Viscous Heating Research Articles

Mixed Convection Flow of a Viscous Dissipative and Heat Generating/Absorbing Fluid Through a Slit Microchannel

This paper is aimed at investigating the consequences of viscous dissipation and super-hydrophobicity on a magnetized mixed convection flow of an electrically conducting fluid across an up facing microchannel influenced by a transverse magnetic field. The plates were alternatively heated and incorporated with heat source/sink effect. The modeled governing equations have been obtained using a semi-analytical (regular perturbation) method. Various line graphs have been plotted to demonstrate the behavior of key parameter dictating the flow. It was found out that the thermal gradient and fluid velocity are significantly enhanced for mounting values of mixed convection Gre, Brickman number Br, Darcy porous number K and heat source S parameters in the superhydrophobic microchannel for constant pressure gradient Г. On the other hand, the velocity deteriorates for increasing levels of magnetic field M and heat sink S factors. Further, the comparison of this present analysis with previously published literature for limiting cases when Г=S=0 and Gre=1 showcased an excellent concurrence, thereby confirming the accuracy and validity of this present investigation. The findings from this research will find relevance in engineering, technological and industrial processes such as for solar collectors, nuclear reactors cooled during emergency shutdown, nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles.

Computational Analysis of Unsteady Oscillatory Flow of Nanofluid with Variable Electric Conductivity: Gear-Generalized Differential Quadrature Approach

Abstract The present study numerically investigated the entropy production in nanofluids' dissipative unsteady oscillatory flow characterised by variable electric conductivity and magnetic heating effects. The imposition of the non-isothermal boundary condition on the oscillatory stretching sheet plays a crucial role in establishing the self-similar solution in the presence of viscous heating. An external magnetic field (uniform in space and time) is imposed perpendicular to the plane of the oscillating stretched boundary. The energy equation, incorporating viscous dissipation effects and momentum equation, is reduced to nonlinear coupled partial differential equations and numerically solved using the Gear-generalized differential quadrature scheme. The Corcione model is implemented to describe the nanofluid's effective viscosity and thermal conductivity. Furthermore, expressions for entropy production and relative irreversibility parameter (Bejan number), considering variable electric conductivity, are derived and computed based on solutions obtained from momentum and energy equations. The impacts of parameters such as magnetic parameter, variable electric conductivity parameter, Eckert number, Strouhal number, Prandtl number and temperature difference parameters on flow, heat transfer, entropy generation, and Bejan number are systematically illustrated and examined. We observed that increasing the variable electric conductivity parameters reduces the velocity profiles while improving the thermal fields. Similar behaviour is found when the strength of a magnetic field is increased. The skin friction coefficient exhibits an augmentation in response to the Eckert number, dimensionless time, Strouhal number, nanoparticle volume fraction, magnetic parameter, and variable thermal conductivity parameter. Conversely, the Nusselt number increases concerning the Strouhal number and nanoparticle volume fraction. At the same time, it declines in association with the magnetic parameter, dimensionless time, Eckert number, and variable electric conductivity parameter. Additionally, to ensure the precision and reliability of the outcomes, the numerical code undergoes a thorough validation process that involves comparing its outputs to the findings of previous available studies. This comprehensive investigation enhances our understanding of nanofluid dynamics and provides valuable insights for optimising thermal management systems across various engineering disciplines.

ANALYTICAL STUDY ON FULLY DEVELOPED MIXED CONVECTION COUETTE FLOW IN A VERTICAL CHANNEL WITH VISCOUS DISSIPATION EFFECT

The study of mixed convection flow in a vertical channel with viscous dissipation has been examined. To solve the governing energy and momentum equations, the Homotopy perturbation method was employed. Graphs were generated to analyze the impact of the governing flow parameters. Numerical values for skin friction, rate of heat transfer, and mass flux were estimated. The study revealed that an increase in mixed convection expands the reverse flow region and raises the critical value of mixed convection that leads to flow reversal. Additionally, both fluid temperature and velocity rise with increased viscous dissipation, as higher viscous dissipative heat elevates temperature, subsequently increasing the buoyancy force.

