Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity

In the present study, we have analyzed the effects of variable viscosity, variable thermocapillarity on the flow and heat transfer in a thin film on a horizontal porous shrinking sheet through a porous medium. The unsteady boundary layer equations for momentum and thermal energy are simplified by using similarity transformations. The resulted, coupled nonlinear differential equations are solved by homotopy analysis method. The results are presented graphically to interpret various physical parameters appearing in the problem.     

Effects of radiation and thermal conductivity on MHD boundary layer flow with heat transfer along a vertical stretching sheet in a porous medium

A steady two-dimensional free convective flow of a viscous incompressible fluid along a vertical stretching sheet with the effect of magnetic field, radiation and variable thermal conductivity in porous media is analyzed. The nonlinear partial differential equations, governing the flow field under consideration, have been transformed by a similarity transformation into a systemof nonlinear ordinary differential equations and then solved numerically. Resulting non-dimensional velocity and temperature profiles are then presented graphically for different values of the parameters. Finally, the effects of the pertinent parameters, which are of physical and engineering interest, are examined both in graphical and tabular form.     

Effect of multipolar interaction on the effective thermal conductivity of nanofluids

Nanofluids or liquids with suspended nanoparticles are likely to be the future heat transfer media, as they exhibit higher thermal conductivity than those of liquids. It has been proposed that nanoparticles are apt to congregate and form clusters, and hence the interaction between nanoparticles becomes important. In this paper, by taking into account the interaction between nearest-neighbour inclusions, we adopt the multiple image method to investigate the effective thermal conductivity of nanofluids. Numerical results show that then the thermal conductivity ratio between the nanoparticles and fluids is large, and the two nanoparticles are close up and even touch, and the point-dipole theory such as Maxwell--Garnett theory becomes rough as many-body interactions are neglected. Our theoretical results on the effective thermal conductivity of CuO/water and Al$_{2}$O$_{3}$/water nanofluids are in good agreement with experimental data.     

Thermal radiation effects on MHD flow of a micropolar fluid over a stretching surface with variable thermal conductivity

In this paper, the effects of variable thermal conductivity and radiation on the flow and heat transfer of an electrically conducting micropolar fluid over a continuously stretching surface with varying temperature in the presence of a magnetic field are considered. The surface temperature is assumed to vary as a power-law temperature. The governing conservation equations of mass, momentum, angular momentum and energy are converted into a system of non-linear ordinary differential equations by means of similarity transformation. The resulting system of coupled non-linear ordinary differential equations is solved numerically. The numerical results show that the thermal boundary thickness increases as the thermal conductivity parameter S$S$ increases, while it decreases as the radiation parameter F$F$ increases. Also, it was found that the Nusselt number increases as F$F$ increases and decreases as S$S$ increases.     

Analytic solution for magnetohydrodynamic boundary layer flow of Casson fluid over a stretching/shrinking sheet with wall mass transfer

In this analysis,the magnetohydrodynamic boundary layer flow of Casson fluid over a permeable stretching/shrinking sheet in the presence of wall mass transfer is studied.Using similarity transformations,the governing equations are converted to an ordinary differential equation and then solved analytically.The introduction of a magnetic field changes the behavior of the entire flow dynamics in the shrinking sheet case and also has a major impact in the stretching sheet case.The similarity solution is always unique in the stretching case,and in the shrinking case the solution shows dual nature for certain values of the parameters.For stronger magnetic field,the similarity solution for the shrinking sheet case becomes unique.     

Bounds on the effective thermal conductivity of two-phase systems

The weighted geometric mean of resistors considered for determining the effective thermal conductivityK _E of two-phase systems has been optimised. Solutions of the equations lead to a useful set of bounds. When compared with other bounds the present bounds give the better results in estimating the upper and lower values of the effective thermal conductivity of a two-phase system.     

Magnetohydrodynamic and radiation effects on the heat transfer of a continuously stretching/shrinking sheet with mass transpiration of the horizontal boundary

The aim of this paper is the investigation of heat transfer regarding the cases of both stretching and shrinking sheets with a sponge-like horizontal wall that allows for mass transpiration. The effects of Prandtl number, radiation and external magnetic field are extensively examined. The Navier-Stokes equations are reduced to partial differential equations, which are eventually become ordinary differential equations and solved analytically. Furthermore, the power-law wall temperature and heat flux boundary conditions are imposed on the boundary layer energy equation for obtaining exact analytical solutions. The results revealed that in both the stretching and shrinking sheet scenarios the thickness of the thermal boundary layer decreases with either increasing of transpiration as well as the Chandrasekhar and Prandtl number numbers or decreasing radiation number. Additionally, the characteristics of the heat transfer regarding a shrinking sheet and those of a stretching sheet are found not to be similar. In fact, a new solution branch appeared which indicates that multiple solutions may emerge under certain circumstances. Finally, by using the present analytical relationships, theoretical guidelines can be given for regulating the procedure.     

