MHD boundary layer micropolar fluid flow over a stretching wedge surface Thermophoresis and brownian

Abstract

Keywords: Chemical Reaction; Micropolar Fluid; Micro-rotation; MHD; Radiation; Shooting Method; Stretching Wedge Surface

Introduction

The micropolar fluid attracted a great deal of interest from the researcher due to its importance in various fluid fields. The micropolar fluid is the sub-class of micro fluids that exhibits the effects and inertia of micro-rotation. Couple stress and body couples are supported by the micropolar fluids. The bar-like elements containing fluids can effectively symbolize the micropolar fluids. Micropolar fluid is a type of anisotropic fluid like liquid crystals composed of dumbbell molecules. Animal blood is also this type of fluid. Eringen [1] has introduced a fluid model which demonstrates several microscopic effects that arise from the micro-motion as well as the local formation of fluid constituents. The fluids which are consisting of inflexible, indiscriminately oriented, or spherical species suspended in the viscous medium are symbolized by the micropolar fluid model and have ignored the particles’ deformation. “Micropolar fluids have been shown to accurately simulate the flow characteristics of polymeric additives, geo-morphological sediments, colloidal suspensions, hematological suspensions, liquid crystals, lubricants, etc.”[2]. Various authors have investigated heat and mass transfer boundary layer flow considering micropolar fluid in different physical situations. Srinivasacharya et al. [3] studied the boundary layer free convective flow considering variable wall temperature as well as a concentration in stratified micropolar fluid on a vertical plate. It was noticed that when stratification is present, the sign of microrotation changes within the boundary layer region from negative to positive values. Khalid et al. [4] have investigated natural convection heat and mass transfer boundary layer flow in a micropolar fluid. For this, they have considered the unsteady flow over an infinite oscillating vertical plate and found that the velocity of Newtonian fluid is larger than the velocity of the micropolar fluid. Many researchers have a great interest in the study of nanofluids because of their several applications in different fields like automotive, electronics, computing technology, communications, optical devices, material processing, lasers, medicine, and material amalgamation, etc., while heat dissipation is required. In 1995, Choi and Eastman [5] first invented the nanofluids notion and they have observed high thermal conductivities demonstrated by nanofluids owing to the existence of solid particles which are nanometer (nm) sized. Some authors [6 -12] have studied mixed convective heat and mass transfer considering different types of nanofluids with diverse conditions as well as their various applications such as Jilte et al.[13] have tested heat sink cooling concert and focused on the geometry of four channels liquid-cooled heat sink configured in an alternate flow passage concentric shape as well as found that the sink of nanofluid cooled heat compared with water rejected a higher heat rate, on the other hand, Jilte et al. [14] have presented two systems of organizing cooling component which is liquid circulated battery cooling systems (LcBS) and liquid-filled battery cooling systems

(LfBS) containing water-based nanofluid (Alumina 40nm diameter nanoparticles) and the results showed that the production and implementation of Li-ion batteries are easier in electric vehicles for the modular nature of both battery cooling system and LfBS uncomplicated in operation. While the turbulent effect is ignorable, the thermophoresis, as well as Brownian diffusion, play an important role in the fluid flow and Buongiorno [15] has recommended a model considering the consequence of thermophoresis and Brownian diffusion and found that an essential decline of viscosity for a heated fluid in the boundary layer which enhanced heat transfer. A few researchers [16-18] investigated the convective boundary layer flow in diverse fields employing the Buongiorno model. Due to its use in a variety of sectors, including MHD power generators, plasma studies, nuclear reactor cooling, the petroleum industry, crystal development, and boundary layer control in aerodynamics, magnetohydrodynamics (MHD) plays a significant role in the study of free convection flow. Pavlov [19] has investigated the MHD boundary layer flow over a stretching surface. In his study, he considered a uniform magnetic field and obtained exact similarity solutions. The viscoelastic fluid flow past a stretching surface has been investigated considering a transverse magnetic field [20]. In this regard, he has obtained a set of the exact solution and found that a transverse magnetic field has a similar effect as the viscoelasticity on the fluid flow. On the other hand, Kasiviswanathan and Gandhi [21] have determined a set of the exact solution for the flow of micropolar fluid between infinite non-coaxial revolving disks in the presence of magnetic field effect and observed that the velocity profile is deformed as well as the magnitude of micro-rotation components vector is diminished for the enhance in Hartmann number. Chamkha et al. [22] have investigated the MHD dominated Cu-water nanofluid flow in a porous enclosure and found that for all values of heat generation parameter inside the porous cavity the entropy generation is decreasing for increasing the nanoparticles volume fraction and miniature volume fraction of nanofluid has a significant consequence on the thermal rendering. The Falkner-Skan equation (F-SE) plays a substantial role in the augmentation of fluid dynamics. “The Falkner– Skan equation, a third-order ordinary differential equation over a semi-infinite domain was originally developed in 1931 by Falkner and Skan [23] from a similarity analysis on the steady, two-dimensional, boundary-layer flow of an incompressible, viscous fluid over a wedge [24]”. Several researchers [25-28] have reviewed flow over a wedge. The flow with a triangular shape is called a wedge flow. In the procedure under consideration, it can be utilized to pull up one object while keeping another in a plane. The wedge transforms the lateral force into a crosswise splitting force. Later, several authors [29-31] investigated the analytical and numerical solutions of the flow past a wedge considering different flow properties. In high-temperature operating systems radiation has a great impact on the heat and mass

