Bivariate Spectral Quasilinearization Method Research Articles

Prediction model based on artificial neural network and bivariate spectral quasi-linearization method for compressible turbulent boundary-layer flow over a smooth flat surface

The compressible two-dimensional turbulent flow solutions at an arbitrary point in time and space by incorporating the mass, momentum, and energy conservation equations over a smooth flat surface and parallel free stream with unfavorable pressure gradient are studied. The Falkner–Skan transformation is applied to the turbulent boundary-layer equations and related boundary conditions, and the resulting nonlinear coupled system of partial differential equations is solved by the bivariate spectral quasi-linearization method. Moreover, to predict the thermal distribution of the flow, an artificial neural network model has been developed with the Nusselt number as target values. Several plots have been depicted, it is evaluated that the mean squared error value is 6.41 × 10−7, the overall coefficient of determination (R) is 0.997 52, and the average error rate is 0.68% for the said model, indicating the attainment of high accuracy for estimation.

Entropy Generation in Double Diffusive Convective Magnetic Nanofluid Flow in Rotating Sphere with Viscous Dissipation

We scrutinize the numerically investigates entropy generation for unsteady magnetic nanofluid with stagnation region of an impulsively rotating sphere with viscous dissipation and chemical reaction effects. Employing the similarity technique, the equations modeling are switched into highly nonlinear differential equations with boundary layer flow. These equations are numerically executed with bivariate spectral quasilinearization method. The entropy generation effect of varying various pertinent different parameters are taken into account and the results analyzed graphically. The rate of entropy generation and energy related temperature profiles increased with increased of thermal radiation parameter, chemical reaction and rotation parameters while increased values of the magnetic number had a negative impact impact on rate of entropy generation. Variables of engineering interest which includes the skin friction coefficient, the Nusselt number and the Sherwood number were also analyzed in a table. These numerical results will be of great value to chemists, biologists, physicists, and engineers interested in the theory and applications of entropy generation.

An Unsteady Nanofluid Flow Past Parallel Porous Plates: A Numerical Study

Background: This study investigates an unsteady, two-dimensional, incompressible viscous boundary layer flow of an electrically conducting nanofluid past parallel plates. The plates are permeable to allow both suction and injection to take place. It is assumed that viscosity, thermal conductivity and mass diffusivity of the nanofluid vary with temperature. The novelty of this study is in consideration of the combined effects of chemical reaction, permeability, externally applied magnetic field, and momentum diffusivity on the flow varibles. The magnetic field force is significant because it provides information regarding the boundary layer characteristics. Methods: The highly nonlinear partial differential equations are solved numerically using the newly developed bivariate spectral quasilinearization method (BSQLM) along with varying thermal and concentration boundary conditions. The BSQLM method is an innovative technique that is more reliable and robust as it demands fewer grid points and has a global approach to solving PDEs. Results: An analysis and comparison of results with existing literature are reported. Excellent agreement has been found between our results and those previously published. Among the findings, we show, inter alia, a significant increase in the profiles for fluid velocity, temperature and concentration with an increase in the chemical reaction, applied magnetic field, and thermal radiation. The BSQLM converges fast and is computationally efficient when applied to boundary layer problems that are defined on a large computational domain. Conclusions: A numerical study on nanofluid flow between parallel porous plates has been carried out, and here are the key findings: 1. Heat flux is directly related to thermal radiation, the applied magnetic field, permeability, and the chemical reaction involved. 2. Mass flux increases with increased chemical reaction, permeability, and the magnetic parameters. 3. The nanofluid concentration is directly related to the Prandtl and magnetic numbers and inversely related to the Reynolds number and chemical reaction. 4. The skin-friction coefficient reduces with higher values of magnetic field and permeability parameters and increases with an increment in thermal radiation and chemical reaction. 5. The BSQLM has a high convergence rate with high accuracy.

