Modified Spectral Quasilinearization Method for Solving Fluid Flow over Stretching Sheet

A three-dimensional hybrid nanofluid flow over a stretching/shrinking sheet is numerically studied. The hybrid nanofluid being considered in this study used water as the base fluid and mixed with two types of solid nanoparticles, namely alumina (Al2O3) and copper (Cu). The main focus of the current study is to examine the effect of magnetic field, Joule heating, and rotating sheet on the velocity, and temperature profiles. In addition, the impact of suction and stretching sheet on the variations of reduced skin friction, , and reduced heat transfer are studied as well. The fluid flow and heat transfer problem presented in this study is governed by a system of nonlinear partial differential equations (PDEs), which is then transformed into the corresponding system of high order nonlinear ordinary differential equations (ODEs) using similarity variables. The resulting system of higher order nonlinear ODEs is solved numerically using a boundary value solver known as bvp4c, which operates on the MATLAB computational platform. Results revealed that dual solutions exist for shrinking sheet while unique solutions are observed for stretching sheet with various values of Cu nanoparticles volume fraction and magnetic parameter. Dual solutions also exist for the value of the suction parameter greater than its critical point with various values of Cu nanoparticles volume fraction. Velocity profile of the hybrid nanofluid increases alongside with the value of magnetic parameter but declination was observed in the profile of and temperature, for both solutions as the value of Cu nanoparticles volume fraction increases. When the value of rotational parameter increases, both velocity and profiles increase for both solutions. This indicates that the momentum boundary layer thickness increases with increasing values of for both solutions, but thermal boundary layer thickness decreases for the first solution and increases for the second solution. Finally, an increment in the value of Eckert number causes the temperature of the hybrid nanofluid to rise as well for both first and second solutions.

Thermal analysis of Non-Newtonian visco-inelastic fluid MHD flow between rotating disks

This research aims to study the 3D magnetohydrodynamics stagnation-point flow (SPF) over a horizontal plane surface (HPS) carrying water-based graphene oxide (GO) nanoparticles caused by an irregular heat source/sink used in heat transfer procedures. In addition, a Tiwari–Das model is used to inspect the dynamics of fluid flow behavior and heat transmission features of the nanoparticles with experiencing the impacts of thermal radiation. The acquired nonlinear set of partial differential equations (PDEs) is transfigured to a system of ordinary differential equations (ODEs) using similarity transformations. The accumulative dimensionless ODEs are then further tackled in MATLAB using the bvp4c solver. Tables and figures are prepared for the execution of several relevant constraints along with nodal/saddle indicative parameter, internal heat source/sink parameter, radiation parameter and nanoparticles volume fraction which divulges and clarify more accurately the posited quantitative data and graphical findings. Also, the velocity profile decelerated in the axial and transverse coordinate axes for a higher value of the nanoparticle volume fraction but the dimensionless temperature distribution is augmented. Additionally, thermal boundary layer thickness and profile of temperature enriches with higher impressions of radiation constraint. However, the internal heat sink factor declines the profiles of temperature while escalating with the superior value of the internal heat source parameter.

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In this research work, we investigate an unsteady flow over a rotating disk. We assign symbols to the selected dependent and independent quantities. Then all physical systems are modeled to mathematical form by applying physical laws for an ionized liquid flow over a rotating disk with nanoparticles from the set of Poisson Nernst–Planck model, Energy equation and Navier–Stokes equations. The set of partial differential equations along with the boundary conditions are transformed to a set of coupled ordinary differential equations for an electro-viscous flow of nanofluid over a rotating disk by using similarity transformations. The unknown physical quantities are investigated through Parametric Continuation Method (PCM). For physical purpose, physical quantities like flow behavior thermal properties, thermal variation, the distribution of ions in the fluid region, skin friction, are analyzed through graphical and tabulated results. As exact solutions are not possible for nonlinear ordinary differential equations (ODEs) system, therefore, such quantities are subjected to numerical calculation following Parametric Continuation Method (PCM) and validated the result through BVP4c package in Matlab.

Nonlinear partial differential equations (PDEs) modelling unsteady boundary-layer flows are solved by the spectral relaxation method (SRM) and the spectral quasilinearization method (SQLM). The SRM and SQLM are Chebyshev pseudospectral based methods that have been successfully used to solve nonlinear boundary layer flow problems described by systems of ordinary differential equations. In this paper application of these methods is extended, for the first time, to systems of nonlinear PDEs that model unsteady boundary layer flow. The new extension is tested on two problems: boundary layer flow caused by an impulsively stretching plate and a coupled four-equation system that models the problem of unsteady MHD flow and mass transfer in a porous space. Numerous simulation experiments are conducted to determine the accuracy and compare the computational performance of the proposed methods against the popular Keller-box finite difference scheme which is widely accepted as being one of the ideal tools for solving nonlinear PDEs that model boundary layer flow problems. The results indicate that the methods are more efficient in terms of computational accuracy and speed compared with the Keller-box.