transfer of the fluid flows. Many processes are carried out at a high temperature in the fluid of engineering. In such circumstances, the investigation of radiative heat transfer plays a vital role in the choice of suitable equipment. There aren’t many industries that produce gas turbines, radioactive and nuclear power plants, manmade satellites, aircraft, weapons, wind turbines, and other things. Dogonchi et al. [32] have investigated the consequence of radiation and heat source on the heat transfer of nanofluid flow and found significant effects of nanofluid volume fraction and radiation parameter on the Nusselt number. Zheng et al. [33] have found a significant consequence of radiation parameters on the heat transfer of nanofluid flow in their investigation. Ahmed et al.[34] have investigated MHD mixed convection water-based micropolar nanofluids flow in trapezoidal areas numerically and found that average Nusselt number enhances for decreasing values of Richardson number as well as dimensionless viscosity, heat source location, Hartmann number have an essential consequence on the average Nusselt number. Bagh et al.[35] have studied the g-jitter effects on mixed convection for the two colloidal liquids flow which are the alone micropolar nanofluid such as copper/water and micropolar hybrid nanofluid such as alumina-copper/water and observed the distinguished increased performance of hybrid nanofluids in the upper space of inclined sheet g-jitter influenced flow. They also found that the variation of skin friction coefficient and heat flux gradient enhanced with larger values of inflection amplitude. Kasmani et al.[36] investigated the double-diffusive convective boundary layer nanofluid flow past a moving wedge with the effects of Soret and Dufour in the presence of suction and found that the temperature enhances with increasing effect of the Soret parameter as well as declines for the increasing effect of Dufour parameter and the opposite trend arises in case of the mass transfer rate. Ashraf et al.[37] have studied the steady MHD convective tangent hyperbolic nanofluid flows across an elastic

surface with the impact of thermal radiation and thermal conductivity. And they observed that for the increasing effects of the Weissengberg number and power-law index, the velocity profile decreased but the temperature profile increased. Ahmed et al. [38] have identified the consequences of thermal conductivity as well as dynamic viscosity on the boundary layer flow of nanofluid and characteristics of heat transfer over a stretching permeable tube with a heat source/sink. The skin friction coefficient decreases, on the other hand, the local Nusselt number increases for the effects of the Reynolds number and suction/injection parameter. The new approach of the present study contributes significantly to the understanding of numerical solutions to nonlinear differential equations in fluid mechanics and micropolar fluid flow. Micropolar fluids are becoming even more of a focus due to the desire for engineering applications in various fields of medical, mechanical engineering, and chemical processing. This study looks into how thermophoresis and Brownian diffusion affect convective heat and mass transfer boundary layer micropolar fluid flow over a stretching wedge-shaped surface in the presence of thermal radiation and chemical reaction.

