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The basic equations for a homogeneous and isotropic thermo-magnetoviscoelastic medium are formulated based on three different theories, i.e., the Green- Lindsay (G-L) theory, the coupled (CD) theory, and the Lord-Shulman (L-S) theory. The variable thermal conductivity is considered as a linear function of the temperature. Using suitable non-dimensional variables, these basic equations are solved via the eigenvalue approach. The medium is initially assumed to be stress-free and subject to a thermal shock. The numerical results reveal that the viscosity, the two-temperature parameter, the gravity term, and the magnetic field significantly influence the distribution of the physical quantities of the thermoelastic medium. [ABSTRACT FROM AUTHOR]

The current work intends to look into the effects of variable liquid properties on the magnetohydrodynamics of peristaltic flow exhibited by Jeffrey fluid through a compliant-walled channel. In order to make realistic approximations for the flow characteristics of blood, the channel is considered to be inclined and porous. Furthermore, convective boundary conditions and concentration slip have been employed in the analysis. The mathematical formulation is established on the grounds of low Reynolds number and long wavelength approximations. Perturbation solution is obtained for the resulting non-linear differential equations of momentum and energy for small values of variable viscosity and variable thermal conductivity, whereas exact solution is found for the concentration field. The impact of various parameters included in the study is displayed graphically. A rise in the parameter for variable viscosity is found to accelerate the fluid flow, hence resulting in an increased bolus size. For variable thermal conductivity, a similar influence on the heat transfer was observed. The behaviour of the skin-friction coefficient, Nusselt and Sherwood numbers have also been plotted for the pertinent parameters. [ABSTRACT FROM AUTHOR]

Numerical analysis has been carried out for a casson fluid flow on a riga-plate with temperature dependent thermal conductivity. The physical model which governs the transport properties is solved numerically. This investigation emphasizes the consequence of variable thermal conductivity and electrically conducting magnetic field on the fluid flow. Rate of heat transfer is elevated, while the flow exposed to constructive case of variable thermal conductivity. The flow speed is enhanced for the improved values of modified Hartmann number. Correlation of the results with the similarity solutions declares the accuracy. [ABSTRACT FROM AUTHOR]

• What is the impact of slip parameters on the velocity profile of Al 2 O 3 / H 2 O nanofluid and Al 2 O 3 - C u / H 2 O hybrid nanofluids? • What is the effect of thermal radiation, and variable thermal conductivityon the heat transfer attributes for the Al 2 O 3 / H 2 O nanofluid and Al 2 O 3 - C u / H 2 O hybrid nanofluids flow? • Which of the analyzed models for the - nanofluid and Al 2 O 3 - C u / H 2 O hybrid nanofluid demonstrates superior heat transport across the governing parameters? This research looks at the stagnation point flow of MHD Al 2 O 3 - C u / H 2 O hybrid nanofluid against a permeable, vertically extending surface that uses thermal radiation and how different models of thermal conductivity affect it. This study is unique because it looks at how changes in thermal conductivity, mixed convection, and the slip velocity of hybrid nanofluids affect the stretching surface. It also looks at how changes in temperature, skin-friction coefficient, Nusselt number, and velocity affect convective thermal boundary conditions. With the use of boundary layer approximations, the complicated system of PDEs is simplified. The dimensionality of these PDEs and the boundary conditions they include are eliminated by applying certain modifications. Combining a local non-similarity approach up to the second truncation level with MATLAB's built-in finite difference code (bvp4c), one can obtain the results of the updated model. The research demonstrates and analyzes the impact of different factors on fluid flow and heat transfer characteristics in the studied flow situations. This is done by comparing the computed data with available literature and presenting the findings in graphical form. Tables are generated to present the numerical fluctuations of the drag coefficient and Nusselt number. This study shows how important thermal conductivity is in the mixed convection of hybrid nanofluids and looks into what happens when the thermal conductivity parameter is changed. Significantly, an augmentation in this parameter results in an elevation in the temperature distribution of the hybrid nanofluid while simultaneously reducing the rate of heat transfer across various models. Furthermore, increasing the nanoparticle volume fraction parameter leads to higher temperature and Nusselt number profiles while reducing skin friction. The mixed convection parameter has a notable impact on increasing the friction coefficient on the stretched vertical surface. However, because these elements function as regulating variables, it decreases when the magnetic, mass suction, and velocity slip parameters are present. The results also demonstrate notable discrepancies in the mean Nusselt values generated by various thermal conductivity models. According to the analysis, the Hamilton-Crosser model has the lowest average Nusselt numbers, followed by the Yamada-Ota model and the Xue model, in that order. [ABSTRACT FROM AUTHOR]

