Two-dimensional Incompressible Flow Research Articles

Exact planetary waves and jet streams

We investigate exact nonlinear waves on surfaces locally approximating the rotating sphere for two-dimensional inviscid incompressible flow. Our first system corresponds to a $\beta$ -plane approximation at the equator, and the second to a $\gamma$ approximation, with the latter describing flow near the poles. We find exact wave solutions in the Lagrangian reference frame that cannot be written down in closed form in the Eulerian reference frame. The wave particle trajectories, contours of potential vorticity and Lagrangian mean velocity take relatively simple forms. The waves possess a non-trivial Lagrangian mean flow that depends on the amplitude of the waves and on a particle label that characterizes values of constant potential vorticity. The mean flow arises due to potential vorticity conservation on fluid particles. Solutions over the entire space are generated by assuming that the flow far from the origin is zonal and there is a region of uniform potential vorticity between this zonal flow and the waves. In the $\gamma$ approximation, a class of waves is found that, based on analogous solutions on the plane, we call Ptolemaic vortex waves. The mean flow of some of these waves, which we can describe in highly nonlinear scenarios due to the exact nature of the solutions, resembles polar jet streams. Several illustrative solutions are used as initial conditions in the fully spherical rotating Navier–Stokes equations, where integration is performed via the numerical scheme presented in Salmon & Pizzo (Atmosphere, vol. 14, issue 4, 2023, 747). The potential vorticity contours found from these numerical experiments vary between stable permanent progressive form and fully turbulent flows generated by wave breaking.

A high-order generalised differential quadrature element method for simulating 2D and 3D incompressible flows on unstructured meshes

In this paper, a high-order generalised differential quadrature element method (GDQE) is proposed to simulate two-dimensional (2D) and three-dimensional (3D) incompressible flows on unstructured meshes. In this method, the computational domain is decomposed into unstructured elements. In each element, the high-order generalised differential quadrature (GDQ) discretisation is applied. Specifically, the GDQ method is utilised to approximate the partial derivatives of flow variables and fluxes with high-order accuracy inside each element. At the shared interfaces between different GDQ elements, the common flux is computed to account for the information exchange, which is achieved by the lattice Boltzmann flux solver (LBFS) in the present work. Since the solution in each GDQ element solely relies on information from itself and its direct neighbouring element, the developed method is authentically compact, and it is naturally suitable for parallel computing. Furthermore, by selecting the order of elemental GDQ discretisation, arbitrary accuracy orders can be achieved with ease. Representative incompressible flow problems, including 2D laminar flows as well as 3D turbulent simulations, are considered to evaluate the accuracy, efficiency, and robustness of the present method. Successful numerical simulations, especially for scale-resolving 3D turbulent flow problems, confirm that the present method is efficient and high-order accurate.

A fourth-order compact finite volume method on unstructured grids for simulation of two-dimensional incompressible flow

This paper introduces a novel and efficient compact reconstruction procedure for high-order finite volume methods applied to unstructured grids. In this procedure, we establish a set of constitutive relations that ensure the continuity of the reconstruction polynomial and its normal derivatives between adjacent elements at control points. The paper delves into the details of the fourth-order compact reconstruction method specifically designed for two-dimensional triangular grids. This method can be considered an extension of the one-dimensional compact scheme and cubic spline interpolation methods to two-dimensional triangular unstructured grids. In the cubic polynomial reconstruction, a two-dimensional cubic polynomial is reconstructed using the cell averages of elements in the standard stencil and the function values at control points. The determination of function values at control points is achieved by adhering to the constraint of normal derivative continuity. This reconstruction is solved using a relaxation iteration method and has been verified to be solvable for triangular grids. Compared to other fourth-order implicit compact polynomial reconstructions, our method requires solving a smaller number of unknowns, which means less computational cost in reconstruction. By applying this method to solve the Poisson equation, the linear convection equation, and incompressible flow benchmarks, it demonstrates that the proposed method exhibits the expected high-order accuracy and performs well in incompressible flow problems.

