Temperature Jump Boundary Conditions Research Articles

Accurate numerical method for solving dual-phase-lagging equation with temperature jump boundary condition in nano heat conduction

Dual-phase-lagging (DPL) equation with temperature jump boundary condition shows promising for analyzing nano heat conduction. For solving it, development of higher-order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nano-scale, using a higher-order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, in this article we present a higher-order accurate and unconditionally stable compact finite difference scheme based on the ratio of relaxation times (0⩽B⩽1 and B>1). The method is illustrated by three numerical examples including a 2D case.

Effect of Finite Diaphragm Rupture Process on Microshock Tube Flows

The study of flow physics in microshock tubes is of growing importance with the recent development of microscale technology. The flow characteristics in a microshock tube is considerably different from that of the conventional macroshock tube due to the boundary layer effects and high Knudsen number effects. In the present study an axisymmetric computational fluid dynamics (CFD) method was employed to simulate the microshock tube flow field with Maxwell's slip velocity and temperature jump boundary conditions, to accommodate the rarefaction effects. The effects of finite diaphragm rupture process and partial diaphragm rupture on the flow field and the wave propagations were investigated, in detail. The results show that the shock propagation distance attenuates rapidly for a microshock tube compared to a macroshock tube. For microshock tubes, the contact surface comes closer to the shock front compared to the analytical macroshock tube case. Due to the finite diaphragm rupture process the moving shock front will be generated after a certain distance ahead of the diaphragm and get attenuated rapidly as it propagates compared to the sudden rupture case. The shock-contact distance reduces considerably for the finite diaphragm rupture case compared to the sudden diaphragm rupture process. A partially burst diaphragm within a microshock tube initiates a supersonic flow in the vicinity of the diaphragm similar to that of a supersonic nozzle flow. The supersonic flow expansion leads to the formation of oblique shock cells ahead of the diaphragm and significantly attenuates the moving shock propagation speed.

Micro shock tube 유동에 대한 유한 격막 파막과정의 영향에 관한 수치 해석적 연구

Recent years have witnessed the use of micro shock tube in various engineering applications like micro combustion, micro propulsion, particle delivery systems etc. The flow characteristics occurring in the micro shock tube shows a considerable deviation from that of well established conventional macro shock tube due to very low Reynolds number and high Knudsen number effects. Also the diaphragm rupture process, which is considered to be instantaneous process in many of the conventional shock tubes, will be crucial for micro shock tubes in determining the near diaphragm flow field and shock formation. In the present study, an axi-symmetric CFD method has been applied to simulate the micro shock tube, with Maxwell’s slip velocity and temperature jump boundary conditions. The effects of finite diaphragm rupture process on the flow field and the shock formation was investigated, in detail. The results show that the shock strength attenuates rapidly as it propagates through micro shock tubes.

Hydrodynamic and Thermal Characteristics of Laminar Slip Flow Over a Horizontal Isothermal Flat Plate

In this paper, hydrodynamic and thermal characteristics of laminar incompressible slip flow over an isothermal semi-infinite flat plate at a relatively low Mach number are considered and revised. The nonsimilar and local similarity solutions of the boundary layer equations with velocity-slip and temperature-jump boundary conditions are obtained numerically for the gaseous slip flow. The numerical calculations are made by assuming no thermal jump for the liquid flow. In addition, the approximate analytical solution of the boundary layer equations for high slip parameter is presented. Results from nonsimilar solution, local similarity approach, and approximate analytical solution are compared. We show that the local similarity approach used by several authors in the last decades produces substantial errors in hydrodynamic and thermal characteristics of the flow. Furthermore, accurate correlations of these characteristics are proposed for gaseous and liquid slip flows. The results of nonsimilar solution show, unlike the previous studies, that the overall skin friction coefficient presents a very slight decrease (indistinguishable) in the interval of the slip flow regime, whereas it decreases significantly as the flow becomes more rarefied. Moreover, with increasing slip condition, the results of overall Nusselt number, for gaseous flow, show that the heat transfer at the plate decreases slightly in the interval of slip flow regime while it increases in the case of liquids flow. This study confirms that for the accurate prediction of characteristics of slip flow, the slip parameter must be treated as a variable rather than a constant in the boundary layer.

