Effect of viscous dissipation and pressure stress work in natural convection along a vertical isothermal plate. New results

The steady laminar boundary layer flow along a vertical stationary isothermal plate is studied taking into account the viscous dissipation and pressure stress work of the fluid. The results are obtained with the direct numerical solution of the boundary layer equations without any approximation. It was found that the variation of wall heat transfer and wall shear stress along the plate is quite different compared to that given by the approximate perturbation method.     

On the effects of viscous dissipation and pressure work in free convection loops

The effects of viscous dissipation and pressure work are examined theoretically for laminar free convection loops. The appropriate governing equations are derived. Whereas previous work has considered only dissipation effects, the present paper shows that dissipation and pressure work effects are of comparable magnitude and must be considered together. Analytical solutions are presented for several open and closed loops. Both constant flux and constant temperature heating conditions are examined. Viscous dissipation and pressure work effects are found to have opposing influences on the flow in a loop. The former can enhance a flow for certain heating orientations, but the latter is usually dominant and retards a flow.     

Mixed convection with viscous dissipation and pressure work in a lid-driven square enclosure

Buoyant laminar flow in a square lid-driven enclosure is analysed. The vertical sides are kept isothermal at different temperatures, while the horizontal sides are insulated. Assisting mixed convection flow due to uniform motion of the top side is considered. The governing balance equations are solved numerically by employing a Galerkin finite element method. The effects of viscous dissipation and pressure work are taken into account. In order to investigate the influence of these effects, the Nusselt number is evaluated with respect to the heat fluxes at both vertical sides, for different values of the Rayleigh number and of the Péclet number based on the lid velocity. Two sample fluids are considered: a gas and a highly viscous liquid. In the framework of the Oberbeck–Boussinesq approximation, a comparison is made between three different energy balance models: (A) enthalpy formulation (pressure work and viscous dissipation are included); (B) internal-energy formulation (viscous dissipation is included); (C) both pressure work and viscous dissipation are neglected. It is shown that, in the absence of a lid motion, the three models yield substantially the same predictions. On the other hand, when the forced flow induced by the lid motion becomes sufficiently large, the three models yield discrepant results, thus implying that pressure work and viscous dissipation are not negligible. Moreover, it is shown that, in this case, model (A) yields unphysical results, while model (B) leads to reasonable predictions.     

Thermodynamics of natural convection in enclosures with viscous dissipation

Consideration of the viscous dissipation effects in natural and mixed convection heat transfer must be taken carefully, both in what concerns the thermodynamics of the problem and the relevance of the dissipation term. This applies equally to external or internal natural and mixed convection, and to spaces filled with a single fluid or to spaces filled with fluid-saturated porous media. The main question is related to the fact that, in natural convection, the work done by the pressure forces must equal the energy dissipated by viscous effects, which is the unique situation compatible with the First Law of Thermodynamics, the net energy generation in the overall domain being zero. If only the (positive) viscous dissipation term is considered in the energy conservation equation, the domain behaves like a heat multiplier, the heat output being higher than the heat input. If this is not taken into consideration, erroneous conclusions about flow and temperature fields and heat transfer results are obtained. In mixed convection problems, part of the viscous dissipation term is equally due to the work of pressure forces. Attention is given mainly to the natural convection problem in a square enclosure, the main conclusions applying for general natural or mixed convection heat transfer problems.     

Effect of viscous dissipation and radiation on natural convection in a porous medium embedded within vertical annulus

The effect of viscous dissipation and thermal radiation on natural convection in a porous medium embedded within a vertical annular cylinder is investigated. The inner surface of the cylinder is maintained at an isothermal temperature $T_w$ and the outer surface is maintained at ambient temperature $T_{\infty }$. The fluid is assumed to obey the Darcy law. Finite element method is used to solve the partial differential equations governing the fluid flow and heat transfer behavior. The study is focused to investigate the combined effect of viscous dissipation and radiation. Results are presented for different values of the viscous dissipation parameter, radiation parameter, radius ratio, aspect ratio and Rayleigh number. It is observed that the viscous dissipation parameter reduces the average Nusselt number at hot surface. However, the average Nusselt number increases at the cold surface due to increased viscous dissipation parameter.     