The purpose of the present study is to characterize the thermal performance of micropolar fluid flows on a vertically elongated permeable sheet in the presence of a heat source/sink. The additional physical aspects include a heat source/sink and the fluid flow across a permeable medium. The mathematical problem is governed by partial differential equations is transformed into a system of nonlinear ordinary differential equations using similarity transformations. The regulating transformed standard ordinary differential equations are subsequently simulated utilizing the Runge‐Kutta fourth order method, accompanied by the shooting technique to evaluate numerical findings of dependent quantities of physical importance through MATLAB. The impact of varied parameters on the fluid momentum, angular momentum, and energy was analyzed and shown graphically. The results demonstrate that the porosity of the medium plays a critical role in regulating both flow resistance and heat transfer: higher porosity enhances permeability, leading to faster fluid motion and improved thermal transport, while low porosity increases drag and suppresses heat transfer. The applied magnetic field introduces Lorentz forces that dampen velocity profiles and increase fluid temperature through Joule heating, thereby thickening the thermal boundary layer. Increasing the micropolar parameter enhances coupling between microrotation and linear motion, resulting in higher velocity and microrotation profiles. The study further provides streamlined and isothermal plots to illustrate the interplay of magnetic and porous effects. These findings underline the novelty of combining porosity, slip, and heat source/sink effects in micropolar fluid dynamics and contribute to optimized designs for thermal management in porous and magnetized environments.

This paper aims to investigate a boundary layer flow with heat and mass transfer of a special third grade fluid over a stretchable surface in a parallel free stream. Using the similarity variables generated by the Lie group analysis, the governing nonlinear partial differential equations (PDEs) are transformed into a system of nonlinear ordinary differential equations (ODEs). The transformed equations are then solved numerically using the shooting technique. In addition, an attempt is made to carry the asymptotic solution behavior for large stretching and suction parameters, and the asymptotic results of skin friction coefficient are then compared with the direct numerical solutions, which shows a good agreement. It is observed that the similarity equations exhibit dual solutions in a certain range of shrinking strength. These two solution branches show different behavior on the skin friction coefficient, local Nusselt number, Sherwood number, velocity, and temperature profiles. Thus, emphasis has been given to carrying out a stability analysis to determine the physically reliable solution. The stability analysis shows that the upper branch solution is stable. It is observed that the suction parameter increases the range of dual solutions and the magnitude of the critical point from where the dual solutions bifurcate; however, the non-Newtonian parameter shows the opposite behavior. The momentum, thermal and concentration boundary layer thicknesses in the upper branch solution are lower than the lower branch solution.

This study investigates heat transportation in slip flow of a non-Newtonian fluid with thermal stratification over a stretched surface, a configuration that closely represents many industrial and engineering processes. The inclusion of slip effects and non-Newtonian rheology provides a more realistic description of flows encountered in polymer extrusion, coating technologies, micro- and nano-scale devices, and biomedical systems. Thermal stratification is incorporated to model non-uniform temperature environments commonly observed in heat exchangers, cooling of electronic components, and energy systems. The governing partial differential equations for momentum, energy, and nanoparticle concentration are transformed into a system of nonlinear ordinary differential equations using appropriate similarity transformations. The resulting boundary value problem is solved numerically to analyze the impact of key parameters such as the slip coefficient, thermal stratification parameter, Sutterby fluid parameter, and Brownian motion on velocity, temperature, and concentration profiles. Results indicate that thermal stratification significantly reduces the temperature distribution within the boundary layer, while velocity slip at the surface diminishes the skin friction coefficient. The non-Newtonian characteristics of the Sutterby fluid substantially influence the heat transfer rate, with shear-thinning fluids demonstrating enhanced thermal performance compared to shear-thickening fluids. This analysis provides valuable insights for thermal engineering applications involving non-Newtonian nanofluids in stratified environments.

