Numerical Computation of Dusty Hybrid Nanofluid Flow and Heat Transfer over a Deformable Sheet with Slip Effect

Abstract

The mathematical modeling of dusty Cu-Al₂O₃/water nanofluid flow driven by a permeable deformable sheet was explored numerically. Rather than no–slip conditions at the boundary, velocity slip and thermal slip were considered. To achieve the system of nonlinear ordinary differential equations (ODEs), we employed some appropriate transformations and solved them numerically using MATLAB software (built–in solver called bvp4c). The influences of relevant parameters on fluid flow and heat transfer characteristics are discussed and presented in graphs. The findings showed that double solutions appeared in the case of stretching and shrinking sheets which contributed to the analysis of stability. The stability analysis, therefore, confirmed that merely the first solution was a stable solution. The addition of nanometer-sized particles (Cu) was found to significantly strengthen the heat transfer rate of the dusty nanofluid. Meanwhile, an upsurge in the velocity and thermal slip was shown to decrease the local Nusselt number. The result also revealed that an increment of fluid particle interaction decreased the boundary layer thickness.

1. Introduction

For a number of years, studies of the heat transfer characteristics of dusty fluid flow (two-phase fluid)—in terms of understanding various real-world problems, especially in atmospheric, physiological and engineering fields—have captivated the attention of numerous researchers. For instance, the application of dust particles can be seen in the petroleum industry, soil erosion by natural winds, purification of crude oil, aerosol and paint spraying, fluidization, dust entrainment during a nuclear explosion in a cloud and waste water treatment [1,2,3]. Saffman [4] first formulated dusty fluid flow equations and evaluated the stability of the laminar flow of a dusty gas wherein particles were evenly scattered. After some years, Chakrabarti [5] conducted a study of dusty gas using boundary layer theory, and soon thereafter, Datta and Mishra [6] and Vajravelu and Nayfeh [7] examined dusty fluid flow over a semi-infinite flat plate and stretching sheet, respectively. Further, a number of mathematicians such as Attia [8], Ajadi [9], Damseh [10], Gireesha et al. [11], Makinde and Chinyoka [12] and Prasannakumara et al. [13] conducted investigations in this field. An exact analytical solution of magnetohydrodynamic (MHD) dusty fluid caused by a stretching sheet was studied by Jalil et al. [14]. Unlike in previous research, Hamid et al. [15] explored the permeable shrinking surface in a dusty fluid and observed the occurrence of dual solutions. Gireesha et al. [16] numerically studied the nonlinear thermal radiation and hall current impact on a dusty fluid in a heated stretching sheet, while Abbas et al. [17] examined dusty fluid flow in a porous medium and took the impact of slip and MHD into consideration. Recently, the dusty fluid flow in a Riga plate with MHD and convective boundary conditions was studied by Prasannakumara et al. [18].

All of the above-mention studies only involved viscous fluids. The inclusion of nanoparticles in a conventional fluid can change its flow and heat transfer capabilities, i.e., boosting its thermal conductivity. It seems that Choi and Eastman [19] first conceived the idea of a nanofluid, i.e., nanoparticles suspended in a base fluid. Since then, nanofluids have been widely used in industrial cooling applications [20,21], biomedical technology [22], solar thermal applications [23] and many others. Numerous researchers, such as Motsumi and Makinde [24], Bachok et al. [25], Pop et al. [26] and Anuar et al. [27,28] have explored the concept of nanofluid flow and heat transfer. However, there have been fewer studies on two-phase fluids. The investigation of MHD flows of a dusty nanofluid in a porous medium over an exponentially stretching sheet was explored by Gorla et al. [29]. Afterwards, Naramgari and Sulochana [30] performed a numerical study of dusty nanofluid flow when the sheet was stretched, and concluded that the rate of heat transfer was enhanced by an upsurge interaction between the nanofluid and the particle phase. An analysis of the MHD flow of dusty Fe₃O₄-ethylene glycol nanofluid was performed by Ghadikolaei et al. [31]. Siddiqa et al. [32] numerically analyzed the natural convective flow of a dusty nanofluid driven by a wavy vertical surface. Recently, Kalpana et al. [33], Mishra et al. [34], Saha et al. [35] and Ibrahim and Gamachu [36] explored the dusty nanofluid flow problem with different surfaces and aspects.

New kinds of nanofluids, known as hybrid nanofluids, which can efficiently improve heat transfer, are being introduced in industry. A hybrid nanofluid is a mixture of two types of nanoparticles suspended in a base fluid. This new kind of fluid, however, provides a great advance in heat conductivity, as proved by the work of Madhesh and Kalaiselvam [37], Tahat and Benim [38] and Devi and Devi [39]. A few researchers have undertaken mathematical investigations of the boundary layer flow of hybrid nanofluids on various surfaces such as in stretching/shrinking sheet [40,41,42], thin needle [43], curved surface [44] and Riga plate [45]. Ghadikolaei et al. [46] studied the behavior of a hybrid nanofluid (Cu-Al₂O₃) in a micropolar dusty fluid. In another study, Raizah [47] conducted a mathematical simulation on a dusty hybrid nanofluid in an enclosure with two inclined, heated fins. Recently, Reddy et al. [48] presented a remarkable study of a Darcy porous medium in a hybrid dusty nanofluid. However, research related to hybrid nanofluids suspended with dust particles is limited. Hence, the main purpose of this investigation is to address the behavior of a hybrid nanofluid flow suspended with uniform distribution of dust particles in a deformable sheet, i.e., stretching and shrinking. It should be pointed out that the deformable sheet is not a new topic among researchers in this field, since such applications are well recognized in processing industries, especially in polymer processing, glass fiber production, the cooling and drying of paper, and many others [49].