In the present article, we introduced a new model of the equations of generalized thermoelasticity for unbounded orthotropic body containing a cylindrical cavity. We applied this model in the context of generalized thermoelasticity with phase-lags under the effect of rotation. In this case, the thermal conductivity of the material is considered to be variable. In addition, the cylinder surface is traction free and subjected to a uniform unit step temperature. Using the Laplace transform technique, the distributions of the temperature, displacement, radial stress and hoop stress are determined. A detailed analysis of the effects of rotation, phase-lags and the variability thermal conductivity parameters on the studied fields is discussed. Numerical results for the studied fields are illustrated graphically in the presence and absence of rotation.

Abstract The aim of this article is to investigate MHD Carreau fluid slip flow with viscous dissipation and heat transfer by taking the effect of thermal radiation over a stretching sheet embedded in a porous medium with variable thickness and variable thermal conductivity. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The constitutive equations of Carreau fluid are modeled in the form of partial differential equations (PDEs). Concerning boundary conditions available, the PDEs are converted to ordinary differential equations (ODEs) by means of similarity transformation. The homotopy analysis method (HAM) is used for solution of the system of nonlinear problems. The effects of various parameters such as Weissenberg number We 2 $\mathit{We}^{2}$ , magnetic parameter M 2 $M^{2}$ , power law index n, porosity parameter D, wall thickness parameter α, power index parameter m, slip parameter λ, thermal conductivity parameter ε, radiation parameter R and Prandtl number on velocity and temperature profiles are analyzed and studied graphically.

Owing to valuable significance of bioconvective transport phenomenon in interaction of nanoparticles, different applications are suggested in field of bio-technology, bio-fuels, fertilizers and soil sciences. It is well emphasized fact that thermal outcomes of nanofluids can be boosted under the consideration of various thermal sources. The aim of current research is to test the induction of induced magnetic force in bioconvective transport of non-Newtonian nanofluid. The rheological impact of non-Newtonian materials is observed by using Casson fluid with suspension of microorganisms. The chemical reaction effected are interpreted. The thermal conductivity of material is assumed to be fluctuated with temperature fluctuation. The flow pattern is endorsed by stretching surface following the stagnation point flow. Under the defined flow assumptions, the problem is formulated. A computational software with shooting technique is used to present the simulations. A comprehensive analysis for problem is presented. It is claimed that the interpretation of induced magnetic force exclusively enhanced the thermal phenomenon., (© 2024. The Author(s).)

The present investigation aims to examine the heat flux mechanism in the hagnetohydrodynamic (MHD) mixed convective flow of Williamson-type fluid across an exponential stretching porous curved surface. The significant role of thermal conductivity (variable), non-linear thermal radiation, unequal source-sink, and Joules heating is considered. The governing problems are obtained using the Navier–Stokes theory, and the appropriate similarity transformation is applied to write the partial differential equations in the form of single-variable differential equations. The solutions are obtained by using a MATLAB-based built-in bvp4c package. The vital aspect of this analysis is to observe the effects of the curvature parameter, magnetic number, suction/injection parameter, permeability parameter, Prandtl factor, Eckert factor, non-linear radiation parameter, buoyancy parameter, temperature ratio parameter, Williamson fluid parameter, and thermal conductivity (variable) parameter on the velocity field, thermal distribution, and pressure profile which are discussed in detail using a graphical approach. The correlation with the literature reveals a satisfactory improvement in the existing results on permeability factors in Williamson fluids.