<div class="section abstract"><div class="htmlview paragraph">In this paper, a new solution method is presented to study the effect of wave propagation in engine manifolds, which includes solving one-dimensional models for compressible flow of air. Velocity, pressure, and density profiles are found by solving a system of non-linear Partial Differential Equations (PDEs) in space and time derived from Euler’s equations. The 1D model includes frictional losses, area change, and heat transfer. The solution is traditionally found by utilizing the Method of Characteristics and applying finite difference solutions to the resulting system of ordinary differential equations (ODEs) over a discretized grid. In this work, orthogonal collocation is used to solve the system of ODEs that is defined along the characteristic curves. Orthogonal polynomials are utilized to approximate velocity, pressure, sound speed, and the characteristic curves along which the system of PDEs reduce to a system of ODEs. The approximation polynomials are defined over the whole manifold domain, transforming Euler’s equations into a system of ODEs that can be solved using a generic ODE solver. This reduction is done symbolically using a computer algebra system (Maple). The method results in a system of ODEs that has a higher spatial order along the whole space compared to methods based on finite differences, reducing the number of nodes required to find an acceptable solution that captures the state dynamics at different locations inside the manifold in real time. The proposed model is compared against the Method of Characteristics (MOC) that is used as a reference model; this comparison includes the states at the inlet, outlet, and midpoint. In summary, a high order method that can calculate solutions of the 1D manifold model equations is developed by finding the respective polynomial approximations along the 1D space and solving the resultant system using a generic solver in real time.</div></div>

SUMMARY The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 × 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how "numerical" basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DCs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless ofthe stability ofthe spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.

Influence of Hall current on flow and heat transfer of dusty fluid over a convectively heated stretching plate in the presence of nonlinear thermal radiation is explored in this paper. The unsteadiness in the flow and temperature fields is because of the time-dependent stretching velocity and surface temperature. Suitable similarity transformations are used to convert the governing partial differential equations of momentum and thermal energy to a system of nonlinear ordinary differential equations. Consequent equations are solved by using shooting method. The details of the velocities, temperatures, local Nusselt number as well as local skin friction for various parameters such as unsteadiness parameter, thermal radiation, Hall effects, Biot number, Eckert number, Prandtl number and magnetic parameter are presented graphically and discussed in detail.

The present perusal is to scrutinize the consequences of free-forced convective heat-mass transportation and n th order chemical reaction on MHD, non-Darcian, incompressible nanofluid fluid flow through an extended lamina, immersed in a permeable medium along with an external constant magnetic field. The fluid flow governing equations (PDEs) has been transformed into a system of nonlinear ODEs with the help of appropriate similarity transformation, and then solved numerically by using SQLM (Spectral quasilinearization method) method. Solution error for temperature and concentration profile against radiation parameter, Brownian motion parameter, and Hartman number also has been depicted diagrammatically. The graphical representation shows that the Hartman number, Brownian motion parameter, and chemical reaction parameter decline in velocity. While, the thermal boundary layer enhances with an enhancement of the Hartman number, Brownian motion parameter, radiation parameter, and chemical reaction parameter. Generation of entropy increases with the Brinkman number, Reynolds number, and Hartman number in the vicinity of the stretching lamina. Brinkman number and Reynolds number reduce the Bejan number. For the engineering interest Skin-friction, the local Nusselt number, and local Sherwood number also have been calculated. Impacts of pertinent parameters on the skin friction, Nusselt number, and Sherwood number are examined graphically and also numerically.

The current work investigates the exact analytical solution for 2D mixed convective flow and heat transfer of Casson hybrid nanofluid over stretching/shrinking sheet linear velocity in porous medium. The nonlinear system of partial differential equations (PDEs) is formed for the defined flow which is then converted into nonlinear system of ordinary differential equations (ODEs) via similarity transformations. Then the resultant dimensionless equation is solved analytically to obtain the exact solution for velocity profile. The hybrid nanofluid Cu – Al2O3/H2O is used in the analysis. Further the effect of porous medium, hybrid nanofluids and Brinkman ratio on the velocity profile is examined with the help of graphs. It found many applications in the fields in solar energy, heating and cooling systems, regenerative heat exchange, chemical catalytic reactors and oil extraction etc.

The current and potential applications of bioconvection renewed drive for theoretical research on synthesis and process control in biofuel cells and bioreactors. Thus, this work devoted to solving the problem of free convection in micropolar boundary layer fluid flow and heat transfer past a vertical flat stretching plate within a porous medium. Scaling group of transformation was performed to achieve the appropriate similarity solutions, which was later applied to modify the governing boundary layer system to a nonlinear ordinary differential equations system. The Runge–Kutta method in association with the shooting technique in the Maple software exercised to attain the numerical solutions. There is a strong dependence of momentum transportation on the increment of the Darcy number, the suction/injection parameter and the Grashof number, respectively. The temperature distribution within the thermal boundary layer aided by augmenting the magnitude of the microrotation density.