Other issues discussed in this work are the simultaneous effect of velocity and thermal slip. Most of the research literature ignores slip boundary conditions; however, the slip effect should be considered in some situations, for example, in a rarefied fluid problem, fluid flow on multiple interfaces, the polishing of artificial heart valves, etc. Andersson [50] and Wang [51] conducted earlier work that took the velocity slip boundary condition driven by a stretching sheet into account. Their concept was utilized by Bhattacharyya et al. [52], who investigated velocity and the thermal slip effect in an unsteady stagnation flow. Afterwards, Mukhopadhyay [53] examined both slip effects on an exponentially stretching sheet and observed that an increase in thermal slip decreased the temperature. By taking into consideration the impact of slips (velocity, thermal and solutal), Ibrahim and Shankar [54] examined the flow of a permeable stretching sheet in a nanofluid. Further, Khan et al. [55] numerically studied the viscous fluid flow induced by a nonlinear stretching cylinder with radiation, MHD, suction/blowing and velocity and thermal slip effects. In a recent study, Kho et al. [56] observed the reduction of boundary layer thickness for increasing velocity and thermal slip parameters in their investigation of Williamson nanofluid flow. Motivated by the aforementioned work, our aim is to examine the effect of both velocity and thermal slip on the heat transfer of a dusty hybrid nanofluid.

In brief, this research paper is an extension of the work of Gireesha et al. [57], who considered the flow of MHD nanofluid embedded with dust particles over a stretching sheet. They studied the dusty nanofluid model using copper as the nanoparticle, analyzed it numerically and obtained a unique solution. Therefore, the present study aims to theoretically study the dusty hybrid nanofluid in stretching/shrinking sheet and take into consideration the effect of slips. The utilization of alumina (Al₂O₃) and copper (Cu) as hybrid nanoparticles with water as a base fluid have not been previously applied in studies on the dusty fluid problem with velocity and thermal slip effects. In addition, this study also comprises a novel attempt to discover the shrinking features of a dusty hybrid nanofluid. The existence of nonunique solutions is acknowledged in the literature [25,27,28,40,42,49]; hence, this paper focuses on obtaining a nonunique solution as well as performing a stability analysis thereof. It is expected that these findings will help engineers and researchers to understand the heat transfer mechanism in dusty hybrid nanofluids. It is worth mentioning that this study is original, and the numerical results presented herein have never been published before.

2. Mathematical Framework

2.1. Basic Equation

The steady two-dimensional flow of a dusty fluid suspended in a hybrid nanofluid driven by a deformable sheet with surface temperature $T_w$ is investigated. The geometry of the problem is depicted in Figure 1. With velocity $U_w(x)=ax$, where $a$ is a positive constant, the surface is stretched or shrunk linearly in the $x$-direction, whilst the $y$-axis is normal to it. Further, the temperature far from the surface is denoted by $T_{\infty }$ and the mass transfer velocity at the surface by $v_w\left(x\right)$.

Under this circumstance, the partial differential equations are as follows (see Devi and Devi [39] and Gireesha et al. [57]):

Fluid phase:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial y^2}+\frac{KN}{{\rho }_{hnf}}\left(u_p-u\right)$$

(2)

$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{hnf}}{{\left(\rho C_p\right)}_{hnf}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{N{C}_{pf}}{{\left(\rho C_p\right)}_{hnf}{\tau }_T}\left(T_p-T\right)+\frac{N}{{\left(\rho C_p\right)}_{hnf}{\tau }_v}{\left(u_p-u\right)}^2$$

(3)

Dust phase:

$$\frac{\partial u_p}{\partial x}+\frac{\partial v_p}{\partial y}=0$$

(4)

$$u_p\frac{\partial u_p}{\partial x}+v_p\frac{\partial u_p}{\partial y}=\frac{K}{m}\left(u-u_p\right)$$

(5)

$$u_p\frac{\partial T_p}{\partial x}+v_p\frac{\partial T_p}{\partial y}=\frac{C_{pf}}{C_{mf}{\tau }_T}\left(T-T_p\right)$$

(6)

Here, $u$ is the velocity components of the dust particle and fluid phases in the $x$ direction, while $v$ is the velocity component along the $y$ axis, $T_p$ and $T$ are the temperature of the dust particle and hybrid nanofluid, respectively, $N$, $m$ and $K$ are the dust particle number, the mass of the dust particle and the Stokes drag coefficient, respectively, $C_{pf}$ and $C_{mf}$ are the specific heat of nanofluid and dust particles, respectively, and ${\tau }_v$ and ${\tau }_T$ are the relaxation time and thermal equilibrium time of the dust particle, respectively, i.e., the time needed by a dust cloud to adapt its velocity and temperature to the nanofluid. Further, ${\rho }_{hnf}$ is the density of the hybrid nanofluids, $k_{hnf}$, ${\mu }_{hnf}$ and ${\left(\rho C_p\right)}_{hnf}$ are the thermal conductivity, dynamic viscosity and heat capacity, respectively, of Cu–Al₂O₃/water.

The boundary conditions are as follows (Ibrahim and Shankar [53]):

$$\begin{array}{c}u=\lambda U_w(x)+L_s\left(\frac{\partial u}{\partial y}\right),v=v_w\left(x\right),T=T_w+L_T\left(\frac{\partial T}{\partial y}\right)\mathrm{at}y=0\\ v_p\to v,u_p\to 0,u\to 0,T_p\to T_{\infty }T\to T_{\infty },\mathrm{as}y\to \infty \end{array}$$

(7)

Here, $\lambda$ corresponds to the stretching and shrinking parameter where $\lambda <0$ and $\lambda >0$ signify the shrinking and stretching sheet, respectively, while $\lambda =0$ signifies that of the static plate. In addition, $L_s$ and $L_T$ symbolize the velocity and thermal slip lengths, respectively. It is important to note that when $L_s=L_T=0$, the no slip condition is indicated.

2.2. Thermophysical Properties

The thermophysical properties of nanofluids and hybrid nanofluids given by Devi and Devi [39] are presented in

Table 1. Physical properties of hybrid nanofluids.

