Mathematical Modeling and Simulation of Antibubble Dynamics

In this study, we propose a mathematical model and perform numerical simulations for the antibubble dynamics. An antibubble is a droplet of liquid surrounded by a thin film of a lighter liquid, which is also in a heavier surrounding fluid. The model is based on a phase-field method using a conservative Allen-Cahn equation with a space-time dependent Lagrange multiplier and a modified Navier-Stokes equation. In this model, the inner fluid, middle fluid and outer fluid locate in specific diffusive layer regions according to specific phase filed (order parameter) values. If we represent the antibubble with conventional binary or ternary phase-field models, then it is difficult to have stable thin film. However, the proposed approach can prevent nonphysical breakup of fluid film during the simulation. Various numerical tests are performed to verify the efficiency of the proposed model.     

Fully Coupled Thermo-Hydrodynamic Simulation Model for Journal Bearings

The isothermal form of Reynolds fluid film equation is used to predict the pressure generation in hydrodynamic journal bearings if temperature effects are neglected. Often, however, temperature effects may be important and cannot be neglected, because oil viscosity significantly varies with temperature. Also, thermal expansion of journal shaft and bearing housing must be taken into account since the bearing clearance changes with increasing temperature. Hence, the Reynolds pressure field equation, the energy equation for the fluid film and the heat transfer equations for journal and bearing housing have to be solved simultaneously. The coupled thermo-hydrodynamic fluid flow problem is mathematically defined by a system of nonlinear integro-differential equations. The governing equations are discretized and solved by a finite element approach. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coupling of the Reynolds Fluid-Film Equation with the 2D Navier-Stokes Equations

The pressure field in thin fluid films can quite precisely be calculated by Reynolds fluid-film equation. In some problems, it may be useful to couple thin fluid-films with general 2D or 3D fluid flows. In the current work, we analyze the fluid flow, pressure and temperature field in a hydrodynamic journal bearing with a rectangular oil groove. Pressure and temperature in the fluid gap are calculated by means of the Reynolds equation and the 2D energy equation. Cavitation effects are taken into account by incorporating a 2-phase cavitation approach. In order to calculate the velocity and pressure field in the oil groove, the 2D Navier-Stokes equations are used; the temperature distribution in the oil groove is computed by means of the 2D energy equation. Appropriate coupling conditions for velocity, pressure and temperature are formulated in order to couple the flow in the fluid gap with the flow in the oil groove. Thermal expansion of journal shaft and bearing housing are also taken into account, since the bearing clearance changes with increasing temperature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The effect of magnetic field on dynamics of gas bubbles in visco-elastic fluids

The effect of magnetic field on nonlinear oscillations of a spherical, acoustically forced gas bubble in nonlinear visco-elastic media is studied. The constitutive equation UCM used for modeling the rheological behaviors of the fluid. By starting from the momentum equations for bubbles considering the magnetic force and considering some simplifying assumptions, the modified bubble dynamics equation (the modified Rayleigh–Plesset equation) has been achieved. Assumptions concerning the trace of the stress tensor are addressed in light of the incorporation of visco-elastic constitutive equations into modified bubble dynamics equations. The governing equations are non-dimesionalized and numerically solved by using 4th order Runge–Kutta method. The accuracy of the calculations and the formulation is compared with the previous works done for models without the presence of magnetic field. Furthermore, the bubble size variations due to acoustic motivations and stress tensor components variations in presence of different magnitudes of magnetic fields are studied. Also, the bubble size dependence on fluid conductivity variations is declared. The relevance and importance of this approach to biomedical ultrasound applications are highlighted. Preliminary results indicate that magnetic field may be an important consideration for the risk assessment of potential cavitations and also it could be possible to damp the bubble oscillations by using magnetic fields or in opposite case amplify the oscillations which could result in higher level light emissions in sonoluminescence approach.     