Numerical Solutions for Laminar Boundary Layer Nanofluid Flow along with a Moving Cylinder with Heat Generation, Thermal Radiation, and Slip Parameter

The investigation of the numerical solution of the laminar boundary layer flow along with a moving cylinder with heat generation, thermal radiation, and surface slip effect is carried out. The fluid mathematical model developed from the Navier-Stokes equations resulted in a system of partial differential equations which were then solved by the multidomain bivariate spectral quasilinearization method (MD-BSQLM). The results show that increasing the velocity slip factor results in an enhanced increase in velocity and temperature profiles. Increasing the heat generation parameter increases temperature profiles; increasing the radiation parameter and the Eckert numbers both increase the temperature profiles. The concentration profiles decrease with increasing radial coordinate. Increasing the Brownian motion and the thermophoresis parameter both destabilizes the concentration profiles. Increasing the Schmidt number reduces temperature profiles. The effect of increasing selected parameters: the velocity slip, Brownian motion, and the radiation parameter on all residual errors show that these errors do not deteriorate. This shows that the MD-BSQLM is very accurate and robust. The method was compared with similar results in the literature and was found to be in excellent agreement.

Numerical analysis of free convection from a spinning cone with variable wall temperature and pressure work effect using MD-BSQLM

Abstract The problem of the numerical analysis of natural convection from a spinning cone with variable wall temperature, viscous dissipation and pressure work effect is studied. The numerical method used is based on the spectral analysis. The method used to solve the system of partial differential equations is the multi-domain bivariate spectral quasi-linearization method (MD-BSQLM). The numerical method is compared with other methods in the literature, and the results show that the MD-BSQLM is robust and accurate. The method is also stable for large parameters. The numerical errors do not deteriorate with increasing iterations for different values of all parameters. The numerical error size is of the order of 1 0 − 10 1{0}^{-10} . With the increase in the suction parameter ξ \xi , fluid velocity, spin velocity and temperature profiles decrease.

Overlapping Multi-Domain Spectral Method for MHD Mixed Convection Slip Flow Over an Exponentially Decreasing Mainstream with Nonuniform Heat Source/Sink and Convective Boundary Conditions

Overlapping multi-domain bivariate spectral quasilinearization method is applied on magnetohydrodynamic mixed convection slip flow over an exponentially decreasing mainstream with convective boundary conditions and nonuniform heat source/sink effects. The method is employed in solving the transformed flow equations. The convergence properties and accuracy of the method are determined. The method gives highly accurate results after few iterations and using few grid points in each space subinterval and the entire interval. The use of minimal numbers of grid points at each subinterval minimizes the effects of round-off errors that can lead to instabilities. The accuracy increases as the number of overlapping subintervals increases. The accuracy improvement is achieved through making the coefficient matrices less dense. The effects of controlling parameters on the flow fields and physical quantities of interest are studied. Results show that increasing Biot number and nonuniform heat source/sink enhances the flow fields while reducing skin friction and heat transfer rate. The fluid properties improve with injection whereas the flow characteristics augment with suction. The considered exponentially decreasing external flows have particular applications in diverging channel flows. This study has practical significance in various boundary layer problems such as in controlling and delaying boundary layer separation on control surfaces and in suppressing recirculating bubbles.

Numerical Investigation of Natural Convection Viscoelastic Jeffrey’s Nanofluid Flow from a Vertical Permeable Flat Plate with Heat Generation, Thermal Radiation, and Chemical Reaction

