Numerical analysis of electro conductive Nano-fluid flow and thermal transport over a moving surface with heat generation and absorption

Introduction

In recent years, research on nanofluidic flows has attracted great interest of researchers due to increased implementations in various fields of technology, science, bio-mechanics, chemistry and nuclear industry. The term nanofluid means adding a smaller amount of nanometer-sized particles nominally below 100 nm to base fluids such as oil, water, bio fluids and ethylene and lubricant. The term was first introduced by Choi et al. [1] ; he added tiny nanoparticles to the base fluid to increase the thermal conductivity. Uddin et al. [2] Research on radiative Convective Nano fluid Flow past a Stretching/ Shrinking Sheet with Slip Effects. The purpose of this study is to improve the thermo physical properties of convection and heat transfer of unsteady nanoliquids through moving Stretching/Shrinking Sheet in the presence of Radiative Convective Nanofluid. B. Mahanthesh et al. [3] analytically investigated the effect of an induced magnetic field on a flat porous plate using an incompressible copper-water nanofluid. The flow should be laminar, stable, and two-dimensional. The plate is subject to normal free flow and suction velocities. Aamir Ali et al. [4] investigated on electro-magneto hydrodynamic nanofluid flow with radiation and variable heat flux impact on a slender tension sheet. It has been observed that linear stretching sheets have been extensively used in heat transfer research. In recent years, MHD fluid flow has attracted the attention of researchers due to controllable heat transfer coefficient. Magnetohydrodynamics plays a decisive role in controlling the cooling rate and melt separation. The term "magnetohydrodynamics" is a combination of his three basic terms: magneto for magnetic fields, hydro for fluids/liquids and dynamics for particle evolution. Vikas Poply [5] analyzed the heat transfer characteristics in a Magnetohydrodynamic (MHD) boundary layer nanofluid flow over a stretching sheet.

They found that the skin friction coefficient increases with increasing magnetic parameter M. Ahmad et al. [6] numerically investigated the effects of thermal radiation and heat generation on the flow of conducting magnetic nanofluid through nonlinear stretchable films through porous media via frictional heating. Study of flow and heat transfer in permeability stretch magnetohydrodynamic nanofluidic sheets under the influence of convection boundary conditions is discussed by B. Srisailam et al. [7] . They observed that the thickness of the boundary layer reduces as the magnetic parameter (M) increases, whereas the temperature and concentration profile increase. Sonika Sharma et al. [8] investigated an exponentially stretched sheet with a magnetic force to explore the Darcy-Forchheimer flow of a two-dimensional blood-based nanofluid and the effect of different relevant parameters of the fluid flow assumption for velocity and temperature profiles. The mixed convection laminar flow of couple stress fluid induced by oscillatory stretching sheet in presence of nanoparticles has been investigated analytically by Shankar et al. [9] . Daniel et al. [10] studied the combined effects of thermal stratification, applied magnetic field, and thermal radiation on a boundary layer flow of electrically conducting nanofluid over a nonlinear stretching sheet. The radiative heat transfer is to be sensitive to an increase in the fluid temperature and thicker thermal boundary layer thickness. Thin layer mechanisms are often used to improve the surface properties of solids. Penetration, degradation, stiffness, illumination, diffusion, absorption, and electrical performance are all properties of bulk media that can be enhanced by thin films. So for this the time-dependent nonlinearly convective flow in thin film nanoliquid over an inclined stretchable sheet with magnetic effects was investigated by Anwar et al. [11] . Iskandar et al. [12] considered the mixed convection and heat transfer across exponentially stretching vertical planes in hybrid nanofluids. They observed that the heat transfer rate enhances in the presence of nanoparticles. The effect of the induced magnetic field on the motion of nanofluid Al2O3 containing rotationally moving microbes through the boundary layer is examined by Nabil et al. [13] .They found that increases in velocity profiles shows increase in M and the Grashof number. While it decreases with an increase in the reciprocal magnetic Prandtl number. Oyier et al. [14] investigated the effect of induced magneticfield on the MHD laminar flow of oldroyd-B nanofluid over elongating surface and heat producing. It was found that the external magnetic field perpendicular to layer affects the fluid flow. Flow and heat transfer induced by exponentially shrinking hybrid films Nano particles are studied by Iskandar et al. [15] . Aluminum oxide (Al2O3) and copper (Cu) nanoparticles are suspended in water to form an Al2O3-Cu/water hybrid nanofluid. Anwar et al. [16] analysied the study on the induced magnetic field applied by the exponentially stretched plates to the flow of second-class fluids with different thermal conductivities. They considered chemical reactions and viscous dissipation effects under the influence of thermophoresis and Brownian motion on exponentially stretched sheets. Alblawi et al. [17] studied on steady flow of Buongiorno's model on the exponentially stretching surface. The aim of the study was to analyze the heat transfer coefficient of nanoliquids on momentum boundary layer thickness, thermal boundary layer thickness, and solution boundary layer thickness are astonishing. Flow and heat transfer through Shrinking Sheet using a modified Buongiorno nanoliquid model by Natalia et al. [18] . They used both nanofluidic flow and hybrid nanofluidic flow (the so-called modified Bongiorno nonliquid model) to study the heat transfer properties across boundary layer flows and permeable isothermal stretch surfaces. Aldabesh et al. [19] studied thermally variable conductivity features in a Buongiorno nanofluid model between parallel stretching discs (energy improvement system efficiency). The study of heat generation or absorption in moving liquids is important for problems related to liquid dissociation. Temperature distribution may change due to heat generation. This determines the deposition rate of particles in nuclear reactors, electronic chips and semiconductor wafers. Kerur et al. [20] investigated on the effects of thermal radiation and chemical reactions heat and mass transfer in Mhd cassion nanofluid flow passes through a stretching sheet under the influence of heat source/sink. For positive values of the parameter n the heat source/sink acts as a heat generator and for negative values of the parameter the heat source/sink acts as a heat sink from the boundary layer. Thiagarajan et al. [21] . Analyzed on nanofluid hydromagnetic boundary layer flow and heat transfer across porous exponentially stretching sheets with effects on heat generation/absorption and ohmic heating. Increasing the magnetic interaction parameter, heat source parameter, Eckert number, thermal radiation parameter, and Brownian motion increases the dimensionless temperature in the boundary layer region, but decreases the heat sink parameter in temperature. Effects of magnetic fields on heat generation and absorption flow of nanofluids across nonlinear stretch films by Santoshi et al. [22] . They observed that heat generation coefficient is responsible for changing the heat gradient of the fluid flow, which has a negligible influence on the concentration levels of the fluid particle. The above studies do not take into account the magnetic induction effect that occurs at higher magnetic Reynolds numbers. MHD nanomaterial processing. Furthermore, thermal radiation is important in high temperature nanomaterials processing. The effect of embedding parameters, ie. Blade velocity parameter λ, magnetic field parameter β, Prandtl number Pr, magnetic Prandtl number Prm, thermal radiation parameter Rd, Lewis number Le, Brownian motion parameter Nb and thermophoretic parameter Nt, heat absorption (sink) parameter for velocity (Φ), induced magnetic field, temperature, nanoparticle concentration, skin friction, local Nusselt number and Sherwood number profiles are visualized. Includes residual error analysis for convergence rate and repetition rate. Therefore, the novelty of this work is that we have considered a large number of different nanoparticles for magnetic nanofluids, including thermal radiation, nanoparticle concentration, and heat source/sink effects. The physical model is shown in Figure 1. A uniform magnetic field of force 0 H is applied to the surface in the normal direction.

