Showing papers in "Continuum Mechanics and Thermodynamics in 2000"

Couette flow with slip and jump boundary conditions

Abstract: The steady plane Couette flow is analyzed within the framework of the five field equations of mass, momentum and energy for a Newtonian viscous heat conducting ideal gas in which slip and jump boundary conditions are considered. The results obtained are compared with those that follow from the direct simulation Monte Carlo method.

Non parabolic band transport in semiconductors: closure of the production terms in the moment equations

Abstract: Closure relations for the production terms of the balance equations describing the motion of electrons in semiconductors are obtained by using the distribution function given, according to 1], by the maximum entropy principle in the case of the Kane dispersion relation for the energy band. Scatterings of electrons with non polar optical phonons (both for intervalley and intravalley interactions), acoustic phonons and impurities are considered. Applications to bulk silicon are presented. In particular the overshoot and saturation eeects are described with good accuracy.

Evolving Microstructure and Homogenization

Abstract: In this article we formulate a mathematical model for the temporally evolving microstructure generated by phase changes and study the homogenization of this model. The investigations are partially formal, since we do not prove existence or convergence of solutions of the microstructure model to solutions of the homogenized problem. To model the microstructure, the sharp interface approach is used. The evolution of the interface is governed by an everywhere defined distribution partial differential equation for the characteristic function of one of the phases. This avoids the disadvantage commonly associated with this approach of an evolution equation only defined on the interface. To derive the homogenized problem, a family of solutions of the microstructure problem depending on the fast variable is introduced. The homogenized problem obtained contains a history functional, which is defined by the solution of an initial-boundary value problem in the representative volume element. In the special case of a temporally fixed microstructure the homogenized problem is reduced to an evolution equation to a monotone operator.

Characteristic waves and dissipation in the 13-moment-case

Abstract: Extended thermodynamics derives dissipative, hyperbolic field equations for monatomic gases. One example is the system of the 13-field-case, which is a dissipative extension of the Euler equations. In this paper the system is investigated by solving a Riemann problem. Additionally some model equations are introduced so as to discuss the main properties in a transparent manner. There arises an interesting interplay of the characteristic waves and the dissipation in the system. For the 13-field-case it turns out that not every Riemann problem has a solution, because of the loss of hyperbolicity of the system.

Temperature jump and velocity slip in the moment method

Abstract: The moment method of the kinetic theory requires boundary conditions for the moments. It is not possible to derive these in an easy manner from the boundary conditions for the phase density. The conservation laws of mass, momentum and energy give only five relations between the moments and the properties of the wall. Additional boundary conditions may be determined from the minimax principle for the entropy production which was recently proposed by Struchtrup & Weiss [1]. These ideas are outlined for the case of 13 and 14 moments and Maxwell’s boundary conditions for the phase density which lead to temperature jumps and velocity slip at walls. In particular, one-dimensional stationary heat transfer between two walls at rest is considered. The temperature jumps at the walls are shown to depend on the values of all moments in front of the wall. The results obtained by the minimax principle are compared with results obtained for the same problem by a minimum principle for the global entropy production and by the so-called kinetic schemes [2].

Nonhomogeneous Initial-Boundary Value Problems for Coercive and Self-Controlling Models of Monotone Type

Abstract: Based on the idea of partial Yosida approximation we prove global in time existence of nonhomogeneous initial-boundary value problems for the class of coercive models of monotone type from the theory of inelastic deformations of metals. Then using this result and the coercive limits idea introduced in [8], we approximate self-controlling problems with nonhomogeneous boundary data by a sequence of coercive models and prove a convergence result.

The cumulant method for computational kinetic theory

Abstract: We propose a new method for numerical simulation of gas dynamics based on kinetic theory. The method is based on a cumulant-expansion-ansatz for the phase space density, which leads to a set of quasi-linear, hyperbolic partial differential equations. The method is compared to the moment method of Grad. Both methods agree for low-order approximations but the method proposed shows additional non-linear terms for high order approximations. Boundary conditions on the cumulants for an ideally reflecting and an ideally rough boundary surface are derived from conditions on the phase space density. A Lax-method is used for numerical analysis of a 2d-BGK fluid, which results in an easy-to-implement algorithm well suited for implementation on massivly parallel computers. The results are found to agree qualitatively with predictions from moment theories.