We have explored multiple solutions for non-Newtonian Casson nanofluid flow past a moving extending sheet under the influence of variable thermal conductivity and nonlinear radiation through a permeable medium with convective boundary conditions. The governing equations are transformed to ODEs by similarity transformations and then solved numerically by the Chebyshev pseudospectral (CPS) method. Dual solutions are obtained for velocity, temperature and nanoparticle concentration distributions with different values of physical parameters. In the present analysis, it was found that, the nonlinearity formula for thermal radiation gives a realistic description of nanofluid mathematical model depending on the existence of nanoscale particles. Furthermore, the concentration of nanoparticles is highly influenced by nonlinear thermal radiation due to the sizes of nanofluid, where linear radiation has a weak effect on the concentration distributions of nanoparticles. These results are very important in medicine, and more specifically for reinforcing the delivery of drugs through the skin, as the nanoparticle entrapment of drugs enhances delivery to, or absorption by, target cells. The transdermal drug delivery system offers huge clinical advantages over other dosage forms. As transdermal drug delivery offers controlled as well as predetermined rate of release of the drug into the patient, it can keep up steady-state nanofluid concentration. [ABSTRACT FROM AUTHOR]

This study investigates the photo-thermo-elastic interaction in an infinite semi-conductor material with cylindrical cavities. The effect of variable thermal conductivity through the photo-thermo-elastic transport process is studied using the coupled models of thermo-elasticity and plasma waves. The internal surface of the cylindrical cavity is loaded by a thermal shock varying heat. The eigenvalue method, under Laplace transform scheme, is used to get the analytical solution of the studied field distribution as parts of this phenomenon. A detailed analysis of the impacts of the variable thermal conductivity on the physical fields is debated. The numerical outcomes of the studied fields are displayed graphically. According to them, the variable thermal conductivity offers finite speed of the mechanical wave and the thermal wave propagations. [ABSTRACT FROM AUTHOR]

Thermophysical and wall‐slip effects arise in many areas of nuclear technology. Motivated by such applications, in this article, the collective influence of variable‐viscosity, thermal conductivity, velocity and thermal slip effects on a steady two‐dimensional magnetohydrodynamic micropolar fluid over a stretching sheet is analyzed numerically. The governing nonlinear partial differential equations have been converted into a system of nonlinear ordinary differential equations using suitable coordinate transformations. The numerical solutions of the problem are expressed in the form of nondimensional velocity and temperature profiles and discussed from their graphical representations. The Nachtsheim‐Swigert shooting iteration technique together with the sixth‐order Runge‐Kutta integration scheme has been applied for the numerical solution. A comparison with the existing results has been done, and an excellent agreement is found. Further validation with the Adomian decomposition method is included for the general model. Interesting features in the heat and momentum characteristics are explored. It is found that a greater thermal slip and thermal conductivity elevate thermal boundary layer thickness. Increasing Prandtl number enhances the Nusselt number at the wall but reduces wall couple stress (microrotation gradient). Temperatures are enhanced with both the magnetic field and viscosity parameter. Increasing momentum (hydrodynamic) slip is found to accelerate the flow and elevate temperatures. [ABSTRACT FROM AUTHOR]

A novel model of variable thermal conductivity of semiconductor medium in two-temperature theory is studied. The Photothermal excitation due to laser pulses is investigated. The governing equations have been studied under three theories in generalized thermoelasticity during Photothermal transport process. Taken into consideration the linearity of thermal conductivity, in which depend on temperature. The normal mode method is used to obtain the exact expressions of some physical fields analytically, in the context of the recombination between thermal-elastic-plasma waves. The silicon (Si) material is used to the numerical simulation. Some comparisons between the results are made as the influence of thermoelectric coupling parameter, variable thermal conductivity and two-temperature parameter of physical fields. The numerical calculations have been discussed and illustrated graphically. [ABSTRACT FROM AUTHOR]

The current paper analyzes the thermal and concentration attributes with the temperature‐dependent mass diffusion coefficient and thermal conductivity for the flow of an Oldroyd‐B nanoliquid over a stretchable configuration using the Buongiorno model under the application of boundary layers. The mechanisms of heat and mass transport are modeled by using the revised definitions of heat and mass fluxes. Mathematical expressions for the conservation laws are transformed into ordinary differential expressions by making the appropriate changes. The resulting complexly structured expressions are handled via an optimal homotopy procedure. The impact of influential variables on the desired solutions is plotted, tabulated, and discussed in detail. Comparative analysis of the thermal wall flux coefficient, concentration flux coefficient, density magnitude of the motile microorganisms, and reduced dimensionless stresses with already published research as a limiting case of this exploration is presented for the validity of the proposed scheme, and an excellent agreement is observed, which confirms the reliability of the homotopic solution. [ABSTRACT FROM AUTHOR]

