Hybrid Nanoliquid Research Articles

Convective flow of hybrid nano particles in combination of TiO2+CuO/engine oil MoS2+ZnO/engine oil and Al2O3+Cu/engine oil with viscous dissipation over vertically moving surface: Numerical and Galerkin approach

Engine oil has a vast application sector as it is the most essential working lubricant for industrial machineries, engines of vehicles, and aircraft combustion processing. In order to find out more effective engine oil-based hybrid nanofluid in terms of heat transfer capabilities and to minimize the production cost, the novelty of our present study focused on three different combinations of nanoparticles (TiO2+CuO, MOS2+ZnO, and Al2O3+Cu) that are dispersed in Engine oil (EO). The fundamental aim is to examine the thermal and flow fluctuation of a convective dissipative engine oil-based hybrid nano liquid to understand the physical quantities and thermal properties over a vertically moving plate by employing the Galerkin Method (GM). Similarity variables are initiated to permute the set of governing equations into the nonlinear coupled ordinary differential equations along with Bc’s, which are solved using both numerical (Maple) and analytical method (GM). This MAPLE software uses the fourth-fifth-order Runge-Kutta-Fehlberg (RFK45) technique as a default function to solve numerically boundary value problems. Validation for this study has been conducted by comparison with previously published results. The results of the present study are given for both opposing and assisting flow with the distribution of fluid motion and temperature profiles, wall shear stress, and rate of heat transfer. The assisting flow has great significance on the flow velocity and temperature with increasing mixed convection parameters. The Brinkmann number and Prandtl number have a larger flow near the wall in temperature distribution and then increase and decrease respectively satisfying the asymptotic boundary conditions. Our results also reveal that the TiO2+CuO/Engine oil has a significant augment in terms of heat transfer than that of MoS2+ZnO/Engine oil and Al2O3+Cu/Engine oil.

Rayleigh–Bénard convection in mono and hybrid nanoliquids in an inclined slot

Linear stability analysis is conducted to investigate the longitudinal and transverse rolls (TRs) generated in Rayleigh–Bénard convection in mono and hybrid nanoliquids confined between two infinite inclined parallel slots. Thermophysical properties of six mono nanoliquids and fifteen hybrid nanoliquids are calculated for different volume fractions (0.5%, 1%, 2%) using phenomenological laws and mixture theory. The shooting method is used to solve boundary eigenvalue problems to obtain the eigenvalues for 16 different boundary conditions. It is observed that as the inclination angle is increased, it delays the onset of longitudinal rolls in the case of all boundary conditions. However, it advances the onset of TRs except when the lower plate is adiabatic. The addition of mono and hybrid nanoparticles results in the advancement of the onset of convection. The addition of SWCNT and SWCNT— accelerates the onset of convection the most while Cu and Cu-Ag accelerates the onset of convection the least amongst the mono and hybrid nanoparticles considered in the study.

A passive control of magnetohydrodynamic flow of a blood‐based Casson hybrid nanofluid over a convectively heated bi‐directional stretching surface

AbstractHybrid nanofluids, which are used in nanotechnology, are advanced fluid classes with enriched thermal properties that produce superior outcomes than nanofluids. There are too many applications of hybrid nanofluids in engineering cosmetics, the automotive industry, the home industry, cancer treatment, textiles, paper plastics, paints, and soaps. The purpose of this study is to investigate the heat transfer rate of magnetohydrodynamic flow of Casson hybrid non‐Newtonian nanofluid across an enlarging surface. The current work focuses on magnetohydrodynamic hybrid nanoliquid flow across an extending 3‐D sheet. Additionally, zero mass flux and an adequate convective heating procedure are used as boundary conditions in this investigation. Blood serves as the base fluid, into which copper and alumina nanoparticles are dissolved to form a hybrid nanofluid. Adjusting the applicable similarity transformation, the present modeled equations are converted into dimensionless form. The Homotopy analysis approach (HAM) computes the resulting systems and illustrates them graphically to explain the flow behavior at the extending electrically conducting surface. Additionally, for changes in the non‐dimensional physical constraint values, the variations in physical quantities such as the skin friction, temperature, Nusselt number and velocity profiles are explained. The results of the current investigation demonstrated that a magnetic field and a non‐Newtonian parameter reduce the hybrid nanoliquid's velocity. The temperature profile goes up with thermophoresis and Brownian motion. The component of velocity is found to fall as the stretching ratio parameter rises, while the component of velocity in the direction experiences the opposite impact. When the parameters of a chemical reaction are adjusted upwards, the concentration profile deteriorates. It is originated that the rate at which heat is transferred by hybrid nanofluids is significantly more progressive than that of nanofluids.