Convective heat transfer in pressure-driven nitrogen slip flows in long microchannels: The effects of pressure work and viscous dissipation

An analytical and numerical study is carried out to examine the convective heat transfer in two-dimensional pressure-driven nitrogen slip flows in long microchannels, whose length-to-height ratios are above 500. The momentum and the energy equations are solved, where variable properties, rarefaction that involves velocity slip, thermal creep and temperature jump, pressure work, and viscous dissipation are all taken into account. Nitrogen is assumed to be a perfect gas. The effects of pressure work and viscous dissipation, which are particularly significant for long microchannels, are examined by analyzing the uniform wall temperature and the uniform wall heat flux cases. It is found that the degree of rarefaction, which is characterized by the Knudsen number, is the key factor that determines the relative importance of pressure work and viscous dissipation. It is demonstrated that, for perfect gases, rarefaction promotes the conversion of internal energy to mechanical energy. Specifically, regardless of the fluid field development, pressure work and viscous dissipation cancel out in the absence of rarefaction, while pressure work is greater than viscous dissipation with rarefaction and its dominance increases as the Knudsen number increases. It is shown that the combination of pressure work and viscous dissipation makes a significant impact on the Nusselt number in both the continuum and the rarefaction cases. Therefore, it is concluded that for convective heat transfer in internal gas flows, both pressure work and viscous dissipation need to be considered in analysis.     

On natural convection in enclosures filled with fluid-saturated porous media including viscous dissipation

Care needs to be taken when considering the viscous dissipation in the energy conservation formulation of the natural convection problem in fluid-saturated porous media. The unique energy formulation compatible with the First Law of Thermodynamics informs us that if the viscous dissipation term is taken into account, also the work of pressure forces term needs to be taken into account. In integral terms, the work of pressure forces must equal the energy dissipated by viscous effects, and the net energy generation in the overall domain must be zero. If only the (positive) viscous dissipation term is considered in the energy conservation equation, the domain behaves as a heat multiplier, with an heat output greater than the heat input. Only the energy formulation consistent with the First Law of Thermodynamics leads to the correct flow and temperature fields, as well as of the heat transfer parameters characterizing the involved porous device. Attention is given to the natural convection problem in a square enclosure filled with a fluid-saturated porous medium, using the Darcy Law to describe the fluid flow, but the main ideas and conclusions apply equally for any general natural or mixed convection heat transfer problem. It is also analyzed the validity of the Oberbeck–Boussinesq approximation when applied to natural convection problems in fluid-saturated porous media.     

Effect of pressure work and viscous dissipation in the analysis of the Rayleigh–Bénard problem

The classical Rayleigh–Bénard problem in an infinitely wide horizontal fluid layer with isothermal boundaries heated from below is revisited. The effects of pressure work and viscous dissipation are taken into account in the energy balance. A linear analysis is performed in order to obtain the conditions of marginal stability and the critical values of the wave number a and of the Rayleigh number Ra for the onset of convective rolls. Mechanical boundary conditions are considered such that the boundaries are both rigid, or both stress-free, or the upper stress-free and the lower rigid. It is shown that the critical value of Ra may be significantly affected by the contribution of pressure work, mainly through the functional dependence on the Gebhart number and on a thermodynamic Rayleigh number. While the pressure work term affects the critical conditions determined through the linear analysis, the viscous dissipation term plays no role in this analysis being a higher order effect. A nonlinear analysis is performed showing that the superadiabatic Rayleigh number replaces Ra in the functional dependence of the excess Nusselt number. Finally, a reasoning is proposed to show how the results obtained may be used as a test on the most appropriate formulation of the Oberbeck–Boussinesq model.     

A correlation for heat transfer by natural convection from horizontal cylinders that accounts for viscous dissipation

Eight previously published correlation equations plus one new correlation for heat transfer by natural convection from horizontal isothermal cylinders are tested against a fairly extensive body of experimental data culled from the literature for 10⁻⁸ < Ra < 10⁸ and 0.7 < Pr < 4 × 10⁴. The new equation, which represents the Nusselt number as a function of the Prandtl and Rayleigh numbers plus an additional dimensionless parameter that accounts for viscous dissipation, is shown to correlate the experimental data more accurately than does any one of the eight previously published equations. It is concluded that viscous dissipation may not be neglected in all cases of natural convection from horizontal cylinders, and further, that the inclusion of a viscous dissipation term in certain related problems, such as natural convection in porous media, may lead to more accurate correlation equations.     

Laminar convection in a vertical channel with viscous dissipation and buoyancy effects

The fully developed and laminar convection in a parallel-plate vertical channel is investigated by taking into account both viscous dissipation and buoyancy. Uniform and symmetric temperatures are prescribed at the channel walls. The velocity field is considered as parallel. A perturbation method is employed to solve the momentum balance equation and the energy balance equation. A comparison with the velocity and temperature profiles in the case of laminar forced convection with viscous dissipation is performed in order to point out the effect of buoyancy. The case of convective boundary conditions is also discussed.     