Mathematical formulation of the problem

The component of the induced magnetic field 2 H disappears when it hits the wall and the parallel component 1 H approaches the value of 0

H . An (x, y) coordinate system is adopted. It is also assumed that the velocity of the free stream is U, and that of the plate (sheet) is

, where λ is a velocity parameter. At the moving surface, the temperature and the nanoparticle concentration take constant values w T and w C , respectively, while the free stream values are prescribed as T  and C  , respectively. Under the boundary layer approximations, using the Buongiorno two-component nanoscale model [23] and Heat sink/source [21] , the conservation equations can be written as follows.

where u and v are the velocity components along the x-axis and the y-axis, respectively, T is the nanofluid temperature,  and ν are the nanofluid density, 0  is the magnetic permeability, 1  is the magnetic diffusivity, / ( )

is the thermal diffusivity of the fluid, K is the thermal conductivity of magnetic nanofluid, ()

is the heat capacity of the base fluid, Where *  and s K are the Stephan-Boltzman constant and the Rosseland mean absorption coefficient, respectively. The temperature differences within the flow are sufficiently small under the assumption that s T may be expressed as a linear combination of the temperature as shown in Chamakha et al. [27] . Expanding 4

T in a Taylor series about T  and neglecting the higher order terms, we obtain

The prescribed boundary conditions at the wall (plate) and in the free stream for equations ( 1)-( 6) are defined as follows:

Similarity Transformations

We now introduce the following similarity transformations, to normalize the boundary layer equations:  is the dimensionless similarity variable, f and g are dimensionless stream and magnetic stream functions, respectively,  is the dimensionless temperature function, and  is the dimensionless Nano particle concentration. Mass conservation and magnetic field continuity are satisfied identically. Substituting the similarity variables into equations ( 3) -( 6) gives the following similarity ordinary nonlinear differential momentum, magnetic, thermal, and species nanoparticles (nanoparticle) boundary layer equations:

The dimensionless boundary conditions for the tenth-order problem now become Sh . These parameters, respectively, characterize the surface drag, wall heat transfer rate, and wall nanoparticle mass transfer rate. The skin-friction coefficient is defined as

Here, w  is the shear stress at the surface of the wall (sheet), which is given by

Using equation ( 18) in (17), we obtain the dimensionless skin friction coefficient (surface drag) as.