AN UNCONDITIONALLY STABLE HYBRID FE-FD SCHEME FOR SOLVING A 3-D HEAT TRANSPORT EQUATION IN A CYLINDRICAL THIN FILM WITH SUB-MICROSCALE THICKNESS

Heat transport at the microscale is of vital importance in microtechnology applications.The heat transport equation is different from the traditional heat transport equation since a second order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study,we develop a hybrid finite element-finite difference (FE-FD) scheme with two levels in time for the three dimensional heat transport equation in a cylindrical thin film with submicroscale thickness. It is shown that the scheme is unconditionally stable. The scheme is then employed to obtain the temperature rise in a sub-microscale cylindrical gold film. The method can be applied to obtain the temperature rise in any thin films with sub-microscale thickness, where the geometry in the planar direction is arbitrary.     

An explicit staggered-grid method for numerical simulation of large-scale natural gas pipeline networks

We present an explicit second order staggered finite difference (FD) discretization scheme for forward simulation of natural gas transport in pipeline networks. By construction, this discretization approach guarantees that the conservation of mass condition is satisfied exactly. The mathematical model is formulated in terms of density, pressure, and mass flux variables, and as a result permits the use of a general equation of state to define the relation between the gas density and pressure for a given temperature. In a single pipe, the model represents the dynamics of the density by propagation of a non-linear wave according to a variable wave speed. We derive compatibility conditions for linking domain boundary values to enable efficient, explicit simulation of gas flows propagating through a network with pressure changes created by gas compressors. We compare our staggered grid method with an explicit operator splitting method and a lumped element scheme, and perform numerical experiments to validate the convergence order of the new discretization approach. In addition, we perform several computations to investigate the influence of non-ideal equation of state models and temperature effects on pipeline simulations with boundary conditions on various time and space scales.     

对流扩散方程的QC3无网格法

对流扩散方程作为偏微分运动方程的分支,在流体力学、气体动力学等领域有着重要应用.为解决对流扩散方程难以通过解析法得到解析解的难题,采用二阶一致3点积分(Quadratically Consistent 3-Point Integration,简称QC3)提高无网格法的计算效率,通过对积分点上形函数导数的修正,改善无网格...     

Rheological effect on thermocapillary flow of a liquid film jet painted on a moving boundary

In the present paper, a liquid (or melt) film of relatively high temperature ejected from a vessel and painted on the moving solid film is analyzed by using the second-order fluid model of the non-Newtonian fluid. The thermocapillary flow driven by the temperature gradient on the free surface of a Newtonian liquid film was discussed before. The effect of rheological fluid on thermocapillary flow is considered in the present paper. The analysis is based on the approximations of lubrication theory and perturbation theory. The equation of liquid height and the process of thermal hydrodynamics of the non-Newtonian liquid film are obtained, and the case of weak effect of the rheological fluid is solved in detail.     

Steady and Unsteady Jet Wiping Processes

The jet wiping process is widely used in continuous coating applications to remove the excess amount of liquid entrained by a sheet moving out of a liquid bath. Typical fields of applications are hot dip galvanization of metal strips and coating of photographic films. The process is based on the impact of a gas jet onto the liquid film carried by the solid substrate. In the present study the process is investigated for the case of strictly two‐dimensional flow. It is assumed that inertia effects on the film flow can be neglected, whereas the effects of the pressure gradient and the shear stress distribution of the impinging jet and the surface tension of the liquid film are taken into account. As a result it is possible to derive a single kinematic wave equation which governs the distribution of the film thickness. Numerical results for representative steady and unsteady processes including the formation of shock discontinuities are presented.     

Thermoelastic rolling contact problem with temperature dependent friction

The paper is concerned with the numerical solution of a thermoelastic rolling contact problem with wear. The friction between the bodies is governed by Coulomb law. A frictional heat generation and heat transfer across the contact surface as well as Archard's law of wear in contact zone are assumed. The friction coefficient is assumed to depend on temperature. In the paper quasistatic approach to solve this contact problem is employed. This approach is based on the assumption that for the observer moving with the rolling body the displacement of the supporting foundation is independent on time. The original thermoelastic contact problem described by the hyperbolic inequality governing the displacement and the parabolic equation governing the heat flow is transformed into elliptic inequality and elliptic equation, respectively. In order to solve numerically this system we decouple it into mechanical and thermal parts. Finite element method is used as a discretization method. Numerical examples showing the influence of the temperature dependent friction coefficient on the temperature distribution and the length of the contact zone are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