Properties	Nanofluid	Hybrid Nanofluid
Density	${\rho }_{nf}=\left(1-\phi \right){\rho }_f+\phi {\rho }_s$	${\rho }_{hnf}=\left(1-{\phi }_2\right)\left[\left(1-{\phi }_1\right){\rho }_f+{\phi }_1{\rho }_1\right]+{\phi }_2{\rho }_2$
Heat capacity	${\left(\rho C_p\right)}_{nf}=\left(1-\phi \right){\left(\rho C_p\right)}_f+\phi {\left(\rho C_p\right)}_s$	${\left(\rho C_p\right)}_{hnf}=\left(1-{\phi }_2\right)\left[\left(1-{\phi }_1\right){\left(\rho C_p\right)}_f+{\phi }_1{\left(\rho C_p\right)}_1\right]+{\phi }_2{\left(\rho C_p\right)}_2$
Dynamic viscosity	${\mu }_{nf}=\frac{{\mu }_f}{{\left(1-\phi \right)}^{2.5}}$	${\mu }_{hnf}=\frac{{\mu }_f}{{\left(1-{\phi }_1\right)}^{2.5}{\left(1-{\phi }_2\right)}^{2.5}}$
Thermal conductivity	$\frac{k_{nf}}{k_f}=\frac{k_s+2k_f-2\phi \left(k_f-k_s\right)}{k_s+2k_f+\phi \left(k_f-k_s\right)}$	$\frac{k_{hnf}}{k_{bf}}=\frac{k_2+2k_{bf}-2{\phi }_2\left(k_{bf}-k_2\right)}{k_2+2k_{bf}+{\phi }_2\left(k_{bf}-k_2\right)}$ where $\frac{k_{bf}}{k_f}=\frac{k_1+2k_f-2{\phi }_1\left(k_f-k_1\right)}{k_1+2k_f+{\phi }_1\left(k_f-k_1\right)}$

. In Table 1, subscript $s$, $nf$, $hnf$ and $f$ denote the nanoparticle, nanofluid, hybrid nanofluid and fluid, respectively, whereas $1$ and $2$ indicate the first and second nanoparticles, respectively. Furthermore, ${\phi }_1$ represents the first nanoparticle volume fraction, while ${\phi }_2$ denotes the second. In this study, copper (Cu) and alumina (Al₂O₃) are taken into consideration as the second and first nanoparticle volume fractions, respectively, and water acts as a base fluid (Oztop and Abu-Nada [58]).

Table 2. Thermo physical properties of Cu, Al₂O₃ and water.

Physical Properties	Cu	Water	Al2O3
$k$ (Wm-1K-1)	400	0.613	40
$C_p$ (J kg-1K-1)	385	4179	765
$\rho$ (kg m-3)	8933	997.1	3970

displays the thermophysical traits of base fluid and nanoparticles. It is important to note that Al₂O₃ is originally disseminated into the water to achieve the desired hybrid nanofluid (Cu-Al₂O₃/water). Afterwards, Cu is disseminated into the Al₂O₃/water nanofluid. In a remarkable study by Turkyilmazoglu [59], the nanofluids were shown to act as non-Newtonian fluids when the nanoparticle volume fraction exceeded 5–6%. Thus, in this study, the volume fraction of Al₂O₃ nanoparticles was set to 1%, and Cu was fluctuated from 0 to 2%, in correspondence with the work of [59]. Since the base fluid was water, the value of the Prandtl number was set to 6.2, i.e., room temperature, or nearly 295.15 K.

2.3. Similarity Solution

The following similarity variables are introduced (Naramgari and Sulochana [30]):

$$\begin{array}{c}u=ax{f}^{\prime }\left(\eta \right),v=-\sqrt{a{\nu }_f}f\left(\eta \right),u_p=ax{f^{\prime }}_p\left(\eta \right),v_p=-\sqrt{a{\nu }_f}{f}_p\left(\eta \right)\\ \theta \left(\eta \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},{\theta }_p\left(\eta \right)=\frac{T_p-T_{\infty }}{T_w-T_{\infty }},\eta =\sqrt{\frac{a}{{\nu }_f}}y\end{array}$$

(8)

where $\eta$ is the similarity variable, ${\nu }_f$ is the kinematic viscosity, $f$ and $\theta$ are the dimensionless function for fluid phase, and $f_p$ and ${\theta }_p$ are the dimensionless function for dust phase. Further, primes denote the differentiation with respect to $\eta$. In order for similarity solutions of Equations (1)–(6) together with Equation (7) to exist, we use

$$v_w=-\sqrt{a{\nu }_f}s\mathrm{and}{T}_w=T_{\infty }+A{\left(x/l\right)}^2$$

(9)

where $s$ is the transpiration rate constant, wherein $s>0$ and $s<0$ signify suction and injection, respectively, while $s=0$ refers to an impermeable plate. $A$ is a constant and $l=\sqrt{{\nu }_f/a}$ is a characteristic length. Invoking Equation (8), Equations (1) and (4) are identically fulfilled, and Equations (2), (3), (5) and (6) transform into the following ordinary differential equation:

Fluid phase:

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{f}^{‴}-{f^{\prime }}^2+f{f}^{″}+\frac{L{\beta }_v}{{\rho }_{hnf}/{\rho }_f}\left({f^{\prime }}_p-f^{\prime }\right)=0$$

(10)

$$\begin{array}{r}\frac{k_{hnf}/k_f}{{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\frac{1}{\mathrm{Pr}}{\theta }^{″}+f{\theta }^{\prime }-2f^{\prime }\theta +\frac{L{\beta }_T}{m{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\left({\theta }_p-\theta \right)\\ +\frac{L{\beta }_{v}Ec}{m{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}{\left({f^{\prime }}_p-f^{\prime }\right)}^2=0\end{array}$$

(11)

Dust phase:

$$f_p{f^{″}}_p-{f^{\prime }}_p{}^2+{\beta }_v\left(f^{\prime }-{f^{\prime }}_p\right)=0$$

