Rosseland Approximation Research Articles

Analytical Study of Magnetohydrodynamic Casson Fluid Flow over an Inclined Non-Linear Stretching Surface with Chemical Reaction in a Forchheimer Porous Medium

This study investigates the steady, two-dimensional boundary layer flow of a Casson fluid over an inclined nonlinear stretching surface embedded within a Forchheimer porous medium. The governing partial differential equations are transformed into a set of ordinary differential equations through similarity transformations. The analysis incorporates the effects of an external uniform magnetic field, gravitational forces, thermal radiation modeled by the Rosseland approximation, and first-order homogeneous chemical reactions. We consider several dimensionless parameters, including the Casson fluid parameter, magnetic parameter, Darcy and Forchheimer numbers, Prandtl and Schmidt numbers, and the Eckert number to characterize the flow, heat, and mass transfer phenomena. Analytical solutions for the velocity, temperature, and concentration profiles are derived under simplifying assumptions, and expressions for critical physical quantities such as the skin friction coefficient, Nusselt number, and Sherwood number are obtained.

Scaling Law of Flow and Heat Transfer Characteristics in Turbulent Radiative Rayleigh-Bénard Convection of Optically Thick Media

Radiative natural convection is of vital importance in the process of energy storage, power generation, and thermal storage technology. As the attenuation coefficients of many heat transfer media in these fields are high enough to be considered as optically thick media, like nanofluids or molten salts in concentrated solar power or phase change thermal storage, Rosseland approximation is commonly used. In this paper, we delve into the impact of thermal radiation on the Rayleigh-Bénard (RB) convection. Theoretical analysis has been conducted by modifying the Grossmann-Lohse (GL) model. Based on turbulent dissipation theory, the corresponding scaling laws in four main regimes are proposed. Direct numerical simulation (DNS) was also performed, revealing that radiation exerts a notable influence on both flow and heat transfer, particularly on the formation of large-scale circulation. By comparing with DNS results, it is found that due to the presence of radiation, the modified Nu scaling law in small Pr range of the GL model is more suitable for predicting the transport characteristics of optical thick media with large Pr. The maximum deviation between the results of DNS and prediction model is about 10%, suggesting the summarized scaling law can effectively predict the Nu of radiative RB convection.

Computational simulation of Casson hybrid nanofluid flow with Rosseland approximation and uneven heat source/sink

This article candidly presents the magnetohydrodynamics Casson hybrid nanofluid flow over a stretching surface. In the present study, we added a nonuniform heat source or sink and non-linear thermal radiation. We considered Al2O3 and copper nanoparticles to have antibacterial and antiviral properties without any harmful impacts and used water as the host fluid. We simplified the governing flow equations by using suitable self-similarity variables, which are used to convert PDEs to ODEs. The mathematical equations are numerically solved by using the bvp5c technique in the MATLAB software. Additionally, with higher values of the magnetic field and Casson fluid parameters the velocity profile decreased. The temperature profile is enhanced by increasing the magnetic field and thermal radiation parameters. Increasing the Casson fluid and radiation parameters enhances the skin friction and Nusselt number profiles. Alumina nanoparticles find applications in cosmetic fillers, polishing materials, catalyst carriers, analytical reagents. Copper nanoparticles have high electrical conductivity, which has many uses in electrical circuits and biosensors.

Heat Transfer and Flow of Natural Convection Past a Semi-Infinite Vertical Plate

This manuscript presents a numerical investigation of the flow of an electrically conducting, incompressible, viscous fluid past a heated vertical channel with thermal radiation influence. The study uses the Rosseland approximation to describe the radiative heat flux and employs Crank Nicolson's implicit finite difference method utilizing the MATLAB programming tool for solving the non-dimensional system of equations. The results demonstrate that an increase in the thermal radiation parameter leads to a decrease in both the velocity and temperature of the fluid. The main findings of the study highlight the impact of thermal radiation on natural convection and heat transfer processes. This research is significant as it contributes to the understanding of convective heat transfer phenomena and provides valuable insights for applications in science and technology.