Rarefaction and Darcy effects on the hydromagnetic flow of radiating and reacting fluid in a vertical channel

An investigation of the combined influence of rarefaction and Darcy effects on the flow of oscillatory MHD viscous, incompressible, chemically reacting, and electrically conducting fluid in a vertical channel is carried out. The fluid is assumed to be gray and absorbing and emitting radiation but to be a nonscattering medium. The flow is assumed to be laminar and hydrodynamically and thermally fully developed. The Navier--Stokes, energy, and concentration equations, which are accompanied by appropriate velocity-slip and temperature-jump boundary conditions, are analytically solved. It is found that rarefaction effects significantly influence the flow and thermal fields such that mass flow and heat transfer rates are increased as compared to the continuum regime.

A robust numerical method for the R13 equations of rarefied gas dynamics: Application to lid driven cavity

In this work we present a finite difference scheme to compute steady state solutions of the regularized 13 moment (R13) equations of rarefied gas dynamics. The scheme allows fast solutions for 2D and 3D boundary value problems (BVPs) with velocity slip and temperature jump boundary conditions. The scheme is applied to the lid driven cavity problem for Knudsen numbers up to 0.7. The results compare well with those obtained from more costly solvers for rarefied gas dynamics, such as the Integro Moment Method (IMM) and the Direct Simulation Monte Carlo (DSMC) method. The R13 equations yield better results than the classical Navier–Stokes–Fourier equations for this boundary value problem, since they give an approximate description of Knudsen boundary layers at moderate Knudsen numbers. The R13 based numerical solutions are computationally economical and may be considered as a reliable alternative mathematical model for complex industrial problems at moderate Knudsen numbers.

The effect of geometrical and flow parameters on mixed convection of developing flow in vertical and inclined microchannels

In this study, mixed convection in vertical and inclined parallel plate microchannels, has been investigated. The flow is steady, laminar, and incompressible and the walls are at constant temperature. A FORTRAN program has been written to solve the governing equations with wall velocity slip and temperature jump boundary conditions, and the results have been presented for Reynolds numbers of 25, 50, and 100, Knudsen numbers of 0, 0.005, 0.02, and 0.09, Richardson numbers of 0, 1, and 1.77, and microchannel inclination angles of 10°, 30°, 50°, and 90°. Results showed that wall velocity slip, friction coefficient, centerline velocity, and heat transfer rate decreases with increasing the Knudsen number. Increasing the Richardson number, for a given Reynolds number, decreases the microchannel centerline velocity because of the increasing of wall adjacent velocity, which is due to the promotion effect of natural convection. Also, the friction coefficient and Nusselt number increases with channel inclination angle.

EFFECTS OF THERMAL CREEP ON COOLING OF MICROFLOWS IN SHORT MICROCHANNELS WITH CONSTANT WALL TEMPERATURE

Fluid flow and heat transfer in the entrance region of rectangular microchannels of various aspect ratios are numerically investigated in the slip-flow regime with particular attention to thermal creep effects. Uniform inlet velocity and temperature profiles are prescribed in microchannels with constant wall temperature. An adiabatic section is also employed at the inlet of the channel in order to prevent unrealistically large axial temperature gradients due to the prescribed uniform inlet temperature as well as upstream diffusion associated with low Reynolds number flows. A control-volume technique is used to solve the Navier–Stokes and energy equations which are accompanied with appropriate velocity slip and temperature jump boundary conditions at the walls. Despite the constant wall temperature, axial and peripheral temperature gradients form in the gas layer adjacent to the wall due to temperature jump. The simultaneous effects of velocity slip, temperature jump and thermal creep on the flow and thermal patterns along with the key flow parameters are examined in detail for a wide range of cross-sectional aspect ratios, and Knudsen and Reynolds numbers. Present results indicate that thermal creep effects influence the flow field and the temperature distribution significantly in the early section of the channel.