(12)

$$f_p{\theta }_p{}^{\prime }-2{f^{\prime }}_p{\theta }_p+{f^{\prime }}_p+\epsilon {\beta }_T\left(\theta -{\theta }_p\right)=0$$

(13)

where ${\beta }_T$ and ${\beta }_v$ are the fluid–particle interaction parameters for temperature and velocity, respectively, $L$ is the mass concentration of dust particles, $\epsilon$ is the ratio of specific heats, and $Ec$ and $\mathrm{Pr}$ are the Eckert and Prandtl numbers, respectively, which are given as:

$${\beta }_v=\frac{1}{a{\tau }_v},{\beta }_T=\frac{1}{a{\tau }_T},L=\frac{Nm}{{\rho }_f},\mathrm{Pr}=\frac{{\left(\mu C_p\right)}_f}{k_f},\epsilon =\frac{C_{pf}}{C_{mf}},Ec=\frac{U_w^2}{C_{pf}\left(T_w-T_{\infty }\right)}=\frac{a^2{l}^2}{C_{pf}A}$$

(14)

The boundary conditions are:

$$\begin{array}{c}f\left(0\right)=s,f^{\prime }\left(0\right)=\lambda +{\sigma }_1{f}^{″}\left(0\right),\theta \left(0\right)=1+{\sigma }_2{\theta }^{\prime }\left(0\right),\\ f^{\prime }\left(\eta \right)\to 0,{f^{\prime }}_p\left(\eta \right)\to 0,f_p\left(\eta \right)\to f\left(\eta \right),\theta \left(\eta \right)\to 0,{\theta }_p\left(\eta \right)\to 0\mathrm{as}\eta \to \infty \end{array}$$

(15)

where ${\sigma }_1=L_s\sqrt{a/{\nu }_f}$ and ${\sigma }_2=L_T\sqrt{a/{\nu }_f}$ are the velocity and thermal slip parameters, respectively.

In this study, the physical quantities of interest are:

$$C_f=\frac{{\mu }_{hnf}}{{\rho }_f{U}_w^2}{\left(\frac{\partial u}{\partial y}\right)}_{y=0},N{u}_x=-\frac{x{k}_{hnf}}{k_f\left(T_w-T_{\infty }\right)}{\left(\frac{\partial T}{\partial y}\right)}_{y=0}$$

(16)

where $C_f$ denotes the skin friction coefficient and $N{u}_x$ is the local Nusselt number. Using Equations (8) and (16), we get:

$$C_f{\mathrm{Re}}_x^{1/2}=\frac{{\mu }_{hnf}}{{\mu }_f}{f}^{″}\left(0\right),N{u}_x{\mathrm{Re}}_x^{-1/2}=-\frac{k_{hnf}}{k_f}{\theta }^{\prime }\left(0\right)$$

(17)

where ${\mathrm{Re}}_x=U_{e}x/{\nu }_f$ is the local Reynolds number.

3. Stability of the Solutions

Due to the occurrence of nonuniqueness in the present research, the stability analysis was performed by referring to the work of Merkin [60], Weidman et al. [61] and Harris et al. [62]. These analyses have been implemented by other researchers too (see, for example, the work of Hamid et al. [15], Anuar et al. [27,28], Waini et al. [43] and Khashi’ie et al. [45]). Some important steps were implemented to identify the stability of solutions, i.e., (i) considering an unsteady governing equation; (ii) introducing new dimensionless time variables and similarity variables; (iii) implementing linear eigenvalue equations, and (iv) relaxing the boundary conditions.

3.1. Unsteady-State Problem

First, Equations (1)–(6) were converted into an unsteady case (dependent of time). So, continuity Equations (1) and (4) remain the same, while the other equations become:

Fluid phase:

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\mu }_{hnf}}{{\rho }_{hnf}}\frac{{\partial }^{2}u}{\partial y^2}+\frac{KN}{{\rho }_{hnf}}\left(u_p-u\right)$$

(18)

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{k_{hnf}}{{\left(\rho C_p\right)}_{hnf}}\frac{{\partial }^{2}T}{\partial y^2}+\frac{N{C}_{pf}}{{\left(\rho C_p\right)}_{hnf}{\tau }_T}\left(T_p-T\right)+\frac{N}{{\left(\rho C_p\right)}_{hnf}{\tau }_v}{\left(u_p-u\right)}^2$$

(19)

Dust phase:

$$\frac{\partial u_p}{\partial t}+u_p\frac{\partial u_p}{\partial x}+v_p\frac{\partial u_p}{\partial y}=\frac{K}{m}\left(u-u_p\right)$$

(20)

$$\frac{\partial T_p}{\partial t}+u_p\frac{\partial T_p}{\partial x}+v_p\frac{\partial T_p}{\partial y}=\frac{C_{pf}}{C_{mf}{\tau }_T}\left(T-T_p\right)$$

(21)

subjected to the following boundary conditions:

$$\begin{array}{l}t<0:u=0,v=0,u_p=0,v_p=0,T=T_w\mathrm{for}\mathrm{any}x,y\\ t\ge 0:u=\lambda U_w(x)+L_s\left(\frac{\partial u}{\partial y}\right),v=v_w\left(x\right),T=T_w+L_T\left(\frac{\partial T}{\partial y}\right)\mathrm{at}y=0\\ v_p\to v,u_p\to 0,u\to 0,T_p\to T_{\infty }T\to T_{\infty },\mathrm{as}y\to \infty \end{array}$$

(22)

where $t$ denotes the time.