First law analysis for viscous dissipation in liquid flows in micro-channels

This paper focuses on temperature rise due to the viscous dissipation in liquids flowing through micro-channels. In the past, equations for the prediction of the temperature rise have been obtained as a function of the friction factor, Reynolds number and Eckert number or a similar form, starting from Navier–Stokes equation and energy equation under the assumption of fully developed laminar flow by researchers. The temperature rises calculated from the equations have been compared with experimental data and the equations have been validated. However, in this paper, a new equation for the prediction of the temperature rise is simply obtained from the first law of thermodynamics without restriction of fully developed laminar flow.     

Effects of viscous dissipation during forced convection of power-law fluids in microchannels

The effect of viscous dissipation in forced convection of power-law fluids through microchannels of different cross-sectional geometries is studied numerically over the ranges of power-law index, 0.8 ≤ n ≤ 1.2 and Brinkman number, 0.001 ≤ Br ≤ 0.1 while keeping Péclet number constant at Pe = 10. Two types of thermal boundary conditions, namely, uniform wall temperature (T₂) and uniform heat flux (H₂), have been employed at the microchannel wall and the results of the temperature fields are expressed in terms of Nusselt number. The interplay between the fluid rheology and viscous dissipation effect gives rise to significant alteration in the net convective transport and thus can be beneficial in the thermal design of biofluidic devices.     

Effects of viscous dissipation on fully developed forced convection in porous media

An analytical study on fully developed forced convection in a homogeneous porous medium is presented. Closed form solutions for the temperature distributions in the transverse direction with the inclusion of frictional heating due to viscous dissipation are obtained, with variations of Darcy number and Brinkman number. The frictional heating effect on the temperature distributions is investigated and analyzed for both heating and cooling processes. Variations of Nusselt number as a function of Darcy number and Brinkman number are examined, and the deviations of the Nusselt numbers with that of the model without incorporating the effect of frictional heating are presented in a contour deviation map for a more holistic comparison.     

Effect of viscous dissipation on finding dual solutions for mixed convection boundary layer flow for nanofluid

The effect of viscous dissipation on mixed convection boundary layer flow for Ag‐water nanofluid under steady‐state condition has been studied numerically for both the buoyancy assisting and opposing flows over a vertical semi‐infinite flat plate. A new co‐ordinate system has been introduced to transform the governing partial differential equations (PDEs) to facilitate the numerical calculations. Then, the local similarity method has been used for approximating the transformed PDEs to ordinary differential equations. Further, the quasi‐linearization method has been introduced to linearize the nonlinear equations and then numerical integration has been carried out using implicit trapezoidal rule along with the principle of superposition. For higher Pr, the coupled differential equations behave like stiff differential equations. To overcome the situation, orthonormalization process has been introduced. The effects of solid volume fraction of nanoparticles , the mixed convection parameter , Prandtl number , and Eckart number have been analyzed on the heat transfer and flow characteristics. It has been observed that the dual solutions are obtained for buoyancy opposing flow only and the range of dual solutions have become wider with the increases in . Further, nanofluids enhance the heat transfer process as compared to conventional heat transfer fluids . Moreover, the addition of viscous dissipation causes less heat transfer in the boundary.     

Forced convection in porous microchannels with viscous dissipation in local thermal non-equilibrium conditions

Fully developed, steady-state forced convection, in parallel-plate microchannels, filled with a porous medium saturated with rarefied gases at high temperatures, in local thermal non-equilibrium (LTNE) condition, is investigated for the first-order slip-flow regime (0 ≤ Kn ≤ 0.1). Both velocity and temperature jumps at the walls are accounted for. An analytic solution is proposed for the Darcy–extended Brinkman flow model with assigned uniform heat flux at the microchannel walls and viscous heat dissipation in the fluid phase. The solution for NTLE includes the shear work done by the slipping effects. A closed-form expression of the Nusselt number is derived. A validation analysis with respect to the case of channels filled with saturated porous medium is accomplished. The results show that the internal dissipation increases as the velocity slip increases. In addition, the heat dissipation strongly affects the fluid temperature profiles. The increases in velocity slip and temperature jump lead to decreases of temperature gradients in the fluid and solid along the sections. The heat transfer at channel walls is enhanced due to an increase in the bulk heat transfer.     