The Nusselt number is defined as

Where the heat transfer rate at the surface is given by '

Using equation ( 21) in (20), the dimensionless wall heat transfer rate is obtained as

The nanoparticle mass transfer rate at the surface is given by '

The Sherwood number is defined as ()

From equations ( 23) and ( 24), the dimensionless nanoparticle mass transfer rate becomes ' 2 (0) Re

Numerical Methods

A nonlinear ordinary derivative boundary value problem defined by equations ( 12)-( 15) can be solved using a number of numerical methods, including finite elements, finite differences, etc. Here, we choose the MATLAB BVP4C routine (boundary value problem), which uses aexceptionally accurate method. This technique has been applied to many multiphysics fluid dynamics problems by Umavathi et al. [23] . In this study, the following fixed parameter values are adopted for computations: Pr = 0.7, Prm = 0.5, λ= 0.1, β= 0.1, Nt = 0.1, Nb = 0.3, Rd = 0.5, and Le = 5. In MATLAB this quadrature is used to obtain solutions for velocity profile, Incuced magnetic stream function, and temperature and Nanoparticle concentration. BVP4C uses stepping formulae which are summarized in Nayema et al. [24] , comparison has been made with Shateyi and Prakash et al. [25] and Shahina Akter et al. [26] . Simulations of the transformed Eqns (12-15) subject to the boundary conditions (16a) -(16b) are found with the help of MATLAB BVP4C numerical method for various values of the flow controlling parameters. In the context of bvp4c function described above we need to transform the governing equations into first order differential equation. At first Eqns (12-15)

Now we need to transform this above equation into first order differential equation.

The boundary conditions become considering a y be the left boundary, b y be the right boundary

Results and Discussion

The analytical solution considers the effects of the nanoparticle sheet velocity parameter, the magnetic field parameter, the Prandtl number, the magnetic Prandtl number, the thermal radiation parameter, the Lewis number, the Brownian motion parameter, the thermophoresis parameter, and the heat source/sink parameters on the heat and mass transfer properties of nanofluids have been examined. Table 2 shows the reduction of skin friction coefficient Cf, reduced Nusselt number Nu, and Sherwood number Sh It is observed that the heat transfer coefficient at the surface increases with increasing Prandtl number (thermal conductivity of magnetic nanofluid y The lower the temperature, the cooler the boundary layer and the better the heat transfer to the wall). In contrast, the mass transport rate decreases with increasing Prandtl number (increasing concentration of nanoparticles in the interfacial region and decreasing mass transport rate of nanoparticles to the wall). Increasing the Lewis number Le and the radiation parameter Rd increase the reduced Sherwood number at the limit. In contrast, the heat transfer coefficient at the boundary is reduced. It is also clear that increasing the moving surface parameter value λ decreases the skin friction on the wall panel, but does so for the Nusselt and Sherwood numbers.

Pr

  Pr = 0.71 Table 3 shows the heat transfer coefficient for various values of the Prandtl number Pr at other limited flow parameters. For negative wall velocities, the reduced Nusselt number decreases, but for positive wall velocities, it decreases steadily for all Prandtl numbers. Table 4 shows the reduced Sherwood number for various values of the Lewis number Le, all other parameters held constant. The decreasing Sherwood number decreases with increasing Lewis number and, for negative wall velocities, increases with a more pronounced Lewis number for positive wall velocities. Table 5 shows the effect of different magnetic force parameter β with different blade speed parameter λ on skin friction coefficient. It can be observed that the coefficient of skin friction decreased very rapidly with increasing values of these parameters. Figure 2 shows the effect on the velocity profile for different values of the blade velocity parameter. It can be seen that an increase in the velocity parameter λ leads to an increase in the flow acceleration. Achieve a smooth profile with a free stream. Figure 3 shows the effect on the induced magnetic current for different values of the blade speed parameter. It can be seen that an increase in the velocity parameter increases the velocity of the flow-induced magnetic field. Asymptotically smooth profiles are obtained for free streams with sufficiently large infinite boundary conditions. Figure 4 shows the effect on the temperature profile for different values of the layer velocity parameter. This indicates that the nature of the curve due to the thermal boundary layer thickness decreases as the wall velocity parameter increases. Figure 5 shows the effect on the concentration profile for different values of the blade speed parameter. As the rate parameter increases, the curve decreases with the boundary layer thickness of the nanoparticle concentration.