Computational assessment of MHD Carreau tri-hybrid nano-liquid flow along an elongating surface with entropy generation : A comparative study

This research looks at the incompressible, viscous, steady-state laminar stagnation-point-flow of a Carreau ternary-hybrid nano-liquid towards a convectively heated expanding surface, taking into account first-order slippage and Ohmic dissipation. Viscous dissipation and the effects of Lorentz forces are also considered. It is discussed how thermal radiation and heat absorption/generation contribute to the heat transport process. Minimizing entropy during the transport of a Carreau ternary hybrid nano-liquid has also been studied. In addition, convective circumstances are used conceptually in the numerical solution of the current model. tiny-particles of silver (Ag), molybdenum disulfide (MoS2), and multi-wall carbon nanotubes (MWCNT) are analyzed in this work’s flow study. As a base fluid, carboxymethyl cellulose (CMC-water) is used. With the use of appropriate similarity trans-formations, the standard model equations are transformed into dimensionless form. Finding solutions for momentum and heat fields using the Runge–Kutta-Fehlberg technique and a shooting strategy. The figures depict a wide range of characteristics, including fluid flow velocity, temperature, skin friction, the Nusselt number, entropy minimization, and the Bejan number. Key results from the present model include the fact that the velocity curve flattens down as Weissenberg number increases.

Generalized thermal properties of hybrid NANOLIQUID composed of aluminum oxide (Al2O3) and silver (Ag) nanoparticles with water (H2O) as base liquid

AbstractHeat transfer rate is numerically analyzed in convective flow of Al2O3‐Ag/ H2O hybrid nanoliquid through a stretching sheet by incorporating induced magnetic field. Results of entropy generation in system are evaluated as well. Considered physical factors associated with heat transfer are heat generation parameter and viscous dissipation. The system of nonlinear partial differential equations is modeled and dimensionally simplified by implementing boundary layer approximation assumption and proper similarity transformations. Adam's Bashforth method is applied to get highly accurate and stable numerical solutions. Numerical results of flow variables, entropy generation number and physical quantities are interpreted by way of graphs and bar charts to perceive the extensive significance of the problem. It is visualized that rise in numeric values of mixed convection parameter λ1 leads to enhance velocity; entropy generation number and Nusselt number while suppress temperature. High magnitude of heat generation parameter δ augments velocity and temperature but reverse behavior is observed for Nusselt number and entropy generation number. Moreover, the factor of viscous dissipation significantly modifies rate of flow and heat transfer under the effect of no‐slip condition on sheet. The present study is useful in different fields of industries, technological processes, mechanical processes, and electrical processes due to the applications of magnetic hybrid nanofluid with improved heat and mass transfer.

Analysis of water conveying aluminum oxide/silver nanoparticles due to mixed convection through four square cavity's variable hot (cold) walled

The present work focuses on the measurement of the general entropy and the heat exchanges that occur inside a square-shaped cavity by changing the cold and hot wall in four cases filled with a hybrid nano-liquid (Al2O3-Ag/water) with a cylinder installed inside it. After validating the model, the parametric study has been conducted based on Rayleigh number (Ra), Hartman number (Ha), Darcy number (Da), Solid volume fraction (φ) and Porosity (ε) aspects. An effective finite element method was employed to solve the problem of flow in a dimensionless form of governing equations towards flow and thermal behaviors of nanofluid which is laminar and incompressible. Using the COMSOL Multiphysics® software computer suite, the equations for energy, motion, and continuity were resolved. Nusselt number calculations are presented to quantify heat transport via mixed convection. In the case studies, the entropy generation improves regardless of where the heated wall is located, except for the third case, where it is higher than the others. The findings are demonstrated with a higher Rayleigh number Ra, average Nusselt number, and entropy production. The heat transfer rate (HTR) can be effectively adjusted by making use of the magnetic field. The process of producing entropy in the third cavity, where the hot wall mediates the right wall, is the crucial observation made through this effort.

