature profile.

Figure 7. Variation of [Forumla omitted. See PDF.] to influence the fluid and dust concentration profile.

Figure 8. Variation of [Forumla omitted. See PDF.] to influence the fluid and dust temperature profile.

Figure 9. Variation of [Forumla omitted. See PDF.] to influence the fluid and dust temperature profile.

Figure 10. Variation of [Forumla omitted. See PDF.] to influence the fluid and dust concentration profile.

Figure 11. Variation of Nusselt number [Forumla omitted. See PDF.] and Sheroowd number [Forumla omitted. See PDF.] along M, [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Figure 12. Variation of skin friction [Forumla omitted. See PDF.] and Nusselt number [Forumla omitted. See PDF.] along M, [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.].

Table 1

The skin friction coefficient was compared for different inputs of ${\beta }_t$ and M while all other parameters were ignored.

M	Gireesha et al. [44] ${\mathit{\beta }}_{\mathit{v}}=0$	Jalil et al. [40] ${\mathit{\beta }}_{\mathit{v}}=0$	Our Results ${\mathit{\beta }}_{\mathit{v}}=0$	Gireesha et al. [44] ${\mathit{\beta }}_{\mathit{v}}=0.5$	Jalil et al. [40] ${\mathit{\beta }}_{\mathit{v}}=0.5$	Our Results ${\mathit{\beta }}_{\mathit{v}}=0.5$
0.2	1.0000	1.000000	1.0000	1.034	1.033505	1.0335
0.2	1.095	1.095445	1.0955	1.126	1.126114	1.1261
0.5	1.224	1.224745	1.2248	1.252	1.252251	1.2523
1.0	1.414	1.414214	1.4142	1.438	1.438101	1.4381
1.2	1.483	1.483240	1.4832	1.506	1.506032	1.5060
1.5	1.581	1.581139	1.5901	1.602	1.602540	1.6026
2.0	1.732	1.732051	1.8301	1.751	1.751609	1.7517

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