Computational Analysis of Magnetohydrodynamic Jeffery–Hamel Flow for Couple Stress Fluids in Stretching/Shrinking Channels via Artificial Neural Network

This study investigates the thermal behavior for Jeffery–Hamel flow of a couple stress fluid under the influence of a magnetic field, employing an artificial neural network (ANN) methodology. The research evaluates how the variations in Reynolds number (Re), Eckert number (Ec), Prandtl number (Pr), couple stress parameter (S), magnetic parameter (M), and channel angle (α) alter the temperature distributions within the fluid. The governing coupled nonlinear differential equations for couple stress fluid in Jeffery–Hamel flow and heat transfer are solved using an artificial neural network (ANN) approach. The ANN is trained to satisfy the boundary value problem by minimizing the residuals of the governing equations and boundary conditions, providing an efficient mesh-free solution. The analysis reveals that increasing the magnetic parameter amplifies the Lorentz force, which leads to an increase in the fluid temperature across both divergent and convergent channels. It is observed that when the couple stress parameter is very low, an increase in channel angle raises fluid temperature. Conversely, for a large value of couple stress parameter, increasing the channel angle decreases fluid temperature in the divergent channel but increases it in the convergent channel. In the absence of a magnetic field (M = 0), enhancing the couple stress parameter results in a decrease in the fluid temperature for a divergent channel, while in a convergent channel, it leads to an increase in the fluid temperature. These insights offer a comprehensive understanding of the interplay between couple-stress fluid properties, magnetic field, heat transfer, and channel geometries, contributing to the optimization of thermal management in systems utilizing couple-stress fluids.

In Jeffery–Hamel flow, the motion of a viscous incompressible fluid between rigid plane walls, unidirectional flow is impossible if the angle between the walls exceeds a critical value of 2α2 which depends on the Reynolds number. In this paper the nonlinear development of the flow near this critical value is studied through numerical solutions of the two-dimensional Navier–-Stokes equations for flow in divergent channels with piecewise straight walls. It is found that if the angle between the walls exceeds 2α2 then Jeffery–-Hamel flow does not occur, and the solution takes the form of a large-amplitude wave with eddies attached alternately to the upper and lower walls. When viewed in the appropriate coordinate system, far downstream the wave has constant wavelength and strength, although, physically, there is a linear increase in wavelength with distance downstream, i.e. the wavelength is proportional to the channel width. If the angle between the walls is less than 2α2, then the existence (or otherwise) of the wave depends on the conditions near the inlet, in particular the local geometry of the channel. Jeffery–-Hamel flow is obtained downstream of the inlet for angles well below 2α2, but close to but below the critical value, solutions have been obtained with the wave extending (infinitely) far downstream. The wavelengths obtained numerically were compared with those from linear theory with spatially developing steady modes. No agreement was found: the wavelengths from the steady Navier–-Stokes solutions are significantly larger than that predicted by the theory. However, in other important aspects the results of this study are consistent with those from previous studies of the development/existence of Jeffery–-Hamel flow, in particular as regards the importance of the upstream conditions and the subcritical nature of the spatial development of the flow near the critical boundary in the Reynolds number–wall angle parameter space.

ABSTRACTThis research is surveying a numerical technique to study effects of CuO nanoparticles in Jeffery–Hamel flow with the high levelized of magnetic field in converging and diverging channels. The innovation of this work is that governing equations in this model is solved using coupling the quasi‐linearization method (QLM) and meshless method, which is based on radial basis functions (RBFs). Also, the RBF method, no need for pre‐defined meshing, reduces the solution of the problem to the solution of a system of algebraic equations by using a set of points within the domain and its boundaries. In addition to, this geometry, the QLM is utilized as a tool for confronting the nonlinearity of the problem and also to reduce the nonlinear boundary problems to a sequence of linear boundary problems which are much simpler to solve. In order to evaluate the convergence analysis of the method, error estimations are made by a residual function denoted. Also, the ability of the present method is shown by comparing it with the numerical method to solve this problem, which is in good agreement. Effects of multivariable parameters are analyzed on magnetic field, nanoparticles volume fraction, and angle of converging and diverging channels. The obtained results show that at angles or Reynolds number of greater in divergent channels, backflow occurs so the high levelized of magnetic field eliminates this phenomenon. Also, the numerical results show that at the angle of channel and which no backflow occurs, the effect of increasing the Hartmann number on the nanofluid velocity from values of to has respectively increased by 35.55 and 66.36%, compare to its initial value.