Investigation of dual-phase-lag heat conduction model in a nanoscale metal-oxide-semiconductor field-effect transistor

This paper investigates the numerical simulation of non-Fourier transient heat transfer in a two-dimensional sub-100nm metal-oxide-semiconductor field-effect transistor (MOSFET). The dual-phase-lag (DPL) model with a specific normalization procedure is introduced for the modeling of nanoscale heat transport. The boundary conditions are selected similar to what existed in a real MOSFET device, both uniform and non-uniform heat generations within the transistor are applied, and the end parts of the top boundary which are in contact with the metallic material are left open. A temperature-jump boundary condition is used on all boundaries in order to consider the boundary phonon scattering at micro and nanoscale. A three-level finite difference scheme has been employed to generate the numerical results which are illustrated for a silicon sub-100nm MOSFET corresponding to the Knudsen number of 10. The results are presented at real times less than 50ps to avoid the three-dimensional effects. It is concluded that the combination of the DPL model with mixed-type temperature boundary condition is able to predict the heat flux and temperature distribution obtained from the Boltzmann transport equation (BTE) more accurate than the ballistic-diffusive equations (BDE). After verification of our results, the thermal field and the hotspot temperature within the bulk silicon are presented and the effect of adding a layer of silicon-dioxide to the transistor on its thermal behavior has been investigated.

Numerical analysis of thermal creep flow in curved channels for designing a prototype of Knudsen micropump

The possibility to generate a gas flow inside a channel just by imposing a tangential temperature gradient along the walls without the existence of an initial pressure difference is well known. The gas must be under rarefied conditions, meaning that the system must operate between the slip and the free molecular flow regimes, either at low pressure or/and at micro/nano-scale dimensions. This phenomenon is at the basis of the operation principle of Knudsen pumps, which are actually compressors without any moving parts. Nowadays, gas flows in the slip flow regime through microchannels can be modeled using commercial Computational Fluid Dynamics softwares, because in this regime the compressible Navier-Stokes equations with appropriate boundary conditions are still valid. A simulation procedure has been developed for the modeling of thermal creep flow using ANSYS Fluent®. The implementation of the boundary conditions is achieved by developing User Defined Functions (UDFs) by means of C++ routines. The complete first order velocity slip boundary condition, including the thermal creep effects due to the axial temperature gradient and the effect of the wall curvature, and the temperature jump boundary condition are applied. The developed simulation tool is used for the preliminary design of Knudsen micropumps consisting of a sequence of curved and straight channels.

Simulation of gas flows in micro/nano systems using the Burnett equations

The gaseous flow characteristics in micro/nano systems are studied using the Burnett equations. The Burnett equations are the second-order approximation of the Chapman–Enskog solution to the Boltzmann equation. This set of equation is appropriate to describe the rarefied gas flows. Velocity slip and temperature jump boundary conditions are applied on solid surface. Several techniques, including the usage of relaxation method on slip values and Burnett terms, are introduced to increase the stability of the Burnett equations. Hence, convergent results at Knudsen number up to 0.5 are achieved for the first time. The results of the Burnett equations are first verified using the corresponding experimental and the DSMC data. The Navier-Stokes and the Burnett equations give almost the same result when the flow is in slip regime (Kn<0.0l). But when the flow is in transition regime (0.1<Kn<0.5), the results of Navier-Stokes equations deviate from DSMC data while those of Burnett equations still agree very well. The Burnett equations are then used to study the gaseous flows in micro/nano systems. The pressure driven plane Poiseuille and the backward-facing step flows, the flows through micro filters are then studied. Different inlet to outlet pressure ratios, size effects, shape effects and boundary conditions are analyzed. The competitive relations of rarefied and compressible effects in micro gas flows are discussed.