Numerical study of binary mixture and thermophoretic analysis near a solar radiative heat transfer

The main focus of the current article is to investigate the two-dimensional boundary layer flow of non-Newtonian fluid induced by a linear stretching surface in the streamwise direction. The flow features of non-Newtonian fluid (i.e. shear-thinning and thickening nature of the fluids) are explored with Carreau fluid model. In addition, thermophoresis, Brownian motion, thermal radiation (with Rosseland approximation), binary mixture and activation enthalpy are also incorporated in fluid model for better analysis of thermal and solutal properties. The mathematical modelling of under consider physical problem yields nonlinear partial differential system. The governing mathematical system is first simplified with scaling transformations for the high Reynolds number and then transformed into non-dimensional ordinary differential system by using the similarity variables. Since the governing mathematical system comprised on nonlinear boundary value problem, thus shooting method is utilized for the numerical solution of the fluid flow governing equations. The obtained results followed the qualitative manners of the previously reported data. The computed results for important physical quantities (i.e. velocity, temperature, and concentration) are presented through graphs and then interpreted physically. It is observed that the boundary layer thickness increased for shear thickening fluids (n>1) while it is reduced for shear thinning fluids (n<1). Also, the temperature of shear thinning fluid is higher than shear thickening fluid. In addition, heat generation and thermal radiations enhance the thermal boundary layer. Thermophoretic viscosity effect reduces the concentration profile significantly in the flow domain.

Dynamics of energy transport on hydromagnetic Casson slippery nanoflow over curved surface

The study of fluid flow over curved surfaces is crucial in various engineering applications, such as designing aircraft wings, turbines, and submarines. Curved surfaces are being explored in various biomedical applications, such as designing stents for blood vessels and implants for bones and joints. Concerning the present applications of curved stretching sheets on fluid dynamics along with trihybrid nanofluids, this study is unique and fills the research gaps and offers solutions to several issues. This work looks at the flow through the boundary layer of an electrically conductive trihybrid nanofluid and the convection heat exchange of a Casson fluid across a curved stretched surface encircled within a circle of radius R. The study considers the effects of thermal radiation using the non-linearized Rosseland approximation, as well as a magnetic field, and hydromagnetic slip. The flow as well as the transfer of heat problem is mathematically described by curvilinear coordinates. Using the combination of the shooting technique and the Runge-Kutta method, similar solutions to the modeled partial differential equations are produced, and the set of non-linear ordinary differential equations with a boundary value solver is implemented through the MATLAB program. The study finds the influence of several limits on critical characteristics such as fluid velocity, coefficient of skin friction, pressure, temperature, and rate of heat exchange over a surface. The findings are shown in tables and graphs. Additionally, a comparative analysis between the current findings and those found in the literature is provided. Blood-containing nanoparticles (GO-SWCNTs-MWCNTs) on curved surfaces could improve drug delivery effectiveness, growth of synthetic tissues or organs with complex structures and effective cancer therapy treatment.

Heat transfer analysis of ternary hybrid Williamson nanofluids with gyrotactic microorganisms across stretching surfaces: Local non-similarity method

Managing and controlling energy amidst rapid industrial and technological advancements pose significant challenges. Enhanced heat transfer processes, like free convection in solar panels and nuclear power plants, offer potential to reduce energy consumption and improve system efficiency. Understanding elements such as magnetic fields, thermal radiation, and nanoparticle morphology can optimize fluid dynamics and heat transfer across various domains including energy, aerospace, and nuclear power. This study aims to develop a two-dimensional Williamson fluid model with ternary hybrid nanofluids and gyrotactic microorganisms to deepen our understanding of heat transfer. Methodological enhancements include integrating viscous dissipation formulations and adopting the Rosseland approximation to address radiation effects accurately. By leveraging non-similarity transformations, equations are rendered nondimensional for efficient numerical solutions using MAT LAB. Graphical overviews and tables aid in comprehending key factors for practical applications and engineering analyses.