Feature-based manifold modeling for the quasiperiodic wake dynamics of a pair of side-by-side cylinders

We propose a feature-based manifold modeling (FeMM) framework for the quasiperiodic wake dynamics of a pair of side-by-side cylinders. The key enabler is to embed the most parsimonious mean-field manifold based on the extracted features, such as force coefficients and probing data from experiments and numerical simulations. The manifold model is then identified under the mean-field constraints of the model structure, ensuring human-interpretability. The FeMM method is demonstrated with a two-dimensional incompressible flow crossing a pair of side-by-side cylinders, exhibiting a flip-flopping wake in quasiperiodic behavior. The transient and post-transient dynamics are characterized by two coupled oscillators associated with vortex shedding and gap flow oscillations. Dynamic mode decomposition analysis reveals significant modal interactions between these two flow mechanisms, posing a serious challenge to projection-based modeling approaches, such as the Galerkin projection method. Nevertheless, the FeMM approach, based on force measurements, yields an interpretable model that accounts for the mechanisms underlying the quasiperiodic dynamics, demonstrating its applicability to higher-order dynamics with multiple scales and invariant sets. This approach is expected to have broad applicability in dynamic modeling and state estimation in various real-world scenarios.

Heat transfer analysis of mixed convection boundary layer blood base nanofluids with the influence of viscous dissipation

In this research study, we delve into the analysis of two-dimensional boundary layer incompressible viscous flow of blood based nanofluids, incorporating a couple stress model. This investigation encompasses considerations of heat transmission and viscous dissipation effects. Specifically, focus on a particular type of nanoparticle, Magnesium oxide (MgO), dispersed in blood, the base fluid. By employing suitable similarity transformations, transform a set of partial differential equations (PDEs) into nonlinear ordinary differential equations (ODEs). To leverage the Homotopy Analysis Approach (HAM) to solve this system of equations. This study discerns the impacts of various factors on temperature and velocity distributions. Visualization through graphs depicting nanoparticle volume concentration, magnetic field strength, couple stress parameter, Eckert number, and Prandtl number aids in presenting our findings. These insights illuminate the intricate interplay among different parameters and offer insights for enhancing heat transfer in blood-based nanofluids. Additionally, scrutinize the thermal performance of the blood-based nanofluid, assessing the Nusselt number and local skin friction coefficient. Such analysis holds significance for ensuring patient safety and treatment efficacy, particularly in medical procedures involving blood circulation, where mixed convection plays a pivotal role. Understanding these dynamics is crucial for the design and optimization of various medical devices and therapies, ensuring optimal temperature maintenance during blood circulation-related operations.

A computational study for simulating MHD duct flows at high Hartmann numbers using a stabilized finite element formulation with shock-capturing

The concern of this manuscript is stabilized finite element simulations of two-dimensional incompressible and viscous magnetohydrodynamic (MHD) duct flows governed by a coupled system of partial differential equations of the convection–diffusion type. In such flows, the high values of the Hartmann numbers (Ha) indicate that the convection process dominates the flow field. As is typical trouble encountered in convection-dominated flow simulations, classical discretization methods fail to function properly for MHD flows at high Hartmann numbers, yielding several numerical instabilities. In this study, the core computational tool we employ in order to overcome such challenges is the streamline-upwind/Petrov–Galerkin (SUPG) formulation. Beyond that, since the SUPG-stabilized formulation requires additional treatment where solutions experience rapid changes, we also enhance the SUPG-stabilized finite element formulation with a shock-capturing operator for achieving better solution profiles around sharp layers. The formulations are derived for both steady-state and time-dependent models. The proposed method, the so-called SUPG-YZβ formulation, techniques used, and in-house-developed finite element solvers are evaluated on a comprehensive set of numerical experiments. The test computations reveal that the SUPG-YZβ formulation yields quite good solution profiles near strong gradients without any significant numerical instabilities, even for quite challenging values of Ha, whereas the SUPG usually fails alone. Furthermore, by using only linear elements on relatively coarser meshes, the proposed formulation achieves this without the need for any adaptive mesh strategy, in contrast to most previously reported studies.

Chaotic Vortex-Induced Vibrations of Rigid Cylinders with Nonlinear Snapping Support