Investigation of thermal properties of ethylene glycol-based Williamson hybrid-nanofluid over stretchable/shrinking flat plate and their effects on solar panels

The global requirement for sustainable energy supply to enhance industrial productivity and reduce production costs has focused researchers’ attention to renewable energy in recent years. Solar energy mitigates the dangers connected with the use of fossil fuels in electricity generation. This work is set to evaluate the heat transmission capacities of Williamson hybrid nanofluid flow across a flat plate with viscous dissipation and heat source. The mathematical model explaining the flow interaction of Williamson hybrid nanofluid, combining viscous dissipation, heat source, and temperature-variation thermal conductivity and viscosity, is created using conservation laws. The specified system of non-linear coupled partial differential equations undergoes non-similarity transformation. The resulting non-dimensional model is solved using the bivariate spectral weighted residual method. The accuracy of the method is proven by comparing obtained results with those in the literature, and a good agreement is observed. Graphs are utilized to explain the thermophysical properties that are being considered. The results show that the fluid temperature rises when there is a source of heating and viscous dissipation. The velocity and the fluid parameter (We) have an inverse connection, whereas the temperature of the fluid has the opposite impact. Moreover, when nanoparticles are present, the thermal boundary layer rises along with the nanoparticles, thickening the velocity boundary layer and decreasing fluid velocity. Findings also show that for the Vd∈[0.1,0.5], the skin drag force and Nusselt number retard by 0.64%,14.06% for the shrinking sheet and 0.21%,6.57% for the stretching sheet respectively. In the same vein, an 100% surge in the porosity parameter escalate the skin friction coefficient by 19.13% and 26.91% and the Nusselt number by 4.92% and 2.06% respectively for both the contracting and elastic sheet. The findings in the research will provide more insight in the design and improvement of solar panel plate efficiency.

Electromagnetic signatures from accreting massive black hole binaries in time domain photometric surveys

We study spectral and time variability of accreting massive black hole binaries (MBHBs) at milli-parsec separations surrounded by a geometrically thin circumbinary disc. To this end, we present the first computation of the expected spectral energy distribution (SED) and light curves (LCs) from 3D hyper-Lagrangian resolution hydrodynamic simulations of these systems. We modelled binaries with a total mass of $10^6$ M$_ eccentricities of $e=0,\, 0.9$, and a mass ratio of $q=0.1,\, 1$. The circumbinary disc has an initial aspect ratio of 0.1, features an adiabatic equation of state, and evolves under the effect of viscous heating, black-body cooling, and self gravity. To construct the SED, we considered black-body emission from each element of the disc and we added a posteriori an X-ray corona with a luminosity proportional to that of the mini-discs that form around each individual black hole. We find significant variability of the SED, especially at high energies, which translates into LCs displaying distinctive modulations of a factor of $ 2$ in the optical and of $ 10$ in UV and X-rays. We analysed in detail the flux variability in the optical band that will be probed by the Vera Rubin Observatory (VRO). We find clear modulations on the orbital period and half of the orbital period in all systems. Only in equal-mass binaries, we find an additional, longer-timescale modulation, associated with an over-density forming at the inner edge of the circumbinary disc (commonly referred to as a lump). When considering the VRO flux limit and nominal survey duration, we find that equal-mass, circular binaries are unlikely to be identified, due to the lack of prominent peaks in their Fourier spectra. Conversely, unequal-mass and/or eccentric binaries can be singled out up to $z 0.5$ (for systems with bol $ erg s$^ $) and $z 2$ (for systems with bol $ erg s$^ $). Identifying electromagnetic signatures of MBHBs at separations of $ $ pc is of paramount importance to understand the physics of the gravitational wave (GW) sources of the future Laser Interferometer Space Antenna, and to pin down the origin of the GW background (GWB) observed in pulsar timing arrays.

Investigating thermal conduction dynamics in ternary Carreau–Yasuda nanofluids under convective boundary conditions and exponential heat source/sink effects

The analysis of the Carreau–Yasuda nanofluid (CYNF) across a stretching surface has practical applications in several fields, such as heat exchange, engineering, and material science. Scholars may discover novel perspectives for increasing heat transfer performance, establishing advanced materials, and enhancing the efficiency of multiple engineering systems by studying the behavior of CYNF in this particular instance. The energy transfer through trihybrid CYNF flow with the effect of magnetic dipole across a stretching sheet is examined in the present study. The ternary hybrid nanofluid (THNF) has been prepared by the addition of ternary nanoparticles (NPs) in the water (50%) and ethylene glycol (50%). Titanium dioxide (TiO2), silicon dioxide (SiO2), and aluminum oxide (Al2O3) are used in base fluid. The fluid velocity and heat transfer are examined under the impact of Darcy–Forchheimer, chemical reaction, convective condition, activation energy, and exponential heat source. The fluid flow has been stated in form of velocity, energy, and concentration equations. The set of modeled equations is simplified to non-dimensional form of ordinary differential equations (ODEs) by using similarity substitutions. Numerically, the system of lowest order ODEs is calculated through the parametric continuation method. It has been noticed that the temperature field augments against the variation of viscous dissipation and heat source term. Furthermore, the magnetic dipole has significant impact to enhance the thermal curve of THNF whereas falloffs the velocity profile. It can be noticed that the energy propagation rate enhances with the rising numbers of ternary NPs, from 1.22% to 5.98% (nanofluid), 2.30% to 9.61% (hybrid nanofluid), and 3.23% to 11.12% (trihybrid nanofluid).

