A Note on the Significance of Quartic Autocatalysis Chemical Reaction on the Motion of Air Conveying Dust Particles

1 Introduction

The spontaneous movement of molecules and particles between two solutions of different concentrations can be categorised as either osmosis or diffusion. Osmosis is the net movement of molecules from a highly concentrated solution through a semipermeable membrane to a low concentrated solution. Meanwhile, diffusion is the migration of particles from a region of high concentration to low concentration. A countable number of reports on these two concepts have shown that the migration of molecules is to establish equality of concentrations on both sides of the membrane. All these reports have indicated that the major types of diffusion are Brownian motion, gaseous diffusion, reverse diffusion, self-diffusion, collective diffusion, surface diffusion, effusion, electron diffusion, facilitated diffusion, momentum diffusion, photon diffusion, Knudsen diffusion, and osmosis. An interaction between two or more chemicals that produces either one or more new chemical compounds is known as a chemical reaction. It is worth remarking that chemical reaction is inevitable during the diffusion process. To some experts, the mixture of solutes and solvents is the fundamental basis of concentration. However, in chemistry, the ratio of mass to volume is known as concentration. This property of fluid refers to the amount of a substance per defined space. The problem of generative and destructive chemical reactions during the separation of chemicals in distillation procedure was modelled by Hayat et al. [1] using two-dimensional stagnation point flow of an upper-convected Maxwell fluid over a stretching surface. It was shown that the concentration of the fluid increases in the case of generative chemical reaction, but it decreases in the case of destructive chemical reaction. Ziaul Haque et al. [2] explored the concentration of a non-Newtonian fluid flow in the presence of suction, Joule heating, mass flux due to a temperature gradient, heat flux due to the mass gradient, and viscous dissipation due to thermal buoyancy. It is seen that the concentration field decreases with an increase in the suction/blowing parameter. At any level of thickness of an object, increase in the velocity index leads to an increase in the concentration of water conveying 29 nm CuO nanoparticles in the presence of thermoelectric effects and gyrotactic microorganisms; see Sivaraj et al. [3]. Mixed convection flow of typical Newtonian fluids over a melting horizontal surface with variable thickness was analysed, and it was revealed that the concentration is a decreasing property of buoyancy [4]. At higher temperatures, the formation of new bounds after breaking of old bonds is more rapid. Scientifically, with an increase in the temperature, the molecules move faster and collide more vigorously (i.e. the likelihood of bond cleavages and rearrangements). This led Kenneth [5] to remark that whether it is only common sense, collision theory, or transition state theory, chemical reactions proceed slower at lower temperatures and faster at higher temperatures.

Many chemical reaction systems that are of interest to scientists, industrialists, chemical engineers, and applied mathematicians can be closely approximated as first-order reactions [6]. This led Sivaraj et al. [7] to model the chemical reaction in the flow of incompressible viscoelastic (Walters liquid-B model) fluid between a vertical long wavy wall and a parallel flat wall saturated with porous medium placed in the plane using first-order reaction model. In a chemical reaction xA+yB→zC, the rate of the chemical reaction using power law is of the form r=k[A]x[B]y [8]. Historically, Max Trautz in 1916 and Lewis William in 1918 proposed almost the same collision theory, which explains chemical reactions between two or more reactants and also provided reliable answers to the open question “Why reaction rates differ for different reactions?” [9], [10]. In a dilute solution, most especially one having a single step with a single transition state is empirically found to obey the law of mass action. According to Connors [11], this implies that the rate of chemical reaction depends only on the concentrations of the reactants, raised to the powers of their stoichiometric coefficients. In the industry, there are times when scientist only needs to deal with the concentration of chemical species in solution rather than the concentration of a chemical compound. Scientifically, a thermodynamic process in which the temperature or changes in temperature of the system remain constant is known as an isothermal process. During this kind of process, transfer of heat energy out or into the system must be at a slow rate such that the thermal equilibrium is maintained. Considering an open system with uniform temperature and concentration in an isothermal continuous stirred tank.

Gray and Scott [12] reported the behaviour of autocatalytic reactions. In the same article, it is stated that in autocatalytic reactions between species A and B in which B is a catalyst, the reaction rate depends on the concentration of the product species B. It can be deduced from the article that the rate of cubic autocatalytic reaction is of the form α[A][B]2. This contribution to the body of knowledge was extended to a case of nonisothermal and also explained the instabilities in the system A+2B→3B, B→C [13]. Scott [14] considered quadratic and cubic autocatalytic together with rates of their chemical reaction and then coupled the equation with the diffusion of the reactants through a permeable boundary from an external reservoir where the concentrations are held constant. Considering all these facts, Lynch [15] reported that isothermal autocatalytic reaction systems can display most if not all of the various types of complex behaviour found in nonisothermal systems. Steinfield et al. [16] stated explicitly that a single chemical reaction is said to have undergone autocatalysis or to be autocatalytic if at least one of the products is a reactant such that the chemical species also act as a catalyst. Recently, Alharthi et al. [17] investigated a mixed quadratic-cubic autocatalytic scheme in a one-dimensional reaction–diffusion cell. Considering the industrial applications of the ignition of homogeneous and heterogeneous reaction systems, Williams et al. [18] reported that this kind of reaction can occur between bulk of fluid and surface of a plate in which the boundary layer is formed upon. Thereafter, Chaudhary and Merkin [19], [20] presented a comprehensive analysis of homogeneous–heterogeneous reactions within the boundary layer flow formed on the surface with uniform thickness. Merkin [21] adopted order of magnitude (scaling analysis) together with the same assumption of large Reynolds number on Navier–Stokes equations as reported by Ludwig Prandtl and then presented a simple model for the interaction between homogeneous (or bulk) and heterogeneous (or surface) reaction involving two chemical species A and B within a thin layer formed on a surface in fluid flow.

It is a well-known fact that there exists a significant difference in the influence of nanoparticles and dust particles on transport phenomena. In the case of nanofluids, the combined action of Brownian motion and thermophoresis cools the channel by affecting the Nusselt number much more than that of the pure fluid case [22]. Physically, an increase in Brownian motions of these tiny particles implies enhancement of haphazard motion of tiny particles [23]. In another related report, Turkyilmazoglu [24] explained the motion of free/circular jets of copper (Cu), copper oxide (CuO), silver (Ag), alumina (Al2O3), and titanium oxide (TiO2) in water. It was found that the motion of water conveying Ag nanoparticles plays the role of the most coolant. Ahmad et al. [25] once remarked in a report on the motion of nanoparticles conveying by the basefluid near Riga plate that nanofluid flow parallel to the plate is assisted by the Lorentz force caused by the Riga plate, and buoyancy forces due to temperature or concentration gradient assist the fluid motion at the expense of increasing wall shear stress. Exergy analysis of heat transfer of magnetic nanoparticles, entropy analysis of nanofluid under the impact of Lorentz force, simulation of nanoparticles application for expediting melting of phase change material inside a finned enclosure, heat transfer of nanoparticles using innovative turbulator, Lorentz forces on water conveying Fe3O4, and comparative analysis between single-walled carbon nanotube and multi-walled carbon nanotube nanoliquids have been reported by Sheikholeslami et al. [26], [27], [28], [29], [30] and Mahanthesh et al. [31]. The literature confirms that it was Roman poet Titus Lucretius Carus (99–55 bc) who discovered the motion of tiny particles in a sunbeam. Carus et al. [32] remarked that the so-called mingling motion of dust particles in sunbeams is only possible due to a significant difference in temperature or/and pressure (i.e. air currents).