Vortex-Induced Vibration (VIV) is a complex fluid–structure interaction in offshore structures. Traditionally, this phenomenon is considered periodic; however, many of its signals show chaotic behavior. The basic model already employed by other researchers is a rigid circular cylinder with linear springs and dampers. In this work, nonlinear snapping support is used to model nonlinearity in the system. To numerically simulate the flow, Reynolds-Averaged Navier–Stokes (RANS) equations for two-dimensional incompressible unsteady flows are applied. The degree of nonlinearity of the system can be changed by manipulating [Formula: see text], which is one of the geometric properties of the spring and takes values between 0 and 1. The 0–1 test, Poincaré section, and Fast Fourier Transform are used to analyze the cylinder and lift force behavior. Also, the Hilbert transform is applied to the signals, and the phase shift between displacement and lift force is obtained. The results show that the system behavior consists of branches: branch I and branch II. The large amplitudes occur in branch II. It is found that chaos emerges at the beginning of branch II, regardless of the value of [Formula: see text]. By raising the [Formula: see text] value, the span of branch II becomes more expansive, and its first point is placed at lower reduced velocities. Also, the wake dynamics becomes more regular at the end of branch I and more irregular at the beginning of branch II with the increase in [Formula: see text]. When the cylinder displacement signal is chaotic, the lift force behavior is also chaotic, but not vice versa.

Fractal structures in the chaotic advection of passive scalars in leaky planar hydrodynamical flows.

The advection of passive scalars in time-independent two-dimensional incompressible fluid flows is an integrable Hamiltonian system. It becomes non-integrable if the corresponding stream function depends explicitly on time, allowing the possibility of chaotic advection of particles. We consider for a specific model (double gyre flow), a given number of exits through which advected particles can leak, without disturbing the flow itself. We investigate fractal escape basins in this problem and characterize fractality by computing the uncertainty exponent and basin entropy. Furthermore, we observe the presence of basin boundaries with points exhibiting the Wada property, i.e., boundary points that separate three or more escape basins.

Parametric analysis of the flow past a square cylinder in the interface of two different-velocity streams

We address the linear stability of the two-dimensional incompressible flow past a square cylinder immersed in the wake of an upstream splitter plate, which separates two streams of different velocities, UT (top) and UB (bottom), to three-dimensional perturbations. The analysis is done in the so-called wake transition regime, across which two-dimensional vortex shedding incorporates spanwise modulation. In addition to the top and bottom stream Reynolds numbers (ReT,B=DUT,B/ν, with D the square cylinder side and ν the kinematic viscosity of the fluid), a parametric analysis is conducted to gauge the effects of splitter plate length (S) and the gap between the trailing edge of the splitter plate and the front face of the cylinder (G) on the leading three-dimensionalizing instabilities. Modes akin to the A- and B-type instabilities that characterize the wake transition regime past circular and square cylinders in homogeneous incoming flow are observed only at very small values of the top-to-bottom Reynolds number ratio R=ReT/ReB, while a third mode, mode C, is ubiquitous beyond more than very mild values of R. Increasing S at constant small G has a stabilizing effect on mode C, whose onset is pushed to higher values of R. Only for long S is mode A observed. Fixing a short S and increasing G results instead in a destabilization of mode C, and mode B is favored over mode A.

Numerical analysis of velocity slip and magnetic Soret effect on dissipative Maxwell liquid motion about a stretching wall with heat source/sink

The main aim of present numerical analysis is to explore the effects of viscous dissipation and velocity slip on the two-dimensional steady-state viscous incompressible flow of Maxwell fluid over a stretching sheet under the infleunce of magnetic field. Novel collective effects of heat source/sink and Soret impacts have been included in the governing equations to explore the salient features of heat and mass transport mechanism. Further, the velocity slip is introduced into the boundary conditions to depict the momentum transport behavior in the flow region. A typical Maxwell fluid model is deployed to separate the non-Newtonian flow behavior from that of classical Newtonian fluids. However, the visco-elastic Maxwell fluid finds its abundant applications in various fields of engineering and medicine such as plastic sheet drawing, metallurgy, polymer sheet extrusion, chemical engineering, electronic cooling, injection molding and nuclear cooling and many others. Inspired by these applications, authors have made an attempt to disclose the magneto-thermo salient features of slip flow of Maxwell fluid over stretching wall under Soret effect. The current physical situation results the highly nonlinear coupled two-dimensional steady-state partial differential equations and which are not amenable to any of the direct techniques. Due to this, the suitable similarity transformations are deployed to reduce the dimensional complexity of the constructed partial differential equations. A robust Matlab-based bvp4c scheme is utilized as a basic tool to produce the similarity solutions of the governing equations. The computer generated numerical data is presented in view of flow, temperature and concentration profiles including physical quantities of interest like skin-friction coefficient, heat and mass transport rates for the various values of physical parameters range such as, 0 ≤ M ≤ 1 , 0 ≤ β ≤ 1 , − 0.3 ≤ λ ≤ 0.6 , − 0.2 ≤ Q ≤ 0.2 , 0 ≤ Ec ≤ 2.2 , 1.2 ≤ Pr ≤ 2.2 , 0 ≤ Sr ≤ 0.5 and 1.2 ≤ Sc ≤ 2.2 . Accordingly, it is noticed that, enhancing magnetic parameter decreases the Maxwell fluid flow and amplifies the temperature and concentration fields. Enlarging Maxwell fluid parameter decreases the velocity field and increases the temperature and concentration profiles. Amplified slip parameter diminished the Maxwell flow inside the flow regime. Increasing heat source/sink parameter amplifies the temperature field. Further, magnifying Soret number amplifies the concentration field. Magnitude of skin-friction coeffiecient enlarged with amplified magnetic and Maxwell fluid parameters. Heat and mass transport rates decayed with enhancement in heat source/sink and Soret parameters. Finally, it is described that, the present similarity solutions showing good agreement with the previously reported solutions in the literature and this fact confirms the accurateness of the used numerical method and the generated similarity solutions.