A Thermodynamic Criterion for the Formation of Circumplanetary Disks

The formation of circumplanetary disks is central to our understanding of giant planet formation, influencing their growth rate during the post-runaway phase and observability while embedded in protoplanetary disks. We use three-dimensional global multifluid radiation hydrodynamics simulations with the FARGO3D code to define the thermodynamic conditions that enable circumplanetary disk formation around Jovian planets on wide orbits. Our simulations include stellar irradiation, viscous heating, static mesh refinement, and active calculation of opacity based on multifluid dust dynamics. We find a necessary condition for the formation of circumplanetary disks in terms of a mean cooling time: When the cooling time is at least 1 order of magnitude shorter than the orbital timescale, the specific angular momentum of the gas is nearly Keplerian at scales of one-third of the Hill radius. We show that the inclusion of multifluid dust dynamics favors rotational support because dust settling produces an anisotropic opacity distribution that favors rapid cooling. In all our models with radiation hydrodynamics, specific angular momentum decreases as time evolves, in agreement with the formation of an inner isentropic envelope due to compressional heating. The isentropic envelope can extend up to one-third of the Hill radius and shows negligible rotational support. Thus, our results imply that young gas giant planets may host spherical isentropic envelopes, rather than circumplanetary disks.

Evaluation of heat transfer for unsteady thin film flow of mono and hybrid nanomaterials with five different shape features

Recent advancement in nanotechnology brings the idea of hybrid nanomaterials which offer distinguish applications in thermal reservoirs, cooling systems, energy applications, chemical engineering, vehicle engines etc. The understating of shape features for hybrid nanomaterials is quite essential as such consequences highly influenced various thermal properties like viscosity, thermal conductivity, optical properties, stability etc. The objective of current work is to examine heat transfer analysis due to thin film unsteady flow of hybrid nanofluid. The properties of hybrid nanofluid are justified for entertaining the copper (Cu), aluminium oxide (Al2O3) nanoparticles with water (H2O) base fluid. Additionally, applications of viscous dissipation, heat source and nonlinear radiated effects are attributed to current flow problem. The thermal properties of nanoparticles are examined in presence of five shape features consisting of blades, platelets, cylinders, bricks and spheres. Numerical simulations of problem are performed via Runge-Kutta-Fehlberg method. Comparative heat transfer is performed for mono nanofluid (Cu/H2O) and hybrid nanofluid (Cu−Al2O3)//H2O. It has been observed that heat transfer enhancement is more stable for cylindrical particles as compared to spherical nanoparticles. The skin friction enhances due to Hartmann number for both mono nanofluid (MNF) and hybrid nanofluid (HNF). Current results claim applications in coating thin films, lubrication systems, improving the thermal efficiency in thermal and industrial systems, heat exchangers, cooling systems etc.

Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model

Abstract The study aims to investigate the heat transfer efficiency in a hybrid nanofluid flow consisting of silver–molybdenum tetra sulphide (Ag–MoS4) with variable magnetism. The Yamada–Ota model is incorporated to account for viscous dissipation and heat source/sink effects, providing a comprehensive understanding of the fluid flow characteristics. However, the dissipative heat along with thermal radiation combined with the hybrid particles enriches the flow properties. The proposed model is simplified to its corresponding non-dimensional form for using proper similarity rules, and the set of transformed problems is handled numerically by employing the in-house MATLAB function bvp5c. The research utilizes a new statistical approach based on response surface methodology (RSM) and sensitivity evaluation to enhance the overall heat transmission performance. The work is conducted to obtain the relevant data on heat transfer rate. The concentration of nanoparticles, thermal radiation, and heat source are selected as the key parameters affecting the heat transfer efficiency. RSM is employed to optimize these parameters and determine the optimal conditions for enhanced heat transfer rate. Furthermore, the sensitivity analysis is performed to evaluate the efficiency of individual parameters on heat transportation. The findings of this study demonstrate that the hybrid nanofluid flow of Ag–MoS4 exhibits improved heat transfer efficiency compared to conventional fluids. Further, the Yamada–Ota conductivity model is also influential in enhancing the heat transfer properties.

