Spectral Relaxation Research Articles

A Spectral Relaxation Approach for Boundary Layer Flow of Nanofluid Past an Exponentially Stretching Surface with Variable Suction in the Presence of Heat Source/Sink with Viscous Dissipation

The present examination is considered to discuss the steady, two-dimensional flow of heat and mass transfer of nanofluid flow along with the variable suction under the influence of heat source/sink alongside the viscous dissipation. Because of the stretchable surface, flow is generated. Spectral relaxation technique is used to acquire the numerical solution for the altered nonlinear group of differential equations. Thereafter, outcomes which were obtained from the above numerical technique are validated by comparing with the available outcomes in the existing literature and also with MATLAB inbuilt solver bvp4c. All the acquired results from the above numerical system are shown through graphs and tables to discuss different resulting parameters related to the present analysis. The range of the physical parameters is taken as $$- \;0.1 \le f_{w} \le 0.5, \;0.1 \le Nt \le 0.7, \;0.1 \le Nb \le 0.7, \;0 \le Ec \le 1.2, \; - 0.5 \le Q \le 0.5, 1 \le Pr \le 10,\; 1 \le Le \le 5.$$ The main findings are as follows: For the effect of suction parameter, the profiles of velocity, temperature and mass friction are decreased, and the coefficient of skin friction and Nusselt number is declined for the impact of thermophoresis parameter.

Thermal Slip Flow of a Three-Dimensional Casson Fluid Embedded in a Porous Medium with Internal Heat Generation

Present assessment is considered to analysis flow as well as heat characteristics of steady, thermal slip flow of three-dimensional Casson fluid embedded in a porous medium with internal heat generation. Geometry of the present analysis is linearly stretched surface. Later, all the PDEs corresponding to the study are altered to set of nonlinear equations ODEs by means of appropriate similarity transformations. An efficient numerical scheme of spectral relaxation method (SRM) is applied to solve the nonlinear ordinary system. Variations of Nusselt number, temperature, velocity, and local skin friction coefficient with fluid parameters exhibited by graphs and tables. Spectral relaxation method gives an exact convergence to the nonlinear boundary value problems compare with general methods. In this study, to improve the precision and accuracy of the SRM successive over-relaxation (SOR) strategy is utilized. Proposed method as well as outcomes is checked with the comparison. A sensible connection is acquired between the current outcomes and accessible outcomes in writing. Some of the observations are skin friction coefficient raises and velocities decreases by the magnetic field strength. Skin friction and Local Nusselt number at the surface is more pronounced for non-Newtonian case than that of Newtonian case.

A Numerical Examination of an Unsteady Nonlinear MHD Flow in the Presence of Thermal Radiation and Heat Generation

In this study, the spectral perturbation method and the spectral relaxation method are used to solve the nonlinear differential equations of an unsteady nonlinear MHD flow in the presence of thermal radiation and heat generation. The SPM is mainly based on series expansion, generating series approximation coupled with the Chebyshev spectral method. The numerical results generated using the spectral perturbation method were compared with those found in the literature, and the two results were in good agreement.

Mixed convection hybrid nanofluids flow of MWCNTs–Al2O3/engine oil over a spinning cone with variable viscosity and thermal conductivity

AbstractThis work analyzed the effects of variable viscosity and thermal conductivity, with mixed convection, thermal radiation and viscous dissipation effects, on multiwalled carbon nanotubes (MWCNTs)–aluminum oxide (Al2O3)/engine oil hybrid nanofluid flow due to a vertically inverted spinning cone embedded in a porous medium. Using suitable similarity transformation, the boundary layer fluid flow governing equations are transformed into dimensionless systems of coupled nonlinear ordinary differential equations. Then, the solutions are obtained numerically employing the spectral relaxation method. The influences of involved parameters are examined, and the results are presented with graphs and tables. The obtained results disclose that both the tangential and azimuthal skin friction coefficients increase with increasing values of temperature‐dependent viscosity and mixed convection parameters. The local heat transfer rate reduces with increasing values of the Eckert number and variable thermal conductivity parameter, whereas it enhances with greater values of the thermal radiation parameter. Generally, hybrid nanofluids of (MWCNTs–Al2O3)/engine oil show better flow distributions with good stability of thermal properties than MWCNTs/engine oil and Al2O3/engine oil mono‐nanofluids.