The current endeavor is communicated to perceive the characteristics of tangent hyperbolic fluid confined by a deformable sheet in the existence of both variable thermal conductivity and magneto hydrodynamics. An exertion is reported to calculate the numerical solutions of tangent hyperbolic fluid. Mathematical model is contrived in the embodiment of PDEs and after that renovated into ODEs by implementing self-felicitous transformations. Furthermore the obtained ODEs are simplified by utilizing Bvp4c (MATLAB package). Numerical computations of Nusselt number, skin friction, temperature and velocity profiles for miscellaneous values of involved parameters are manifested via graphs and tables. A comparison of contemporary outcomes with existing published work has been taken into account which shows excellent agreement of results. [ABSTRACT FROM AUTHOR]

This article presents an inverse problem of determination of a space-dependent heat flux in steady-state heat conduction problems. The thermal conductivity of a heat conducting body depends on the temperature distribution over the body. In this study, the simulated measured temperature distribution on part of the boundary is related to the variable heat flux imposed on a different part of the boundary through incorporating the variable thermal conductivity components into the sensitivity coefficients. To do so, a body-fitted grid generation technique is used to mesh the two-dimensional irregular body and solve the direct heat conduction problem. An efficient, accurate, robust, and easy to implement method is presented to compute the sensitivity coefficients through derived expressions. Novelty of the study is twofold: (1) Boundary-fitted grid-based sensitivity analysis in which all sensitivities can be obtained in only one direct solution (at each iteration), irrespective of the number of unknown parameters, and (2) the way the measured temperatures on part of boundary are related to a variable heat flux applied on another part of boundary through components of a variable thermal conductivity. The conjugate gradient method along with the discrepancy principle is used in the inverse analysis to minimize the objective function and achieve the desired solution. [ABSTRACT FROM AUTHOR]

This study addresses the impact of variable thermal conductivity and induced magnetic field on an unsteady two‐dimensional channel flow of an incompressible laminar mixed convective and chemically reacted Jeffrey fluid embedded in a non‐Darcy porous medium with an appropriate convective type boundary conditions. The suction/injection velocity distribution has been assumed to be in an exponential form. The set of transport equations is reduced into coupled ordinary differential equations by using appropriate similar variables, which are solved by shooting technique with Runge‐Kutta fourth‐order algorithm. The investigation is carried out for various emerging nondimensional parameters on the axial, radial velocities, temperature distribution, concentration, and induced magnetic fields and also with skin friction coefficient are discussed through graphs. The value of the local Sherwood and Nusselt numbers are analyzed numerically. We noticed that the effect of the induced magnetic field is increased with Strommer's number while it decreases for high magnetic Reynolds number. [ABSTRACT FROM AUTHOR]

The purpose of this research article is to investigate the impact of magnetic dipole on the flow of nano-fluids over the stretching surface with variable viscosity and thermal conductivity. Fluid fills the porous space. Heat transfer analysis is carried out in the presence of Newtonian heating. Water and ethylene glycol are used as base fluid while gamma alumina acts as a nanoparticle. System of ordinary differential equations is obtained by appropriate transformations. Numerical solutions are computed for the resulting nonlinear differential system by applying shooting method with RK-4 scheme. Impact of different parameters on the velocity and temperature profiles is examined. Computations for surface drag force and heat transfer rates are presented and examined for the influences of pertinent parameters. It is observed that fluid velocity reduces for larger viscosity parameter while opposite behavior is observed for temperature. Increasing values of ferro hydrodynamic interaction parameter and Curie temperature correspond to the enhancement in temperature. Surface drag force increases for larger values of nanoparticles volume fraction. Gamma alumina-ethylene glycol nano-fluid produce less surface drag force than gamma alumina-water nano-fluid. Heat transfer rate is increasing function of Prandtl number. [ABSTRACT FROM AUTHOR]