在纵向自由薄膜凝固期内温度分布的数值分析

考虑到薄膜表面的热通量主要是来自辐射，因而采用一个依赖时间的拟二维拟线性扩散方程的Ｓｔｅｆａｎ问题（混合初边值问题）作为该问题的数学模型。用一种具有Ｃｒａｎｋ－Ｎｉｃｈｏｌｓｏｎ格式的无条件稳定的有限差分析来求解抛物型偏微分方程的定解问题。用追赶法求解离散化的三对角方程组，然后用预估校正法求解拟线性扩散方程，从而避免了示解非线性差分方程组，琥珀亚硝酸酯在纵向自由薄膜凝固期内的温度分布数值计算结果和     

An unconditionally stable finite difference scheme for solving a 3D heat transport equation in a sub-microscale thin film

Heat transport at the microscale is of vital importance in microtechnology applications. The heat transport equation is different from the traditional heat diffusion equation since a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we develop a finite difference scheme with two levels in time for the 3D heat transport equation in a sub-microscale thin film. It is shown by the discrete energy method that the scheme is unconditionally stable. The 3D implicit scheme is then solved by using a preconditioned Richardson iteration, so that only a tridiagonal linear system is solved for each iteration. The numerical procedure is employed to obtain the temperature rise in a gold sub-microscale thin film.     

非牛顿流体内流传热研究

在本文中,研究了注入轴对称模腔非牛顿流体非定常流动.本文的第二部份研究了上随体Maxwell流体管内热流动.对于注入模腔流动.其本构方程采用幂律流体模型方程.为了避免在表现粘度中温度关系引起的非线性.引进了一特征粘度的概念.描述本力学过程的基本方程是,本构方程、定常状态的运动方程、非定常能量方程及连续方程.该方程组在空间是二维问题,在数学上是三维问题.采用分裂差分格式求得本方程组的数值解答.分裂法曾成功应用于求解牛顿流体问题.在本文中,首次将分裂法成功地应用解决非牛顿流体流动问题.对于圆管内热流,给出了差分格式,使基本方程组化为一个三对角方程组.其结果,给出了不同时刻的模腔内二维温度分布.     

Numerical study of melting in an enclosure with discrete protruding heat sources

In order to explore the capability of a solid–liquid phase change material (PCM) for cooling electronic or heat storage applications, melting of a PCM in a vertical rectangular enclosure was studied. Three protruding generating heat sources are attached on one of the vertical walls of the enclosure, and generating heat at a constant and uniform volumetric rate. The horizontal walls are adiabatic. The power generated in heat sources is dissipated in PCM (n-eicosane with the melting temperature, T_m = 36 °C) that filled the rectangular enclosure. The advantage of using PCM is that it is able to absorb high amount of heat generated by heat sources due to its relatively high energy density. To investigate the thermal behaviour and thermal performance of the proposed system, a mathematical model based on the mass, momentum and energy conservation equations was developed. The governing equations are next discretised using a control volume approach in a staggered mesh and a pressure correction equation method is employed for the pressure–velocity coupling. The PCM energy equation is solved using the enthalpy method. The solid regions (wall and heat sources) are treated as fluid regions with infinite viscosity and the thermal coupling between solid and fluid regions is taken into account using the harmonic mean of the thermal conductivity method. The dimensionless independent parameters that govern the thermal behaviour of the system were next identified. After validating the proposed mathematical model against experimental data, a numerical investigation was next conducted in order to examine the thermal behaviour of the system by analyzing the flow structure and the heat transfer during the melting process, for a given values of governing parameters.     