An Efficient Spectral Method-based Algorithm for Solving a ‎High-dimensional Chaotic Lorenz System

In this paper, we implement the multidomain spectral relaxation method to numerically study high dimensional chaos by considering the nine-dimensional Lorenz system. Chaotic systems are characterized by rapidly changing solutions, as well as sensitivity to small changes in initial data. Most of the existing numerical methods converge slowly for this kind of problems and this results in inaccurate approximations. Spectral methods are known for their high accuracy. However, they become less accurate for problems characterised by chaotic solutions, even with an increase in the number of grid points. As a result, in this work, we adopt the multidomain approach which assumes that the main interval can be decomposed into a finite number of subdomains and the solution obtained in each of the subdomains. This approach remarkably improves the results as well as the efficiency of the method.

Numerical Analysis of MHD Stagnation Point Flow Towards a Radially Stretching Convectively Heated Disk

The steady stagnation point flow and heat transferof an electrically conducting incompressible viscous fluid isextended to the case where the disk surface is convectivelyheated and radially stretching. The fluid is subjected to anexternal uniform magnetic field perpendicular to the planeof the disk. The governing momentum and energy balanceequations give rise to non-linear boundary value problem.Using a spectral relaxation method with a Chebyshev spectralcollocation method, the numerical solutions are obtained overthe entire range of the physical parameters. Emphasis hasbeen laid to study the effects of viscous dissipation and Jouleheating on the thermal boundary layer. Pertinent results on theeffects of various thermophysical parameters on the velocityand temperature fields as well as local skin friction and localNusselt number are discussed in detail and shown graphicallyand/or in tabular form.

On Spectral Relaxation Approach for Thermal Diffusion and Diffusion Thermo Effects on Viscous Dissipative Casson Fluid Through a Stretched Surface

An investigation has been performed to study the influences of Soret and Dufour on steady, viscous dissipative Casson fluid flow along a linear stretching surface. Numerical solution of this problem is obtained by Spectral Relaxation Method. The outcomes are enumerated in terms of profiles of velocity, temperature, concentration, coefficients of skin friction, Nusselt and Sherwood numbers with reference to certain physical parameters. The results show that velocity reduces when there is an increase in Casson parameter. The Nusselt number enhances with Soret number whereas it decreases by enhancing Dufour number. Also, Nusselt number as well as Sherwood number increase with ratio parameter. Further, concentration profiles decrease because of increment in ratio parameter and Schmidt number, but they raise for higher Casson parameter and Soret number values.

Asymmetric broadening and blue shift of the stimulated Raman scattering spectrum in water under chirped picosecond laser pulse train excitation

For the first time, we have revealed an asymmetric broadening and a 35 cm−1 blue shift (up to 3400 cm−1) of the stimulated Raman scattering OH band in water under chirped picosecond pulse train excitation (pulse duration ∼60 ps with 8 ns interval between pulses) while increasing the train length to 25 pulses. From the observed broadening and shift of the OH band centroid, a lower estimate (≥8 ns) of molecular complexes spectral states relaxation time at room temperature was obtained.

Soret–Dufour mechanism on an electrically conducting nanofluid flow past a semi‐infinite porous plate with buoyancy force and chemical reaction influence

AbstractThis study explored the flow of an electrically conducting and chemically reacting nanofluid over a semi‐infinite porous vertical plate under the influence of Soret–Dufour and nonlinear buoyancy force. The flow analysis consists of partial differential equations (PDEs). These equations were nondimensionalized via nondimensional quantities into coupled nonlinear PDE and solved numerically using spectral relaxation method. Numerical computations of all flow parameters for velocity, temperature along with concentration are presented in diagrams. The result shows that an upsurge in the value of the radiation parameter enhances both velocities along with temperature profiles, respectively. This is true because the temperature at the plate is much when there is an increase in the radiation parameter. The present results were compared with the previously published result and were found to be in excellent agreement.