[Display omitted] • Peristalsis of hybrid nanofluid via a curved channel with curvature dependent channel walls is examined. • The hybrid nanofluid is composed of A l 2 O 3 and MWCNTs. • Thermal conductivity is considered to vary with temperature. • The lubrication assumptions are utilized to reduce the consider physical phenomenon. • The resulting model is treated through numeric scheme. This study focused on the peristaltic motion of hybrid nano-liquid via a curved geometry with curvature-dependent channel walls. The hybrid nanofluid is composed of alumina (A l 2 O 3) nanoparticles mixed with multi-wall carbon nanotubes (MWCNTs) and suspended in thermal oil. This combination is based on an experimental examination conducted by Asadi et al. [58]. Keeping in view the industrial applications of such flows, the hybrid nanofluid thermal conductivity is considered to vary with temperature. Furthermore, heat sink/source is also taken into account in the energy equation. Lubrication approximation is utilized to simplify the flow problem. The resulting model is treated using a numerical scheme NDSolve based on the shooting method ("a built-in command in Mathematica"). The primary focus remains on analyzing the impacts of the curvature parameter, flow rate, and variable thermal conductivity parameter. Outcomes indicate that there is a critical "flow rate" value for such a type of channel. The thermal transfer rate increases with an improvement in the values of the curvature parameter. Furthermore, graphical results reveal that hybrid nanofluid's temperature decreases for higher values of flow rate, while it rises with heat generation/absorption parameter. It is observed that temperature of the hybrid nanofluid increases as the values of curvature parameter improve. A higher flow rate reduces the rate of heat transfer. It is also noted that velocity profile remains constant at a flow rate value of 7.40. Moreover, the variable thermal conductivity has a remarkable impact on temperature profile. [ABSTRACT FROM AUTHOR]

Flows configured by moving surface holds considerable importance in fluid mechanics. The moving surface finds widespread usages in distinct real-world scenarios, for illustration rubber sheets, hot rolling, plastic drawing, paper manufacturing, glass blowing and wire fabrication. The rate of stretching/cooling during these processes play a vital role in ensuring the production of high-excellence end-products. This attempt features entropy optimized slip flow featuring rate type (Jeffrey) liquid subjected to Cattaneo–Christov (CC) dual diffusion and nonlinear convection aspects. Variable fluid aspects (i.e., thermal conductivity and mass diffusivity) along with thermal source and first order chemical reaction are scrutinized. The problem is formulated by employing fundamental laws of fluid dynamics. The non-linear partial differential equations are transformed into the ordinary ones using suitable transformations. Subsequently, optimal homotopic methodology (OHAM) is employed to achieve computational results. Developed nonlinear problems are determined for velocity, temperature and concentration fields. Bejan number, skin friction and the graphs of entropy generation parameters have been plotted and argued. To validate the problem, the comparison for different parameters has been addressed through Shooting and OHAM techniques. Total residual error is calculated. Main outcomes are presented in conclusions. [ABSTRACT FROM AUTHOR]

This article is concerned with the two-dimensional flow of Powell–Eyring fluid with variable thermal conductivity. The flow is caused due to a stretching cylinder. Temperature dependent thermal conductivity is considered. Both numerical and analytic solutions are obtained and compared. Analytic solution is found by homotopy analysis method. Numerical solution by shooting technique is presented. Discussion to different physical parameters for the velocity and temperature is assigned. It is observed that the velocity profile enhances for larger magnetic parameter. It is also further noted that for increasing the value of Prandtl number temperature profile decreases.

In this paper, heat transfer and entropy of steady Williamson nanofluid flow based on the fundamental symmetry is studied. The fluid is positioned over a stretched flat surface moving non-uniformly. Nanofluid is analyzed for its flow and thermal transport properties by consigning it to a convectively heated slippery surface. Thermal conductivity is assumed to be varied with temperature impacted by thermal radiation along with axisymmetric magnetohydrodynamics (MHD). Boundary layer approximations lead to partial differential equations, which are transformed into ordinary differential equations in light of a single phase model accounting for Cu-water and TiO2-water nanofluids. The resulting ODEs are solved via a finite difference based Keller box scheme. Various formidable physical parameters affecting fluid movement, difference in temperature, system entropy, skin friction and Nusselt number around the boundary are presented graphically and numerically discussed. It has also been observed that the nanofluid based on Cu-water is identified as a superior thermal conductor rather than TiO2-water based nanofluid.

At present, with the development in nanotechnology, nanostructures with temperature-dependent properties have been used in nano-electromechanical systems (NEMS). Thus, introducing an accurate mathematical model of nanobeams with temperature-dependent properties is a major and important topic for the design of NEMS. This paper aims to discuss nonlocal nanobeams analysis depending on the theories of Euler–Bernoulli and modified couple-stress (MCS). It also is assumed that the thermal conductivity of the nanobeam is dependent on the temperature. Physical fields of the nanobeam are obtained utilizing Laplace transform and state-space techniques. The effects of the size and nonlocal parameters, variability of thermal conductivity and couple stress on various distributions are presented graphically and studied in detail. Numerical results are presented as application scales and the design of nanoparticles, nanoscale oscillators, atomic force microscopes, and nanogenerators, in which nanoparticles as nanobeams act as essential and basic elements.