Novel design of intelligent Bayesian networks to study the impact of magnetic field and Joule heating in hybrid nanomaterial flow with applications in medications for blood circulation

In current study, the effects of magnetic field and Joule heating in flow of (Cu+CuO) hybrid nanomaterial model with blood as a base fluid is analyzed effectively by exploiting novel design of intelligent Bayesian regularization neural networks (IBRNN) for several important physical aspects. Mathematical models of the systems are derived by considering the small Eckert and Prandtl numbers with thermo-physical characteristics of Cu+CuO/blood hybrid nanoliquids while the heat emission/absorption is added in energy equation to maintain the temperature for blood flow. The effects of various parameters of interest like wall deformation rate, Joule heating parameter, magnetic parameter, volume fraction and permeability parameter on velocity and temperature distribution have been studied using viable numerical approach of Adams method as well as presented intelligent Bayesian networks. The reliability of the Bayesian networks is endorsed by convergence plots of MSEs, effective outputs of adaptive controlling parameters of optimization algorithm, error based histograms, and regression statistics for exhaustive simulations studies in case of sundry scenarios.

Falkner-Skan aspects of a radiating (50% ethylene glycol + 50% water)-based hybrid nanofluid when Joule heating as well as Darcy-Forchheimer and Lorentz forces affect significantly

Falkner-Skan aspects are revealed numerically for a non-homogeneous hybrid mixture of 50% ethylene glycol-50% water, silver nanomaterials Ag, and molybdenum disulfide nanoparticles MoS2 during its motion over a static wedge surface in a Darcy-Forchheimer porous medium by employing the modified Buongiorno model. The Brownian and thermophoresis mechanisms are included implicitly along with the thermophysical properties of each phase via the mixture theory and some efficient phenomenological laws. The present simulation also accounts for the impacts of nonlinear radiative heat flux, magnetic forces, and Joule heating. Technically, the generalized differential quadrature method and Newton-Raphson technique are applied successfully for solving the resulting nonlinear boundary layer equations. In a limiting case, the obtained findings are validated accurately with the existing literature outcomes. The behaviors of velocity, temperature, and nanoparticles volume fraction are discussed comprehensively against various governing parameters. As crucial results, it is revealed that the temperature is enhanced due to magnetic field, linear porosity, radiative heat flux, Brownian motion, thermophoresis, and Joule heating effects. Also, it is depicted that the hybrid nanoliquids present a higher heat flux rate than the monotype nanoliquids and liquids cases. Moreover, the surface frictional impact is minimized via the linear porosity factor. Furthermore, the surface heat transfer rate receives a prominent improvement due to the radiative heat flux inclusion.

Thermal convection and entropy generation analysis of hybrid nanofluid slip flow over a horizontal poignant thin needle with an inclined magnetic field: A numerical study

Hybrid nanofluids have emerged as a promising field of research in recent years, finding applications in various industries and sectors. The entropy generation and impact of an inclined magnetic field on a water-Ethylene Glycol (50:50) mixture-based nanofluid over a poignant thin needle has been investigated under slip conditions. The flow is swotted using thermal and velocity slip boundary conditions. The numerical effects of an inclined magnetic field on the flow of ferrous and aluminum oxide nanoparticles in a hybrid nanoliquid with as base fluids have been examined. The non-dimensional ODEs along with boundary conditions are solved by numerical technique based on the Runge–Kutta fourth-order method. Velocity profile has been analyzed for the magnetic parameter, inclined angle and volume fractions. The Bejan number provides insights into the balance between heat transfer and entropy generation in a system. From the numerical values of skin friction coefficient and Nusselt number, it is analyzed that [Formula: see text]O3 possesses 1.01% and 1.06% of high heat transfer rate and high surface drag force than [Formula: see text]O4. Heat transfer profile has been analyzed for entropy generation and the Bejan number. The applications of hybrid nanofluids are vast, and one specific area where they can be utilized is in horizontal needle systems.