Steady Boundary Layer Flow of Casson Fluid over a Nonlinear Stretched Sheet in Presence of Viscous Dissipation Using the Spectral Relaxation Method

Steady Boundary Layer Flow of Casson Fluid over a Nonlinear Stretched Sheet in Presence of Viscous Dissipation Using the Spectral Relaxation Method

Hall and Ion Slip Effects on Mixed Convection Flow of Eyring-Powell Nanofluid over a Stretching Surface

The purpose of this research is to inspect the mixed convection flow of Eyring-Powell nanofluid over a linearly stretching sheet through a porous medium with Cattaneo–Christov heat and mass flux model in the presence of Hall and ion slip, permeability, and Joule heating effects. Proper similarity transforms yield coupled nonlinear differential systems, which are solved using the spectral relaxation method (SRM). The story audits show that the present research problem has not been studied until this point. Efficiency of numerous parameters on velocity, temperature, and concentration curves is exposed graphically. Likewise, the numerical values of skin friction coefficients, local Nusselt, and Sherwood numbers are computed and tabulated for some physical parameters. It is manifested that fluid velocities, skin friction coefficients, local Nusselt, and Sherwood numbers promote with the larger values of Eyring-Powell fluid parameter ε. It is also noticed that primary velocity promotes with larger values of mixed convection parameter λ, Hall parameter βe, and ion slip parameter βi, while the opposite condition is observed for secondary velocity, temperature, and concentration. Furthermore, comparative surveys between the previously distributed writing and the current information are made for explicit cases, which are examined to be in a marvelous understanding.

Internal heat generation on bioconvection of an MHD nanofluid flow due to gyrotactic microorganisms

The principal aim of this paper is to analyse the two-dimensional magnetohydrodynamic flow and heat and mass transfer phenomena of water-based nanofluid containing gyrotactic microorganisms over a vertical plate by means of heat generation or absorption. Set of nonlinear ordinary differential equations are derived from the governing partial differential equations of the two-dimensional flow of MHD nanofluid and nanoparticles by utilizing appropriate similarity transformations. The numerical results are obtained with the proposed novel spectral relaxation method. The results revealed that in the heat and mass transfer, the motile microorganism flux rates as well as the velocity profiles are decreased throughout the fluid medium with the impact of magnetic field strength. Moreover, the drag stress rate and motile microorganism profiles are increased with the enhancement of magnetic field. Even though the problem of nanofluid has been broadly investigated, limited discoveries can be found through a gyrotactic microorganisms. Indeed, this paper managed to obtain the numerical analysis is performed. Furthermore, the authors also considered the MHD phenomena, heat generation or absorption effects. Very few studies in the fluid with gyrotactic microorganisms embedded in this parameter in their problems.

Spectral relaxation method analysis of Casson nanofluid flow over stretching cylinder with variable thermal conductivity and Cattaneo–Christov heat flux model

AbstractThis article deals with non‐Newtonian Casson nanofluid flow and heat transfer over stretching cylinder in a porous medium. The mode of heat transfer is presented considering temperature‐dependent thermal conductivity by integrating the Cattaneo–Christov heat flux and mass flux models. Boundary layer theory is applied to develop the governing partial differential equations from the physical problem. Employing proper similarity transformation, the governing boundary layer equations are transformed into dimensionless system of nonlinear ordinary differential equations. Then, the resulting problem is numerically solved by means of spectral relaxation method. The convergence analysis of the proposed numerical scheme is presented via a table, which confirms almost the 10th order of approximation is enough for the convergence of the skin friction coefficient, local heat transfer, and mass transfer rates. The effects of various embedded parameters on velocity, temperature, and concentration profiles as well as skin friction coefficient, surface heat and mass transfer rates are examined through graphs and tables. The findings reveal that the growth of permeability and velocity slip parameters appears to decelerate the velocity distributions of fluid. Thermal boundary layer thickness tends to develop with greater values of permeability and Brownian motion parameters. Also, the local heat transfer rate is less with Fourier's law of heat conduction than Cattaneo–Christov heat flux model. Furthermore, the validity and accuracy of the present result is checked with the available literature, and very sound agreement has been obtained.