In the flow of a laminar fluid conveying particles through a tube, Starkey [33], [34] observed that suspended particle follows a path adjacent to the tube axis and moves with a component of translation normal to the streamlines and directed toward the axis. However the case, there exist lateral forces that act on the suspended particles. This report is a fundamental fact on the dynamics of fluids conveying dust. Addition of dust is a way of reducing resistance coefficient in the case of the turbulent flow of air through pipes [35]. Indirectly, this theory by Starkey was employed by Saffman [36], [37] to describe the motion of gas carrying small dust particles and the flow of a dusty gas between rotating cylinders. The rate at which the velocity of dust particles adjusts to changes in the gas velocity is known as the relaxation time. The effects of dust are best described using the concentration of the dust and the relaxation time. In addition, the relaxation time is strongly dependent on the size of the individual particles. As the force exerted on the dust by the gas is equal and opposite to the force exerted on the gas by the dust, continuity equation of the dust and equation of motion of the dust are required [36]. Marble [38] deliberated on the dynamics of gases containing small solid particles. In the report, it is remarked that the motion of a gas containing solid particles is dependent on the fluid–particle interaction and particle–particle interaction. Also, Stokes’ law is suitable to model the force accelerating the particle in the fluid. If a fluid contains particles of different radii, these particles move relative to each other with different velocities, and collisions between the two particles are bound to occur. Michael and Miller [39] investigated the motion of a dusty gas containing a uniform distribution of dust in a flowing fluid on the semi-infinite plane that moves parallel in simple harmonic motion and impulsively from rest with uniform velocity. It is found that the shear force is maximum at an initial time. The dynamics of dusty fluid near a solid circular cylinder was investigated by Miller [40]. In the study, it is assumed that the presence of the particles does not significantly alter the fluid motion. It was established that the mode1 could be used to design a centrifuge for separating particles of different types. According to Marble [41], the force exerted by a single particle moving through the gas is 6πσμ(up−u). For n noninteracting particles in a unit volume of space, the effective volumetric force Fp=n∗6πσμ(up−u). Also, the heat transferred from a single particle to the gas is 4πσk(Tp−T), and the heat exchange per unit volume is Qp=n∗4πσk(Tp−T). All these and many others were considered to present suitable governing equation of dusty fluid on the 40^th page of the report.

Suspension of many particles of uniform size in gas was investigated by Sone [42] as it flows over a body. It was discovered that the velocity of dusty fluids is almost exactly the same as that of the incompressible inviscid fluid (i.e. having no or negligible viscosity), which does not contain any dust particle. The two-phase flow of a dusty fluid over an asymmetric body at zero angles of attack when the mass fraction of suspended particles is sufficiently large and the two-way interaction between particle phase and gas phase is highly considered was presented by Barron and Wiley [43]. It is shown that the presence of the dust increases the pressure and decreases the temperature of the gas along the wedge surface. The significance of drag force due to slip and the transverse force due to slip-shear on the boundary layer flow of a dusty fluid over a semi-infinite flat plate was presented by Datta and Mishra [44]. It is observed that skin friction, shearing stress, and dimensionless drag coefficient are increasing properties of the concentration parameter. The motion of unsteady magnetohydrodynamics flow of dusty fluid on nonconducting plates when the lower plate oscillates with time period less than the upper, both the plates oscillate with the same time period, and the lower plate oscillates with time period greater than the upper was presented by Debnath and Ghosh [45]. The study shows that the speed of the dust particles decreases with the increase of the magnetic field. Also, an increase in the concentration of the particles retards the fluid motion when the time period of the oscillation of the plates is small.

Theoretical analysis of a squeezing fluid conveying dusty on a cylinder was presented by Hamdan and Barron [46]. It is shown that the resistance to motion produced by squeezing dusty fluids can be traced to the fluid viscosity, the fluid inertia, and the dust inertia. The dusty plasma, which is an ionised gas containing micron-size charged condensed grains (i.e. dust), led Fortov et al. [47] to carry out an experimental study of the motion of dust grains in the plasmas of capacitive discharge and glow discharge. It is remarked that the loss of momentum by the particles in the central part of the flow of radius may be due to the interaction of particles (internal friction of the gas component) or due to the shear viscosity caused by the Coulomb interaction in the dust subsystem. The motion of a viscoelastic fluid conveying dust particles during rocket boosters, film vapourisations, and cross-hatching on ablative surfaces was pointed out by Sivaraj and Kumar [48]. This led to the illustration of unsteady dusty viscoelastic fluid between two irregular porous plates in which one is moving relative to another when Joule heating, thermal radiation, viscous dissipation, and chemical reaction are significant. The multistep differential transform method was adopted by Rasekh et al. [49] to investigate the effect of Lorentz force on dusty fluid flow on a convectively heated hollow cylinder. It was observed that the temperature of the fluid and that of the dust increase with Eckert number. Also, the relaxation time of the dust phase decreases the velocities of the fluid and dust. Ramesh and Gireesha [50] deliberated on the effects of thermal radiation on the boundary layer flow of a dusty fluid over a stretchable surface. It is observed that the Nusselt number decreases with an Eckert number. The temperature of the fluid and that of the dust particle decrease with fluid–particle interaction parameter.

In a study on the dynamics of a dusty fluid over an exponentially stretchable surface in the presence of internal heat source by Pavithra and Gireesha [51], it was discovered that the fluid phase temperature is higher than that of the dust phase. Also, with an increase in the viscous dissipation term, the temperature of the fluid and that of the dust particle increase. In addition, the temperature increases with an increase in the magnitude of fluid–particle interaction parameter. The effect of melting heat transfer on the flow of incompressible viscous dusty fluid was investigated by Prasannakumara et al. [52]. Fluid phase velocity is found to be greater than that of the particle phase. Fluid–particle interaction parameter for velocity decreases fluid phase velocity and increases dust phase velocity. Meanwhile, fluid–particle interaction parameter for temperature increases the fluid phase temperature and decreases the dust phase temperature. Microstructures and particles affect significantly not only the mechanical properties of materials but also the thermal property. Experimental investigation of thermal conductivity of nanocomposites made of poly(2-vinyl pyridine) (97k) polymer containing 14- and 50-nm diameters of SiO₂ and different volume fractions (10 %, 20 %, 30 %, 40 %, 45 %, and 50 %) was presented by Tessema and Kidane [53]. It was shown that the particle size influences the thermal conductivity of the nanocomposites, although the effect of particle size on the thermal conductivity is minimal when compared with the volume fraction. Also, the thermal conductivity of the nanocomposites increases with the temperature at lower volume fractions, but this thermophysical property decreases with the temperature at higher volume fraction. The problem of unsteady flow of a dusty fluid in the presence of Lorentz force was investigated by Manjunatha and Gireesha [54]. Fluid–particle interaction parameter decreases the fluid phase velocity negligibly and increases the dust phase velocity. Two-dimensional Beltrami flow through the hexagonal channel of a dusty fluid was presented by Madhura et al. [55]. Beltrami effect decreases the velocity profiles of both the main fluid and dust phase. It was remarked that if the dust is very fine, i.e. mass of the dust particles is negligibly small, then the relaxation time of dust particle decreases.