In this work, the generalized photo-thermo-elastic model with variable thermal conductivity is presented to estimates the variations of temperature, the carrier density, the stress and the displacement in a semiconductor material. The effects of variable thermal conductivity under photo-thermal transport process is investigated by using the coupled model of thermoelastic and plasma wave. The surface of medium is loaded by uniform unit step temperature. Easily, the analytical solutions in the domain of Laplace are obtained. By using Laplace transforms with the eigenvalue scheme, the fields studied are obtained analytically and presented graphically.

An upper-convected Maxwell (UCM) fluid flow over a melting surface situated in hot environment is studied. The influence of melting heat transfer and thermal stratification are properly accounted for by modifying the classical boundary condition of temperature to account for both. It is assumed that the ratio of inertia forces to viscous forces is high enough for boundary layer approximation to be valid. The corresponding influence of exponentially space dependent internal heat generation on viscosity and thermal conductivity of UCM is properly considered. The dynamic viscosity and thermal conductivity of UCM are temperature dependent. Classical temperature dependent viscosity and thermal conductivity models are modified to suit the case of both melting heat transfer and ther- mal stratification. The governing non-linear partial differential equations describing the problem are reduced to a system of nonlinear ordinary differential equations using similarity transformations and completed the solution numerically using the Runge-Kutta method along with shooting technique (RK4SM). The numerical procedure is validated by comparing the solutions of RK4SM with that of MATLAB based bvp4c. The results reveal that increase in stratification parameter corresponds to decrease in the heat energy entering into the fluid domain from freestream and this significantly reduces the overall temperature and temperature gradient of UCM fluid as it flows over a melting surface. The transverse velocity, longitudinal velocity and temperature of UCM are increasing func- tion of temperature dependent viscous and thermal conductivity parameters. At a constant value of melting parameter, the local skin-friction coefficient and heat transfer rate increases with an increase in Deborah number.

The present work inspects the entropy generation on radiative heat transfer in the flow of variable thermal conductivity optically thin viscous Cu–water nanofluid with an external magnetic field through a parallel isothermal plate channel. Our approach uses the power series from the governing non-linear differential equations for small values of thermal conductivity variation parameter which are then analysed by various generalizations of Hermite- Padé approximation method. The influences of the pertinent flow parameters on velocity, temperature, thermal conductivity criticality conditions and entropy generation are discussed quantitatively both numerically and graphically. A stability analysis has been performed for the rate of heat transfer which signifies that the lower solution branch is stable and physically acceptable, whereas the upper solution branch is unstable.

The problem of unsteady as well as steady hydromagnetic natural convection and mass transfer flow of viscous reactive, incompressible and electrically conducting fluid between two vertical walls in the presence of uniform magnetic field applied normal to the flow region is studied. Thermal diffusion, temperature dependent variable viscosity and thermal conductivity are assumed to exist within the channel. The governing partial differential equations are solved numerically using implicit finite difference scheme. Results of the computations for velocity, temperature, concentration, skin friction, rate of heat and mass transfer are presented graphically to study the hydrodynamic behavior of fluid in the channel.

The suspension of non-Newtonian materials with nanoparticles is important to enhance the thermal phenomenon in various engineering and industrial processes. The versatile research in nanomaterials provide different applications in thermal processes, heat exchangers, thermoelectric devices, HVAC systems, energy processes etc. Following to such novel motivations in mind, current research endorsed the enhancement in heat transfer due to suspension of Jeffrey nanofluid comprising the variable thermal conductivity. The cause of flow is associated to two disks attaining fixed distance. The modified developed relations for Fourier's hypothesis are utilized to model the problem. The flow problem is modeled with appliance of fundamental novel laws. By applying suitable transformations, corresponding differential equations are renovated into dimensionless forms which are solved with applications of analytic homotopic algorithm. The behavior of temperature and velocity due to various parameters is discussed. The numerical calculations have been done for wall shear force and Nusselt number. The results show that the velocity profile boosted due to variation of stretching ratio constant. The enhancement in heat transfer is observed due to Reynolds number. Moreover, the increasing observations for wall shear force in upper and lower disk surfaces are obtained against larger material parameter. The simulated results may find applications in improving heat transfer phenomenon, manufacturing systems, recovery processes, cooling systems, chemical phenomenon, fuel cells etc., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author.)