The stability and bifurcations associated with the loss of azimuthal symmetry of planar flows of a viscous incompressible fluid, such as vortex-source and Jeffery–Hamel flows, are studied by employing linear, weakly nonlinear and fully nonlinear analyses, and features of new solutions are explained. We address here steady self-similar solutions of the Navier–Stokes equations and their stability to spatially developing disturbances. By considering bifurcations of a potential vortex-source flow, we find secondary solutions. They include asymmetric vortices which are generalizations of the classical point vortex to vortical flows with non-axisymmetric vorticity distributions. Another class of solutions we report relates to transition trajectories that connect new bifurcation-produced solutions with the primary ones. Such solutions provide far-field asymptotes for a number of jet-like flows. In particular, we consider a flow which is a combination of a jet and a sink, a tripolar jet, a jet emerging from a slit in a plane wall, a jet emerging from a plane channel and the reattachment phenomenon in the Jeffery–Hamel flow in divergent channels.

Purpose – The purpose of this paper is to demonstrate the eligibility of the Weighted Residual Methods (WRMs) applied to magneto hydro dynamic (MHD) nanofluid flow in divergent and convergent channels. Selecting the most appropriate method among the WRMs and discussing about Jeffery-Hamel flow's treatment in divergent and convergent channels are the other important purposes of the present research. Design/methodology/approach – Three analytical methods (Collocation, Galerkin and Least Square Method) and numerical method have been applied to solve the governing equations. The reliability of the methods is also approved by a comparison made between the fourth-order Runge-Kutta numerical method. Findings – The obtained solutions revealed that WRMs can be simple, powerful and efficient techniques for finding analytical solutions in science and engineering non-linear differential equations. Originality/value – It could be considered as a first endeavor to use the solution of the MHD nanofluid flow in divergent and convergent channels using these kinds of analytical methods along with the numerical approach.

Purpose This paper aims to model and analyze Jeffery Hamel’s channel flow with the magnetohydrodynamics second-grade hybrid nanofluid. Considering the importance of studying the velocity slip and temperature jump in the boundary conditions of the flow, which leads to results close to reality, this paper intends to analyze the mentioned topic in the convergent and divergent channels that have significant applications. Design/methodology/approach The examination is conducted on a EG-H_2 O <30%–70%> base fluid that contains hybrid nanoparticles (i.e. SWCNT-MWCNT). To ensure comprehensive results, this study also considers the effects of thermal radiation, thermal sink/source, rotating convergent-divergent channels and magnetic fields. Initially, the governing equations are formulated in cylindrical coordinates and then simplified to ordinary differential equations through appropriate transformations. These equations are solved using the Explicit Runge–Kutta numerical method, and the results are compared with previous studies for validation. Findings After the validation, the effect of the governing parameters on the temperature and velocity of the second-grade hybrid nanofluid has been investigated by means of various and comprehensive contours. In the following, the issue of entropy generation and its related graphical results for this problem is presented. The mentioned contours and graphs accurately display the influence of problem parameters, including velocity slip and temperature jump. Besides, when thermal radiation is introduced (Rd = +0.1 and Rd = +0.2), entropy generation in convergent-divergent channels decreases by 7% and 14%, respectively, compared to conditions without thermal radiation (Rd = 0). Conversely, increasing the thermal sink/source from 0 to 4 leads to an 8% increase in entropy generation at Q = 2 and a 17% increase at Q = 4 in both types of channels. The details of the analysis of contours and the entropy generation results are fully mentioned in the body of the paper. Originality/value There are many studies on convergent and divergent channels, but this study comprehensively investigates the effects of velocity slip and temperature jump and certainly, this geometry with the specifications presented in this paper has not been explored before. Among the other distinctive features of this paper compared to previous works, the authors can mention the presentation of velocity and temperature results in the form of contours, which makes the physical analysis of the problem simpler.

This article examines the heat and mass transfer capabilities of a constitutive model in a thermally evolving steady laminar Jeffery–Hamel flow through a convergent-plate channel, including streamwise conduction with step changes in uniform wall temperature. A Jeffery–Hamel problem with a simple shear flow is used to undertake a comparative computational analysis of the thermal behavior of a viscoelastic fluid subjected to autocatalytic processes. The flow is tracked in a purely radial orientation with the deployment of coupled stresses in momentum conservation. The computational solutions for the flow, temperature and concentration distribution, and heat and mass transfer coefficient of a viscoelastic fluid obeying the complex Oldroyd-B constitutive equation in laminar converging channel flows are established. The analysis of the impacts of the thermal radiation, the heat source, and the chemical reaction as an autocatalytic process is included in the model, which is valid for fully developed thermal and hydrodynamic flow conditions with a constant heat and mass flux imposed at the wall. In the diverging part of the channel, where vortex compression is the predominant flow topology, there exist patches of local flow compression. On the flow field, the modified relaxation and retardation parameters show an opposing behavior. An Oldroyd-B fluid exhibits higher interactions with nearby vortices in the divergent channel, allowing a complex flow structure. The viscoelastic characteristics are anticipated to change the homogeneous–heterogeneous reaction transport processes, offering tremendous potential for applications in associated sectors. The deceleration flow in the diverging channel and the acceleration flow in the converging channel augment the average Nusselt numbers.