Animasaun et al. [56] deliberated on the flow of water containing dusty, Cu and CuO particles over an inclined nonisothermal stretching sheet in the presence of induced magnetic field. It is observed from the figures that an increase in the fluid–particle interaction parameter depreciates the velocity, induced magnetic field, and temperature profiles. The outcome of the analysis of electrically conducting dusty fluid over a stretching surface shows that fluid velocity and the velocity of dust particles decrease with increasing Lorentz force [57]. Turkyilmazoglu [58] examined the momentum and temperature distribution during the motion of magnetohydrodynamic two-phase dusty fluid over a stretching/shrinking permeable sheet and discovered that two-phase velocity and temperature fields of either unique or dual solutions are suppressed by the presence of particle loading, fluid–particle interaction, and uniform magnetic field. In another study on the heat transfer from particles confined between two parallel walls and thermal dipole induced by a sphere placed between two walls with unequal temperatures, it was remarked that the rate of either heat conduction or mass transfer by diffusion from the cylindrical/spherical particle is a function of position and the radius of particles [59]. The significance of Hall current on the time-dependent flow of nanofluid (Cu–water) and dusty fluid was reported by Gireesha et al. [60]. It was concluded that the presence of dust particles reduces the growth of velocity and temperature boundary layer. Also, the flow and thermal fields of nanofluid phase are higher than the dust particle phase. Recently, Kalpana et al. [61] presented the motion of dusty fluid through a permeable medium with temperature-dependent viscosity and thermal conductivity. It is found that the velocity, temperature, and concentration distributions of dust phase behave the same as fluid phase. It was also reported that the presence of dust particles does not make much significant change on the characteristics of the fluid flow.

Searching through published articles on the subject matter, it is worth remarking that there exist a countable number of reports on the effects of internal heat source and buoyancy on the dynamics of dusty fluid over surfaces with uniform thickness. Meanwhile, the flow of a fluid on movable objects such as the bonnet of cars, pointed edge of bullets, pointed edge of aircraft, and umbrella is inevitable. In fact, rubbing of body creams on human body is another suitable example of fluid flow on an object with nonuniform thickness. Based on these facts, it is of importance to deliberate extensively on the dynamics of dusty fluid, velocity of the dust, and temperature of the dust over an upper horizontal surface of an object with nonuniform thickness when the transmission of energy in the form of electromagnetic is nonlinear in the presence of quartic autocatalytic kind of chemical reaction.

2 Mathematical Formulation of the Transport Phenomena

The motion of dust particles of diameter greater than that of nanoparticles is assumed to be undeformable and spherical in shape having a single radius suspended in air within the domain A(x+b)1−m2⩽y<∞. In this transport phenomenon, the particles are noninteracting, all the particles in a local volume have the same velocity vector, and the interaction between the solid particles in the motion of air is described by the Stokes drag Stg=6πμa1v and the transverse force due to slip-shear. However, the number density of the particles is taken to be very large, but the volume fraction of the particle cloud is considered to be so small that the interaction between individual particles may be neglected or highly simplified. The immediate layers of the dusty fluid on nonporous and nonmelting upper horizontal surface of a paraboloid of revolution are stretched parallel with velocity Uw=Uo(x+b)m. It is assumed that quartic autocatalytic chemical reaction with catalyst decay between reactant A and reactant B also occurs in the motion of steady two-dimensional dusty fluid. This chemical reaction is described as a kind in which the homogeneous (bulk fluid) react with the heterogeneous (three molecules of the catalyst at the surface) by isothermal quartic autocatalator kinetics and first-order kinetics, respectively. In this case, x axis is taken along the direction of the horizontal surface and y axis is normal to it.

The origins of x axis and y axis are not the starting point of the flow. Hence, the last layer of the flowing fluid at all points in the horizontal direction is a function y=A(x+b)1−m2. Consequently, the velocity along x direction u(x, y), velocity along y direction v(x, y), velocity of the particles along x direction u_p(x, y), velocity of the particles along y direction v_p(x, y), temperature of the fluid T(x, y), temperature of the dust particle T_p(x, y), the concentration of the bulk dusty fluid a(x, y), and the concentration of the catalyst ℓ(x,y) are the major dependent variables incorporated in the mathematical formulation. Also, the radius of each particle is a₁, the number density of the particles is n_p, the mass of each particle is m_p, specific heat capacity of dust particles is C_mf, thermal equilibrium time is τ_T, and relaxation time of the dust particle is T_v. As shown in Figure 1, the temperature of the horizontal surface with variable thickness is of the form Tw(x)=A1(x+b)1−m2; here parameter “m” is known as velocity power index, “b” is known as parameter related to stretching sheet, and A₁ is a temperature-related parameter. It is worth remarking that the free stream temperature is a constant function of temperature. Following the formulation of Saffman [37], Michael and Miller [39], Marble [41], and Gireesha et al. [60], the governing boundary-layer equations are of the form

Figure 1:

Coordinate system of a dusty fluid flow on an upper horizontal surface of a paraboloid of revolution.

For this case, the continuity equation and momentum equation for the motion of dust particles are

The energy equation in which nonlinear thermal radiation and space-dependent internal heat source are incorporated is of the form

Temperature of the dusts/particles in the viscous medium is of the form

Homogeneous–heterogeneous reaction model for the mixture between concentration of dusty fluid A and concentration of the catalyst B as stated in the reaction scheme is

Considering the fact that reactants A and B undergo chemical changes at the wall, hence homogeneous–heterogeneous catalytic reaction at the boundary is properly accounted for. Suitable boundary conditions governing the flow are

The transformation, nondimensionalisation, and parameterisation of governing (1–8) and boundary conditions 9 and 10 can be easily achieved using

It is worth deducing that thickness parameter χ, fluid–particle interaction for velocity β₁, mass concentration of the dust particles α, buoyancy parameter G_rm are of the form

Relaxation time of the dust particles τ_v, space-dependent parameter γ, and Prandtl number P_r are defined as

Also, introducing Eckert number E_c, mass concentration of the dust particles α, fluid–particle interaction for velocity β₁, and local fluid–particle interaction parameter for temperature β₂

It is very important to remark that the advective transport u2=[Uo(x+b)m]2, heat dissipation potential is CpΔT=Cp[Tw(x)−T∞], the first length scale is l1=(x+b)−m+1, and the second length scale is l2=(x+b)1/2.