Figure 6 shows the effect on the velocity profile for different values of the magnetic blade parameters. It is observed that the velocity field decreases, the thickness of the momentum boundary layer increases and the thickness of the magnetic boundary layer decreases. Figure 7 shows the effect on the induced magnet current for different values of the magnetic sheet parameters. It is observed that the induced magnetic current decreases, while the thickness of the pulse boundary layer increases and the thickness of the magnetic boundary layer decreases.

Figure 8 shows the effect on the temperature profile for different values of the Brownian motion parameter Nb. It can be seen that the temperature profile decreases as the Brownian motion parameter Nb increases. Achieve a smooth profile with a free jet. Figure 9 shows the effect on the concentration profile for different values of the Brownian motion parameter Nb. It can be seen that the Brownian motion parameter Nb decreases the velocity of the concentration profile. Figure 10 shows the effect on the temperature profile for different values of the thermophoretic parameter Nt. It can be seen that the temperature profile decreases as the thermophoretic parameter Nt increases. Achieve a smooth profile with a free jet. Figure 11 shows the effect on the concentration profile for different values of the thermophoretic parameter Nt. It can be seen that the thermophoretic parameter Nt leads to a slowing down of the concentration profile.

Figures 12 and13 show the variation of temperature and nanoparticle concentration distribution as the Prandtl number Pr is varied. An increase in the Prandtl number Pr significantly reduces the temperature of the nanofluid and suppresses the thickness of the thermal boundary layer. The nanoparticle concentration profile shows a monotonic response to the Prandtl number. An increase in H. Pr promotes diffusion of nanoparticles and also increases the thickness of the boundary layer of nanoparticles concentration. Figure 14 shows the effect on the velocity profile for different values of the magnetic Prandtl number Prm. It can be seen that the flow acceleration increases as the magnetic Prandtl number Prm increases. Figure 15 shows the effect on the induced magnetic current for different values of the magnetic Prandtl number Prm. It can be seen that the velocity of the flowinduced magnetic field increases with an increase in the magnetic Prandtl number Prm.

Figure 16 shows the effect on the temperature profile for different values of the radiation parameter Rd, and it can be seen that an increase in the radiation parameter Rd leads to a decrease in the temperature profile. Figure 17 shows the effect on the nanoparticle concentration profile for different values of the radiation parameter Rd, showing that the radiation parameter Rd leads to a decrease in the concentration profile. Figure 18 shows the change in nanoparticle concentration profile with the Lewis number Le. At Le = 1, these velocities are equal and the boundary layer thicknesses of momentum and nanoparticle concentration are similar. Figure 19 shows the variation of the temperature profile with heat source/sink parameters. It can be seen that the temperature profile decreases as the heat source/sink parameters increase.

Conclusion

A mathematical study of Two-Dimensional MHD steady-state boundary laminar flow by thermal convection of a conducting nanoliquid across a smooth free-flow moving surface with heat source/sink is presented. The Buongiorno model was used, which takes into account Brownian diffusion and thermophoretic results. The flow and heat transfer partial differential equations were transformed into coupled nonlinear ordinary differential equations using appropriate similarity transformations. The coupled nonlinear momentum and thermal boundary layer equations are computed numerically using MATLAB bvp4c. The solutions obtained are linked to the literature and conclusions are drawn. Changes in skin friction coefficient, heat transfer coefficient, mass transfer coefficient, velocity, and induced magnetic field, temperature, and nanoparticle volume fraction profiles are displayed in tabular and graphical form for various values of physical parameters. From this study, some key findings are summarized as follows.

1. It is observed that the Velocity and induced magnetic field profiles decrease with an increase in magnetic body force parameters. 2. Increasing in the magnetic parameter, thermophoresis parameter, and radiative parameter thickness is enhanced in boundary layer. In contrast, they are diminished with more incredible wall velocity, Brownian motion, and Prandtl numbers. 3. Induced magnetic field stream function is affected by reciprocating the magnetic Prandtl number Prm compared with the skin-friction and heat transfer coefficients. 4. The nanoparticle concentration improves with greater values of thermophoresis and Prandtl number, but it is suppressed with increasing moving wall, Brownian motion, radiative, and Lewis numbers. 5. The heat source/sink parameter suppresses the temperature profile and decrease with higher values.

The work can be continued by selecting several more models of different non-Newtonian liquids and nanoliquids made with different combinations of nanoparticles. However, attention is limited to Newtonian nanofluids. This study, which will be published soon, explores the rheological behavior of nanofluids, including viscoelastic [28] and viscoplastic [29] models. The results obtained in such studies are of great benefit in dealing with nanofluidic coating streams, energy conservation, magneto therapy and biomedical fields.

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