A linear mass and energy conserving numerical scheme for two-phase flows with thermocapillary effects

A thermodynamically consistent phase-field model and an entropy stable numerical method for simulating two-phase flows with thermocapillary effects

Partial differential equations (PDEs) are used in the real world to model physical phe- nomena such as heat, wave, Laplace, and Poisson equations. For regular shape domains, the heat equation can be solved analytically; however, for irregular domains, the computation of the solu- tion is difficult and numerical methods like Finite Difference Method (FDM) and Finite Element Method (FEM) can be used. FEM provides approximate values at discrete points in the domain. It breaks down a large problem into smaller finite elements. These element’s equations are combined into a system representing the whole problem. We show the comparison between analytic solution, solutions by FDM and FEM. The impact of heat on the material is examined at various positions and multiple positions. We compare the analytical and numerical (by FEM) solution considering several homogeneous materials with various diffusivity values (α). Finally, the simulation results of different non-homogeneous materials were compared. Science and engineering fields that use heat equations can be evaluated using the numerical method applied here.

We consider the heat equation on the N‐dimensional cube (0, 1)N and impose different classes of integral conditions, instead of usual boundary ones. Well‐posedness results for the heat equation under the condition that the moments of order 0 and 1 are conserved had been known so far only in the case of , for which such conditions can be easily interpreted as conservation of mass and barycenter. In this paper we show that in the case of general N the heat equation with such integral conditions is still well‐posed, upon suitably relaxing the notion of solution. Existence of solutions with general initial data in a suitable space of distributions over (0, 1)N are proved by introducing two appropriate realizations of the Laplacian and checking by form methods that they generate analytic semigroups. The solution thus obtained turns out to solve the heat equation only in a certain distributional sense. However, one of these realizations is tightly related to a well‐known object of operator theory, the Krein–von Neumann extension of the Laplacian. This connection also establishes well‐posedness in a classical sense, as long as the initial data are L2‐functions.

In this paper, we present the elementary principles of nonlinear quantum mechanics (NLQM), which is based on some problems in quantum mechanics. We investigate in detail the motion laws and some main properties of microscopic particles in nonlinear quantum systems using these elementary principles. Concretely speaking, we study in this paper the wave-particle duality of the solution of the nonlinear Schrodinger equation, the stability of microscopic particles described by NLQM, invariances and conservation laws of motion of particles, the Hamiltonian principle of particle motion and corresponding Lagrangian and Hamilton equations, the classical rule of microscopic particle motion, the mechanism and rules of particle collision, the features of reflection and the transmission of particles at interfaces, and the uncertainty relation of particle motion as well as the eigenvalue and eigenequations of particles, and so on. We obtained the invariance and conservation laws of mass, energy and momentum and angular momentum for the microscopic particles, which are also some elementary and universal laws of matter in the NLQM and give further the methods and ways of solving the above questions. We also find that the laws of motion of microscopic particles in such a case are completely different from that in the linear quantum mechanics (LQM). They have a lot of new properties; for example, the particles possess the real wave-corpuscle duality, obey the classical rule of motion and conservation laws of energy, momentum and mass, satisfy minimum uncertainty relation, can be localized due to the nonlinear interaction, and its position and momentum can also be determined, etc. From these studies, we see clearly that rules and features of microscopic particle motion in NLQM is different from that in LQM. Therefore, the NLQM is a new physical theory, and a necessary result of the development of quantum mechanics and has a correct representation of describing microscopic particles in nonlinear systems, which can solve problems disputed for about a century by scientists in the LQM field. Hence, the NLQM built is very necessary and correct. The NLQM established can promote the development of physics and can enhance and raise the knowledge and recognition levels to the essences of microscopic matter. We can predict that nonlinear quantum mechanics has extensive applications in physics, chemistry, biology and polymers, etc.