In order to transform the domain from [χ,∞) to [0,∞), it is valid to use F(ς)=F(η−χ)=f(η), Ω(ς)=Ω(η−χ)=ω(η), Θ(ς)=Θ(η−χ)=θ(η), G(ς)=G(η−χ)=g(η), H(ς)=H(η−χ)=h(η), and Θp(ς)=Θp(η−χ)=θp(η) (Fig. 2). The final dimensionless governing equation for the case of unequal diffusion coefficients of chemical reactant A (homogeneous bulk fluid of the dusty fluid) and reactant B (heterogeneous catalyst at the surface) is

Figure 2:

Conversion of domain from η∈[χ,∞) to ς∈[0,∞).

Subject to

3 Research Methodology and Method of Solution

The corresponding boundary value problems were converted to an initial value problem (IVP) using the method of superposition and solved numerically. Numerical solutions of the IVP were obtained using classical Runge–Kutta method with shooting techniques and MATLAB package (bvp4c). First, the set of coupled nonlinear ordinary differential equations along with boundary conditions is reduced to a system of IVP using the method of superposition in Na [62]. The boundary value problem cannot be solved on an infinite interval, and it would be impractical to solve it for even a very large finite interval. In this study, the infinite boundary condition at a finite point ς is 2.5. In order to integrate the corresponding IVP, the values of the unknown dependent variables at ς = 0 are required. However, such values do not exist after the nondimensionalisation of the boundary conditions. In this case of unequal diffusivity, at specific value of δ and Λ, the correct values for G′(ς=0) and H′(ς=0) can be easily obtained once G(ς=0) is known. The suitable guess values are chosen, and then numerical integration was carried out. The calculated values of the unknown functions at the infinity ς∞=2.5. For more information, see [63], [64], [65], [66]. The reliability and validity of the numerical solution are established by comparing the results obtained from the two methods for a limiting case. As shown in Table 1, the accuracy of the results was also established and presented by comparing the limiting case of the present study with that of Ramesh and Gireesha [50] when Ec=δ=Λ=ξ=ℏ=β2=K=Sca=Grm=γ=α=β1=θw=0, mm+1=1, 1−mm+1=1, 2m+1=1 and Ra=∞.

4 Analysis and Discussion of Results

The significance of increasing the strength of the homogeneous reaction and strength of the heterogeneous reaction was investigated. It was discovered that the strength of homogeneous reaction and strength of heterogeneous reaction have no effect on the dynamics and concentration of dusty fluid flow because of the thermal diffusivity of air. The effects of Prandtl number were investigated using m = 0.5, α = 0.12, β1=0.13, Grm=1, Ec=0.01, n = 1.8, Sca=0.62, K = 0.4, δ = 1.2, ℏ = 5, β2=0.5, γ = 1.5, θw=1.2, Ra=8, and Λ = 0.4 when the thickness of the object is small and large. Table 2 shows that with an increase in the magnitude of Prandtl number the motion of air conveying dust particles along the surface of the object decreases at ς = 1. Meanwhile, the shear stress decreases significantly near the wall (i.e. ς = 0.6). Also, the temperature distribution decreases at the wall, at the free stream, and concentration of the bulk fluid decreases within the fluid domain (i.e. ς = 1.1). It is worth remarking that the concentration of the bulk fluid increases within the fluid layers near the upper horizontal surface of a paraboloid of revolution (ς = 0.1) that is thin (i.e. χ = 0.1). Also, it is confirmed that the shear stress across the flow of dusty fluid away from the wall experiences a decrease due to an increase in the magnitude of Prandtl number. However, when the magnitude of thickness is increased (i.e. in the case of the same fluid flow on an object with larger thickness), changes in the effects of Prandtl number are observed and presented as Table 3. In fact, this analysis shows that it is possible for the concentration of the bulk fluid to decrease with Prandtl number near the wall (ς = 0.1) at the rate of 0.019. Meanwhile, at ς = 0.1, the same property (concentration) increases with Prandtl number at the rate of 0.06075. It is worth deducing that the observed decrease in the concentration due to an increase in Prandtl number is in good agreement with the results illustrated as figure eight by Dogonchi and Ganji [68].

There is variation in the concentration of the homogeneous bulk fluid (dusty fluid) and catalyst at the wall due to an increase in the magnitude of buoyancy, internal space–dependent heat source γ, thickness parameter χ, and fluid–particle interaction for velocity β₁ using m = 0.5, α = 0.12, β1=[0.01,0.13], Grm=1 and Grm=3, Ec=0.01, Pr=0.72, n = 1.8, Sca=0.62, K = 0.4, δ = 1.2, ℏ = 5, β2=0.5, γ = 0.1 and γ = 1.5, θw=1.2, Ra=8, Λ = 0.4, and χ = 0.1, χ = 1, and χ = 2. It is shown in Table 4 that the concentration of the fluid conveying dust decreases with thickness of the paraboloid of revolution object. The results of this study further show that decrease in the concentration is achievable even when internal heat source is more enhanced (herein γ = 1.5). In physics, it is a well-known fact that there exists friction between the immediate fluid’s layer adjacent to the wall and the surface of an upper horizontal surface of a paraboloid of revolution. Gravity is the unbalanced force acting against the motion of dusty fluid as it climbs the object with nonuniform thickness. This study unravels the combined influence of friction and gravity on the flow of dusty fluid when the thickness is small and large. Physically, a positive increase in the magnitude of χ corresponds to an increase in the thickness of the upper horizontal surface of a paraboloid of revolution. Fluid flow on an object with a larger thickness is practically possible because of stretching, buoyancy, and heating of the fluid’s molecules.

When stretching, buoyancy, and heating of the fluid’s molecules are fixed (i.e. constant), and the magnitude of χ is small, lesser gravitational force is experienced on the motion of dusty fluid, while significant gravitation force is felt on the fluid flow when the magnitude of χ is large. When the effect of gravitation force on the motion is negligible χ = 0.1, the flow of dusty fluid is faster, while mixing of the species is negligible because of insufficient steady time frame for the quartic autocatalysis reaction. In the case of motion on larger thickness (i.e. χ = 2), the motion is slow because of the effect of gravity and leads to slower diffusion, while the reaction between the three molecules of the catalyst at the surface and one molecule of the bulk fluid is intense. This explains the occurrence of maximum concentration of dusty fluid when χ = 0.1 and minimum concentration in the flow on an object with a larger thickness χ = 2. Larger values assigned to fluid–particle interaction for velocity are seen to be a yardstick to negligibly enhance the concentration of dusty fluid with thickness parameter as presented in Table 4. Slope linear regression through the data point ascertains that due to the higher level in fluid–particle interaction, utmost decrease in the concentration of bulk fluid with thickness parameter occurs when β1=0.13. Based on the outcome of the analysis of dusty fluid, it is important to remark that utmost maximum decrease in the concentration of the dusty fluid with thickness parameter is ascertained at larger values of fluid–particle interaction for velocity β₁. It is found that the concentration is minimum when β₁ is large at various values of thickness parameter and when space-dependent internal heat source is small and large. This can be traced to the fact that fluid–particle interaction for velocity is a decreasing property of fluid’s velocity.