Numerical heat and mass transfer in magnetized Williamson liquid motion over a sensor wall engulfed in a micro-cantilever system: An application to thin-film fluidic cells

Central focus of this numerical analysis is to investigate the influence of magnetic body force and thermal radiation on the time-dependent electrically conducting two-dimensional viscous incompressible boundary layer flow of non-Newtonian Williamson fluid over a microcantilever sensor surface suspended in a squeezed regime numerically. A novel heat source/sink and Soret effects are included in the respective governing equations. The constructed time-dependent nonlinear coupled partial differential equations are solved by using a bobust RK-4 technique. Increasing magnetic number increases the velocity distribution and decreases the temperature and concentration profiles in the flow regime.Enhancing Soret number increases the mass diffusion profile. Increasing magnetic number suppress the skin-friction coefficient. Heat transport rate booted with rising radiation number and Sherwood number decreased with enlarged Schmidt number in the channel regime. Finally, the numerical accuracy of the present similarity solutions and used numerical method is validated with existing results and observed a remarkable agreement.

An MHD nanofluid flow with Marangoni laminar boundary layer over a porous medium with heat and mass transfer

ABSTRACT The current work considers two-dimensional incompressible fluid flow with heat transfer over a porous medium with mass transpiration and Marangoni boundary condition. The nanoparticles of graphene are mixed in water to improve thermal efficiency. It behaves as a heater upon increasing its solid volume fraction, Eckert number, and as a cooler upon increasing suction. This flow has remarkable interest due to its applications in glues, silicon wafers, heat exchangers, paints, crystal growth in space, etc. The ordinary differential equations (ODEs) are obtained from the considered partial differential equations (PDEs) using similarity transformation technique. Then the solutions are analytically recovered from the system of ODEs. The solutions of temperature and concentration fields are obtained in terms of Laguerre polynomial. The impacts of different parameters, such as Marangoni number, inverse Darcy number, Prandtl number, and Eckert number are analysed using graphical observations. Specially mentioned is Marangoni convection, which contains many industrial and engineering applications such as in nanotechnology, atomic reactors, soap films, semiconductor processing, and atomic reactors. This work adds the effect of graphene water nanofluid which enhance the thermal efficiency in a better way than the base fluid and explains the physical problem on the basis of chemically radiative Marangoni flow.

A numerical investigation of the two-dimensional magnetohydrodynamic water-based hybrid nanofluid flow composed of Fe3O4 and Au nanoparticles over a heated surface

Abstract In this research article, the viscous, steady, and incompressible two-dimensional hybrid nanofluid flow composed of Fe3O4 and Au nanoparticles on an extending sheet has been presented. An inclined magnetic field impact is used for evaluating the impacts of various factors in that case. Furthermore, the influences of porosity, Brownian motion, thermophoresis, thermal and space-dependent heat sources, and thermal radiation factors are also used in this work. The numerical analysis is done by using the bvp4c technique. Validation of the present results confirms that the present analysis is valid. The outcomes show that the higher magnetic factor reduces velocity distribution while increasing the frictional force at the surface due to Lorentz forces which oppose the fluid flow. The friction force at the sheet’s surface is higher when the sheet stretches as compared to the case when the sheet shrinks. Increase in the magnetic factor increases the skin friction of sheet’s surface which consequently increases the rate of thermal transmission at the surface along with thermal distribution. The higher values of thermal radiation and thermal-dependent heat source increase the thermal transportation rate of sheet’s surface. Insights from this investigation can improve electronics cooling systems, vital for devices prone to overheating. Optimizing heat transfer with magnetohydrodynamic water-based hybrid nanofluids containing Fe3O4 and Au nanoparticles ensures efficient heat dissipation, enhancing device performance and longevity.