3.2. New Similarity Transformation

Next, a new time-dependent, dimensionless variable had to be introduced. Hence, we have (Hamid et al. [15]):

$$\tau =at$$

(23)

while Equation (8) is replaced by:

$$\begin{array}{c}u=ax\frac{\partial f}{\partial \eta }\left(\eta ,\tau \right),v=-\sqrt{a{\nu }_f}f\left(\eta ,\tau \right),u_p=ax\frac{\partial f_p}{\partial \eta }\left(\eta ,\tau \right),v_p=-\sqrt{a{\nu }_f}{f}_p\left(\eta ,\tau \right)\\ \theta \left(\eta ,\tau \right)=\frac{T-T_{\infty }}{T_w-T_{\infty }},{\theta }_p\left(\eta ,\tau \right)=\frac{T_p-T_{\infty }}{T_w-T_{\infty }},\eta =\sqrt{\frac{a}{{\nu }_f}}y\end{array}$$

(24)

By substituting Equations (23) and (24) into Equations (18)–(22), we get:

Fluid phase:

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}\frac{{\partial }^{3}f}{\partial {\eta }^3}+f\frac{{\partial }^{2}f}{\partial {\eta }^2}-{\left(\frac{\partial f}{\partial \eta }\right)}^2+\frac{L{\beta }_v}{{\rho }_{hnf}/{\rho }_f}\left(\frac{\partial f_p}{\partial \eta }-\frac{\partial f}{\partial \eta }\right)-\frac{{\partial }^{2}f}{\partial \eta \partial \tau }=0$$

(25)

$$\begin{array}{r}\frac{k_{hnf}/k_f}{{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\frac{1}{\mathrm{Pr}}\frac{{\partial }^2\theta }{\partial {\eta }^2}+f\frac{\partial \theta }{\partial \eta }-2\frac{\partial f}{\partial \eta }\theta +\frac{L{\beta }_T}{m{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\left({\theta }_p-\theta \right)\\ +\frac{L{\beta }_{v}Ec}{m{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}{\left(\frac{\partial f_p}{\partial \eta }-\frac{\partial f}{\partial \eta }\right)}^2-\frac{\partial \theta }{\partial \tau }=0\end{array}$$

(26)

Dust phase:

$$f_p\frac{{\partial }^2{f}_p}{\partial {\eta }^2}-{\left(\frac{\partial f_p}{\partial \eta }\right)}^2+{\beta }_v\left(\frac{\partial f}{\partial \eta }-\frac{\partial f_p}{\partial \eta }\right)-\frac{{\partial }^2{f}_p}{\partial \eta \partial \tau }=0$$

(27)

$$f_p\frac{\partial {\theta }_p}{\partial \eta }-2\frac{\partial f_p}{\partial \eta }{\theta }_p+\frac{\partial f_p}{\partial \eta }+\epsilon {\beta }_T\left(\theta -{\theta }_p\right)-\frac{\partial {\theta }_p}{\partial \tau }=0$$

(28)

and the boundary conditions are:

$$\begin{array}{c}f\left(0\right)=s,\frac{\partial f}{\partial \eta }\left(0\right)=\lambda +{\sigma }_1\frac{{\partial }^{2}f}{\partial {\eta }^2}\left(0\right),\theta \left(0\right)=1+{\sigma }_2\frac{\partial \theta }{\partial \eta }\left(0\right),\\ \frac{\partial f}{\partial \eta }\left(\eta \right)\to 0,\frac{\partial f_p}{\partial \eta }\left(\eta \right)\to 0,f_p\left(\eta \right)\to f\left(\eta \right),\theta \left(\eta \right)\to 0,{\theta }_p\left(\eta \right)\to 0\mathrm{as}\eta \to \infty \end{array}$$

(29)

3.3. Introducing Linear Eigenvalue Equations

The stability of the steady flow solutions can be explored by setting $f\left(\eta \right)=f_0\left(\eta \right)$, $f_p\left(\eta \right)=f_{p_0}\left(\eta \right)$, $\theta \left(\eta \right)={\theta }_0\left(\eta \right)$ and ${\theta }_p\left(\eta \right)={\theta }_{p_0}\left(\eta \right)$, which satisfies the boundary value Equations (10)–(13) and Equation (15). Thus, the following equations are introduced (Weidman et al. [61]):

$$\begin{array}{c}f\left(\eta ,\tau \right)=f_0\left(\eta \right)+e^{-\gamma \tau }F\left(\eta ,\tau \right),f_p\left(\eta ,\tau \right)=f_{p_0}\left(\eta \right)+e^{-\gamma \tau }{F}_p\left(\eta ,\tau \right),\\ \theta \left(\eta ,\tau \right)={\theta }_0\left(\eta \right)+e^{-\gamma \tau }H\left(\eta ,\tau \right),{\theta }_p\left(\eta ,\tau \right)={\theta }_{p_0}\left(\eta \right)+e^{-\gamma \tau }{H}_p\left(\eta ,\tau \right)\end{array}$$

(30)

where $F\left(\eta ,\tau \right)$, $F_p\left(\eta ,\tau \right)$, $H\left(\eta ,\tau \right)$, $H_p\left(\eta ,\tau \right)$ and their derivatives are relatively smaller than $f_0\left(\eta \right)$, $f_{p_0}\left(\eta \right)$, ${\theta }_0\left(\eta \right)$ and ${\theta }_{p_0}\left(\eta \right)$. In addition, $\gamma$ is the unknown eigenvalue which will be used to specify the stability of the solutions. Substituting Equation (30) into Equations (25)–(29) and letting $\tau \to 0$, in which $F\left(\eta \right)=F_0\left(\eta \right)$, $F_p\left(\eta \right)=F_{p_0}\left(\eta \right)$, $H\left(\eta \right)=H_0\left(\eta \right)$ and $H_p\left(\eta \right)=H_{p_0}\left(\eta \right)$, we have:

Fluid phase:

$$\frac{{\mu }_{hnf}/{\mu }_f}{{\rho }_{hnf}/{\rho }_f}{F}_0{}^{\prime \prime \prime }+f_0{F}_0{}^{\prime \prime }+F_0{f}_0{}^{\prime \prime }-2f_0{}^{\prime }{F}_0{}^{\prime }+\frac{L{\beta }_v}{{\rho }_{hnf}/{\rho }_f}\left(F_{p_0}{}^{\prime }-F_0{}^{\prime }\right)+\gamma F_0{}^{\prime }=0$$