The flow of dusty fluid on an object with larger thickness is very slow; hence, the maximum concentration of the catalyst situated on an object with a larger thickness χ = 2 is ascertained and illustrated in Table 5. Further revealed in Table 5 is that the concentration of the catalyst in the flow over various thicknesses when internal heat source is more enhanced is lower in magnitude compared to when the internal heat source is negligibly small. For instance, in the flow over a larger thickness χ = 2, when γ = 0.1, the concentration of the catalyst in the case of smaller fluid’s particle interaction for velocity is 0.3034. But, when γ = 1.5 and χ = 2, the concentration of the catalyst in the same flow when β1=0.01 is 0.2990. According to the kinetic theory, molecules of the homogeneous bulk fluid of dusty fluid are in constant motion. Meanwhile, both molecules and kinetic energy increase with temperature, which is ascertained with an increase in wall temperature at larger values of γ. Scientifically, this leads to greater collisions between the molecules of the bulk fluid and that of inconsumable three molecules of the catalyst at the surface. The concentration of the bulk fluid decreases because its molecules are gradually used up in the process. In addition, the concentration of the catalyst increases because the three molecules of the catalyst only speed up the reaction through the provision of a lower energy path between reactants and product. This is the major reason why the concentration of homogeneous bulk fluid decreases and the concentration of the catalyst increases with thickness parameter and space-dependent internal heat source in Tables 4 and 5. Scientifically, internal heat energy generation is a well-known method of generating heat energy within a body by a chemical, electrical, or nuclear process. Meanwhile, the rate at which heat energy is being generated is higher at the wall where skin friction occurs. This explains the observed decrease in the concentration of dusty fluid with buoyancy in Table 4 and increase in the catalyst at the wall with buoyancy in Table 5. With an increase in the heat source, strong destabilisation is ascertained Matta [69]. As the flow of dusty fluid on an object with larger thickness is very slow, a larger portion of the catalyst is used up during the chemical reaction. Thereafter, the species/constituents of the three molecules of the catalyst are able to spread at the wall and hence lead to maximum concentration of the catalyst at the wall when χ = 2 as reported in Table 5. Reverse is the case χ = 0.1 where a small portion of the catalyst was used during quartic autocatalytic chemical reaction because the flow was very fast. This fact justifies the minimum concentration of the catalyst situated on an object with a smaller thickness χ = 0.1 in Table 5. Physically, fluid flow is faster when the molecules are heated. This reduces viscosity and boosts the transport phenomena of dusty fluid. In such cases, the constituents of the dusty fluid are still unable to mix thoroughly with the constituents of the catalyst and hence lead to a decrease in the concentration of the catalyst.

In Tables 6 and 7, the significance of dust particles and buoyancy is presented. Here, the concentration of the homogeneous bulk fluid increases negligibly due to an increase in the thickness of paraboloid of revolution when buoyancy is small. Reverse is the case when buoyancy is sufficiently large as concentration of the bulk fluid at the wall decreases. Table 7 shows that similar decreasing effects in the concentration of the catalyst at the wall due to an increase in the thickness of paraboloid of revolution are seen when buoyancy is small. It is evident that buoyancy is suitable to influence not only the concentration of the bulk fluid, but also the catalyst. This is true because fluid flow is more enhanced at larger values of buoyancy. This is strongly dependent on the mass of the conveyed dust particles. Physically, an increase in the magnitude of buoyancy parameter dependent on thermal volumetric expansion of the bulk fluid corresponds to an increase in wall temperature at all points in x direction. In this case, with an increase in G_rm, the transport phenomena are increased and consequently lead to faster rubbing of immediate fluid’s layer and the surface of the object. When two surfaces are in contact, such that one is moving relative to another, conversion of kinetic energy to thermal energy is inevitable due to friction. This process generates more heat and accounts for the observed increase in the local skin friction with G_rm as presented in Table 8. Without any doubt, this leads to maximum local skin friction at each value of G_rm when χ = 0.1 in the flow of dusty fluid compared to the case χ = 2 where the fluid flow is very slow as shown in Table 8. It is also seen that when χ = 0.1, skin friction coefficients F′′(0) increase with buoyancy G_rm at the rate of 0.483118616. In addition, when χ = 2 in Table 8, the same physical quantity increases with buoyancy G_rm at the rate of 0.506162704. In real life, temperature distribution increases at the wall with x and model as Tw(x). At this stage, the effect of gravity on the dust particles is influenced as their temperature being boosted due to the increase in wall temperature. This fact justifies why −Θ′(0) increases with G_rm when χ = 0.1 but decreases significantly when χ = 2 as shown in Table 8. In other words, the temperature of the dust is a decreasing property of buoyancy due to heating of the surface, which tends to push the dust far away to free stream where its temperature is dissipated to the cool environment.

Effects of thermal radiation R_a on the flow of dusty fluid were investigated using m = 0.5, α = 0.1, β1=0.01, Grm=5, Ec=0.01, Pr=0.72, n = 1.8, Sca=0.62, K = 0.5, δ = 1.2, ℏ = 5, β2=0.5, γ = 5, θw=1.2, and Λ = 0.4, when thickness of the horizontal paraboloid of revolution is small χ = 1. It is established in this study that with an increase in the magnitude of thermal radiation, the flow of dusty fluid in y direction decreases. Near the wall, thermal radiation has no effect on the flow of air conveying dust along y direction, but a decrease in the flow parallel to the surface within 0.5⩽ς⩽1.8 is ascertained. These facts can be deduced in Table 9 where it is shown that the vertical velocity at the free stream (ς=2.5) decreases with thermal radiation at the rate of −0.014475. At ς, the horizontal velocity decreases with thermal radiation at the rate of −0.00695. With an increase in the thermal radiation, shear stress between successive layers within 0.4⩽ς⩽0.8 decreases. Reverse is the case at the wall. As shown in Table 9, F′′(0) increases with R_a at the rate of 0.021525. In this study, it is observed that with an increase in the thermal radiation across the flow of air conveying dusty, velocity of the dust particles Ω(ς) decreases, temperature distribution increases near the wall 0⩽ς⩽0.51 with a significant decrease near the free stream, temperature gradient decreases within the interval 0.25⩽ς⩽0.8, there is no significant effect on the concentration of the homogeneous bulk fluid and the concentration of the catalyst, and temperature of the dust Θp(ς) increases near the wall 0⩽ς⩽0.41. For brevity, graphical representations of these observed effects are not shown. However, it can be deduced from Table 9.