Transient thermal behavior and efficiency in fully wet porous longitudinal fin: The influence of shape-dependent hybrid nanofluid and internal heat generation

The main focus of this study is to analyze the transient heat transfer characteristics of a convective–radiative longitudinal porous fin fully wetted in hybrid nanofluid along with internal heat generation. The Darcy formulation is considered for analyzing the velocity of the fluid passing through the fin and the Rosseland approximation determines the radiation heat flux. The hybrid nanofluid is obtained by immersing GO and Mo S 2 nanoparticles in the hybrid base fluid C 2 H 6 O 2 ‐ H 2 O . Furthermore, three different shapes of nanoparticles namely spherical, platelet, and blade have been selected for the investigation. The derived nonlinear partial differential equation is nondimensionalized and solved numerically by employing the finite difference method. The effect of shape factor, nanoparticle volume fraction, dimensionless time, wet porous parameter, radiation–conduction parameter, heat generation parameters, and other relevant parameters on the transient thermal analysis of the fin structure has been graphically analyzed and discussed. Furthermore, the transient thermal fin efficiency has been modeled and its variation with the significant parameters has been examined. One of the major outcomes is that efficiency increases with nanoparticle volume fraction. The examination has resulted in a novel outcome that the presence of hybrid nanofluid enhances the fin efficiency and also the high fin efficiency is achieved in the presence of blade-shaped nanoparticles. The findings of the investigation play a prominent role in the heat transfer enhancement of industrial processes.

¬¬Chemical Reaction and Nonlinear Rosseland Approximation Effects On Double-Diffusive MHD Sisko Nanofluid Flow Over a Nonlinear Stretching Sheet in A Porous Medium with Concentration-Dependent Internal Heat Source

This study investigates the two-dimensional magnetohydrodynamics laminar flow of a Sisko nanofluid over a nonlinear stretching sheet in a porous medium, considering chemical reactions, nonlinear Rosseland approximation, and an internal heat source based on concentration. The governing boundary layer equations and associated boundary conditions are transformed into non-dimensional form using appropriate variables. The resulting non-dimensional equations are solved using the modified Adomian decomposition technique, implemented with the symbolic computing software MATHEMATICA. The influence of various parameters, including Prandtl number, thermophoresis parameter, Brownian motion parameter, chemical reaction parameter, Sisko fluid parameters, heat source parameter, radiation parameter, Darcy number, mass Grashof number, magnetic field, and Grashof number, on the flow fields of velocity, temperature, and nanoparticles volume fractions is depicted through graphical representations. The study reveals that the velocity profile is accelerated by the first value of the Sisko fluid parameter and Darcy number, but it diminishes in the presence of a magnetic field and the second Sisko fluid parameter. The temperature profile decreases with an increment in the Prandtl number, while it improves in proportion to increases in the heat source, thermophoresis, Brownian motion, and radiation parameters respectively. Additionally, an increase in the chemical reaction parameter leads to an increase in the concentration profile of the fluid.

Effect of the Non-Linear Radiative Unsteady Mixed Convective Flow over a Curved Stretching Surface with Soret and Dufour Effects: A Numerical Study

The subject of unsteady convective flow with non-linear thermal radiation has become an important issue of research, due to its implications in advanced energy conversion systems operating at high temperature, solar energy technology and chemical process at high operation temperature. Due to the importance of this issue, a time dependent incompressible viscous fluid flow, heat and mass transfer over a curved stretching surface has been numerically analysed by taking into account the heat flux due to concentration gradient and mass flux due to temperature gradient. Together with this the Rosseland approximation is being employed for the nonlinear thermal radiation impact in presence of thermal slip. With the aid of non-dimensional variables and the corresponding physical boundary conditions, the leading nonlinear momentum, energy, and species equations are converted into a set of coupled ordinary differential equations. These equations are then resolved using the MATLAB bvp4c solver. The stability of the numerical technique has been verified and compared with available literatures. The resultant parameters of engineering interest and the boundary layer flow field parameters and have been presented using tables and graphically plots. The study concludes that for lesser curvature parameter (0.5≤K≤0.7) the surface drag force, heat and mass transfer rates can improve by about 9.59%, 2.87% and 1.67% each respectively. The presence of the temperature ratio parameter and the non-linear thermal radiation are found to greatly influence the temperature profile and the heat transfer rate of the system. Results show that the heat transfer rate improves by about 24.39% and 16.66% for varying non-linear thermal radiation (1≤Rd≤1.5) and temperature ratio parameter (1.2≤θw≤1.4) respectively. Results obtained also show that improving the thermal slip parameter (0.4≤L≤0.6) can reduce heat transfer rate by about 13.62% and reduce the surface temperature profile.