Unsteady problem of magnetohydrodynamic heat plus mass transfer convective flow over a moveable plate with effects of thermophoresis and thermal radiation

AbstractThis paper investigates the problem of unsteady magnetohydrodynamic heat plus mass transfer convective flow over a moveable vertical plate with the influence of thermophoresis and thermal radiation. The physical problem is governed by a set of partial differential equations. These sets of equations are coupled and are nonlinear. They were transformed into a dimensionless form of equations by introducing appropriate nondimensional quantities. An iterative method called the spectral relaxation method was used to linearize and decouple the set of dimensionless equations. Results were presented both in graphs and tables. It was found out that thermophoresis parameter has a significant effect on velocity and concentration fields. The thermal radiation is seen to have a significant effect on velocity and temperature fields. The skin friction is seen to increase the moment thermal Grashof number is increased. The model of Newtonian fluid flow over a moveable vertical plate is considered. The plate was considered moving toward the ‐direction and the radiative heat flux is only with respect to . This study considered effects of viscous dissipation, thermophoresis, and radiation on heat plus mass transfer. This, to the best of our knowledge, has not been considered in the literature.

Three-Dimensional MHD Mixed Convection Flow of Casson Nanofluid with Hall and Ion Slip Effects

The intention of the present study is to scrutinize the three-dimensional MHD mixed convection flow of Casson nanofluid over an exponentially stretching sheet using the impacts of Hall and ion slip currents. Moreover, the impacts of thermal radiation and heat source are considered in this study. The governing partial differential equations are transformed into a system of joined nonlinear ordinary differential equations using similarity transformations, and they are solved numerically employing a spectral relaxation method (SRM). The obtained results are contrasted with existing specific cases, and a reasonable harmony is established. The impacts of noteworthy physical parameters on the velocities, thermal and concentration distributions, skin friction coefficients, local Nusselt number, and local Sherwood number are investigated graphically. It is found that the rise in Casson fluid and magnetic field parameters reduce the velocity profiles along bothx−andy−directions while the reverse tendency is observed with an increment in Hall, ion slip, and mixed convection parameters. Moreover, the increase in both radiation and heat source parameters enhances the temperature profile. It is also observed that both the skin friction coefficients reduced with an increase in Casson fluid, Hall, and ion slip parameters. Furthermore, the local Nusselt number enhances with an augment in radiation parameter, whereas the opposite trends of local Nusselt and Sherwood numbers are found with an increase in heat source parameter.

MHD Bioconvection Flow and Heat Transfer of Nanofluid through an Exponentially Stretchable Sheet

Recently, bioconvection phenomenon has gained great importance in research for its use in many engineering and biological applications. Therefore, this work investigates the magnetohydrodynamic flow of a dissipative nanofluid, including gyrotactic microorganisms along an exponentially moving sheet. Since the governing equations that describe the problem are nonlinear and more complicated, similarity transformations are used to get a reduced mathematical model in which all the differential equations are ordinary and asymmetric. The computational analysis for the reduced mathematical model is carried out, employing the spectral relaxation technique (SRM) via software called MATLAB. Comparison results are also validated by using the boundary value problem solver (bvp4c) in MATLAB. The obtained results were compared with previously published researches, and a high degree of compatibility and accuracy were found symmetric. The implications of pertinent parameters on velocity, temperature, nanoparticles volume fraction, and density of the microorganism profiles are graphically presented. A decline was seen in the velocity field with augmentation in the magnetic parameter, but certain enhancement was noticed in the temperature field for augmented values of the magnetic parameter, thermophoresis, and Brownian motion parameters. A significant reduction was also noticed in the behavior of the concentration profile for augmented values of the Brownian motion parameter and Lewis number, while it was enhanced with the boost in the thermophoresis and magnetic parameters. The results also indicated that the density of the motile microorganism decreases with bioconvection Lewis number, Prandtl number, Lewis, and Peclet numbers.

Mixed convection flow of a Maxwell nanofluid with Hall and ion‐slip impacts employing the spectral relaxation method