Magneto-thermogravitational convective flow and thermal behavior of hybrid nanoliquid in a novel T-shaped wavy chamber considering various shapes of nanoparticles

Magneto-thermogravitational convective flow and thermal behavior of hybrid nanoliquid in a novel T-shaped wavy chamber considering various shapes of nanoparticles

Second law analysis on cross flow of hybrid nanoliquid in a Darcy–Forchheimer medium with thermal radiative flow

This article’s objective is to examine the optimization of entropy in Darcy–Forchheimer cross-hybrid nanofluids as they flow toward a stretchy surface. There is a current because the surface is being stretched. The energy equation is broken down into its component parts including thermal radiation, heat source sink, and convective flow. In this case, copper oxide ( CuO ) and titanium dioxide ( TiO 2 ) are taken into consideration as nanoparticles, while engine oil (EO) is taken into consideration as a continuous phase fluid. In addition, we carried out an investigation into the relative performance of copper oxide ( CuO ) and titanium dioxide ( TiO 2 ) while they were suspended in water engine oil. The application of the second law of thermodynamics allows for the calculation of the rate of entropy optimization. Using a suitable transformation, nonlinear partial differential equations can be transformed into a standard system. In this work, we use the numerical built-in BVP4c solution method to generate numerical results for the derived nonlinear flow equation. Both copper oxide ( CuO ) and titanium dioxide ( TiO 2 ) undergo a graphical analysis of the effects of varying technical factors on entropy optimization, velocity, the Bejan number, and temperature. Numerical computations of the skin friction coefficient and Nusselt number for a range of fascinating parameters are performed for both nanoparticles ( TiO 2 and CuO ) . It is clear from the data that entropy optimization improves as the size of the radiation and porosity estimates reduces. The porosity parameter exhibits direct relationships to both temperature and velocity. Tabular comparisons of the current study with the previously published literature show a high degree of agreement. Copper and titanium oxide nanoparticles are used to increase Engine oil ( EO ) thermal enactment, making it a more useful base fluid. Further, some significant industrial and engineering applications are related to the present problem discourse.

Influence of an induced magnetic field and flow behavior of (AA7072–AA7075/water) hybrid nanoliquid in a vertical channel with suction velocity

Influence of an induced magnetic field and flow behavior of (AA7072–AA7075/water) hybrid nanoliquid in a vertical channel with suction velocity

Gyrotactic microorganism hybrid nanofluid over a Riga plate subject to activation energy and heat source: numerical approach

The current article aims to examine the magnetohydrodynamics (MHD) impact on the flow of MgO–Ag/water-based hybrid nanoliquid with motile microorganisms and the fluid is allowed to flow over a Riga plate subject to slip effects and activation energy. Furthermore, the presence of a uniform heat source/sink is also addressed in the energy equation. In addition to this, the thermophoresis effect is highlighted in the concentration equation. From the present proposed model, we get a non-linear system of the governing equations. The obtained system of partial differential equations (PDEs) is converted to the dimensionless system of ordinary differential equations (ODEs) using the similarity transformation. The obtained high non-linear system of equations has been solved numerically, using the parametric continuation method (PCM). In the present analysis, the main motivation is to highlight the heat transfer rate of MgO–Ag/water-based hybrid nanofluid flow over a Riga plate. The second motivation of the present research is to highlight the impact of slip conditions on the velocity, energy, and mass profiles. From the graphical analysis, it is depicted that the slip conditions reduce the velocity, energy, and mass outlines. From the present analysis, we concluded that volume friction reduced the flow profile while increasing the temperature of the fluid flow over a Riga plate. All the parameters of the present research are highlighted in velocity temperature and concertation of the fluid. In addition to this in all the figures we have compared the hybrid nanofluid with mono nanofluid and the also the comparison between slip and no-slip conditions have carried out through graphs for velocity, temperature, and concentration.

Motile microorganisms hybrid nanoliquid flow with the influence of activation energy and heat source over a rotating disc

The present article examines the consequences of a magnetic field, Hall current, and thermal radiation on the spinning flow of hybrid nanofluid (HNF) across a revolving disc. The core objective of the study is to improve the energy transference rate through hybrid nano liquid for industrial and engineering operations. The HNFs have advanced thermophysical characteristics. Therefore, in the current study, a superior class of nanomaterials (carbon nanotubes (CNTs) and Al2O3) are added to the base fluid. The modeled equations are demoted to a dimensionless set of Ordinary differential equations (ODEs) through similarity conversion and are analytically solved by engaging the homotopy analysis method. The physical constraints’ effect on energy, velocity, motile microorganism, and mass profiles have been drawn and discussed. For accuracy, the results are compared to the published studies, which ensures the accuracy and reliability of the technique and results. It is observed that the energy communication rate lessens with the flourishing values of thermal radiation and for Hall current. Furthermore, it is noted that due to its carbon–carbon bonding in CNTs, it has a greater tendency for energy propagation than Al2O3 nanoparticles.