Mathematical Model With FLOW Configuration

The two-dimensional steady convective boundary layer flow of viscous, incompressible, and electrically conducting fluid and heat transfer of micropolar fluid over a stretching wedge-shaped surface considering chemical reaction and radiation effects are demonstrated in this study. Coordinate system and configuration of flow as shown in Figure 1. The Cartesian coordinate x- and y-axes are determined along and normal to the surface of the wedge, respectively. Fluid velocities such as u and v act along x- and y- axes, correspondingly. Uw(x) = axm, where a > 0, is the stretching

wedge velocity and U(x) = bxm is the free stream velocity outside the boundary layer of the micropolar fluid. In Figure 1, Ω = πβ is the wedge angle where β is the Hartree pressure gradient. Under the above assumptions, the basic equations are considered as [39]:

where is the Stefan-Boltzmann constant as well as k* the mean absorption coefficient, and T 4 stand for the linear sum of the temperature. Applying Taylor series expansion T 4 can be expanded T∞ as:

The temperature differences are considered very small, so ignoring the higher order terms involving (T − Too , Eq. (7) one can be written as:

To transfer the set of PDEs (Eqs. (2), (3), (5), and (10)) into a set of dimensionless ODEs introduce the following similarity transformations:

1. At the wedge surface (i.e. at y = 0), N = 0, while s = 0.

It represents the flow of concentrated particles in which the microelements closed to the wall surface are unable to rotate. This case is also known as a strong concentration of microelements. The antisymmetric part of the stress tensor will vanish when s = 0.5 consequently concentration of microelements becomes weak. Whereas, the case s = 1 is used for the modeling of turbulent boundary layer flows [41]”. Considering Rossel and approximation the radiative heat flux can be written as (6)

where Cw is the concentration at the stretching surface and Tw is convective fluid temperature. Using Eq. (11) in Eqs. (2), (3), (5), and (10), one can be determined the following ODEs: (12)

for the diverse values of concerned physical parameters employing the Runge-Kutta fourth-order method by exploiting symbolic software MATLAB. In this regard Eqs. (12) - (15) along with boundary conditions are converted into a set of first-order ODEs and then are situated into the MATLAB numerical software code to find the solutions. Let us consider into account

In the case of A > 0 and A < 0 the wedge moves in the same and opposite directions, respectively. On the other hand, the wedge will be static while A = 0 [42]. The skin friction coefficient, Nusselt number, Sherwood number, and couple stress coefficient are the physical quantities of engineering interest. One can be defined as:

(21) The numerical results of velocity (f '), temperature (θ), and nanoparticle volume fraction (ϕ), angular velocity (h) profiles are calculated for diverse values of obtained physical parameters and presented graphically, as well as discussed physical viewpoint.

Results And Discussion

, and indicate the shear stress, the transfer rate of heat, and mass per unit area of the surface, respectively. The three quantities are defined as (16) With the aid of the above equation (Eq. (16)), one can determine the one-dimensional form of Cf, Nu and Sh, Cr which are proportional to and respectively. Solution Procedure It is impossible to find the exact solution of higher-order nonlinear coupled ordinary differential equations (CODEs). Therefore the numerical solution of the CODEs Eqs. (12) - (15) with boundary conditions are obtained

The Investigations are conducted on the impacts of the relevant physical parameters on the velocity, temperature, and concentration profiles. Equations (11) through (14) are dimensionless coupled nonlinear ordinary differential equations having boundary conditions (Eq.21) that are numerically solved using the fourth-order Runge-Kutta (RK) technique and the shooting approach. This results in the conversion of Eqs. (11) through (14) into nine first-order ordinary differential equations including boundary conditions, described in Section 3. The typical values of the relevant physical parameters (especially the Prandtl number and Grashof number are taken for the water and the cool surfaces, respectively) are taken into consideration for numerical solutions as coupling constant parameter (0.01 ≤ B1 ≤ 0.05), magnetic parameter (1.0 ≤ M ≤ 15.0), Grashof number (0.3 ≤ Gr ≤ 0.9), modified Grashof number (0.3 ≤ Gm ≤ 0.8), micropolar parameter (2.0 ≤ G2 ≤ 7.5) vortex viscosity constraint (0.02 ≤ G1 ≤ 0.2), Prandtl number (7.0 ≤ Pr ≤ 15.0), thermal radiation parameter (0.25 ≤ R ≤ 0.50), Brownian

motion parameter (0.2 ≤ Nb ≤ 0.62), thermophoresis parameter (0.04 ≤ Nt ≤ 0.10), heat generation parameter (0.1 ≤ Q ≤ 0.5), Biot number (0.65 ≤ Bi ≤ 1.0), stretching parameter (0.2 ≤ A ≤ 0.5), Lewis number (3.0 ≤ Le ≤ 7.0), and chemical reaction parameter (0.2 ≤ K ≤ 0.7). In this section, the impacts of the relevant parameters on velocity, temperature, concentration, and angular velocity are discussed in detail. Effect of the Magnetic Parameter (M) Figure 2 depicts the velocity (f'), temperature (θ), concentration (ϕ), and angular velocity (h) profiles for the effect of the magnetic parameter (M). For higher values of the magnetic parameter, a substantial resistive force called the Lorentz

force opposes the fluid motion, and as a result, the rate of heat transfer within the boundary layer increases. It indicates that one of the sources of fluid ruggedness and heat transport is symbolized by the magnetic field. Once higher values of M the velocity (concentration) profile enhance in the range 0 < η <