The heat and moisture transfer balance theory of garment simulation

Purpose The purpose of this paper is to propose an efficient space-time operator-splitting method for the high-dimensional vector-valued Allen–Cahn (AC) equations. The key of the space-time operator-splitting is to devide the complex partial differential equations into simple heat equations and nolinear ordinary differential equations. Design/methodology/approach Each component of high-dimensional heat equations is split into a series of one-dimensional heat equations in different spatial directions. The nonlinear ordinary differential equations are solved by a stabilized semi-implicit scheme to preserve the upper bound of the solution. The algorithm greatly reduces the computational complexity and storage requirement. Findings The theoretical analyses of stability in terms of upper bound preservation and mass conservation are shown. The numerical results of phase separation, evolution of the total free energy and total mass conservation show the effectiveness and accuracy of the space-time operator-splitting method. Practical implications Extensive 2D/3D numerical tests demonstrated the efficacy and accuracy of the proposed method. Originality/value The space-time operator-splitting method reduces the complexity of the problem and reduces the storage space by turning the high-dimensional problem into a series of 1D problems. We give the theoretical analyses of upper bound preservation and mass conservation for the proposed method.

Solving Allen-Cahn equations with periodic and nonperiodic boundary conditions using mimetic finite-difference operators

Two energy balance equations widely used to describe simultaneous transfer of heat and mass in porous media are inconsistent with control volume energy conservation. Potential energy, enthalpy, and internal energy terms are involved in the discrepancies. Energy within a volume is properly counted as the sum of internal, potential, and kinetic energy. However, one equation uses enthalpy where internal energy should have been used. In the other, potential energy and shifts in internal energy associated with heat of wetting are not included. Energy conservation for a control volume dictates summing convective fluxes of internal, potential, and kinetic energy at the control volume surface along with conducted heat and work crossing the boundary. The pressure–volume (pv) work at the volume surface may be combined with internal energy convection so that flow of enthalpy is used in the flux term. Examples of energy change versus work input in adiabatic processes illustrate the error introduced when enthalpy rath...

This thesis presents a computational method for seamlessly bridging the atomistic and the continuum realms at finite temperature. The theoretical formulation is based on the static theory of the quasicontinuum and extends it to model non-equilibrium finite temperature material response. At non-zero temperature, the problem of coarse-graining is compounded by the presence of multiple time scales in addition to multiple spatial scales. We address this problem by first averaging over the thermal motion of atoms to obtain an effective temperature-dependent energy on the macroscopic scale. Two methods are proposed to this end. The first method is developed as a variational mean field approximation which yields local thermodynamic potentials such as the internal energy, the free energy, and the entropy as phase averages of appropriate phase functions. The chief advantage of this theory is that it accounts for the anharmonicity of the interaction potentials, albeit numerically, unlike many methods based on statistical mechanics which require the quasi-harmonic approximation for computational feasibility. Furthermore, the theory reduces to the classical canonical ensemble approach of Gibbs under the quasi-harmonic approximation for perfect, isotropic, infinite crystals subjected to uniform temperature. In the second method, based on perturbation analysis, the internal energy is derived as an effective Hamiltonian of the atomistic system by treating the thermal fluctuations as perturbations about an equilibrium configuration. These energy functionals are then introduced into the quasicontinuum theory, which facilitates spatial coarse-graining of the atomistic description. Finally, a variational formulation for simulating rate problems, such as heat conduction, using the quasicontinuum method is developed. This is achieved by constructing a joint incremental energy functional whose Euler-Lagrange equations yield the equilibrium equations as well as the time-discretized heat equation. We conclude by presenting the results for numerical validation tests for the thermal expansion coefficient and the specific heat for some materials and compare them with classical theory, molecular dynamics results, and experimental data. Some illustrative examples of thermo-mechanical coupled problems such as heat conduction in a deformable solid, adiabatic tension test, and finite temperature nanoindentation are also presented which show qualitative agreement with expected behavior and demonstrate the applicability of the method.