Irreversibility analysis for unsteady slip flow of heat absorptive liquid with multiple mass diffusion of nanomaterials over a permeable vertical moving plate: A comparative study

This study looks at how thermal radiation and magnetic fields affect the generation of entropy in the unsteady flow of a tri-hybridized nanofluid across a plate that is moving vertically and has surface slippage and suction that changes over time. The movement of the plate aligns with fluid flow, and the far-away velocity increases exponentially following the small perturbation rule. The unsteadiness of the flow is attributed to the varying free stream velocity. A homogeneous magnetic flux acting transversely to the porous wall regulates the laminar and incompressible two-dimensional flow. The presence of tri-hybridized nanoparticles makes it possible to study how adding nanoparticles with different thermal properties, such as Cu, Fe3O4 and TiO2 changes the way base fluids like water and kerosene flow and how heat moves through them. Using dimensionless variables, the mathematical model of the problem is transformed into a form amenable to an analytical solution. The solution is achieved with the help of two-term harmonic and non-harmonic functions. Several factors that influence fluid flow and heat transmission are examined and explored in depth. These include plate velocity, slippage parameter, nanoparticle volume fraction, thermal radiation and heat absorption. The findings show that ternary nanoparticles raise the temperature and speed of both water and kerosene more than hybrid nanoparticles. The increase in the volume fraction of nanomaterials also increases the temperature and velocity of both fluids. These results might be used to improve the efficiency of thermal management systems in a wide range of technical and industrial operations having heat transfer and fluid movement.

A numerical study of thermal and diffusion effects on MHD Jeffrey fluid flow over a porous stretching sheet with activation energy

The primary aim of this research work is to develop a mathematical model for the two-dimensional steady incompressible flow of a Jeffrey fluid over a vertical stretching sheet immersed in a Darcy porous medium. The combined effects of Soret and Dufour numbers, activation energy, viscous dissipation, thermal radiation, mixed convection, and Joule heating are included in the energy and concentration equations within this model. Using appropriate similarity variable transformations, the governing partial differential equations were altered into nonlinear ordinary differential equations and then solved numerically using the Runge-Kutta-Fehlberg method along with the shooting technique. The impact of various physical parameters on concentration, velocity, and temperature profiles is deployed through graphs. The velocity increased with a gradual increment in the values of the viscous ratio, thermal, and solutal stratification parameters, whereas the reverse trend was noticed for the magnetic strength. The temperature diminishes as the Prandtl number increases, whereas the opposite scenario exists for Eckert and Dufor numbers and radiation. The fluid concentration rises for a monotonic increment of the Soret number, while it reduces as the Schmidt number, activation energy, and reaction constraint escalate. The amplification of skin friction is observed with an increase in magnetic field strength, while it decreases with an increasing viscous ratio factor. The Nusselt number decreases as radiation increases and the Eckert number rises, but the opposite scenario occurs with increases in the Prandtl number. The Sherwood number is heightened by an increase in the Schmidt number, reaction, and activation energy factors. Moreover, the numerical values of skin friction and the Nusselt number are compared with the previously published work, demonstrating excellent agreement between the results.

Illustration of thermal radiation on the mixed convection of Williamson nanofluid over an inclined cylindrical surface

This study emphasizes the two-dimensional incompressible flow of Williamson nanofluid caused by an inclined, infinite, expanding/contracting cylinder that is inclined at an angle with the horizontal axis. The focus is on the behavior of non-Newtonian fluid on the flow, heat, and mass transfer during the process. The properties of the fluid are modeled using a set of partial differential equations in association with the characterizing parameters obtained during the presentation of non-dimensional form. The problem is solved numerically using the shooting-based Runge–Kutta fourth-order technique. The results are shown graphically for the behavior of the several contributing parameters. The effect of various pertaining parameters is analyzed henceforth. The physical quantities of interest are also discussed in the tables. The study provides useful insight into the behavior of the fluid, i.e. suction/injection shows opposite behavior on the fluid velocity profiles where suction decelerates the profile and injection enhances it significantly.