Thermal distribution and viscous heating of electromagnetic radiative Eyring–Powell fluid with slippery wall conditions

The research examines the intricate between Eyring-Powell microfluidic heat transfer, electromagnetic radiation and viscous heating in a Riga slippery device as applied in heat exchangers, cooling systems, biomedical devices, energy generation, polymer processing, and others. The viscoelastic property of the fluid is characterized by the Eyring-Powell Cauchy fluid model with shear-thickening and shear-thinning phenomena that are useful in several heat transport processes and thermal management. The developed governing model is taken from the constitutive relations and conservation principles and solved via an adaptive partition weighted residual method. The essential fluid term sensitivities are systematically investigated on the flow and thermal distribution characteristics. An appropriate validation and comparison of results is done and found to agree quantitatively, this confirmed the correctness of the presented outcomes. The results reveal the significant impact of the thermofluidic terms interaction on the viscous flowing fluid and thermal behaviour. As seen, slip conditions momentously increase the flow rate gradients for about 1.7% to 2.4% close to the boundary wall and consequently raise the microfluidic thermal propagation rates with 3.7% non-Newtonian fluid and thickness of the boundary layer. Moreover, the thermal gradient and distribution complexities are modulated within the fluid regime due to radiation and Lorentz force.

Self-similar flow behind a shock wave in a gas under the effect of viscosity, heat conduction, and variable ambient density

Self-similar solutions have been investigated to describe the propagation of planar shock waves in a non-ideal gas generated by a piston under viscous stress and heat flux. The equation of state for non-ideal gas incorporates the correction in pressure and volume of the gas. The piston position and ambient density vary exponentially with time. Newton’s law of viscosity is used for the viscous stress and Fourier’s law of heat conduction is taken for heat flux. The viscosity coefficient is taken as constant whereas the thermal conductivity coefficient varies with temperature and density following the power law. The shock jump conditions have been derived for the viscous non-ideal gas using integral form of conservation laws. The shock Reynolds number Re s has been introduced to study the effect of viscosity on shock propagation in non-ideal gas. It is found that similarity solution exists only in an ideal gas under the condition that the ambient density exponent is equal to twice the shock position exponent. This study shows that shock Reynolds number Re s and heat conduction parameter Γ c can be used to control the variation of the flow variables and piston position significantly. The shock strength decreases with increase in the value of shock Reynolds number Re s but is independent of the heat conduction parameter Γ c . The pressure, density, and adiabatic compressibility have significant deviations from high to low viscous flow of ideal gas but the velocity and heat flux undergo negligible change. The results do not support the claim of negligible effect of viscosity in earlier studies and establish the impact of viscosity and heat flux on shock propagation in an ideal gas.

Temperature Changes of Cryogenic Fluid Flow in Pipe Bends due to Viscous Heating Effect

Temperature Changes of Cryogenic Fluid Flow in Pipe Bends due to Viscous Heating Effect

Temperature Changes of Cryogenic Fluid Flow in Pipe Bends due to Viscous Heating Effect

Temperature Changes of Cryogenic Fluid Flow in Pipe Bends due to Viscous Heating Effect

Thermal analysis of MHD unsteady Darcy‐Forchheimer thin film flow in a porous system

AbstractThis study has investigated a Darcy‐Forchheimer thin film flow over an extended horizontal surface with thermal radiation and chemical reaction effects. The governing time‐dependent equations have been non‐dimensionalized using similarity transformations and solved numerically using the fourth‐order Runge‐Kutta method and the shooting technique. The influence of magnetohydrodynamics, non‐uniform heat sourcing, viscous heat radiation, and chemical reactions on temperature, velocity, skin friction, Nusselt, and Sherwood numbers has been examined. Results have shown that porous media, magnetic field, and transient effects decrease the velocity profile, while thermal radiation and variable thermal properties enhance temperature distributions. Findings have indicated that the magnetic field and porosity enhance the skin friction coefficient whereas the heat transfer rate increases with Eckert number and Prandtl number. Rising the chemical reaction parameter from 0.2 to 0.5 rises the mass transfer rate by approximately 9.85%. The thermal analysis of MHD Darcy‐Forchheimer thin film flow in a porous system has been crucial for understanding heat transfer and fluid dynamics in complex environments. It helped in optimizing various engineering processes, such as cooling systems, filtration, and energy conversion, by providing insights into temperature distribution, convective heat transfer, and fluid behavior. This analysis has aided in designing efficient and reliable systems with improved performance and reduced energy consumption.