Physically, an increase in the magnitude of thermal radiation corresponds to an increase in the electromagnetic radiation that was generated by the thermal motion of particles in flowing liquids. Provided the temperature of any matter is above absolute zero, the motion of these particles sets in either by dipole oscillation or charge–acceleration, and emission is guaranteed. This process affects the motion of fluid because thermal radiation is a mode of heat transfer. With an increase in this partial or sometimes full conversion of thermal energy into electromagnetic energy, the viscosity is increased leading to the observed decrease in the flow along the vertical axis and the horizontal axis as reported in Table 9. Definitely, as the motion is barricaded due to an increase in the magnitude of thermal radiation, a decrease in local skin friction is bound to occur. This is only true because of the presence of nanoparticles. Table 9 indicates that the local skin friction coefficients in the flow of air conveying dust increase with thermal radiation. As shown in Table 9, when temperature parameter (θ_w) is increased, two things are involved. It is either the wall temperature increases at a fixed small free stream temperature or the free stream temperature decreases suddenly at a fixed higher value of wall temperature. Based on the physical configuration of the flow, free stream temperature is known to be very small in quantity (i.e. T∞≈0). Sequel to this fact, exposure of dust particles to a significant increase in the wall temperature explains why the vertical velocity and horizontal velocity of the flow in x direction decrease with an increase in local skin friction coefficients when θ_w = 1.2 compared to when θ_w = 5. In the study of matter, most especially fluids with a specific level of temperature comprise particles (made up of molecules and atoms) that not only possess kinetic energy but also interact. The level of kinetic energy of molecules and atoms involved in random movements can be used to measure thermal energy. Whatever the case, the collision of nanoparticles with these particles (made up of molecules and atoms) before, during, and after collision is a major factor. This is the major reason why the concentration of the homogeneous bulk fluid and catalyst decreases and increases negligibly with thermal radiation when θw=1.2, as shown in Table 9. In the motion of dusty fluid, decrease in the concentration of the homogeneous bulk fluid was estimated using the slope of linear regression through the data points of G(0) and R_a as −0.0003. Above all, the temperature of the dust increases with thermal radiation at the wall ς = 0 and a few distances away from the upper horizontal surface of a paraboloid of revolution ς = 0.2. In addition, the velocity of the dust is found to be a decreasing function of thermal radiation.

Comparative analysis between the flow along an object with smaller thickness and larger thickness at various values of buoyancy was simulated using 5⩽Grm⩽10, m = 0.5, α = 0.12, β1=0.01, Ec=0.01, Pr=0.72, n = 1.8, Sca=0.62, K = 0.4, δ = 1.2, ℏ = 5, β2=0.01, γ = 5, θw=1.2, Ra=8, and Λ = 0.4. Figures 3 and 4 show that the flow of the dusty fluid is seen to be an increasing function of buoyancy parameter G_rm. As unusual, the flow of the dusty fluid near the free stream increases significantly with no significant effect at the wall. However, an utmost vertical velocity is ascertained in the flow of dusty fluid over an object with larger thickness (Fig. 4). Also, increasing buoyancy parameter improves the transport phenomenon in x direction (i.e. over the horizontal paraboloid of revolution) (Figs. 5 and 6). These results corroborate with the results illustrated as Figure 3 by Dogonchi and Ganji [70], where it was stated that with rising buoyancy parameter the fluid velocity increases and the temperature profile decreases. With an increase in the magnitude of buoyancy depending on the volumetric thermal expansion of two-dimensional flow of dusty fluid along an upper horizontal surface of a paraboloid of revolution, wall temperature increases. In view of this, the effect cannot be feasible near the wall in the flow along y direction during the motion due to the direction of the motion. Besides, thermal expansion is always small but never insignificant. Due to these facts, the effect of stretching is minimal near the free stream where buoyancy has a better effect on the flow along y direction. Shear stress between successive layers near the wall increases with G_rm. At ς = 0.5 in the case of χ = 0.1, all the profiles turn, and it is seen that G_rm decreases shear stress between successive layers within the interval 0.5⩽ς⩽1.8 (Figs. 7 and 8). Maximum local skin friction is seen in the flow of dusty fluid over an object with smaller thickness (χ=0.1). Using the slope linear regression through the data point suggested by Shah et al. [71], F′′(0) increases with G_rm at the rate of 0.483118616 when χ = 0.1 and 0.506162704 when χ = 2 (Table 8). The increase of buoyancy parameter G_rm causes a rising of the velocity of the dust particles (Figs. 9 and 10). It is seen that the velocity of the dust particles at the wall is almost the same as the velocity at the free stream. This study shows that the velocity of dust particles in the flow over an object with larger thickness is greater than that of flow of dusty fluid on a thin object (Fig. 9). Local fluid acceleration force and buoyancy force are major contributions of fluid–particle interaction force. With an increase in buoyancy force, local fluid acceleration force is also enhanced. This justifies the observed increase in the velocity of the dust particles due to buoyancy. Maximum increase in the velocity of the dust particles due to buoyancy is seen in the flow along a larger thickness because of virtual mass force and drag force. This is true as virtual mass force is parallel to the relative acceleration between the phases Jamshidi and Mazzei [72]. The temperature across the flow of dusty fluid is found to be decreasing due to the increase in the magnitude of buoyancy parameter G_rm (Figs. 11 and 12). However, maximum temperature distribution is ascertained in the flow over an object with a smaller thickness. The temperature gradient shows a depressing behaviour near the wall and increases within the fluid domain (Figs. 13 and 14). The results of this study, Figures 15 and 16, indicate that the concentration of homogeneous bulk fluid increases with buoyancy parameter G_rm. Buoyancy parameter G_rm alters the concentration gradient of the homogeneous bulk fluid far away from the wall (Figs. 17 and 18). The results of this study indicate that the concentration of the catalyst is a decreasing function of buoyancy (Figs. 19 and 20). In the motion of dusty fluid, buoyancy is found to be suitable to enhance/increase the concentration gradient near the free stream (Figs. 21 and 22).

Figure 3:

Effects of G_rm on the velocity along y direction when χ = 0.1.

Figure 4:

Effects of G_rm on the velocity along y direction when χ = 2.

Figure 5:

Effects of G_rm on the velocity along x direction when χ = 0.1.

Figure 6:

Effects of G_rm on the velocity along x direction when χ = 2.

Figure 7:

Effects of G_rm on the shear stress when χ = 0.1.

Figure 8:

Effects of G_rm on the shear stress when χ = 2.

Figure 9:

Effects of G_rm on the velocity of the dust particles when χ = 0.1.

Figure 10:

Effects of G_rm on the velocity of the dust particles when χ = 2.

Figure 11:

Effects of G_rm on the temperature distribution when χ = 0.1.

Figure 12:

Effects of G_rm on the temperature distribution when χ = 2.

Figure 13:

Effects of G_rm on the temperature gradient when χ = 0.1.

Figure 14:

Effects of G_rm on the temperature gradient when χ = 2.

Figure 15:

Effects of G_rm on the concentration of the homogeneous bulk fluid when χ = 0.1.

