Dynamics of Marangoni Convection in Hybrid Nanomaterials Flow with Dust Particles Random Motion

: Here we are working on the flow of dust particles in hybrid nanofluid. Marangoni convective flow of hybrid nanofluid is accounted by considering silver and copper as nanoparticles and water as base fluid. Dust particles and nanoparticles are used in this flow are spherical type. For thermal conductivity we have considered the Maxwell model. Porous medium is placed over a stretching sheet. Flow is generated via stretching sheet. MHD effects are also considered. Nonlinear equation of fluid phase and dust phase are converted in to ODE's by suitable transformations. These ordinary differential equations are solved numerically. Effect of involved dimensionless variables against velocity and temperature of hybrid nanofluid and dust phase, skin friction and Nusselt number of hybrid nanofluid is studied through graphs and tables. It is observed that temperature and velocity is more in case of hybrid nanofluid as compared to dust phase. Velocity of Ag-Cu water hybrid nanofluid enhances for greater mass concentration of dust particles. Velocity in both phase decay for higher porosity variable. Good match of results are seen by comparing current situation to earlier study in particular case.


1: Introduction
Heat and momentum transfer finds wide range of applications in sciences and engineering. In the last few years, different researchers have worked on studying the heat and mass transfer in dusty and nanofluids. In this study the variation of mass and heat transfer of nanofluids on the stretching surfaces, volume fraction of nanofluids is taken in consideration which area embedded with dust particles. The purpose of this study is to find the metallic oxides and metals while they are Mohankrishna et al. [10] discussed the effects of radiation on unsteady convective flow of nanofluids over a vertical plate as heat source. Akbar et al. [11], [12] discussed the flow of CNT Prandtl and suspended nano fluids on different sheets. Nadeem et al. [13] studied the partial slip effect on non-aligned stagnation flow on convective stretching surfaces. Malvandi et al. [14] studied the flow of nanofluids on the stretching surfaces slip effects was also considered. Prasad et al. [15] studied the variation in viscosity on MHD flow on stretching surfaces. Various studies are carried on dusty nanofluids which can be seen in literature [16][17][18][19][20][21]. Sheikholeslami et al. [22] studied the heat transfer and physical properties of nanofluids. Rashidi et al. [23] discussed the effect of buoyancy on MHD flow of nanofluids in radiations. Hayat et al. [24] studied the MHD flow of nanofluids on a stretching sheet. Sheikholeslami and Ganji [25] studied the effect of nanofluids in rotating frame. Elbashbeshy et al. [26] studied different properties of radiation on nanofluids on a moving surface with different thicknesses. Khader et al. [27] studied the boundary layer flow and slip boundary conditions. Abdel-Wahed et al. [28] simulated the effect of different thicknesses on the heat source of nanofluids. Hayat et al. [29] discussed the characteristics of different thicknesses on the behavior of flow and the parameters that are considered are temperature and velocity. The Brownian motion also plays an important role in nanofluids where real BC are imposed [30]. Rout and Mishra [31] studied the heat energy carrying on nanofluids on the stretching surfaces. Some more studies about these type of flows are presented in Refs. [32][33][34][35][36][37][38][39].
Main purpose of this study is to analyze the flow of dust particles in hybrid nanofluid. In previous literature there is no study in Marangoni convective flow of hybrid nanofluid is accounted by considering silver and copper as nanoparticles and water as base fluid. Porous medium is placed over a stretching sheet. Flow is generated via stretching sheet. MHD effects are also considered.
Effect of pertinent variables are studied in detail against velocity, temperature, Nusselt number and surface drag force. Also the present study is validated by comparing values of heat transfer with [32], [33] and [34].

2: Modeling
Two dimensional, steady MHD flow of hybrid nanofluid is considered. Marangoni convective boundary conditions are considered for momentum equation. Dust particles flow in hybrid nanofluid is examined in detail. Comparative results of dust phase and fluid phase are presented.
Silver and Cu are nanoparticles and water is base fluid. Porous medium is considered. Cartesian coordinates are used where sheet is place along x  axis and flow is towards positive y  axis.
() w ux is the stretching velocity of the sheet along x  axis (See Fig. 1).

. 1:
Required equations of dust phase and fluid phase are designed as [8], [32], [34] and [39]: where   respectively. The volume fraction of the nanoparticles (silver and copper) and dust particle are 1 , T T  denote the kinematic viscosity, number density of dust particles, density, Stokes resistance, dust particle of mass, magnetic field strength, dynamic viscosity, density, heat capacitance of hybrid nanofluid, respectively, particles phase density, thermal equilibrium and relaxation time of dust particles, specific heat for fluid and dust particles, porosity constant, the porosity parameter, wall temperature and ambient temperature respectively. Surface tension is defined as Transformations are considered as: , , .
After implementation of Eq. (8), Eqs. (1) and (2) are satisfied and Eqs. (3), (4), (5), (6) and (7) take the form where Re, , Nusselt number and skin friction for given flow system are defined as    A are expressed as:  is directly proportional to Lorentz force which produces resistance between the particles to flow hence velocity decays. Furthermore it is also seen that decrease in dust phase is more as compared to hybrid nanofluid because dust particles are denser than nanoparticles. Due to this reason velocity of dust particle is less than nanoparticles.

 
Since density of base fluid is inversely proportional to    so due to less density velocity of the hybrid nanofluid enhances. Influence of interaction parameter    on fluid phase and dust phase is presented in Fig. 4. Decrease in   , fF  is seen for higher estimation of   0,1, 2,3 .

 
Impact of volume fraction of dust particles on hybrid nanofluid flow is seen in Fig.   5. Increment in velocity of hybrid nanofluid is seen for greater estimation of  

 
Physically motion of the fluids slow down because when there is more saturation of nanoparticles in the fluid than nanofluid will be more dense due which velocity reduces. It is also observed that velocity in fluid phase is more than dust phase. Impact of interaction parameter   T  on temperature field of hybrid nanofluid and dusty nanofluid is portrayed in Fig. 9 Table 4 is designed for validation of our problem in particular case with [32], [33] and [34]. It is noted that there is approximately equal values of (0)    in present study with previous literature. . 7: ′ (ƞ) ′ (ƞ) 1 .