Utilization of the magnetogranulometric analysis to estimate the thermal conductivity of magnetic fluids

In this study, the semi-empirical equation for the effective thermal conductivity of the Holotescu-Stoian model was applied to a set of four dilutions of a transformer oil based magnetic fluid with magnetite nanoparticles as magnetic phase, using the results obtained for the size distributions from the magnetogranulometry analysis, followed by a comparison with the measured values of the effective thermal conductivity obtained by the hot ball method. The link between the size distribution by number and by volume used in the magnetogranulometry analysis and the Holotescu-Stoian model adaptation to the lognormal distribution were presented. The comparison between the results given by the model and the corresponding experimental data showed that by using the approximated size distribution to calculate the effective thermal conductivity the analytical results much closer to the experimental ones are obtained, compared to those given by the Maxwell classical model.     

Variable viscosity and thermal conductivity effects on MHD flow and heat transfer in viscoelastic fluid over a stretching sheet

The problem of flow and heat transfer of an electrically conducting viscoelastic fluid over a continuously stretching sheet in the presence of a uniform magnetic field is analyzed for the case of power-law variation in the sheet temperature. The fluid viscosity and thermal conductivity are assumed to vary as a function of temperature. The basic equations comprising the balance laws of mass, linear momentum, and energy modified to include the electromagnetic force effect, the viscous dissipation, internal heat generation or absorption and work due to deformation are solved numerically.     

Effect of thermal fluctuations on magnetic phase separation and on the parameters of spin-spiral waves

Effects of temperature on magnetic phase separation and on the parameters of spin-spiral waves are studied. The study is performed using the two-dimensional single-band Hubbard model and the Hubbard-Stratonovich transformation. Both commensurate (antiferromagnetic, ferromagnetic) and incommensurate (spiral) magnetic phases are considered. The problem is solved using the static approximation with allowance for transverse fluctuations of the magnetic moment. It is shown that the temperature significantly affects the collinear and spiral magnetic phases. With an increase in the temperature, the phase-separation region near the half-filling is sufficiently reduced and substituted by the antiferromagnetic phase. The results are used for the interpretation of the magnetic properties of cuprates.     

Stability analysis of unsteady MHD stagnation point flow and heat transfer over a shrinking sheet in the presence of viscous dissipation

The unsteady MHD stagnation-point flow and heat transfer over a shrinking sheet was carried out. This study also was conducted in the existence of suction and viscous dissipation. In order to convert the governing partial differential equations to an ordinary differential equation, an appropriate similarity transformation was applied in this study. Then, the resulting equations are worked out by Bvp4c solver in Matlab. The impacts of the parameters involved in this study towards skin friction, Nusselt number, velocity and temperature profile are showed graphically and thoroughly discussed. Remarkably, there were dual solutions present in this study which made us continue deeper in performing the stability analysis. As expected, our study proves that the solution is stable only in the first one while not in the second solution.     

Unsteady boundary layer flow of a nanofluid over a stretching sheet with variable fluid properties in the presence of thermal radiation

In this paper, we investigated numerically an unsteady boundary layer flow of a nanofluid over a stretching sheet in the presence of thermal radiation with variable fluid properties. Using a set of suitable similarity transformations, the governing partial differential equations are reduced into a set of nonlinear ordinary differential equations. System of the nonlinear ordinary differential equations are then solved by the Keller-box method. The physical parameters taken into consideration for the present study are: Prandtl number Pr, Lewis number Le, Brownian motion parameter N _b, thermophoresis parameter N _t, radiation parameter N _r, unsteady parameter M. In addition to these parameters, two more new parameters namely variable thermophoretic diffusion coefficient parameter e and variable Brownian motion diffusion coefficient parameter β have been introduced in the present study. Effects of these parameters on temperature, volume fraction of the nanoparticles, surface heat and mass transfer rates are presented graphically and discussed briefly. To validate our method, we have compared the present results with some previously reported results in the literature. The results are found to be in a very good agreement.     