Figure 16:

Effects of G_rm on the concentration of the homogeneous bulk fluid when χ = 2.

Figure 17:

Effects of G_rm on the concentration gradient of the homogeneous bulk fluid when χ = 0.1.

Figure 18:

Effects of G_rm on the concentration gradient of the homogeneous bulk fluid when χ = 2.

Figure 19:

Effects of G_rm on the concentration of the catalyst when χ = 0.1.

Figure 20:

Effects of G_rm on the concentration of the catalyst when χ = 2.

Figure 21:

Effects of G_rm on the concentration gradient of the catalyst when χ = 0.1.

Figure 22:

Effects of G_rm on the concentration gradient of the catalyst when χ = 2.

The results of this analysis show that temperature of the dust is a decreasing property of buoyancy. As shown in Figures 23 and 24, the observed decrease in the temperature of the dust can be greatly minimised when the thickness of the object is large. The corresponding effects of increasing the fluid–particle interaction for velocity β₁ on the motion of air conveying dust over the horizontal paraboloid of revolution with small thickness χ = 0.1 and larger thickness χ = 2 were investigated using m = 0.5, α = 0.12, Grm=5, Ec=0.01, Pr=0.72, n = 1.8, Sca=0.62, K = 0.4, δ = 1.2, ℏ = 5, β2=0.01, γ = 5, θw=1.2, Ra=8, and Λ = 0.4. When the parameter ranges from 0.1⩽β1⩽0.13, its effect is only significant on dust’s velocity Ω(ς) and velocity gradient of the dust in the viscous fluid Ω′(ς). It is seen that Ω(ς) is a decreasing function of β₁, while Ω′(ς) increases with β₁. These can be deduced from Figures 25–28. It is worth noticing that utmost velocity of the dust occurs at a minimum value of fluid–particle interaction for velocity β₁ in the flow over a larger thickness of paraboloid of revolution. Physically, the viscosity of the viscous fluid conveying the dust increases due to an increase in the magnitude of fluid–particle interaction for velocity β₁. This fact explains the reason why fluid–particle interaction has no effect on the transport phenomena when β₁ increases negligibly and decreases the velocity of the dust particles. This is true because the rate of increase in viscosity is minimal and not sufficient to boost the temperature distribution. The observed increase in temperature distribution can be traced to the fact that fluid–particle interaction for velocity β₁ is a multiplier of viscous dissipation. When the fluid–particle interaction for velocity was greatly enhanced 0.5⩽β1⩽4.5, although not significant, but it is noticed that the vertical velocity increases negligiblly within the fluid domain 0.5⩽ς⩽1.8, it is ascertained that the horizontal velocity increases with β₁ in the interval 0.1⩽ς⩽0.63, the shear stress between successive layers of dusty fluid near the wall increases with β₁, and the temperature of the dust increases with fluid–particle interaction for velocity β₁; all these reports are not shown graphically for the flow on small and larger thickness of paraboloid of revolution. It is observed in Figures 29–36 that with an increase in the magnitude of the parameter that represents fluid–particle interaction for velocity for the motion on small and large thicknesses of paraboloid of revolution χ = 0.1 and χ = 2, the velocity of the particles decreases (Figs. 29 and 30), the velocity gradient of the particles increases (Figs. 31 and 32), and temperature distribution across the flowing fluid increases (Figs. 33 and 34), while the temperature gradient decreases within the interval 0.25⩽ς⩽1 (Figs. 35 and 36). Comparing Figures 33 and 34, it is noteworthy that maximum temperature distribution is noticed at all the magnitudes of fluid–particle interaction for velocity in the flow of dusty fluid over a smaller thickness of a paraboloid of revolution. Physically, there exists a significant difference in the dissipation of internal energy in the flow over an object with smaller thickness and larger thickness of paraboloid of revolution. This explains why an increase in fluid–particle interaction for velocity leads to maximum distribution of temperature distribution in the flow along a smaller thickness of paraboloid of revolution where dissipation of energy is minimal. As shown in Figures 35 and 36, maximum temperature gradient occurs in the flow over larger thickness of paraboloid of revolution as −Θ′(0.6)≈−1.55.

Figure 23:

Effects of G_rm on the temperature of the dust when χ = 0.1.

Figure 24:

Effects of G_rm on the temperature of the dust when χ = 2.

Figure 25:

Effects of β₁ on the velocity of the dust particles when χ = 0.1.

Figure 26:

Effects of β₁ on the velocity of the dust particles when χ = 2.

Figure 27:

Effects of β₁ on the velocity gradient of the dust particles when χ = 0.1.

Figure 28:

Effects of β₁ on the velocity gradient of the dust particles when χ = 2.

Figure 29:

Effects of β₁ on the velocity of the dust particles when χ = 0.1.

Figure 30:

Effects of β₁ on the velocity of the dust particles when χ = 2.

Figure 31:

Effects of β₁ on the velocity gradient of the dust particles when χ = 0.1.

Figure 32:

Effects of β₁ on the velocity gradient of the dust particles when χ = 2.

Figure 33:

Effects of β₁ on the temperature distribution when χ = 0.1.

Figure 34:

Effects of β₁ on the temperature distribution when χ = 2.

Figure 35:

Effects of β₁ on the temperature gradient when χ = 0.1.

Figure 36:

Effects of β₁ on the temperature gradient when χ = 2.

Next stage is to unravel the effects of local fluid–particle interaction for temperature β₂ on the dynamics of dusty fluid using m = 0.5, α = 0.12, β1=0.01, Grm=5, Ec=0.01, Pr=0.72, n = 1.8, Sca=0.62, K = 0.4, δ = 1.2, ℏ = 5, γ = 5, θw=1.2, Ra=8, and Λ = 0.4 over the transport phenomenon over the horizontal paraboloid of revolution with small thickness χ = 0.1 and larger thickness χ = 2. With an increase in the magnitude of local fluid–particle interaction for temperature β₂, it is seen that the velocity of the flow in y direction increases, the velocity of the flow in x direction (parallel to the stretching on upper horizontal surface of a paraboloid of revolution) increases near the wall, the shear stress increases within successive layers of dusty fluid in the interval 0⩽ς<4, concentration of the homogeneous bulk fluid increases, and concentration of the catalyst at the surface decreases. All these observed effects are not reported. The outcome of the analysis shows that the velocity of the dust particles Ω(ς) increases with β₂. In this case, reported as Figures 37 and 38, Ω(ς=0)<Ω(ς=2.5). It is worth remarking that the maximum velocity of the dust particles occurs at larger values of β₂ when χ = 2. More so, the temperature distribution across the flow decreases with β₂. Minimum temperature distribution occurs in the flow on a larger thickness of paraboloid of revolution χ = 2 (Figs. 39 and 40). The temperature of the dust Θp(ς) is seen to be an increasing property of the local fluid–particle interaction for temperature. The flow of air conveying dust particles under consideration is in thermal equilibrium with itself because the temperature within the flow is temporally constant and spatially uniform. This is true because there is no net flow of thermal energy between fluid phase and dust phase. However, as β₂ grows, τ_T is fixed, while heat capacity ρCp decreases. This justifies the results illustrated as Figures 37, 38, 41, and 42, which show clearly that temperature and velocity of the dust are increasing properties of local fluid–particle interaction for temperature. This fact can be further established in Figures 39 and 40, where the temperature distribution is found to be a decreasing property of β₂. At various values of β₂, Figures 41 and 42 show that the temperature of the dust is more enhanced in the case of χ = 0.1 compared to that of χ = 2. It is further discovered that when the local fluid–particle interaction for temperature is more enhanced 0.5⩽β2⩽4.5, vertical velocity and horizontal velocity are influenced positively (Figs. 43–46).