The boundary layer flow of an incompressible viscoelastic Jeffrey’s nanofluid from a vertical permeable flat plate is investigated. We consider the effects of heat generation, thermal radiation, and chemical reaction on the fluid flow. The nonlinear transformed coupled differential equations that describe the transport processes are solved numerically using a multidomain bivariate spectral quasilinearization method (MD-BSQLM). This innovative method involves blending the quasilinearization idea with the bivariate Lagrange interpolation. The solutions of the resulting system of equations are then obtained sequentially on multiple intervals using the Chebyshev spectral collocation method. The method is shown to give accurate solutions for boundary layer-type equations. The influence of various physical parameters on velocity, temperature, and nanoparticle concentration fields, as well as on the skin friction and heat and mass transfer coefficients, is shown and discussed in detail. The range of the values of the governing parameters considered in this study is between 0 , 4 . For qualitative validation of the results and the numerical method used, calculations were carried out to graphically obtain the velocity, temperature, and nanoparticle concentration fields for selected physical parameter values. The results obtained were found to correlate with the results from published literature. For quantitative verification of our findings, the MD-BSQLM numerical solutions were again confirmed against published results reported in the literature, and the results were observed to be in perfect agreement. This study’s findings indicate that the Deborah number and suction parameter have related effects on the velocity profile, which is to suppress both the flow velocity and the momentum boundary layer thickness. Increasing the heat generation and thermal radiation parameters enhances both the temperature and thermal boundary layer depths. In contrast, an increase in the chemical reaction parameter causes a decrease in the fluid concentration.

MHD mixed convective radiative flow of Eyring‐Powell fluid over an oscillatory stretching sheet using bivariate spectral method on overlapping grids

AbstractThe bivariate spectral quasilinearization method (BSQLM) on overlapping grids is presented and applied in the analysis of unsteady magnetohydrodynamic mixed convection flow of Eyring‐Powell fluid over an oscillatory stretching sheet embedded in a non‐Darcy porous medium with nonlinear radiative heat flux and variable thermophysical properties. The fluid properties, namely the fluid viscosity, thermal conductivity, and mass diffusivity, are assumed to vary with temperature. It is assumed that the first‐order chemical reaction with heat generation/absorption takes place in the flow. The flow domain is subject to uniform transverse magnetic field perpendicular to the stretching surface. The transformed flow equations are solved numerically using BSQLM on overlapping grids. The convergence properties and accuracy of the method are assessed. The proposed method is computationally efficient, and it gives stable and highly accurate results after few iterations and using few grid points in each subinterval. The improved accuracy rests upon the use of the overlapping grid, which produces sparse coefficient matrices that are easy to invert and have small condition numbers. The effects of physical parameters on the flow fields, local skin friction, the Nusselt number, and the Sherwood number are exhibited through graphs and tables. Amongst other findings, we found that the amplitude of the fluid flow along with flow characteristics may efficiently improve through the utilization of variable fluid viscosity. Heat and mass transportation processes enhance with the inclusion of nonlinear radiative heat flux, temperature‐dependent thermal conductivity, and mass diffusion coefficient, whereas they diminish with the increase in the local inertia coefficient. The current flow analysis can be useful in various engineering applications including paper production, polymer solution, glass blowing, extrusion of thermal system manufacturing process, and heat transportation enhancement.

A multivariate spectral quasi-linearization method for the solution of (2+1) dimensional Burgers’ equations

AbstractIn this paper, we introduce the multi-variate spectral quasi-linearization method which is an extension of the previously reported bivariate spectral quasi-linearization method. The method is a combination of quasi-linearization techniques and the spectral collocation method to solve three-dimensional partial differential equations. We test its applicability on the (2 + 1) dimensional Burgers’ equations. We apply the spectral collocation method to discretize both space variables as well as the time variable. This results in high accuracy in both space and time. Numerical results are compared with known exact solutions as well as results from other papers to confirm the accuracy and efficiency of the method. The results show that the method produces highly accurate solutions and is very efficient for (2 + 1) dimensional PDEs. The efficiency is due to the fact that only few grid points are required to archive high accuracy. The results are portrayed in tables and graphs.

Rotational nanofluids for oxytactic microorganisms with convective boundary conditions using bivariate spectral quasi-linearization method