Hydrostatic equilibrium solutions for the hot gas in a cluster of galaxies have been studied assuming the polytropic relation. The final equilibrium state is related to the initial state of the gas in the cluster using the mass and energy conservation laws between the above two states. Results are applied to the X-ray source in Coma cluster. We have obtained the central temperature, the central density and the mass of the intracluster gas such as 1.8X10' OK, 2.8X10- 27 gm/cm' and 5.7X101'vf0, respectively. As for the X-ray emission mechanism of these sources, inverse compton and thermal bremsstrahlung theories have been proposed. However, recent spectral analyses of soft X-ray data2l tend to show a strong preference for the thermal model. Moreover, tailed radio sources3l and the excess of SO galaxies4l found in these clusters may be well interpreted, if hot intracluster gases exist. Concerning the thermal model, a wind model, sl infall models6l, 7) and static modelsn,sl have been studied. Lea8l has calculated the hydrostatic equilibrium solu­ tion for a polytropic gas in a cluster of galaxies under the boundary condition that the temperature of the gas approaches to zero at infinity. By comparing that solution with observations for the Coma cluster, she shows that the intracluster gas is distributed more widely than the galaxies by a factor of about 2--'2.5 and the temperature of the gas given from a best-fit model is 1.5 X 108 oK. Takahara et al. 9l have investigated the dynamical collapse of gas in a cluster of galaxies. As a result of numerical calculations, they find that a hot plasma is formed after the bounce and it attains an equilibrium state. In § 2 the hydrostatic equilibrium solution for a polytropic gas in a cluster of galaxies is derived. In § 3 the hydrostatic equilibrium state which is attained after the dynamical collapse is discussed using the energy and mass conservation laws which relate the final state of the gas to the initial state. In § 4 the central temperature, the central density and the mass of the intracluster gas are obtained

Heat equation is a partial differential equation used to describe the temperature distribution in a heat-conducting body. The implementation of a numerical solution method for heat equation can vary with the geometry of the body. In this study, a three-dimensional transient heat conduction equation was solved by approximating second-order spatial derivatives by five-point central differences in cylindrical coordinates. The stability condition of the numerical method was discussed. A MATLAB code was developed to implement the numerical method. An example was provided in order to demonstrate the method. The numerical solution by the method was in a good agreement with the exact solution for the example considered. The accuracy of the five-point central difference method was compared with that of the three-point central difference method in solving the heat equation in cylindrical coordinates. The solutions obtained by the numerical method in cylindrical coordinates were displayed in the Cartesian coordinate system graphically. The method requires relatively very small time steps for a given mesh spacing to avoid computational instability. The result of this study can provide insights to use appropriate coordinates and more accurate computational methods in solving physical problems described by partial differential equations.

A new DNS formalism dedicated to turbulent two-phase flows with phase change

The present study examines the electroosmosis peristaltic motion with magnetohydrodynamics for the Casson fluid. Internal energy contained within a thermodynamic system can generate heat . Internal energy and activation energy are necessary to commence a chemical reaction within a system. The effects are appealing in bounded systems to highlight their heat transfer aspects. Thus heat equation is modeled under the two effects. Using a large wavelength and low Reynolds number, the equations for momentum, mass, and temperature are simplified and solved. Convective conditions for temperature and concentration are considered. The effects of various parameters are graphically analyzed.

A composite grid solver for conjugate heat transfer in fluid–structure systems

The purpose of this study is to design a ventilated window, forming a dual skin facade, used in the promotion of natural ventilation in occupied spaces. The study will be conducted in an experimental chamber that will be equipped with three ventilated windows. All constructive details and materials used will be analyzed. The project will be carried out from a system of integral equations of energy and mass conservation, in the thermal component. In the fluid dynamics component, they will be considered a system of second-order nonlinear energy conservation equations and a system of first-order linear mass flow conservation equations. In this project heat and mass flow equations will be considered in the thermal component equations and localized and continuous load losses equations will be considered in the fluid dynamics component. In this work, 5 cases were studied. The best ventilating rate was obtained for the case where the air insufflation is done in a duct and the air extraction is done in 3 ventilated windows. In this case, a good indoor air quality can be guaranteed for at least 5 occupants.