Irreversibility analysis of Eyring Powell nanofluid flow in curved porous channel

AbstractThe aim of this study is to minimize entropy in the MHD Eyring‐Powell fluid through a semi‐porous curved channel. The flow phenomena are examined under the consideration of joule heating, viscous heating, thermophoresis, and Brownian motion. The motivation of this research is to minimize entropy production in curved porous channel because thermal systems become more efficient as a result of decreased energy consumption, operational expenses, and experimental costs. This approach is useful in designing industrial equipment that requires efficient thermal management, such as fuel cells, chemical processing, and advanced refrigeration systems. The coupled boundary layer equations of the problem are highly nonlinear PDEs, which are transformed into systems of coupled ODEs by using similarity variables. The solution of a coupled system of ODEs is obtained numerically via Bvp4c. The effect of several physical parameters for entropy analysis, Bejan number, concentration, and velocity/temperature are illustrated and analyzed using graphs. Furthermore, the computational outcomes of physical quantities, for example, heat and mass transfer rate are also presented. Results indicated that the increasing value of the Reynolds number increases the entropy generation rate, while the reverse tendency is noticed for the Eyring‐Powell parameter. The rising values of the Brownian parameter increase the Bejan number after its decrease, while reverse behavior is observed for the thermophoresis parameter.

Quantitative analysis of energy transfer and separation in vortex tubes: Factors and empirical insights

As a device with a simple structure for energy separation, the low efficiency of the vortex tube is the primary obstacle hindering its widespread application. No existing energy separation theory has gained wide acceptance due to inconsistencies with actual processes. Therefore, drawing on prior studies concerning the flow field, the formation process of flow distribution is presented. Then a quantitative analysis of the energy transfer and separation processes in vortex tubes driven by velocity gradient, temperature gradient, and pressure gradient is proposed. The viscous shear work, heat transfer, and turbulent heat transfer processes were analyzed separately. It was found that the main factors of vortex tube energy separation originated from the expansion of the inlet nozzle and the turbulent heat transfer process. At the same time, the viscous shear work and heat transfer processes also have a specific effect on the energy separation, in which the heat transfer process suppressed the energy separation effect of the vortex tube. In addition, its empirical results indicate that interactions among the turbulent heat transfer process, frictional dissipation, angular momentum conservation, and heat transfer should be considered in further studies.

Disc novae: thermodynamics of gas-assisted binary black hole formation in AGN discs

ABSTRACT We investigate the thermodynamics of close encounters between stellar mass black holes (BHs) in the gaseous discs of active galactic nuclei (AGNs), during which binary black holes (BBHs) may form. We consider a suite of 2D viscous hydrodynamical simulations within a shearing box prescription using the Eulerian grid code athena++. We study formation scenarios where the fluid is either an isothermal gas or an adiabatic mixture of gas and radiation in local thermal equilibrium. We include the effects of viscous and shock heating, as well as optically thick cooling. We co-evolve the embedded BHs with the gas, keeping track of the energetic dissipation and torquing of the BBH by gas and inertial forces. We find that compared to the isothermal case, the minidiscs formed around each BH are significantly hotter and more diffuse, though BBH formation is still efficient. We observe massive blast waves arising from collisions between the radiative minidiscs during both the initial close encounter and subsequent periapsis periods for successfully bound BBHs. These ‘disc novae’ have a profound effect, depleting the BBH Hill sphere of gas and injecting energy into the surrounding medium. In analysing the thermal emission from these events, we observe periodic peaks in local luminosity associated with close encounters/periapses, with emission peaking in the optical/near-infrared (IR). In the AGN outskirts, these outbursts can reach 4 per cent of the AGN luminosity in the IR band, with flares rising over 0.5–1 yr. Collisions in different disc regions, or when treated in 3D with magnetism, may produce more prominent flares.