Purpose – The purpose of this paper is to demonstrate the eligibility of the weighted residual methods (WRMs) applied to Jeffery-Hamel Flow. Selecting the most appropriate method among the WRMs and discussing about Jeffery-Hamel flow's treatment in divergent and convergent channels are the other important purposes of the present research. Design/methodology/approach – Three analytical methods (collocation, Galerkin and least square method) have been applied to solve the governing equations. The reliability of the methods is also approved by a comparison made between the forth order Runge-Kutta numerical method. Findings – The obtained solutions revealed that WRMs can be simple, powerful and efficient techniques for finding analytical solutions in science and engineering non-linear differential equations. Originality/value – It could be considered as a first endeavor to use the solution of the Jeffery-Hamel flow using these kind of analytical methods along with the numerical approach.

In this paper, the MHD Jeffery–Hamel flow with cu-water nanofluid between two smooth rectangular walls with the transverse magnetic field is studied. Differential transform method (DTM) is used to obtain the velocity profile of Jeffery–Hamel flow in both convergent and divergent channels for different values of Reynolds number and Hartmann number. Finally, to examine the accuracy and the validity of the method, the obtained results have been compared with the available collation method results.

Couple Stress Fluid (CSF) incorporates a new material constant whose, in contrast to a Newtonian fluid, regulates both couple stress and lubricant viscosity. Due to the fourth‐order spatial derivative term that this material constant introduces into the momentum equation, this fluid (CSF), even for classical fluid problems, has received limited research. The aim of this article is to examine the joint impact of heat transmission and magnetic field on a time‐fractional model of two‐phase free convection flow of dusty fluid that is electrically conducting. Among parallel plates, one of which is stationary and the other of which is oscillating with constant velocity, the CSF is supposed to flow. Heat is transferred by free convection and buoyant force, which generates the flow. Furthermore, all spherical dust particles are uniformly dispersed through the fluid. The flow is modeled mathematically in terms of PDEs. The provided derived system of PDEs is generalized by applying a recently invented fractional derivative, the Caputo‐Fabrizio fractional derivative. Finite sine Fourier and Laplace transformations are jointly applied to handle the problem. The velocity and temperature profiles have closed form solutions. The CSF outcomes for stimulating fluid parameters are shown in numerous graphs for CF time fractional derivatives. Additionally, the influence of various parameters has been discussed. Mathcad‐15 is used to plot the graphical outcomes for the CSF, dust particle, and temperature profiles. Furthermore, the skin friction and Nusselt number are calculated. Table 1 demonstrates how the rate of heat transmission reduces as the Peclet number's value rises. Similarly, Table 2 demonstrates that the skin friction increases as the magnetic parameter and couple stress parameter are raised. Table 3 and 4, shows the Regression analysis that the variation in the velocity for Couple stress and dusty fluid parameter are statistically significant. By increasing the couple stress parameter λ, retarde the both velocities profile.

An approximation of the analytical solution of the Jeffery–Hamel flow by decomposition method

Investigation of cross-diffusion effect on radiative Jeffery-Hamel flow in convergent/divergent stretchable channel with Lorentz force and Joule heating

Heat transfer in the Jeffery-Hamel flow of a yield-stress fluid

The combined effects of nanoparticle and magnetic field on the nonlinear Jeffery-Hamel flow are analyzed in the present study. The basic governing equations are solved analytically to nonlinear ordinary differential equation using perturbation method together with a semi-numerical analytical technique called Hermite- Pade approximation. The obtained results are well agreed with that of the Adomian decomposition method (ADM). The velocity profiles are presented in divergent channel for various values of nanoparticle solid volume fraction, Hartmann number, Reynolds number and channel angle. The relations between velocity field with Reynolds number and channel angle with the effect of nanoparticle solid volume fraction and Hartmann number are also performed qualitatively.

Many researchers have been interested in application of mathematical methods to find analytical solutions of nonlinear equations and for this purpose, new methods have been developed. One of the newest analytical methods to solve nonlinear equations is Reconstructio n of variational Iteration Method (RVIM) which is an accurate and a rapid convergence method in finding the approximate solution for nonlinear equations. By applying Laplace Transform, RVIM overcomes the difficulty of the perturbation techniques and other variational methods in case of using small parameters and Lagrange multipliers, respectively. In this study RVIM is applied for the effects of magnetic field and nano particle on the Jeffery-hamel flow. The traditional Navier-Stokes equation of fluid mechanics and Maxwell’s electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. The flow field inside the divergent channel is studied for various values of Hartmann number and angle of channel. Finally the effect of nano particle volume fraction in the absence of magnetic field is investigated, too. The validity of RVIM method is ascertained by comparing our results with numerical (Runge Kutta method) results.