Entropy generation analysis of multi-mass diffusion in a nanofluid-interfaced three-phase viscous fluid in an inclined channel

The primary aspect of the present study is to analyze the velocity and thermal slip effects on the entropy in the three-phase flow of viscous liquid amid nano liquids (NFs), hybrid nano liquids (HNFs) and tri-hybrid nano liquids (THNFs) with different shaped nanomaterials in an inclined channel. Moreover, entropy generation (EG) analysis is conducted for all three fluids i.e., viscous fluid sandwiched between Fe2O3 water NFs, viscous fluid sandwiched between Fe2O3−Cu water HNFs, and viscous fluid sandwiched between Fe2O3−Cu−MoS2 water THNFs. The uniqueness of the present study is the inclined nature of the channel as no study is present on three-phase immiscible flow in an inclined channel. In oil recovery systems, an immiscible mixture of nanofluids and viscous fluid can be used to recover the remaining oil from reservoirs. It can be used in a reservoir through an inclined channel to improve displacement efficiency. Furthermore, nanoparticles present in the above-mentioned immiscible mixture can reduce interfacial tension amid oil and injected fluid for increased oil recovery. The solution of the mathematical model (systems of PDEs) of three immiscible fluids flow is obtained via the perturbation method. The momentum and heat transport results are presented via graphical and tabular illustrations using the MATHEMATICA program. Our findings show that the velocity and heat fall when the inclination angle is varied. So, the inclined behavior of a channel can be used for a controlled flow of heat and fluid. The enhanced momentum and heat flow can be seen when we consider an immiscible mixture of Fe2O3−Cu−MoS2 water THNF and viscous fluid as compared to Fe2O3−Cu water HNF and Fe2O3 water NF. Moreover, heat flow is more significant in the case when differently shaped nanoparticles are considered as compared to the same spherical-shaped nanoparticles.

Significance of Arrhenius activation energy and binary chemical reaction in ternary hybrid nanofluid flow over a convectively heated porous curved surface: A passive control strategy

The convective flow of Jeffrey ternary hybrid nanoliquid over a curved stretching sheet, water-based aluminum oxide and graphene nanoparticles and single wall carbon nanotubes had been employed in this work to explore the hydrothermal variation. In the process of modeling the mechanism of mass transfer, activation energy and binary chemical processes are taken into consideration. The expressions of energy and mass are computed with the extra influence of Brownian diffusion and the thermophoresis characteristics. The initial step in evaluating leading equations involves transforming them into dimensionless forms through similarity transformations. To supplement the assessment, a diverse range of graphs and tables are utilized. The exploration and discussion of various parameters and their impact on involved fields are also presented. In ternary hybrid nanoliquids, an increase in thermophoresis and Brownian motion parameters positively impacts heat transfer capacity. The Biot number exhibits a positive influence on heat transport, while the Schmidt number reduces mass transfer. The curvature parameter has a beneficial impact on the heat transmission rate. Furthermore, as the activation energy parameter rises, the mass transport increases, while it decreases as the chemical reaction rate parameter increases.

Entropy Generation in a Magnetohydrodynamic Hybrid Nanofluid Flow over a Nonlinear Permeable Surface with Velocity Slip Effect

The current study is designed to model the hydrothermal feature of a hybrid nano liquid slip flows over a permeable expanding/contracting surface with entropy generation. The model incorporates Cu-Al2O3 nanoparticles with water as the host liquid to simulate the flow. Additional impacts incorporated into the novelty of the model are viscous dissipation and Joule heating. The model is transformed appropriately to its dimensionless form using similarity quantities and the solution is numerically obtained using the spectral quasi-linearization method (SQLM). The impact of pertinent factors on the flow characteristics is communicated through graphs for the hybrid nano-suspension to discuss the hydrothermal variations. The friction factor and the rate of heat transport are also discussed with sensible judgment through tables. To ensure the code validity, a comparison with earlier studies is conducted and excellent consensus is accomplished. The result explored that diminution in the irreversibility ratio is witnessed for rising magnetic field strength along the free stream, distance away from the permeable surface as the heat dissipation to the surrounding decelerates. Also, the augmented nonlinearity parameter intensified the heat transfer rate for about 2.79% of the hybrid nano-suspension.