So the expression HT ht in eqs. 12-14 is not correct and has to be replaced by HT ht (constant viscosity of the fluid at HT ht ). [Extracted from the article]

This work considers the two-dimensional steady MHD boundary layer flow of heat and mass transfer over a flat plate with partial slip at the surface subjected to the convective heat flux. The particular attraction lies in searching the effects of variable viscosity and variable thermal diffusivity on the behavior of the flow. In addition, non-linear thermal radiation effects and thermophoresis are taken into account. The governing nonlinear partial differential equations for the flow, heat and mass transfer are transformed into a set of coupled nonlinear ordinary differential equations by using similarity variable, which are solved numerically by applying Runge–Kutta fourth–fifth order integration scheme in association with quasilinear shooting technique. The novel results for the dimensionless velocity, temperature, concentration and ambient Prandtl number within the boundary layer are displayed graphically for various parameters that characterize the flow. The local skin friction, Nusselt number and Sherwood number are shown graphically. The numerical results obtained for the particular case are fairly in good agreement with the result of Rahman [6].

In this paper we study the effects of variable viscosity and thermal conductivity on the heat transfer in the pressure-driven fully developed flow of a slurry (suspension) between two horizontal flat plates. The fluid is assumed to be described by a constitutive relation for a generalized second grade fluid where the shear viscosity is a function of the shear rate, temperature and concentration. The heat flux vector for the slurry is assumed to follow a generalized form of the Fourier’s equation where the thermal conductivity k depends on the temperature as well as the shear rate. We numerically solve the governing equations of motion in the non-dimensional form and perform a parametric study to see the effects of various dimensionless numbers on the velocity, volume fraction and temperature profiles. The different cases of shear thinning and thickening, and the effect of the exponent in the Reynolds viscosity model, for the temperature variation in viscosity, are also considered. The results indicate that the variable thermal conductivity can play an important role in controlling the temperature variation in the flow.

The motion of temperature dependent viscosity and thermal conductivity of steady incompressible laminar free convective (MHD) non-Newtonian Casson fluid flow over an exponentially stretching surface embedded in a thermally stratified medium are investigated. It is assumed that natural convection is induced by buoyancy and exponentially decaying internal heat generation across the space. The dimensionless temperature is constructed such that the effect of stratification can be revealed. Similarity transformations were employed to convert the governing partial differential equations to a system of nonlinear ordinary differential equations. The numerical solutions were obtained using shooting method along with the Runge-Kutta Gill method. The behaviour of dimensionless velocity, temperature and temperature gradient within the boundary layer has been studied using different values of all the controlling parameters. The numerical result show that increase in the magnitude of temperature dependent fluid viscosity parameter leads to an increase in velocity, decrease in temperature, decrease in temperature gradient near the wall and increase in temperature gradient far from the wall. The velocity profile increases, temperature distribution increases and temperature gradient increases near the wall only by increasing the magnitude of temperature dependent thermal conductivity parameter.

In this study, we considered the flow of a micropolar fluid over a vertical Riga sheet. The non linear stretching sheet is considered. The effects of variable thermal conductivity and radiation on the Riga sheet are taken into account. Additionally, we also debated the Brownian motion and thermophoretic. To simplify the partial differential equations, we converted them into dimensionless ordinary differential equations using suitable similarity variables and solved dimensionless system numerically using the bvp4c function. The impact of some intended parameters on the dimensionless velocity, microrotation, temperature, and concentration distributions graphically are presented and the numerical outcomes of physical quantities like skin friction, Nusselt number, Sherwood number, and couple stress have been presented in tabular form. The micropolar parameter increased which increased the couple stress and friction at surface. Because, the fluid rotation increased which increased friction at surface and also increased the couple stress. The transfer of mass decayed and transfer of heat heightened by larger values of variable thermal conductivity. Thermal conductivity improved which improved the heat transfer phenomena, so transfer of heat at surface becomes larger while also reducing the transfer of mass., Competing Interests: The authors declare no conflict of interest., (© 2023 The Authors.)