Conclusion

Under consideration of the impact of the concerned physical parameters, a computational examination of the MHD boundary layer of micropolar fluid flow across a stretched wedge is carried out. The Buongiorno model is taken into account for this. The significant findings of this investigation are listed as : • The magnetic parameter act as a controlling parameter in this study because fluid velocity and the rate of temperature enhance but the rate of mass transfer decreases within the boundary layer region consequence of the magnetic parameter. • The Brownian motion parameter acts as a source of cooling but the thermophoresis parameter acts as a heat source parameter because, within the boundary layer region, the temperature of the fluid decreases for the effect of the Brownian motion parameter on the other hand the reverse results arises for thermophoresis parameter. • The stretching or velocity ratio parameters act as a formation of the boundary layer. While the surface velocity is less than the free stream velocity, the structure of the boundary layer will be inverted but the opposite shape is formed when the surface velocity is greater than the free stream velocity. In the study, one can find the reverse inverted shape boundary layer.

Nomenclature

Latin alphabet Specific heat at constant pressure [Jkg−1 K −1] Cp Strength of transverse magnetic field [kg s−2 A−1] B0 u Velocity component of micropolar along x- direction [ms-1] v Velocity component of micropolar along y- direction [ms-1] Ω Wedge angle β The gradient of Hartree pressure [Nm-3] σ* Stefan-Boltzmann constant [Wm-2K-4] k* Mean absorption coefficient The stretching velocity of the surface [ms-1] Uw U Free stream velocity of the fluid [ms-1] f (η) Dimensionless stream function f '(η) Dimensionless velocity f ''(η) Dimensionless velocity gradient a Stretching rate constant b The free stream rate constant Heat generation Q0

Chemical reaction Volumetric coefficient of thermal expansion [K−1] Volumetric coefficient expansion of nanoparticle volume fraction Acceleration due to gravity vector [ms-2] The dimensional temperature of the fluid [K] Convective fluid temperature [K] Ambient nanofluid temperature [K] The dimensional nanoparticle volume fraction [kg m-3] Nanoparticle volume fraction at the stretching surface [kg m-3] Free stream nanoparticle volume fraction [kg m-3] Brownian diffusion coefficient [cm2 s -1] Thermophoresis diffusion coefficient Radiative heat flux [Wm-2] Coefficient of heat transfer Coupling constant micro-rotation or angular velocity [rads-1] Thermal conductivity [Wm-1K] Microinertia density

Greek alphabet γ Spin gradient τ The ratio of nanoparticle effective heat capacity and base fluid heat capacity τw Shear stress [Nm-2] σ Electrical conductivity [Sm-1] ρ The density of the base fluid [kgm-3] (ρCp)f Micropolar heat capacity α Thermal diffusivity of base fluid [m2s-1] ν Kinematic viscosity of the base fluid [kg m-3] J Stream function [m-2s-1] θ(η) Dimensionless temperature θ'(η) Rate of temperature ϕ(η) The dimensionless nanoparticle volume fraction ϕ'(η) Rate of nanoparticle volume fraction Dimensionless numbers/parameters M Magnetic parameter Gr Grashof number Gm Modified Grashof number Pr Prandtl number R Thermal radiation parameter Nb Brownian motion parameter Nt Thermophoresis parameter Q Heat generation parameter Sc Schmidt number Bi Biot number A Stretching parameter Le Lewis number K Chemical reaction parameter B1 Coupling constant parameter G1 Vortex viscosity constraint G2 Micropolar parameter

Transfer rate of heat per unit area of the surface Transfer rate of mass per unit area of the surface skin friction coefficient Nusselt number Sherwood number

Abbreviations

BL Boundary layer MHD Magnetohydrodynamic PDE’s Partial Differential Equations ODEs Ordinary Differential Equations CODEs Coupled Ordinary Differential Equations BCs Boundary conditions

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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