AbstractThis article investigates the Hall and ion‐slip impacts on the mixed convection flow of a Maxwell nanofluid over an expanding surface in a permeable medium. The impacts of Brownian movement and thermophoresis parameters, Soret, Dufour, viscous dissipation, chemical reaction, and suction parameters, are, moreover, considered. Using the similitude changes, the partial differential equations with regard to the momentum, energy, and concentration equations are transformed to an arrangement of nonlinear ordinary differential equations, which are handled numerically utilizing a spectral relaxation method (SRM). The impacts of noteworthy physical parameters on the velocities, thermal, and concentration distributions are investigated graphically. Moreover, the numerical values of skin‐friction coefficients, local Nusselt number, and Sherwood number for different values of the mixed convection parameter Deborah number Hall parameter ion‐slip parameter Dufour number (Du), and Soret number are computed and tabulated. It is discovered that ascent in Deborah number reduces both the stream and transverse velocity profiles, while the inverse pattern is seen with augmentation in the mixed convection parameter. In addition, inverse patterns of the stream and transverse velocity profiles are seen with expansion in magnetic, Hall, and ion‐slip parameters. Besides this, the temperature and concentration disseminations decline with augmentation in Dufour number and chemical reaction parameters, respectively. It is likewise seen that both the skin‐friction coefficients lessen with expansion in Deborah number, and they ascend with upgrade in blended convection and ion‐slip parameters, while the opposite condition is noticed with augmentation in Hall parameter. Furthermore, the reverse trends of local Nusselt and Sherwood numbers are discovered with expansion in the Dufour and Soret numbers.

Buoyancy effect on mixed convection boundary layer flow of Casson fluid over a non linear stretched sheet using the spectral relaxation method

The principal idea of this study is to examine the buoyancy effect on two-dimensional flow of Casson fluid and mixed convection over a nonlinear stretching sheet. The governing equations of the present study along with suitable boundary conditions are transformed into non dimensional differential equations along with appropriate boundary conditions by using similarity transformations. Spectral relaxation method is applied to study the considered problem as numerically. Moreover in case of linear stretching Exact solution is also performed. To improve the effectiveness and accuracy of the proposed method spectral over relaxation method is used. The effects of various emerging parameters are studied to analyze temperature, local Nusselt number, friction factor and velocity by means of graphs and tables. Numerical outcomes are validated by comparing the outcomes obtained and existing literature. Mainly, noticed that, enhancement in the velocity and decrement in temperature for the increasing augments of the buoyancy parameter.

Similarity solution of MHD slip with energy mass transport through chemically reacting stretching permeable surface in porous media with variable properties

In this paper, a mathematical model is presented for magnetohydrodynamic viscous incompressible boundary layer flow of an electrically conducting chemically reacting fluid over a moving permeable flat vertical surface in a porous medium with variable fluid properties. A linear momentum slip boundary condition is included. Temperature and concentration difference between the wall and the environment are assumed to vary with the axial distance according to power law forms. Furthermore, a concentration-dependent mass diffusivity is featured to provide for a more realistic simulation of relevance to materials processing systems. The governing equations with associated boundary conditions are transformed into similarity equations using appropriate variables furnished with Lie group algebraic methods. The resulting nonlinear differential ordinary boundary value problem is then solved numerically with a Runge–Kutta–Fehlberg fourth–fifth order numerical method. The effects of the governing parameters on the boundary layer flow characteristics (velocity, temperature, concentration, wall shear stress, heat and mass transfer rates) have been investigated, displayed graphically and discussed. The computations indicate that variable mass diffusivity plays a significant role on transport characteristics. It is also shown that concentration magnitudes increase whereas the rate of mass transfer at the wall decreases with increasing mass diffusivity. Comparison of numerical and analytical results with results from the open literature show excellent agreement. Further verification of the general model is conducted with the Spectral Relaxation Method (SRM). The study is relevant to electroconductive polymeric sheet processing in manufacturing operations and smart coating systems.

Femtosecond Fluorescence Spectra of NADH in Solution: Ultrafast Solvation Dynamics.

The ultrafast solvation dynamics of reduced nicotinamide adenine dinucleotide (NADH) free in solution has been investigated, using both a femtosecond upconversion spectrophotofluorometer and a picosecond time-correlated single-photon counting (TCSPC) apparatus. The familiar time constant of solvent relaxation originating in "bulk water" was found to be ∼1.4 ps, revealing ultrafast solvent reorientation upon excitation. We also found a slower spectral relaxation process with an apparent time of 27 ps, suggesting there could either be dissociable "biological water" hydration sites on the surface of NADH or internal dielectric rearrangements of the flexible solvated molecule on that timescale. In contrast, the femtosecond fluorescence anisotropy measurement revealed that rotational diffusion happened on two different timescales (3.6 ps (local) and 141 ps (tumbling)); thus, any dielectric rearrangement scenario for the 27 ps relaxation must occur without significant chromophore oscillator rotation. The coexistence of quasi-static self quenching (QSSQ) with the slower relaxation is also discussed.