Study of twisting of β-Ga2O3 crystals based on optical absorption and thermal conductivity anisotropy in the crystals grown by the Czochralski method

The effects of internal radiation and thermal conductivity anisotropy in β-gallium oxide crystals on heat and mass transfer in the Czochralski method were numerically studied. Specifically, we calculated the temperature and velocity distributions in a crystal and/or melt and the interface shape between the crystals and melt for the Czochralski method. We used the Rosseland approximation to include the internal radiation in the crystals to calculate the heat transfer in the crystal. Moreover, the thermal conductivity anisotropy in the crystals was considered to study the asymmetric temperature distribution.The results indicated that the temperature gradient in the semitransparent crystal was smaller than that in the opaque crystal. This result is attributed to the absorption of the outgoing heat flux from the interface between the crystal and melt in the crystal. We studied the asymmetric temperature distributions in the crystals caused by thermal conductivity anisotropy as a function of the crystal absorption coefficient.The results also indicated that the interface shape between a semitransparent crystal and the melt was convex to the melt, whereas the shape between an opaque crystal and the melt was almost flat.

Investigation of the thermal analysis of a wavy fin with radiation impact: an application of extreme learning machine

The combined impact of radiation and convection on the heat transfer of a wavy fin is scrutinized in the present analysis. The novelty of this research work is that it proposes a deterministic machine learning model known as an extreme learning machine to address the heat transfer problem of a wavy fin. The effect of radiation on convective heat transfer and the Rosseland approximation for the radiation heat exchange have been considered in the investigation. The nonlinear ordinary differential equation (ODE) is converted to its nondimensional form using the appropriate dimensionless variables. Runge-Kutta-Fehlberg's fourth-fifth order technique (RKF 45) is used to solve the nondimensional ODE numerically. The roles of convection-conduction, radiation-conduction, thermal conductivity, and radiation parameters have been discussed for satisfying a prescribed temperature distribution in rectangular and wavy fins with graphical visualization. A rise in convection-conduction and radiation-conduction variables decreased the thermal distribution of both the wavy fin and rectangular fin. Further, ANSYS simulation analyzes the variation of temperature and total heat flux in both rectangular and wavy fins. The study demonstrates the effectiveness of the model selected through the obtained results, which indicate the potential of the regression model for providing an accurate prediction.

Thermal radiation and heat generation on three-dimensional Casson fluid motion via porous stretching surface with variable thermal conductivity

Abstract The impact of variable thermal conductivity plays a key role in the analysis of fluid mechanics. The applications of liquid are significant in nuclear reactors, automobiles, technical and manufacturing industry, electronic appliances, and so on. The novelty of this analysis is to exhibited the variable thermal conductivity in motion of Casson liquid via porous stretching sheet. In energy equation is consider thermal radiation and heat generation. Rosseland approximation plays a key role in the current work. Under the liquid motion assumptions, BL approximation is applied on numerical model and developed partial differential equations (PDE). The similarity transportation variable is taken by transporting PDE’s to ordinary differential equations. Numerical model is explored with the help of fourth-order boundary value problem with R–K–F procedure via shooting technique. The main key points noticed are: the heat transfer rate is more effective in the presence of heat generation than that in the absence of heat generation parameter. The temperature is enhanced in presence of variable thermal radiation while comparing absence of variable thermal radiation for large numerical values of Biot Number.