Effect of training level and blood flow restriction on thermal parameters: Preliminary study

Training with blood flow restriction could lead to an effect on skin temperature. Additionally, this effect could be higher in people with lower physical fitness level due to their lower capacity of heat loss. The aim of this preliminary study was therefore to evaluate the effects of training experience on the acute and chronic thermal skin responses after performing exercise with and without blood flow restriction. The study included ten men, of these, five were trained. All subjects performed tests and re-tests for maximum strength (1 repetition maximum) through unilateral leg extensions (right thigh at 45 ± 6.7 kg and left thigh at 45.5 ± 8.1 kg, p > 0.05). The protocol consisted of four sets to concentric failure, with one-minute rest intervals between sets at an intensity corresponding to 40% of 1 RM. There were 7-day intervals between experimental sessions (150 mmHg versus unrestricted flow restriction). The thermal images were made before the protocol (pre), immediately after the end of the series (post), and 24 h afterward (post 24 h). When comparing temperature variation (Δ exercise and Δ 24 h) between groups, it was observed that the trained participants showed a greater drop in temperature 24 h after exercise with 150 mmHg restriction (confidence interval: 95% of Δ 24 h [−0.2 to −0.9 °C]) compared to untrained subjects (p = 0.006 and ES > 1.5, confidence interval: 95% Δ 24 h [−0.1 to 0.6 °C].) In conclusion, this preliminary study showed that training experience interferes with the chronic cutaneous thermal temperature of the anterior thigh when strength training associated with blood flow restriction 150 mmHg was performed.     

Effect of dispersion on the phonon focusing and anisotropy of thermal conductivity of silicon single crystals in the boundary scattering regime

The effect of dispersion on the focusing of thermal phonons and on the thermal conductivity of silicon single crystals in the boundary scattering regime has been investigated. Analysis of the spectra of acoustic modes obtained for silicon single crystals from inelastic neutron scattering data has demonstrated that, upon transition from long-wavelength phonons to short-wavelength phonons, the directions of their focusing change. With an increase in temperature, this leads to a change in the anisotropy of thermal conductivity of phonons with different polarizations and, consequently, to a change in the anisotropy of the total thermal conductivity. Analysis of the temperature dependence of the thermal conductivity has revealed that the presence of extended flattened sections in the spectrum of short-wavelength transverse phonons indicates anomalously low values of the group velocity and, accordingly, a significant decrease in the contribution from these phonons to the thermal conductivity with increasing temperature. The contribution from longitudinal phonons to the thermal conductivity also significantly increases even at temperatures higher than 110 K and becomes dominant.     

MHD flow and mass transfer of a upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species

This article looks at the mass transfer of the steady two-dimensional magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid past a porous shrinking sheet in the presence of chemical reaction. The resulting nonlinear partial differential equations are reduced to the system of nonlinear ordinary differential equations by means of similarity transformations. Expressions of velocity and the concentration fields are obtained using the homotopy analysis method (HAM). The convergence of the obtained series solutions is explicitly discussed. The influences of sundry parameters on the velocity and the concentration fields are made and discussed in detail. The values of the skin friction coefficient and the surface mass transfer for various interesting parameters are also tabulated.     

MHD boundary layer flow of Casson fluid passing through an exponentially stretching permeable surface with thermal radiation

This article numerically examines the boundary layer flow due to an exponentially stretching surface in the presence of an applied magnetic field. Casson fluid model is used to characterize the non-Newtonian fluid behavior. The flow is subjected to suction/blowing at the surface. Analysis is carded out in presence of thermal radiation and prescribed surface heat flux. In this study, an exponential order stretching velocity and prescribed exponential order surface heat flux are accorded with each other. The governing partial differential equations are first converted into nonlinear ordinary differential equations by using appropriate transformations and then solved numerically. The effect of increasing values of the Casson parameter is to suppress the velocity field. However the temperature is enhanced when Casson parameter increases. It is found that the skin-friction coefficient increases with increasing values of suction parameter. Temperature also increases for large values of power index n in both suction and blowing cases at the boundary. It is observed that the thermal radiation enhances the effective thermal diffusivity and hence the temperature rises.     

Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet

This Letter endeavours to complete an earlier numerical analysis for flow and heat transfer in a viscous fluid over a sheet nonlinearly stretched by extending the investigation in two directions. On one side, the effects of thermal radiation are included in the energy equation, and, on the other hand, the prescribed wall heat flux case (PHF case) is also analyzed. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation. The variations of dimensionless surface temperature as well as flow and heat-transfer characteristics with the governing dimensionless parameters of the problem, which include a nonlinearly stretching sheet, thermal radiation, viscous dissipation and power-law index of the wall temperature parameters, are graphed and tabulated.     

Influence of structural characteristics on the thermal conductivity of polycrystalline diamond films

By analyzing the signal formed by the photoacoustic effect as a function of the light modulation frequency, it is shown that this effect may be used to determine the thermal conductivity of diamond materials. The method is checked experimentally for two types of polycrystalline diamond films grown by chemical vapor deposition with the gaseous medium activated by a dc discharge and a microwave discharge. The data obtained on the thermal conductivity of the films are discussed with reference to the results of an investigation of the optical absorption, Raman light scattering, and cathodoluminescence of similar films. It is shown that the thermal conductivity of polycrystalline diamond films depends on the structural characteristics, which are determined by the deposition conditions. Fiz. Tverd. Tela (St. Petersburg) 40, 1221–1225 (July 1998)