In this study, we considered the three-dimensional flow of a rotating viscous, incompressible electrically conducting nanofluid with oxytactic microorganisms and an insulated plate floating in the fluid. Three scenarios were considered in this study. The first case is when the fluid drags the plate, the second is when the plate drags the fluid and the third is when the plate floats on the fluid at the same velocity. The denser microorganisms create the bioconvection as they swim to the top following an oxygen gradient within the fluid. The velocity ratio parameter plays a key role in the dynamics for this flow. Varying the parameter below and above a critical value alters the dynamics of the flow. The Hartmann number, buoyancy ratio and radiation parameter have a reverse effect on the secondary velocity for values of the velocity ratio above and below the critical value. The Hall parameter on the other hand has a reverse effect on the primary velocity for values of velocity ratio above and below the critical value. The bioconvection Rayleigh number decreases the primary velocity. The secondary velocity increases with increasing values of the bioconvection Rayleigh number and is positive for velocity ratio values below 0.5. For values of the velocity ratio parameter above 0.5, the secondary velocity is negative for small values of bioconvection Rayleigh number and as the values increase, the flow is reversed and becomes positive.

Unsteady Casson fluid flow in a porous medium with inclined magnetic field in presence of nanoparticles

A theoretical study on the flow of Casson nanofluid past a linear stretching sheet in a non-Darcian porous medium under the influence of inclined magnetic field is performed. Heat transfer characteristics along with Joule and viscous dissipation are analysed by considering the effect of non-uniform heat source/sink. The governing partial differential equations are non-dimensionalized and put into similar form by using a set of transformations. The highly non-linear coupled partial differential equations are solved using bivariate spectral quasi linearization method. The modification in flow, temperature and concentration profiles due to variations of parameters, namely, space- and temperature-dependent heat source/sink, Casson fluid parameter, Brownian motion, thermophoresis, Eckert number, Lewis number and the magnetic parameter are discussed by plotting the numerical result in tabular and graphical forms. Inclination angle parameter thins momentum boundary layer whereas it thickens thermal boundary layer.

Bioconvection in Casson nanofluid flow with Gyrotactic microorganisms and variable surface heat flux

This paper presents a two-dimensional unsteady laminar boundary layer mixed convection flow heat and mass transfer along a vertical plate filled with Casson nanofluid located in a porous quiescent medium that contains both nanoparticles and gyrotactic microorganisms. This permeable vertical plate is assumed to be moving in the same direction as the free stream velocity. The flow is subject to a variable heat flux, a zero nanoparticle flux and a constant density of motile microorganisms on the surface. The free stream velocity is time-dependent resulting in a non-similar solution. The transport equations are solved using the bivariate spectral quasilinearization method. A grid independence test for the validity of the result is given. The significance of the inclusion of motile microorganisms to heat transfer processes is discussed. We show, inter alia, that introducing motile microorganisms into the flow reduces the skin friction coefficient and that the random motion of the nanoparticles improves the rate of transfer of the motile microorganisms.

Heat and Mass Transfer in an Unsteady Second Grade Nanofluid with Viscous Heating Dissipation

This study presents the boundary layer flow of a second grade fluid in a rotating reference frame. The model equations are solved numerically using the bivariate spectral quasi-linearization method to obtain the fluid properties. We investigate the effects of measurable quantities such as the rate of rotation, thermal diffusivity, energy dissipation, viscoelastic parameter and other material properties of the fluid on the flow. The results show that slow rotation leads to a quick decline in the thicknesses of the momentum, thermal and concentration boundary layers. There is also a region of reverse flow for fluids with high Prandtl number.

Entropy generation in MHD radiative viscous nanofluid flow over a porous wedge using the bivariate spectral quasi-linearization method

We study the viscous nanofluid flow over a non-isothermal wedge with thermal radiation. The entropy due to irreversible processes in the system may degrade the performance of the thermodynamic system. Studying entropy generation in the flow over a porous wedge gives insights into how the system is affected by irreversible processes, and indicate which thermo–physical parameters contribute most to entropy generation in the system. The bivariate spectral quasi-linearization method is used to find the convergent solutions of the model equations. The impact of significant parameters such as the Hartmann number, thermophoresis and Brownian motion parameter on the fluid properties is evaluated and discussed. The Nusselt number, skin friction coefficients and Sherwood number are determined. An analysis of the rate of entropy generation in the flow for various parameters is presented, and among other results, we found that the Reynolds number and thermal radiation contribute significantly to entropy generation.