Impact of Joule heating on magnetohydrodynamic dissipative flow above a slendering surface with thermophoresis and Brownian movement with slip/no-slip conditions

ABSTRACT Purpose Hybrid nanofluids meet the necessities of several technological and production industries. Owing to these ample applications, a numerical investigation is accomplished to examine the thermal transmission feature of magneto-hybrid nanofluid flow due to an extending surface of varied thickness with multiple slip effects. Design/methodology The arising system of Partial Differential equations (PDEs) are exercised by adopting suitable similarities to attain the non-dimensional Ordinary Differential equations (ODEs) and the solutions are obtained by implementing bvp5c built in MATLAB package. The energy equation is loaded with Brownian motion, thermophoresis and Joule heating effects. The implication of pertinent constraints upon the flow features is discussed graphically. Findings The outcomes imply that the energy transmission rate is considerably high in no-slip case as equated with the slip case. Further, the power-law index parameter effectively uplifts the thermal and concentration gradients in both the situations. Originality The prime motive of this research is to present the relative analysis of heat transmission of hybrid nanoliquid in two different situations, namely slip and no-slip cases.

Flow and heat transfer mechanism of engine-oil based hybrid nanofluid due to a nonlinearly extending surface: A comparative study

ABSTRACT The present report emphasizes on the thermo-physical feature of the radiative hydrodynamic stream of hybrid nanofluid above a nonlinearly extending surface with uneven heat sink or source effects. The equations yielded by mathematical modeling are accessed numerically by employing bvp5c MATLAB package. The uniqueness of the current scrutiny is to deliberate the simultaneous results of the radiative hydrodynamic flow along extending surface for nanofluid (ethylene glycol + CuO) and hybrid nanofluid (ethylene glycol + CuO + Al2O3) instances. Furthermore, the influences of sundry physical parameters on the flow and thermic gradients are graphed, and wall friction and heat transmission rate are illustrated via tables for nanofluid and hybrid nanofluid situations. The outcomes reveal that the energy transport rate in a hybrid nanofluid case is substantially larger than the nanofluid case. Moreover, the viscous dissipation parameter performs a major role in enriching the temperature gradient of the hybrid nanoliquid than the nanoliquid. The prime application of the present work can be found in industrial processing, where control over heat transport is required.

Natural convection and entropy generation in a trapezoidal region with hybrid nanoliquid under magnetic field

PurposeThis paper aims to study numerically the steady natural convective heat transfer of a hybrid nanosuspension (Ag-MgO/H2O) within a partially heated/cooled trapezoidal region with linear temperature profiles at inclined walls under an effect of uniform Lorentz force. This investigation is useful for researchers studying in the area of cavity flows to know features of the flow structures and nature of hybrid nanofluid characteristics. In addition, a detailed entropy generation analysis has been performed to highlight possible regimes with minimal entropy generation rates.Design/methodology/approachThe governing equations formulated using the Oberbeck–Boussinesq approach and single-phase nanoliquid model are transformed to a non-dimensional form by using non-dimensional variables. The obtained equations with appropriate boundary conditions are resolved by the finite difference technique. The developed code has been validated comprehensively. Analysis has been performed for a wide range of governing parameters, including Rayleigh number (Ra = 105), Prandtl number (Pr = 6.82), Hartmann number (Ha = 0–100), magnetic field inclination angle (φ = 0–?/2) and nanoparticles volume fraction (φhnf = 0 and 2%).FindingsIt has been shown that inclined magnetic field can be used to manage the energy transport performance. An inclusion of nanoparticles without Lorentz force influence allows forming more stable convective regime with descending heat plume in the central zone, while such a regime was performed for clear fluid only for moderate and high Hartmann numbers. Moreover, the average overall entropy generation can be decreased with a growth of the Hartmann number, while an addition of hybrid nanoparticles allows reducing this parameter for Ha = 30 and 50. The average Nusselt number can be increased with a growth of the nanoparticles concentration for low values of the magnetic field intensity.Originality/valueGoverning equations written using the conservation laws and dimensionless non-primitive variables have been resolved by the finite difference approach. The created numerical code has been verified by applying the grid independence test and computational outcomes of other researchers. The comprehensive analysis for various key parameters has been performed.