The Role of Quadratic-Linearly Radiating Heat Source with Carreau Nanofluid and Exponential Space-Dependent Past a Cone and a Wedge: A Medical Engineering Application and Renewable Energy

A mathematical analysis for a steady, incompressible, laminar Carreau nanofluid flow over a porous cone and wedge is considered. The novelty of this research work is to scrutinize an exponential space-dependent Cattaneo–Christov heat flux and quadratic Rosseland approximation effects. Two-phase nanofluid model, i.e., Brownian motion, and thermophoretic effects are included. This study also employs non-linear thermal radiation and convective surface boundary conditions to investigate heat transport phenomena. The Carreau nanofluid boundary layer equations are derived using the standard boundary layer approximations. By applying the similarity transformations, the managing set of partial differential equations (PDEs) becomes a system of connected non-linear ordinary differential equations (ODEs). The resultant non-linear ODEs are solved numerically utilizing the numerical method, particularly the bvp4c function in MATLAB. The significances of the fluid motion are visualized by sketching several Carreau nanofluid flow’s relevant parameters using graphic outputs. The effects of a wide range of thermophysical variables on liquid properties including velocity, temperature, concentration, skin-friction, Nusselt number, and Sherwood number are investigated and discussed. This study’s findings are compared to previously announced results within a limited range, where strong validation was seen. The results demonstrate that the opposite mechanism is observed with repercussion of Weissenberg number (We) on velocity profile and concentration, the enhancement of We increases the momentum boundary layer while it reduces the concentration boundary layer. The outcomes could be applied to the cooling of equipment, electronics, and various industrial units.

Magnetohydrodynamic micropolar nanofluid flow over a vertically elongating sheet containing gyrotactic microorganisms with temperature-dependent viscosity

Magnetohydrodynamic bioconvective micropolar nanofluid flow with migrating microorganisms opens up an intriguing new field in materials science. The distribution of particles during processing, such as metal casting, polymer extrusion, etc., can affect material characteristics and final product quality. So, the current research examines magnetohydrodynamic micropolar nanofluid flow over a vertically extending material in the presence of swimming microorganisms. The mathematical model also considers the temperature dependence of dynamic viscosity with thermal radiation using the Rosseland approximation, Brownian diffusion, and the thermo-migration of microscopic particles. Similarity variables are used to convert the partial derivatives to ordinary derivatives. Then the Runge-Kutta-Gill method is used with the shooting technique to obtain a numerical solution to the resulting set of equations. Various tables and graphs are included to illustrate the impact of the emerging parameters on the flow fields. Results show that an increase in the material parameter improves fluid mobility but decreases the skin friction coefficient, whereas the magnetic field term exhibits the opposite behaviour due to friction induced by the Lorentz force. The analysis shows that the thermal field significantly expands as the thermo-migration of tiny particles and Brownian motion get stronger.

Numerical heat transfer of non-similar ternary hybrid nanofluid flow over linearly stretching surface

The primary objective of this research is to construct a 2D mathematical model to understand the heat transfer phenomena in a ternary hybrid nanofluid across a stretched surface. We endorsed viscous dissipation formulations in energy equations and the radiation impacts may also be precisely handled by using Rosseland approximation. Nanoparticles and their effects are also considered including and The equations used in the model are made nondimensional through non-similarity transformation. In addition, the local non-similarity approach is employed to convert non-similar partial differential equations into ordinary differential equations. These equations can subsequently be solved using the bvp4c MATLAB tool. Key factors have been thoroughly mapped out in graphical form for easy comprehension. In a flow regime, increasing nanoparticle concentration, magnetic number, and Eckert number lowers heat transmission and raises the ternary hybrid nanofluid's temperature profile. The effective skin friction coefficient and Nusselt number are tabulated concerning the aforementioned important factors.