A model for fluid flow in non-fully filled tribological interfaces – Part 1: Basics and Numerics

In literature, most contributions on starved lubrication focus on the occurring pressures in macroscopic devices. Hereby, usually the Reynolds equation is modified in different ways. In contrast to this proceeding, this paper's intention is the general investigation of this tribological regime to get a fundamental comprehension on the transition from boundary lubrication to mixed lubrication. The respective model describes the flow of the fluid through two rough surfaces moving relative to each other. The lack of fluid is regarded by the fact that elements may not be fully filled with the fluid. Only elements where the fluid fully fills the gap, generate a pressure. This effect is considered by a type of unilateral constraint in combination with a penalty function. The fluid flow is computed according to the Navier-Stokes equation. In combination with the continuity equation, a set of implicit nonlinear equations has to be solved. Its potential and basic application fields are finally discussed. A further paper will show applications of the algorithm towards different scenarios. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An analytical approach to the unified solution of kinetic equations in rarefied gas dynamics. III. Evaporation and condensation problems

An analytical version of the discrete-ordinates method, the ADO method, is used here to solve two problems in the rarefied gas dynamics field, that describe evaporation/condensation between two parallel interfaces and the case of a semi-infinite medium. The modeling of the problems is based on a general expression which may represent four different kinetic models, derived from the linearized Boltzmann equation. This work is an extension of two other previous works, devoted to rarefied gas flow and heat transfer problems, where the complete development of the ADO solution, which is analytical in terms of the spatial variable, is presented in a way, such that, the four kinetic models are considered, in an unified approach. A series of numerical results are showed in order to establish a general comparative analysis between this consistent set of results provided by the same methodology, based on kinetic models, and results obtained from the linearized Boltzmann equation. In particular, the temperature and density jumps are evaluated.     

On the linearization of the equations of motion of a rotating disk

In this paper, we present a consistent approach to reduce the fully nonlinear equations of a rotating disk to the classical linear equation derived by Lamb and Southwell and the nonlinear equations derived by Nowinski. The approach recognizes the fact that the out-of-plane deflection and the in-plane deflections are of different orders of magnitude. By using the ratio between the plate thickness and the outer radius as a measurement and carefully examining the reasonable magnitudes of all the variables involved, the fully nonlinear equations can be non-dimensionalized with all the terms being sorted according to their orders of magnitude. It is found that the classical linear equation derived by Lamb and Southwell can be recovered if all the terms of the lowest order of magnitude in the fully nonlinear equations are retained. If all the terms of the lowest two orders of magnitude are retained, Nowinski’s equations can then be recovered. Furthermore, the terms arising from in-plane deformation and rotary inertia are of the highest order and can be ignored in most of the applications.     

A semianalytical approach to nonlinear fluid film forces of a hydrodynamic journal bearing with two axial grooves

A semianalytical approach to nonlinear fluid film forces of a hydrodynamic journal bearing with two axial grooves under the cavitation boundary condition is proposed. The pressure distribution of the Reynolds equation of a finitely long journal bearing with axial grooves is expressed as a particular solution and a homogeneous solution. The particular solution and the homogeneous solution are separated, respectively, into an additive form and a multiplicative form by the method of separation of variables. The circumferential separable function of the homogeneous solution can be expanded on the basis of the infinite series of trigonometric functions. The pressure distribution of the particular solution is obtained by the Sommerfeld transformation. The termination positions of the fluid film are determined by the continuity condition. The analytical expressions for the nonlinear fluid film forces of a finitely long journal bearing with two axial grooves are obtained. The fluid film forces calculated by the proposed method agree well with the results obtained by the finite-difference method. The effects of the bearing parameters on the nonlinear fluid film forces are analyzed.     

Rimming flow of a power-law fluid: Qualitative analysis of the mathematical model and analytical solutions

Rimming flow of a non-Newtonian fluid on the inner surface of a horizontal rotating cylinder is investigated. Simple lubrication theory is applied since the Reynolds number is small and liquid film is thin. For the steady-state flow of a power-law fluid the mathematical model reduces to a simple algebraic equation regarding the thickness of the liquid film. The qualitative analysis of this equation is carried out and the existence of two possible solutions is rigorously proved. Based on this qualitative analysis, different regimes of the rimming flow are defined and analyzed analytically. For the particular case, when the flow index in a power-law constitutive equation is equal to 1/2, the problem reduces to the fourth order algebraic equation which is solved analytically by Ferrari method.