Identification of material parameters for inelastic constitutive models: stochastic simulations for the analysis of deviations

Abstract: For parameter identification a distance function between the measured and the simulated data has to be minimized. Therefore, the influence of three different norms used in the definition of such a distance function is investigated. The nonlinear optimization problem is solved using a modified random search algorithm originally proposed by Price (1978). Next a stochastic model for the generation of artificial test data is presented. This model is used for a stochastic simulation of test data (constant strain rate tension with relaxation and creep). From these artificial data the material parameters of the model of Chan, Bodner and Lindholm are identified. To measure the quality of the identified material parameters their mean values and empirical standard deviations are computed. Furthermore, the coefficients of the empirical correlation matrix for the material parameters are computed. The model responses for tensile tests with the parameter vector generated from all tests and with the estimated parameters (from stochastic simulations) differ not considerably. However, for the creep tests the different parameter estimations lead to quite different model responses.

Wave-induced liquefaction of a saturated sand layer

Abstract: The paper presents a theoretical study of the dynamic liquefaction of a saturated sand layer. The motion of the sand is induced by a periodic disturbance at the lower boundary of the layer, which simulates the influence of a plane excitation wave coming from below. The initial boundary value problem is solved numerically with the use of the hypoplastic constitutive equation for particular sand. Both horizontal and vertical disturbances are considered. The repeated deformation caused by a strong dynamic disturbance results in the reduction of the effective pressure and in the liquefaction of the sand. The degree of liquefaction as a function of the depth is nonuniform. The liquefaction patterns produced by vertical and horizontal disturbances are completely different.

Phenomenological modeling of electrorheological fluids with an extended CASSON-model

Abstract: In this paper we propose a phenomenological theory for electrorheological fluids. In general these are suspensions which undergo dramatic changes in their material properties if they are exposed to an electric field. In the context of continuum mechanics these fluids can be modeled as non-Newtonian fluids. Recalling the governing equations of rational thermodynamics and electrodynamics of moving media (Maxwell-Minkowski-equations), we derive suitable governing equations of electrorheology using essentially two assumptions concerning magnetic quantities. Furthermore we introduce a 3-dimensional nonlinear constitutive equation for the Cauchy stress tensor which is an extension of the model proposed by Ružicka (see [14]). Assuming a viscometric flow, we compare the shear stress of our model with other well known models and fit the parameters by using measurements that were obtained in a rotational viscometer. Excellent agreement between model and measurements is achieved. On the basis of these results we propose a 3-dimensional model, the so-called extended Casson -model. This model is investigated further for a channel flow configuration with a homogeneous electric field. We determine analytical solutions for the electric field, the velocity and the volumetric flow rate and illustrate the velocity profiles and the predicted pressure drop. The velocity profiles are flattened compared to parabolic profiles and become more flat if the electric field increases.

Nonlinear convection induced by inclined thermal and solutal gradients with mass flow

Abstract: The energy method is developed for the convection problem induced by inclined thermal and solutal gradients, with horizontal mass flow, in a horizontal layer of a saturated porous medium. A non-linear stability analysis is performed and compound matrix method is employed for numerical calculations. For representative parameter values the critical vertical thermal Rayleigh number and wave number are calculated. It is noted that the effect of horizontal thermal or solutal gradient is to switch from stabilizing to destabilizing as their magnitude increases, for zero or small values of mass flow rate. For higher values of mass flow rate the effect is always destabilizing. It is also noted that the horizontal concentration gradient has a stronger destabilizing effect as compared to the horizontal temperature gradient.

Long-time behavior for the full one-dimensional Fremond model for shape memory alloys

Abstract: This note deals with a well-posed one-dimensional initial-boundary value problem related to the Fremond thermomechanical model of structural phase transitions in shape memory materials. The long-time behavior is investigated and it is proved that the omega-limit set in a suitable topology only contains stationary states.

Tales of Thermodynamics and Obscure Applications of the Second Law

Abstract: Today, firmly based on statistical mechanics, everything appears to be totally simple and clear. A macroscopic body, for example a gas, a piece of iron, or a rubber band, consists of microscopic particles that constantly perform a more or less irregular motion. In a gas these micro-particles consist of atoms or molecules that irregularly traverse the space that is at their disposition. Molecules can additionally vibrate and rotate. In iron, the iron atoms are embedded regularly in a crystal lattice but vibrate around their assigned lattice sites with variable amplitudes. And in rubber’s long chain structure, isoprene molecules play the role of chain links and perform a macroscopically invisible, irregular motion that constantly leads to new configurations.