A thermal resistance analysis on forced convection with viscous dissipation in a porous medium using entransy dissipation concept

The analytical solution of a two-equation model presented in an earlier study is examined. Heat transfer characterization is classified into two regimes which are dominated by fluid conduction or solid conduction and interstitial heat exchange, respectively by using the entransy dissipation concept. The computed pattern of variation of thermal resistance with shape factor S at a fixed Brinkman number for a low ratio of the fluid to solid effective thermal conductivities implies the occurrence of temperature gradient bifurcation as S decreases. Therefore, the thermal diffusion term in the fluid phase in the two-equation model is not negligible for both regimes.     

Routes to chaos of natural convection flows in vertical channels

The aim of the present study is the analysis of the transition to turbulence of natural convection flows between two infinite vertical plates. For the study of the problem, a number of Direct Numerical Simulations (DNSs) have been performed. The continuity, momentum and energy equations, cast under the Boussinesq assumption, are tackled numerically by means of a pseudospectral method, through which the three-dimensional domain is decomposed with Chebychev polynomials in the wall-normal direction and with Fourier modes in the wall-parallel directions. For low Rayleigh number values, the predictions of the flow regimes are consistent with the classical analytical results and linear stability analyses. In particular, the first bifurcation (Ra ≈ 5800) from the so-called laminar conduction regime to steady convection is correctly captured. By increasing the Rayleigh number beyond a second critical value (Ra ≈ 10200), the flow regime becomes chaotic. This transition to chaos is found to be related with the amplification of spanwise instabilities occurring at scales larger than the channel gap, H. The study of the return of the system from the chaotic regime to the laminar base flow reveals a phenomenon of hysteresis, i.e. the chaotic regime persists even at Ra-values lower than the second critical value. From a numerical point of view, the predicted flow regimes appear to be extremely sensitive to the domain size, grid resolution and perturbation amplitude. These aspects are shown to be of crucial importance for the prediction of the heat transfer performance, and, hence, should be taken into consideration when numerical methods are used for the simulation of real-world problems.     

Limiting Nusselt numbers for laminar forced convection in asymmetrically heated annuli with viscous dissipation

Analytical expressions for the Nusselt number at the inner and outer pipes, kept at unequal temperatures, for laminar forced convection through annuli including viscous dissipation have been obtained in the conduction limit. This article examines the dependence of the limiting Nusselt numbers on the Brinkman number and the degree of asymmetry in inner and outer pipe temperatures. Further, the limiting temperature profile obtained when viscous dissipation is included serves the purpose of providing the downstream boundary condition needed in solving the elliptic form of conservation of thermal energy equation that arises when axial conduction is included.     

Forced convection heat transfer inside an anisotropic porous channel with oblique principal axes: Effect of viscous dissipation

An analytical study on laminar and fully developed forced convection heat transfer in a parallel-plate horizontal channel filled with an anisotropic permeability porous medium is performed. The principal axis of the anisotropic porous medium is oriented from 0 to 90 degrees. A constant heat flux is applied on the outer wall of the channel. Both clear (Newtonian) fluid and Darcy viscous dissipations are considered in the energy equation. Directional permeability ratio parameter A¹ is defined to combine both the effect of the dimensionless permeability ratio parameter K¹=(K₁/K₂) and orientation angle φ into one parameter. The effects of the parameter A¹, the Darcy number Da and the modified Brinkman number Br¹ on the heat transfer and fluid flow characteristics in the channels are investigated and presented in graphs. The obtained results show that the parameters A¹, Da and Br¹ have strong effects on the dimensionless normalized velocity and temperature profiles as well as on the Nusselt number. It is found that for a particular value of A¹, called as critical value A_cr¹, the external heat applied to the surface of the channel is balanced by the internal heat generation due to viscous dissipation and the bulk mean temperature approaches the wall temperature. Hence, the Nusselt number approaches infinity for the critical values A_cr¹.     

Effects of pressure work and radiation on natural convection flow around a sphere with heat generation

The effects of pressure work and radiation on natural convection flow around a sphere in presence of heat generation have been investigated in this paper. The governing equations are transformed into dimensionless non-similar equations by using set of suitable transformations and solved numerically by the finite difference method along with Newton's linearization approximation. Attention has been focused on the evaluation of shear stress in terms of local skin friction and rate of heat transfer in terms of local Nusselt number, velocity as well as temperature profiles. Numerical results have been shown graphically and also in tabular form for some selected values of parameter set consisting of heat generation parameter Q, radiation parameter Rd, pressure work parameter Ge and the Prandtl number Pr.