On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem

AbstractIn this paper, a bivariate spectral quasi-linearization method is used to solve the highly non-linear two dimensional Bratu problem. The two dimensional Bratu problem is also solved using the Chebyshev spectral collocation method which uses Kronecker tensor products. The bivariate spectral quasi-linearization method and Chebyshev spectral collocation method solutions converge to the lower branch solution. The results obtained using the bivariate spectral quasi-linearization method were compared with results from finite differences method, the weighted residual method and the homotopy analysis method in literature. Tables and graphs generated to present the results obtained show a close agreement with known results from literature.

Unsteady Mixed Convection in Nanofluid Flow Through a Porous Medium with Thermal Radiation Using the Bivariate Spectral Quasilinearization Method

Unsteady Mixed Convection in Nanofluid Flow Through a Porous Medium with Thermal Radiation Using the Bivariate Spectral Quasilinearization Method

Cross-Diffusion Effects on Unsteady Bioconvective Flow Past a Stretching Sheet

This paper presents a mathematical analysis of bioconvection heat, mass, and motile microorganisms transfer over a stretching sheet in a medium filled with a fluid containing gyrotactic microorganisms. Cross-diffusion is taken into account in the medium. Using the boundary layer approximations, the set of unsteady partial differential equations governing the fluid flow is transformed into nonlinear PDEs form and then solved numerically using bivariate spectral relaxation method (BSRM) and bivariate spectral quasi-linearization method (BSQLM). A comparison between BSRM and BSQLM is made for the first time in this work. The accuracy and convergence analysis of the methods are also discussed. The methods are found to be convergent and give very accurate results with very few grid points in the numerical discretization procedure. A parametric study of the entire flow regime is carried out to illustrate the effects of various governing parameters on the fluid properties and flow characteristics. The results obtained show a significant effect of cross-diffusion on the fluid properties and flow characteristics. The Dufour number was found to increase the local Sherwood number and density number of motile microorganisms while decreasing the Nusselt number, and the reverse effect is true for the Soret number. Furthermore, the Nusselt number, Sherwood number, and density number of motile microorganisms are highly influenced by buoyancy and bioconvection parameters.

A numerical study of unsteady non-Newtonian Powell-Eyring nanofluid flow over a shrinking sheet with heat generation and thermal radiation

In this paper we investigate the unsteady boundary-layer flow of an incompressible Powell-Eyring nanofluid over a shrinking surface. The effects of heat generation and thermal radiation on the fluid flow are taken into account. Numerical solutions of the nonlinear differential equations that describe the transport processes are obtained using a multi-domain bivariate spectral quasilinearization method. This innovative technique involves coupling bivariate Lagrange interpolation with quasilinearization. The solutions of the resulting system of equations are then obtained in a piecewise manner in a sequence of multiple intervals using the Chebyshev spectral collocation method. A parametric study shows how various parameters influence the flow and heat transfer processes. The validation of the results, and the method used here, has been achieved through a comparison of the current results with previously published results for selected parameter values. In general, an excellent agreement is observed. The results from this study show that the fluid parameters ε and δ reduce the flow velocity and the momentum boundary-layer thickness. The heat generation and thermal radiation parameters are found to enhance both the temperature and thermal boundary-layer thicknesses.

Unsteady boundary layer flow and heat transfer of Oldroyd-B nanofluid towards a stretching sheet with variable thermal conductivity

This paper presents a time dependent boundary layer flow and heat transfer of an incompressible Oldroyd-B nanofluid past an impulsively stretching sheet. Heat transfer analysis is carried out by taking thermal conductivity as a function of temperature. The non-dimensionalized partial differential equations are solved using bivariate spectral quasi-linearization method). The employs the concept of quasi-linearization to obtain a linear system of partial differential equations which is subsequently solved using a spectral collocation method that uses bivariate Lagrange interpolating polynomials as basic functions. This method is found to converge rapidly and is very effective in yielding accurate results. Numerical results have been presented graphically to illustrate the details of flow and heat transfer characteristics and their dependence on some of the physical parameters.