On the role of surface shape in a micro-scale heat conduction problem

The present study investigates the importance of the surface shape in a micro-scale heat conduction problem. A heated infinitely-thin cylindrical shell is positioned in the middle of two concentric cylinders, and the heat transfer through a rarefied gas between the shell and the confining inner (or outer) cylinder is investigated. The study initially considers the solution of the first- and second-order temperature-jump models (i.e. the conventional heat equation with temperature-jump boundary conditions). The study then examines the numerical solution of the nonlinear Shakhov model kinetic equation subject to the Maxwell boundary condition using the discrete velocity method (DVM). The variable-hard-sphere molecular interaction model is taken into account in the temperature-jump models allowing the presence of significant temperature differences between surfaces to be considered. Anomalous temperature profiles near the convex (or concave) side of the shell are attributed to the effects of surface shape.

Molecular dynamics modeling of thermal resistance at argon-graphite and argon-silver interfaces

Heat conduction between two parallel solid walls separated by liquid argon is investigated using three-dimensional molecular dynamics (MD) simulations. Liquid argon molecules confined in silver and graphite nano-channels are examined separately. Heat flux and temperature distribution within the nano-channels are calculated by maintaining a fixed temperature difference between the two solid surfaces. Temperature profiles are linear sufficiently away from the walls, and heat transfer in liquid argon obeys the Fourier law. Temperature jump due to the interface thermal resistance (i.e., Kapitza length) is characterized as a function of the wall temperature. MD results enabled development of a phenomenological model for the Kapitza length, which is utilized as the coefficient of a Navier-type temperature jump boundary condition using continuum heat conduction equation. Analytical solution of this model results in successful predictions of temperature distribution in liquid argon confined in silver and graphite nano-channels as thin as 7 nm and 3.57 nm, respectively.

Slip flow fluid-structure-interaction

While many microscale systems are subject to both rarefaction and fluid-structure-interaction (FSI) effects, most commercial algorithms cannot model both, if either, of these for general applications. This study modifies the momentum and thermal energy exchange models of an existing, continuum based, multifield, compressible, unsteady, Eulerian-Lagrangian FSI algorithm, such that the equivalent of first-order slip velocity and temperature jump boundary conditions are achieved at fluid-solid surfaces, which may move with time. Following the development and implementation of the slip flow momentum and energy exchange models, several basic configurations are considered and compared to established data to verify the resulting algorithm's capabilities.

Investigation of 2D Transient Heat Transfer under the Effect of Dual-Phase-Lag Model in a Nanoscale Geometry

Analytical and numerical solutions of the 2D transient dual-phase-lag (DPL) heat conduction equation are presented in this article. The geometry is that of a simplified metal oxide semiconductor field effect transistor with a heater placed on it. A temperature jump boundary condition is used on all boundaries in order to consider boundary phonon scattering at the micro- and nanoscale. A combination of a Laplace transformation technique and separation of variables is used to solve governing equations analytically, and a three-level finite difference scheme is employed to generate numerical results. The results are illustrated for three Knudsen numbers of 0.1, 1, and 10 at different instants of time. It is seen that the wave characteristic of the DPL model is strengthened by increasing the Knudsen number. It is found that the combination of the DPL model with the proposed mixed-type temperature boundary condition has the potential to accurately predict a 2D temperature distribution not only within the transistor itself but also in the near-boundary region.