Thermodynamically Consistent Coefficient Calibration in Nonlinear and Anisotropic Closure Models for Turbulence

Abstract: At this first order level of closure, the scalars, turbulent kinetic energy and its dissipation rate, emerge as natural basic variables for the turbulence modeling in contrast to the considerations in Wang [8] and others.

Full and reduced model solutions of steady axi-symmetric ice sheet flow over small and large bed topography slopes

Abstract: Two numerical methods for solving the full steady ice-sheet equations in axi-symmetric flow are described. The free-boundary problem is treated by transforming the problem to a fixed domain using either an orthogonal co-ordinate transformation or a variant of a transformation proposed by Landau, and difficulties with the former, more sophisticated, method are demonstrated. The simpler Reduced Model is also presented, and accurate solutions for flows over bed topography with moderate to large slopes are generated by an inverse method for comparison with the numerical solutions of the full equations. The reduced model is not valid for such bed slopes, and the comparisons demonstrate the extent and nature of the errors arising from the use of the simpler model.

Discussion of Finite Deformation Viscoelasticity Laws with Reference to Torsion Loading

Abstract: In a previous paper two classes of finite deformation viscoelasticity laws, referred to as Model A and Model B, respectively, have been introduced. They were derived as finite deformation counterparts of two well-known spring-dashpot linear solids, the first one being a spring in parallel with a Maxwell element and the second model consisting of a spring in series with a Kelvin element. In particular, two special forms of the free energy function related to Model A (respectively to Model B) were considered, implying two different finite deformation viscoelasticity laws referred to as Model A1 and Model A2 (respectively Model B1 and Model B2). In the present paper we discuss predicted responses of these models with reference to simple torsion as well as torsion with free ends. Generally, the investigations of the paper have fundamental character in what concerns the basic concepts in formulating viscoelastic models of rate type.

The Burnett equations for a relativistic gas

Abstract: The Burnett constitutive equations for the dynamic pressure, heat flux and pressure deviator are obtained from the relativistic Boltzmann equation by using fourteen moment equations and an iteration method akin to the Maxwellian iteration procedure of the non-relativistic theory. The non-relativistic and ultra-relativistic limits of all Burnett coefficients are given and it is shown that the non-relativistic limit of the Burnett constitutive equations agree with those obtained from the non-relativistic Boltzmann equation.

Balance equation of the generalised sub-grid scale (SGS) turbulent kinetic energy in a new tensorial dynamic mixed SGS model

Abstract: A new dynamic model is proposed in which the eddy viscosity is defined as a symmetric second rank tensor, proportional to the product of a turbulent length scale with an ellipsoid of turbulent velocity scales. The employed definition of the eddy viscosity allows to remove the local balance assumption of the SGS turbulent kinetic energy formulated in all the dynamic Smagorinsky-type SGS models. Furthermore, because of the tensorial structure of the eddy viscosity the alignment assumption between the principal axes of the SGS turbulent stress tensor and the resolved strain-rate tensor is equally removed, an assumption which is employed in the scalar eddy viscosity SGS models. The proposed model is tested for a turbulent channel flow. Comparison with the results obtained with other dynamic SGS models (Dynamic Smagorinsky Model, Dynamic Mixed Model and Dynamic K-equation Model) shows that the tensorial definition of the eddy viscosity and the removal of the local balance assumption of the SGS turbulent kinetic energy considerably improves the agreement between results obtained with Large Eddy simulation (LES) and Direct Numerical Simulations (DNS), respectevely.

Adhesion of oriented nematic droplets

Abstract: We propose and analyze a two-dimensional model for the equilibrium of oriented droplets of nematic liquid crystals that may adhere to a rigid substrate, while surrounded by an isotropic environment. We obtain the contact condition at the edge where the liquid crystal, the substrate, and the environment come together. We further develop a fairly general method to arrive at the equilibrium shapes of a drop, which is then applied to the case where the surface tension at the liquid crystal interface is given by Rapini and Papoular's expression. In this case, we also predict the existence of concave equilibrium shapes. Here is indeed the main difference between this method and Wulff's construction, which always yields convex equilibrium shapes for a drop free from adhesion.