(31)

$$\begin{array}{c}\frac{k_{hnf}/k_f}{{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\frac{1}{\mathrm{Pr}}{H}_0{}^{\prime \prime }+f_0{H}_0{}^{\prime }+F_0{\theta }_0{}^{\prime }-2{\theta }_0{F}_0{}^{\prime }-2H_0{f}_0{}^{\prime }+\frac{L{\beta }_T}{m{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\left(H_{p_0}-H_0\right)+\gamma H_0\\ +\frac{L{\beta }_{v}Ec}{m{\left(\rho C_p\right)}_{hnf}/{\left(\rho C_p\right)}_f}\left(2f_{p_0}{}^{\prime }{F}_{p_0}{}^{\prime }-2f_{p_0}{}^{\prime }{F}_0{}^{\prime }-2f_0{}^{\prime }{F}_{p_0}{}^{\prime }+2f_0{}^{\prime }{F}_0{}^{\prime }\right)=0\end{array}$$

(32)

Dust phase:

$$f_{p_0}{F}_{p_0}{}^{\prime \prime }+F_{p_0}{f}_{p_0}{}^{\prime \prime }-2f_{p_0}{}^{\prime }{F}_{p_0}{}^{\prime }+{\beta }_v\left(F_0{}^{\prime }-F_{p_0}{}^{\prime }\right)+\gamma F_{p_0}{}^{\prime }=0$$

(33)

$$f_{p_0}{H}_{p_0}{}^{\prime }+F_p{\theta }_{p_0}{}^{\prime }-2{\theta }_{p_0}{F}_{p_0}{}^{\prime }-2H_{p_0}{f}_{p_0}{}^{\prime }+\epsilon {\beta }_T\left(H_0-H_{p_0}\right)+\gamma H_{p_0}=0$$

(34)

The conditions now take the following form:

$$\begin{array}{c}{F}_0\left(0\right)=0,F_0{}^{\prime }\left(0\right)={\sigma }_1{F}_0{}^{\prime \prime }\left(0\right),H_0\left(0\right)={\sigma }_2{H}_0{}^{\prime }\left(0\right),\\ F_0{}^{\prime }\left(\eta \right)\to 0,F_{p_0}{}^{\prime }\left(\eta \right)\to 0,F_{p_0}\left(\eta \right)\to F_0\left(\eta \right),H_0\left(\eta \right)\to 0,H_{p_0}\left(\eta \right)\to 0\text{ as }\eta \to \infty \end{array}$$

(35)

3.4. Relaxation of Boundary Condition

To solve the stability model, we needed to relax the boundary conditions as proposed by Harris et al. [62]. For that reason, the conditions $F_0{}^{\prime }\left(\eta \right)\to 0$ as $\eta \to \infty$ were replaced by new conditions $F_0{}^{\prime \prime }\left(0\right)=1$. It should be pointed out that the linearized boundary value Equations (31)–(35), together with new conditions $F_0{}^{\prime \prime }\left(0\right)=1$, yielded the unlimited set of unknown eigenvalues $\left({\gamma }_1<{\gamma }_2<{\gamma }_3<\dots \right)$. If the smallest eigenvalues $\gamma$ showed a positive sign, the solutions observed an initial decay of perturbation, and accordingly indicated a stable solution. On the other hand, as the smallest eigenvalues $\gamma$ showed a negative sign, an early growth of disruption was noticed, which signified an unstable solution.

4. Numerical Solutions and Discussions

To solve the boundary value Equations (10)–(13) with the boundary conditions given by Equation (15), we adopted a built-in function called bvp4c from Matlab package. Bvp4c is a finite difference code that applies the three stage Lobato IIIa formula (Shampine et al. [63], Kierzenka and Shampine [64]). In this research, the appropriate finite values of ${\eta }_{\infty }$ was set to be 10. Further, to ensure the precision of the current algorithm, the current results of skin friction coefficient $f^{″}\left(0\right)$ were compared with previously reported solutions of Hayat et al. [65] and Ibrahim and Shankar [53]. These comparative solutions are revealed in

Table 3. Values of $f^{″}\left(0\right)$ for extended sheet $\left(\lambda =1\right)$ when ${\phi }_1={\phi }_2=L=B_v=0$ and $s=0$.

${\mathit{\sigma }}_1$	Hayat et al. [65] (Homotopy Analysis Method)	Ibrahim and Shankar [53] (Shooting)	Present Result (Bvp4c)
0	−1.000000	−1.0000	−1.000000
0.1	−0.872082	−0.8721	−0.872083
0.2	−0.776377	−0.7764	−0.776377
0.5	−0.591195	−0.5912	−0.591196
2.0	−0.283981	−0.2840	−0.283981
5.0	−0.144841	−0.1448	−0.144842

for selected values of velocity slip parameter ${\sigma }_1$. Additionally, a comparison of the values of heat transfer $-{\theta }^{\prime }\left(0\right)$ for the present method with those obtained by Gireesha et al. [57] and Naramgari and Sulochana [30] was made; see

Table 4. Comparative values of $-{\theta }^{\prime }\left(0\right)$ for stretching sheet $\left(\lambda =1\right)$ when ${\phi }_1={\phi }_2=L=B_T=B_v=Ec=0$ and ${\sigma }_2=0$.

$\mathbf{Pr}$	Gireesha et al. [57]	Naramgari and Sulochana [30]	Present Result
0.72	1.0885	1.088561	1.088527
1	1.3333	1.333333	1.333333
10	4.7968	4.796817	4.796873

. It can be observed from these tables that good agreement with these methods was achieved, thereby confirming the consistency of the present approach.