Modal analysis for incompressible fluid flow

This paper presents a numerical method for incompressible fluid flow. A difficulty in analyzing incompressible fluid flow is that the continuity equation has no time evolution term. In the marker and cell (MAC) method, Poisson’s equation is solved iteratively, which takes most of the computation time, and in the artificial compressibility method (ACM), pseudo-time iteration is necessary to solve for unsteady solutions. Here, modal analysis that uses the velocity eigenvectors corresponding to zero eigenvalues is proposed for analyzing two-dimensional incompressible fluid flow. The proposed method involves only about one third of the number of variables needed in the MAC method and the ACM, and it does not require iterative calculation of Poisson’s equation or pseudo-time iteration. Numerical results for a simple flow system and a cavity flow obtained using the proposed method are compared with those obtained using the ACM and the simplified MAC method. The results agree well, thereby validating the proposed modal analysis.

NUMERICAL STUDY OF CARREAU FLUID FLOW ALONG AN EXPONENTIAL CURVED STRETCHING SURFACE

A two-dimensional incompressible boundary layer Carneau fluid flow with heat-transfer analysis over a curved stretching surface is analyzed. The energy equation with the inclusion of thermal radiation and viscous dissipation effects is considered. The governing partial differential equations which govern such flow phenomena are transformed into suitable form of ordinary differential equations for integration by using stream function formulation. The developed non-linear problem has been solved by computational approach based on shooting technique using sixth-order Runge-Kutta method and Matlab built-in function bvp4c program. The effects of non-dimensional controlling parameters on temperature and velocity profile are analyzed with the aid of tables and figures. The surface drag force and Nusselt numbers are studied for the different values of the governing parameters. It is predicted that velocity of the fluid and boundary layer thickness is decreased when radius of curvature parameter δ is increased. Furthermore, the temperature profile dwindles for the growing values of δ. Other important information is that for shear-thinning fluid the velocity profile shows its increasing nature, whereas for shear-thickening fluid the opposite trend has been observed. For increasing values of curvature parameter δ from 2 to 1000, the temperature distribution and velocity profile is decreased. The radiative heat flux is included to enhance the temperature of the system, so, for the increasing values of radiation parameter <i>R<sub>d</sub></i> from 0.2-0.5 the temperature distribution is increased. Further, as the Biot number and Eckert number are increased from 0.2-2 and 0.1-1, respectively, the temperature distribution is increased.

Bifurcation analysis of double cavity flows

The first few bifurcations in a two-dimensional incompressible double cavity flow are investigated using the linear stability analysis, the Floquet analysis, and the nonlinear direct numerical simulations (DNS). The prediction of the critical Reynolds number and the type of bifurcation (Hopf, pitchfork, inverse pitchfork, and Neimark–Sacker), which depend on cavity configuration, by the linear stability analysis and the Floquet analysis is consistent with nonlinear DNS. The nonlinear DNS results show that the state of the system passes through multiple intermediate (unstable) states before it reaches the stable attractor (heteroclinic chain), and the type of intermediate states depends on initial conditions. The intermediate states are reported as the asymptotic state in the literature for some flow conditions because it is not known a priori how long it will take to reach the asymptotic state in nonlinear simulations. The present study reports the actual asymptotic state for those flow conditions.

Analytical Approach to Nonlinear Oscillatory Flow Through a Semi-Infinite Vertical Permeable Wall in a Porous Medium with MHD Mixed Convection

The present work illustrates mixed MHD convection flow of an unstable two-dimensional viscous incompressible fluid flow in a porous material under the following conditions: semi-infinite vertical permeable wall support (i) There is a constant mean oscillation in the suction velocity normal to the wall throughout time. (ii) A constant mean velocity of the free stream; (iii) a constant wall temperature; and (iv) a somewhat high temperature differential between the wall and the free stream, which generates free convection currents. Conservation of mass, energy and momentum are used to obtain the governing equations. These equations are non-linear in nature and connected with one another. The nonlinear differential equations are non-dimensionalized employing non-dimensionalized parameters such as Prandtl number Pr, Grashof number Gr, Eckert number Ek, Hartmann number Mh, and Viscous ratio. An analytical method is used to solve approximate solutions for nonlinear PDEs utilizing the double regular perturbation technique. The numerical results for temperature, skin friction, heat flow, velocity, and skin friction are computed for various parameters and are demonstrated to be in good agreement with previous findings.