Computational analysis of nonsimilar electromagnetohydrodynamic radiative nanofluid flow across Darcy–Forchheimer–Brinkman porous media

This study develops nonsimilar analysis for the electrically conducting nanofluid flow over a vertically placed stretching surface adjacent to porous medium with transverse magnetic field assumption. The features of thermal radiations, viscous dissipation, and heat source factors are also incorporated to scrutinize the thermal prospective. The Brinkman–Forchheimer who extended Darcy model is considered for the porous medium and a Rosseland approximation is employed for thermal radiations. Titanium oxide and silica are considered nanoparticles, while kerosine oil and silicon ceramic serving as the base fluids in porous medium, respectively. Tiwari–Dass approach is used to derive the governing system. Appropriate transformations are deployed to render the governing partial differential equations into a dimensionless system. To address the transformed system, the local nonsimilarity technique in conjunction with the bvp4c (fourth-order method boundary value problem) is used. Variations in velocity and thermal distributions of nanofluids have been investigated in relation to developing dimensionless parameters such as nanoparticle concentration, Darcy porosity parameter, Darcy number, electric parameter, Hartmann number, radiation parameter, Richardson number, heat source/sink parameter, and Eckert number. Numerical findings are found to interpret the role of the emerging parameters on physical quantities. The obtained findings reveals that the surge in the estimations of Richardson number enhances the nanofluid velocity while it diminishes the temperature profile. The concentration of nanoparticles improves the thermal profile of both nanofluids under consideration, whereas the enhancement in Darcy porosity parameter reduces velocity and improves temperature distributions. Results found to be in excellent agreement with published results on a special case of the considered problem.

Exploration of diffusion-thermo and thermo-diffusion on the nonlinear radiative heat flow of a conducting fluid over a permeable surface

The featured problem explores the impact of cross-diffusion on the two-dimensional electrically conducting flow of a viscous liquid over a nonlinearly stretching sheet through a permeable medium. An inclusion of radiative heat energizes the heat transport phenomenon whereas the solutal transfer enriches by the conjunction of the chemical reaction. To justify the behavior of electromagnetic radiation, the Rosseland approximation is used by considering nonlinear thermal radiation. Further, the convective boundary conditions also affect the flow properties. The approachable transformations are employed to get a suitable non-dimensional form of the governing equations for the formulated problem. Due to the complex nature of the distorted equations, the system of equations is solved using an in-built code bvp5c predefined in MATLAB. The computation is carried out for the involvement of the suitable values of contributing parameters on the flow characteristics and along with the simulations of the rate coefficients. Further, the assigned particular parameters present an outcome that validates with a good correlation. Finally, the important outcomes are — enhanced suction due to the permeability of the surface augments the fluid velocity whereas the trend is reversed in the case of injection. The augmentation in the fluid temperature is exhibited for the radiating heat but the reacting species attenuates the fluid concentration.

Role of particle shape considering three-dimensional flow of water-based ternary hybrid nanofluids for the interaction of magnetic field

In developing current flow phenomena, the proposed study is carried out by the consideration of ternary hybrid nanofluid flow past a stretching surface. The flow became electrically conducting due to the inclusion of transverse magnetic field along the normal direction of the flow. Further, the energy profile is enhanced for the interaction of the radiative heat using Rosseland approximation. The innovation behind this approach is due to the assumption of the various shapes of the particles, i.e. spherical, cylindrical along with platelet-shaped nanoparticles and the corresponding physical properties are demonstrated. In view of physical phenomena, carbon nanotube, graphene, and aluminum oxide nanoparticles with water as a base liquid are used to prepare the hybrid nanofluid. Because of the complexity of the formulated model, the governing non-dimensional set of equations is handled by using traditional numerical technique following shooting-based “Runge–Kutta fourth-order” technique. Further, the significant role of the pertinent factors involved in the flow phenomena is the case of ternary nanofluid overshoots the fluid velocity in comparison to the case of nanofluid and hybrid nanofluid and the shear rate enhances for the increasing magnetization whereas the heat transfer rate attenuates significantly.

MHD ternary hybrid nanofluid flow over a porous stretching sheet with various effects of Boussinesq and Rosseland approximations

MHD ternary hybrid nanofluid flow over a porous stretching sheet with various effects of Boussinesq and Rosseland approximations