In this section, the results of local skin friction $C_f{\mathrm{Re}}_x^{1/2}$, Nusselt number $N{u}_x{\mathrm{Re}}_x^{-1/2}$, velocity profiles for fluid phase $f^{\prime }\left(\eta \right)$ and dust phase $f_p{}^{\prime }\left(\eta \right)$, as well as the temperature profiles for the fluid phase $\theta \left(\eta \right)$ and dust phase ${\theta }_p\left(\eta \right)$ are illustrated graphically to examine the impact of some governing parameters, namely, Cu nanoparticle volume fraction $\left(0\le {\phi }_2\le 0.02\right)$, suction parameter $\left(s>1\right)$, velocity slip parameter $\left(0\le {\sigma }_1\le 0.2\right)$, thermal slip parameter $\left(0\le {\sigma }_2\le 0.2\right)$, fluid interaction parameter for velocity $\left(0\le B_v\le 1\right)$, fluid interaction parameter for temperature $\left(0\le B_T\le 1\right)$ and mass concentration $\left(0\le L\le 1\right)$. Figure 2 and Figure 3 depict the influence of the Cu nanoparticle volume fraction parameter ${\phi }_2$ toward suction parameter $s$ on the local skin friction $C_f{\mathrm{Re}}_x^{1/2}$ and Nusselt number $N{u}_x{\mathrm{Re}}_x^{-1/2}$ for the shrinking sheet $\left(\lambda =-1\right)$, as given in Equation (17). From these figures, it is interesting to see that two solutions are likely to occur for a particular range of suction parameters. For instance, when ${\phi }_2=0$, the solutions are bound to exist when $s>1.54382$, while as ${\phi }_2$ increases to $0.01$ and $0.02$, the range of suction increases, i.e., $s>1.52027$ and $s>1.49934$. It should also be mentioned that the increment of the volume fraction of Cu nanoparticle ${\phi }_2$ from $0$ to $0.02$ enhances both $C_f{\mathrm{Re}}_x^{1/2}$ and $N{u}_x{\mathrm{Re}}_x^{-1/2}$. Moreover, the values of $C_f{\mathrm{Re}}_x^{1/2}$ and $N{u}_x{\mathrm{Re}}_x^{-1/2}$ for the first solution increase as suction parameter $s$ increases, while the opposite was observed for the second solution.

The effects of Cu nanoparticle volume fraction ${\phi }_2$ against stretching/shrinking parameter $\lambda$ on the local skin friction $C_f{\mathrm{Re}}_x^{1/2}$ and Nusselt number $N{u}_x{\mathrm{Re}}_x^{-1/2}$ are shown in Figure 4 and Figure 5. It is apparent from these figures that for all values of stretching/shrinking parameter $\lambda$, dual solutions occurred. However, it should be noted that no solution exists when $\lambda <{\lambda }_c,$ which indicates that the boundary layer separates from the surface and the principles of boundary layer theory are no longer valid. Moreover, ${\lambda }_c$ is the critical point that connects the first and second solutions. From Figure 4, it may be seen that an increase in the Cu nanoparticle volume fraction ${\phi }_2$ enhances the local skin friction $C_f{\mathrm{Re}}_x^{1/2}$ when the sheet is shrinking $\left(\lambda <0\right)$ in both solutions, while it decreases when the sheet is stretched $\left(\lambda >0\right)$ in both solutions. In addition, it is evident from Figure 5 that the enhancement in a Cu nanoparticle volume fraction ${\phi }_2$ improves the local Nusselt number in the first and second solutions. This finding proves that the increment of Cu nanoparticle volume fraction ${\phi }_2$ can improve the heat transfer efficiency. This also implies that the dusty hybrid nanofluid provides better heat performance than the dusty nanofluid. Furthermore, an enhancement of the Cu nanoparticle volume fraction ${\phi }_2$ on $C_f{\mathrm{Re}}_x^{1/2}$ and $N{u}_x{\mathrm{Re}}_x^{-1/2}$ can delay the boundary layer flow separation. Figure 6 and Figure 7 demonstrate the velocity $f^{\prime }\left(\eta \right),f_p{}^{\prime }\left(\eta \right)$ and temperature $\theta \left(\eta \right),{\theta }_p\left(\eta \right)$ profiles when the Cu nanoparticle volume fraction ${\phi }_2$ varies from $0$ to $0.02$ for both the fluid and dust phases. It reveals that increasing the Cu nanoparticle volume fraction ${\phi }_2$ decreases the momentum and thermal boundary layer thickness in the first solution of both phases. However, the opposite was observed with the second solution. Furthermore, we can see that the boundary layer thickness of the first solution and dust phase was slimmer than that of the fluid phase. In addition, all the published profiles are asymptotically satisfied with Equation (15) when ${\eta }_{\infty }=10$ is used in the bvp4c function (MATLAB); this supports the findings in Figure 4 and Figure 5.

Figure 8 and Figure 9 show the impact of the no-slip parameter $\left({\sigma }_1={\sigma }_2=0\right)$ and slip parameters $\left({\sigma }_1={\sigma }_2=0.1\mathrm{and}{\sigma }_1={\sigma }_2=0.2\right)$ on the local skin friction $C_f{\mathrm{Re}}_x^{1/2}$ and Nusselt number $N{u}_x{\mathrm{Re}}_x^{-1/2}$ toward the permeable stretching/shrinking parameter $\lambda$. The occurrence of slip parameters (velocity, ${\sigma }_1$ and thermal, ${\sigma }_2$) consequently diminishes the local skin friction $C_f{\mathrm{Re}}_x^{1/2}$ when the sheet is shrinking $\left(\lambda <0\right)$, but increases it for a stretching sheet $\left(\lambda >0\right)$ in both solutions. In Figure 9, the local Nusselt number shows a declining trend in both solutions, along with a simultaneous increment of slip parameters $\left({\sigma }_1,{\sigma }_2\right)$. A simultaneous increase in both the velocity and thermal slip parameters postpones boundary layer separation, as shown in Figure 8 and Figure 9. A diminished momentum boundary layer thickness was observed in the first solution when the velocity slip parameter ${\sigma }_1$ increased on the dusty hybrid nanofluid for both phases, as shown in Figure 10; however, the opposite occurred for the second solution. Note that when ${\sigma }_1=0,{\sigma }_2=0.2$, it corresponds to thermal slip only, while when ${\sigma }_1=0.2,{\sigma }_2=0$, it corresponds to velocity slip only. On the other hand, Figure 11 and Figure 12 illustrate the variation of the temperature profile of the fluid $\theta \left(\eta \right)$ and dust phases ${\theta }_p\left(\eta \right)$ when velocity ${\sigma }_1$ and thermal ${\sigma }_2$ slip parameters fluctuate from $0$ to $0.2$, respectively. It is interesting to observe from these figures that for the first and second solutions, the thermal boundary layer thickness decreased with an increase in velocity and thermal slip parameters. However, we did not plot the velocity profiles for thermal slip parameter ${\sigma }_2$, as this parameter caused no changes in the velocity profiles.