Rarefaction effects in hypersonic flow about a blunted leading edge

The rarefaction effects in the problem of hypersonic flow around a profile with blunted leading edge are studied in the flow regimes when the edge bluntness radius is comparable with the mean free path in the free stream. The flow around a cylindrically blunted thick plate at zero incidence was modelled numerically in the transitional regime by using the direct simulation Monte Carlo method, the finite-difference solution of the kinetic equation of the relaxation type (the ellipsoidal statistical model), and the solution of the Navier — Stokes equations. It is shown that for the Knudsen numbers in terms of the bluntness radius below 0.1, the Navier — Stokes equations can be applied successfully for viscous flow description behind the shock wave provided that the initial rarefaction effects are taken into account via the slip and temperature jump boundary conditions on the plate surface. For Knudsen number of about 0.5, the rarefaction effects are more appreciable; in particular, a substantial anisotropy of the distribution function takes place, but the Navier — Stokes equations yield, as before, a qualitatively correct result. The initial stage of the boundary layer development in the edge vicinity has been studied. In the considered range of Knudsen numbers, the entropy layer near the edge is comparable with the boundary layer thickness. As the distance from the leading edge increases one observes the absorption of the entropy layer by the boundary layer. In the studied parameter range, the interaction between the boundary and entropy layers leads to a flow stability increase.

Numerical Study of Flow and Heat Transfer of High Knudsen Number Flow Regimes in Nanochannels Filled with Porous Media

Considering the dependency of viscosity on Kn, a unified flow model for all flow regimes with different Kn was obtained. Applying the Dary Brinkman – Forchheimer flow model with the slip boundary condition, finite difference solutions for fully developed velocity distribution in a nanochannel of circular cross section, filled with porous media was presented. Convection heat transfer of the system, reflected in Nu was analyzed using the temperature jump boundary condition. It is shown that despite of the fact that in most of previous researches, Kn was assumed constant along the channel, the variations of Kn due to the pressure variations, have considerable effects on heat transfer and temperature distribution across the channel cross section.

New velocity-slip and temperature-jump boundary conditions for Navier–Stokes computation of gas mixture flows in microgeometries

Recently, several kinetic descriptions have been utilized for the simulation of gas mixture flows in microgeometries. Although such developments are important, Navier–Stokes computation can find extended use in engineering applications. In the present contribution, a rarefied flow of a gas mixture in the vicinity of a wall is concerned and new velocity-slip and temperature-jump boundary conditions are derived for the whole mixture. Appealing to these new boundary conditions, Navier–Stokes computation of microscale gas mixture flows becomes feasible. Consequently, the proposed conditions in conjunction with the Navier–Stokes equations and an equation for the conservation of species are solved for a He–Ne flow in a microchannel and suitability of the derived boundary conditions is demonstrated in the slip flow regime.

Data on the Velocity Slip and Temperature Jump on a Gas-Solid Interface

The present review is dedicated to the velocity slip and temperature jump coefficients applied to modeling of gas flows. Such coefficients are used when a moderate gas rarefaction must be taken into account. In this case, calculations of gas flows can be performed on the basis of continuum mechanics equations applying the velocity slip and temperature jump boundary conditions. Thus, the velocity slip and temperature jump coefficients have the same importance in gas dynamics as the transport coefficients such as viscosity, thermal conductivity, and diffusion coefficients. A critical analysis of theoretical and experimental data on the slip and jump coefficients available in the open literature is presented in an accessible form so that it can be easily understandable for nonspecialists in rarefied gas dynamics. The most reliable results are selected and tabulated. The results cover a single gas with the complete and noncomplete accommodation on a solid surface, gaseous mixtures, and polyatomic gases. Many examples of applications of the slip and jump boundary conditions are given. The review will be useful as a reference for mathematicians, physicists, and engineers dealing with flows of moderately rarefied gases.

Evaluation of Nusselt number for a flow in a microtube using second-order slip model

In this paper, the fully-developed temperature profile and corresponding Nusselt value is determined analytically for a gaseous flow in a microtube with a thermal boundary condition of constant wall heat flux. The flow assumed to be laminar, and hydrodynamically and thermally fully developed. The fluid is assumed to be constant property and incompressible. The effect of rarefaction, viscous dissipation and axial conduction, which are important at the microscale, are included in the analysis. Second-order slip model is used for the slip-flow and temperature jump boundary conditions for the implementation of the rarefaction effect. Closed form solutions for the temperature field and the fully-developed Nusselt number is derived as a function of Knudsen number, Brinkman number and Peclet number.