Figure 37:

Effects of β₂ on the velocity of the dust particles when χ = 0.1.

Figure 38:

Effects of β₂ on the velocity of the dust particles when χ = 2.

Figure 39:

Effects of β₂ on the temperature distribution when χ = 0.1.

Figure 40:

Effects of β₂ on the temperature distribution when χ = 2.

Figure 41:

Effects of β₂ on the temperature of the dust when χ = 0.1.

Figure 42:

Effects of β₂ on the temperature of the dust when χ = 2.

Figure 43:

Effects of β₂ on the velocity along y direction when χ = 0.1.

Figure 44:

Effects of β₂ on the velocity along y direction when χ = 1.5.

Figure 45:

Effects of β₂ on the velocity along x direction when χ = 0.1.

Figure 46:

Effects of β₂ on the velocity along x direction when χ = 1.5.

Also, the velocity of the dust particles illustrated in Figures 47 and 48 and temperature of the dust increase, while the temperature distribution decreases with β₂ (Figs. 49 and 50). When the local fluid–particle interaction for temperature is more enhanced 0.5⩽β2⩽4.5, increasing β₂ boosts the temperature of the dust when χ = 0.1 and χ = 2, as shown in Figures 51 and 52. It is worth noticing that maximum Θp(ς) is found near the wall 0⩽ς⩽0.5. Increase in the local fluid–particle interaction for temperature corresponds to an increase in thermal diffusivity; this fact justifies the reason why Θp(ς) increases with β₂. However, the increase in thermal diffusivity enhances the number density of the particles in the flow over an object with a smaller thickness (χ = 0.1) due to less elevation compared to (χ = 2) with a substantial elevation where minimum temperature of the dust is found in Figure 52. The effects of mass concentration of the dust particles α on the velocity of the dust particles Ω(ς) were simulated using m = 0.5, β1=0.01, Grm=5, Ec=0.01, Pr=0.72, n = 1.8, Sca=0.62, K = 0.4, δ = 1.2, ℏ = 5, β2=0.5, γ = 5, θw=1.2, Ra=8, and Λ = 0.4 when the thickness of the horizontal paraboloid of revolution is small and large. It is seen that increasing the magnitude of mass concentration of the dust particles α has no effect on all the profiles except the velocity of the dust particles Ω(ς). It is observed in Figures 53 and 54 that Ω(ς) decreases with α. In the transport phenomenon of viscous Newtonian fluid conveying dust particles over the horizontal paraboloid of revolution with small thickness χ = 0.1, maximum velocity of the dust particles Ω(ς=2.5)≈2.295. In the flow over a larger thickness (i.e. χ = 2), Ω(ς=2.5)≈2.536. Literally, increase in the magnitude of mass concentration of the dust particles (α) implies an increase in the product of number of particles and mass particles. This fact justifies the observed decrease in the velocity of the dust particles due to an increase in α. The outcome of the results presented as Figures 3 and 4 can be traced to the property of dust particles, presence of particle loading, and fluid–particle interaction, which are affected by temperature. It is worth remarking that increase in the thermal conductivity depends on shape and size of the solid nanoscale particles Malik and Nayak [73].

Figure 47:

Effects of β₂ on the velocity of the dust particles when χ = 0.1.

Figure 48:

Effects of β₂ on the velocity of the dust particles when χ = 2.

Figure 49:

Effects of β₂ on the temperature distribution when χ = 0.1.

Figure 50:

Effects of β₂ on the temperature distribution when χ = 2.

Figure 51:

Effects of β₂ on the temperature of the dust when χ = 0.1.

Figure 52:

Effects of β₂ on the temperature of the dust when χ = 2.

Figure 53:

Effects of α on the velocity of the dust particles when χ = 0.1.

Figure 54:

Effects of α on the velocity of the dust particles when χ = 2.

First, the observed increase in the velocity due to buoyancy parameter dependent on the volumetric expansion of the dusty fluid corroborates with the conclusion of Shah et al. [71], which says that, with an increase in the magnitude of buoyancy parameter, increase in the velocity profile is guaranteed. In addition, maximum vertical velocity is ascertained when either thickness or internal heat source is considerably large in magnitude. Considering the fact that the temperature of the horizontal surface with variable thickness is of the form Tw(x)=A1(x+b)1−m2, it increases as x grows large. The corresponding input of heat energy due to the increase in T_w(x) is very less in the flow of fluid along a small thickness of paraboloid of revolution compared to the significant input of heat energy injected into the boundary layer owing to T_w(x). It is noteworthy that this is responsible for maximum velocity of fluid flow along a surface with larger thickness as seen in Figures 4, 10, 38, 44, 48, and 54. The results of this study presented as Figures 5 and 6 corroborate with Figure 6 reported by Seth et al. [74] on the effects of rotation and Hall current on buoyancy-induced flow and impulsively moving vertical surface with ramped temperature. In the report, the velocity increases with buoyancy parameter not only for ramped temperature case but also for isothermal case. Physically, increase in buoyancy-related parameter corresponds to an increase in the wall temperature T_w(x), and this makes the bond(s) between the base fluid to become weaker and reduces internal friction and the gravity to become stronger enough. It is worth pointing out that the significance of heat source in the boundary layer generates heat energy, which causes the temperature of the fluid to increase. The effect of increase in the temperature causes an increase in the flow velocity and temperature distribution within the boundary layer. The process was explained by Malik and Nayak [73], p. 804] as follows, “the fluid close to the heat source gets heated quickly due to absorption of heat, the molecules of fluid move upward due to buoyancy force and cold fluid replaces the empty region.” It is also seen in Figures 5 and 45 that the flow of dusty fluid over an object with smaller thickness is faster than that of larger thickness. This fact corroborates the observation by Anjali-Devi and Prakash [75], which says, “increasing in wall thickness barricades the motion to slow down.” Without any doubt, maximum horizontal velocity is guaranteed in the flow of dusty fluid along a smaller thickness of a paraboloid of revolution object at various values of buoyancy parameter.

5 Conclusion and Recommendation

In this report, the dynamics of dusty fluid with a great emphasis on the significance of buoyancy, nonlinear thermal radiation, space-dependent internal heat source, fluid–particle interaction for velocity, and temperature have been explored.