Computation of SWCNT/MWCNT-doped thermo-magnetic nano-blood boundary layer flow with non-Darcy, chemical reaction, viscous heating and Joule dissipation effects

CNTs have been shown to exhibit exceptional electrical and thermal conductivities, chemical and mechanical stability and physiochemical reliability in wide-ranging applications covering biomedicine, energy and aerospace. Motivated by emerging applications in nano-drug delivery in medicine and radiative ablation therapy, a steady-state mathematical model is established for magnetohydrodynamic boundary-layer transport of chemically reactive carbon nanotubes (CNTs)-doped electrically conducting blood along a porous stretching wall of a blood vessel. Multiple and single walls CNTs are considered using the model developed by Xue (2005). Heat generation, radiative flux, wall mass (concentration) slip, viscous heating and Joule magnetic dissipation are incorporated. Chemical reaction effects are addressed utilizing homogenous first-order model. The Darcy-Forchheimer drag force model is utilized for bulk matrix and inertial drag effects in the porous medium. Radiative heat-transport is studied considering Rosseland's diffusion model. The governing partial differential conservation equations for mass, momentum, energy and species are formulated in a two-dimensional Cartesian coordinate system with allied wall and free-stream boundary conditions. Via scaling transformations, a nonlinear boundary value problem is derived. Numerical computations are achieved through bvp4c scheme. The impact of selected parameters on dimensionless quantities (velocity, temperature, concentration, skin friction, local Nusselt and Sherwood numbers) are computed and elaborated through tables and graphs. A good correlation of the numerical solutions with earlier simpler models from the literature is included. The simulations show that with augmentation in Hartmann magnetic number and Forchheimer inertial drag number, significant damping in the nano-doped blood flow is produced for both SWCNTs and MWCNTs. Temperature is elevated with increment in dissipation effect i. e. Eckert number. Higher values of mass (solutal) slip parameter induce a depletion in the concentration. Nusselt number i. e. heat transfer rate to the blood vessel wall is improved with greater values of solid volume fraction for both CNTs. The present model is relevant to radiative ablation therapy combined with nano-medical treatments in blood vessels wherein constant mechanical loading via blood pressure and flow can elevate internal stresses (circumferential wall stress and endothelial shear stress) and induce morphological alterations in blood vessel wall (stretching) and endothelium in conjunction with biochemical reactions.

External velocity and dissipative flow of clay nanoparticles on the lubricity of drilling fluids across a vertical surface in a Darcy-Brinkman porous medium with thermal radiation

Drilling fluids are important in the extraction of oils and gases through rocks and soil. Clay nanoparticles are essential for enhancing drilling fluid efficiency. The thermal conductivity, viscosity, and boiling point of drilling fluids increase when clay nanoparticles are incorporated, providing resistance to high temperatures and regulating fluid costs. This article illustrates the convection heat transfer in drilling nanofluid while taking into account the significant presence of clay nanoparticles in the fluid with viscous dissipation, thermal radiation, and heat source/sink. The efficient thermophysical characteristics of clay nanofluid are expressed mathematically using Maxwell-Garnett and Brinkman’s formulas. The partial differential equations (PDEs) with physical boundary conditions that control the flow phenomena are predetermined. The similarity technique is employed to transmute these PDEs into ordinary differential equations (ODEs) and then an efficient boundary value problem fourth-order (bvp4c) solver is utilized to find dual solutions. The Nusselt number and skin friction are calculated and displayed in tabular form as well as graphical form along with the velocity and temperature profiles. Multiple solutions are observed in the shrinking sheet as well as the buoyancy assisting flow. The findings demonstrate that when volume concentration increases, the Nusselt number rises noticeably. In addition, the permeability parameter expands the boundary layer thickness in the lower solution, while the contrary behavior is observed in the upper solution.

Existence of Regular Solutions for a Class of Incompressible Non-Newtonian MHD Equations Coupled to the Heat Equation

We consider a system of PDE’s describing the steady flow of an electrically conducting fluid in the presence of a magnetic field. The system of governing equations composes of the stationary non-Newtonian incompressible MHD equations coupled to the heat equation wherein the influence of buoyancy is taken into account in the momentum equation and the Joule heating and viscous heating terms are included. We proved the existence of C1,γ(Ω¯)×W2,r(Ω)×W2,2(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\gamma }({\\bar{\\Omega }})\ imes W^{2,r}(\\Omega )\ imes W^{2,2}{(\\Omega )}$$\\end{document} solutions of the systems for 1<p<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1< p<2$$\\end{document} corresponding to a small data and we show that this solution is unique in case 6/5<p<2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$6/5< p < 2$$\\end{document}. Moreover, we also proved the higher regularity properties of this solution.