Micro polar fluids have a wide variety of applications in biomedical, manufacturing, and technical activities, such as nuclear structures, biosensors, electronic heating and cooling, etc. The aim of this study is to investigate the properties of heat transfer on a magnetohydrodynamic free convection movement of micro polar fluid over an exponentially stretchable curved surface. The flow is non-turbulent and steady. The effects of Joule heating, varying thermal conductivity, irregular heat reservoir, and non-linear radiation are anticipated. The modelled PDEs are converted to ODEs via transformation, and the integration problems are then addressed using ND-Solve method along with bvp4c package. It is observed that velocity is reduced and the micro rotation field is increased as the micro rotation parameter is increased. It is witnessed that the temperature of the fluid enhances as the Eckert number is augmented. The velocity is increasing function of the curvature parameter while the decreases with increasing magnetic factor. The distribution of temperature is improved by a rise in temperature-dependent thermal conductivity characteristic. It is investigated that as the values of temperature ratio, Prandtl number, and the Biot number are increased the temperature distribution is enhanced. For the stability of the numerical results, the mean square residue error (MSRE) and total mean square residue error (TMSRE) are computed. For the confirmation of the present analysis, a comparison is done with the published study and excellent settlement is found., Competing Interests: No conflict of interest exist about this manuscript., (© 2023 The Authors. Published by Elsevier Ltd.)

The present work examines the combined influence of variable thermal conductivity and viscosity on the irreversibility rate in couple stress fluid flow in between asymmetrically heated parallel plates. The dimensionless fluid equations are solved by using homotopy analysis method (HAM) and validated with Runge‐Kutta shooting method (RKSM). The convergent series solution is then used for the irreversibility analysis in the flow domain. The effects of thermal conductivity and viscosity variation parameters, couple stress parameter, Reynolds number, Grashof number, Hartmann number on the velocity profile, temperature distribution, entropy production, and heat irreversibility ratio are presented through graphs, and salient features of the solutions are discussed. The computations show that the entropy production rate decreases with increased magnetic field and thermal conductivity parameters, whereas it rises with increasing values of couple stress parameter, Brinkman number, viscosity variation parameter, and Grashof number. The study is relevant to lubrication theory. [ABSTRACT FROM AUTHOR]

White crystalline powder (aluminum oxide-Al2O3) and water are the products often formed after the heating of aluminum hydroxide. In this report, boundary layer flow of two different nanofluids (i.e., 36 nm Al2O3-water and 47 nm Al2O3-water) over an upper horizontal surface of a paraboloid of revolution under the influence of magnetic field is presented. The combined influence of magnetic field strength, electric current density, electric charge, electron collision time, and the mass of an electron in the flows are considered in the governing equations. Three-dimensional transport phenomenon was considered due to the influence of the Lorentz force (F→) along the z-direction as in the case of Hall currents. In this study, the dynamic viscosity and density of the nanofluids are assumed to vary with the volume fraction ϕ. The dimensional governing equations were non-dimensionalization and parametrization using similarity variables. The corresponding boundary value problem was transformed into initial value problem using the method of superposition and solved numerically using fourth-order Runge-Kutta method with shooting technique (RK4SM). Magnetic field parameter is seen to have dual effects on the cross-flow velocity profiles of both nanofluids. The maximum cross-flow velocity is attained within the fluid domain when 36 nm nanoparticles alumina is used. The cross-flow velocity gradient at the wall increases with magnetic field parameter (M) and also increases significantly with Hall parameter at larger values of M. [ABSTRACT FROM AUTHOR]

An analysis is introduced to investigate the salient features of nonlinear convective flow of thixotropic fluid in the version of Cattaneo-Christov heat flux theory. The stagnation point flow is present. The flow phenomenon is by an impermeable stretching sheet. The energy expression is modeled through the theory of Cattaneo-Christov heat flux. Characteristics of heat transfer phenomenon are described within the frame of variable thermal conductivity. Suitable variables reduced to the nonlinear partial differential expressions to the ordinary differential expressions. Series solutions of resulting systems are acquired within the frame of homotopy theory. Convergence analysis is achieved and suitable values are determined by capturing the so-called ℏ−curves. Graphical results for velocity and temperature are displayed and argued for sundry physical variables. Expression of skin friction coefficient is calculated through numerical values. Higher values of mixed convection parameter, Prandtl number, and thermal relaxation time lead to decay the temperature and layer thickness. [ABSTRACT FROM AUTHOR]