The distribution of velocity $f^{\prime }\left(\eta \right),f_p{}^{\prime }\left(\eta \right)$ and temperature $\theta \left(\eta \right),{\theta }_p\left(\eta \right)$ profiles for selected values of fluid particle interaction parameters for velocity $B_v$ and temperature $B_T$ are plotted in Figure 13 and Figure 14. For increasing velocity fluid particle interaction parameter $B_v$, the momentum boundary layer thickness decreased for the fluid phase, while the opposite occurred for the dust phase in the first solution. Nevertheless, the thickness of the momentum boundary layer increased in the second solution for both phases. Additionally, the temperature profile $\theta \left(\eta \right)$ in Figure 14 indicates that the thermal boundary layer thickness for the first and second solutions was diminished with increasing values of $B_T$. Meanwhile, the temperature profile in the dust phase ${\theta }_p\left(\eta \right)$ showed the opposite result from the fluid phase in both solutions. It is evident from Figure 13 and Figure 14 that a very high value of $B_v$ and $B_T$ will eventually cause the velocity and temperature for both phases to be the same.

Figure 15 and Figure 16 are plotted to indicate the impact of mass concentration of dust particles $L$ on the velocity $f^{\prime }\left(\eta \right),f_p{}^{\prime }\left(\eta \right)$ and temperature $\theta \left(\eta \right),{\theta }_p\left(\eta \right)$ profiles for the fluid and dust phases in a permeable shrinking sheet $\left(\lambda =-1\right)$. When the mass concentration increases, the thickness of the momentum boundary layer decreases for the fluid and dust phases in the first solution, while it increases in the second. This finding can be explained by the fact that the drag force between the hybrid nanofluid and dust particle will increase as the mass concentration increases; this will consequently slow down the fluid motion. On the other hand, thickening of the thermal boundary layer was noticed for the fluid and dust phases as the mass concentration increased in both solutions.

The boundary value Equations (10)–(13), together with Equation (15), provided nonunique solutions for some governing parameters. The phenomenon of nonunique solutions, namely, the first and second solutions, was proven; see Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17. Accordingly, a stability analysis was performed to identify the most stable solutions. Linearized Equations (31)–(34), along with Equation (35), were numerically with the aid of the bvp4 function in MATLAB. The smallest eigenvalues $\gamma$ on the selected parameter ${\phi }_2$ and $\lambda$ from Figure 4 and Figure 5 are listed in

Table 5. Smallest eigenvalues $\gamma$ for selected values of ${\phi }_2$ and $\lambda$.

${\mathit{\phi }}_2$	$\mathit{\lambda }$	1st Solution	2nd Solution
0	−1.6121	0.0026	−0.0027
	−1.612	0.0131	−0.0130
	−1.61	0.0590	−0.0580
0.01	−1.6677	0.0115	−0.0114
	−1.667	0.0363	−0.0359
	−1.66	0.1158	−0.1120
0.02	−1.7207	0.0085	−0.0085
	−1.72	0.0359	−0.0355
	−1.7	0.1924	−0.1825

. The values of $\gamma$, however, drew closer to zero in both (first and second) solutions as the stretching/shrinking parameter approached its critical value. Apart from this, it is shown that the second solution displayed negative values of $\gamma$, whereas the first demonstrated positive values. The smallest eigenvalues $\gamma$ against $\lambda$ when ${\phi }_1={\phi }_2=0.01$ are plotted in Figure 17. This figure supports the findings presented in Table 5. In reference to the previous literature, we propose that the first solution is stable while the second is unstable. It is worth noting that this analysis is important in identifying the stable solution when nonunique solutions exist, so that the flow behavior can be accurately predicted.

5. Conclusions

The flow of a dusty hybrid nanofluid over a permeable deformable sheet with velocity and thermal slip was investigated numerically. The similarity solutions were produced by utilizing the bvp4c function in the MATLAB software (R2018a, MathWorks, Natick, MA, USA). The impact of emerging parameters was examined and illustrated graphically. The conclusions can be summarized as follows:

• The presence of double solutions is noticeable for a stretching and shrinking sheet when suction parameter $\left(s>s_c\right)$ is imposed.
• A stability analysis was carried out and the first solution proved to be stable, whereas the other solution was not.
• An increase in the Cu nanoparticle volume fraction ${\phi }_2$ in the dusty nanofluid has a tendency to improve the local Nusselt number for all range of $\lambda$, and to increase the local skin friction for shrinking sheet; however, the opposite is true for a stretching sheet.
• The simultaneous increase of velocity ${\sigma }_1$ and thermal ${\sigma }_2$ slip parameters decrease the local Nusselt number for fluid phase.
• The similarity solutions can be widened with an increase in Cu nanoparticle volume fraction ${\phi }_2$ and slip parameters (velocity ${\sigma }_1$ and thermal ${\sigma }_2$), thereby delaying boundary layer separation.
• The momentum thickness in fluid phase decreases and dust phase increases as velocity fluid interaction parameter $B_v$ increases.
• An upsurge of fluid interaction for temperature parameter $B_T$ decreases the thermal boundary layer thickness of the fluid phase, while it does the opposite in the dust phase.
• The mass concentration of dust particle $L$ decreases the momentum thickness but increases the thermal thickness in both phases.