Linear Momentum Balance Equations Research Articles

Nemato-capillarity theory and the orientation-induced Marangoni flow

The macroscopic equations of nemato-capillarity, including the interfacial linear momentum balance equation and the interfacial director torque balance equation, are presented. The interfacial linear momentum balance equation for isotropic fluid-nematic liquid crystals involves the surface divergence of the surface stress tensor. It is shown that the surface stress tensor for isotropic fluid-nematic interfaces is, in most cases of interest, dominated by elastic modes. It is found that the anisotropic elastic contribution to the surface stress tensor gives rise to bending stresses, not observed in interfaces between isotropic fluids. In addition it is found that the anisotropic contribution to the surface elasticity also gives rise to tangential forces. Thus when the director orientation deviates from the easy axis of an isotropic fluidnematic interface and the deviation has surface gradients, an orientation-driven Marangoni flow can exist. The strength of this novel effect is proportional to the anchoring energy of the interface, and the direction of flow is from low energy regions towards high energy regions, that is, from regions where the director is aligned along the easy axis towards regions where the director deviates from the easy axis.

Elastoplastic consolidation at finite strain part 2: finite element implementation and numerical examples

A mathematical model for finite strain elastoplastic consolidation of fully saturated soil media is implemented into a finite element program. The algorithmic treatment of finite strain elastoplasticity for the solid phase is based on multiplicative decomposition and is coupled with the algorithm for fluid flow via the Kirchhoff pore water pressure. A two-field mixed finite element formulation is employed in which the nodal solid displacements and the nodal pore water pressures are coupled via the linear momentum and mass balance equations. The constitutive model for the solid phase is represented by modified Cam—Clay theory formulated in the Kirchhoff principal stress space, and return mapping is carried out in the strain space defined by the invariants of the elastic logarithmic principal stretches. The constitutive model for fluid flow is represented by a generalized Darcy's law formulated with respect to the current configuration. The finite element model is fully amenable to exact linearization. Numerical examples with and without finite deformation effects are presented to demonstrate the impact of geometric nonlinearity on the predicted responses. The paper concludes with an assessment of the performance of the finite element consolidation model with respect to accuracy and numerical stability.

THE EFFECTIVE STRESS PRINCIPLE: INCREMENTAL OR FINITE FORM?

Different expressions of the effective stress principle can be found in the literature, in particular some are written in finite form and others in incremental form. For the purpose of the paper we take for granted that stress–strain relationships exist or can be obtained for the effective stress coming from both formulations. We investigate the consequences of the choice of particular finite or differential forms when they are introduced in a weak form of the linear momentum balance equation of two- of three-phase porous media for its numerical solution. For partially saturated geomaterials the importance of the capillary pressure–saturation relationship is pointed out.

A penalty-hybrid finite element analysis of stokes flow

A type of penalty-hybrid variational principle is suggested for the analysis of Stokesian flow. On such a basis, a finite element model is formulated featuring, among others, a priori satisfaction of the deviatoric stress and hydrostatic pressure on linear momentum balance equations. Also in the present scheme the hydrostatic pressure is successfully eliminated at the element level, leaving only nodal velocities as solution unknowns. A series of 4-node and 8-node quadrilateral elements are derived and examined. Numerical examples demonstrating their characteristic behaviors are also included.

A continuum approach to high velocity flow in a porous medium

A general macroscopic linear momentum balance equation is derived in the form of a constitutive relation for high velocity fluid flow in a porous medium. It shows the nonlinearity in Forchheimer's formula for nonDarcy flow arising primarily from microscopic inertial phenomena, and expresses the inertial force in terms of the macroscopic velocity in an anisotropic and nonlinear manner. The point of departure is Euler's first law of motion, valid at any point in the fluid phase which is assumed to completely occupy the void space. The geometry of the void space, i.e., of the solid matrix, is taken as arbitrary. By introducing an alternative description of the microscopic kinematic field, namely deviations of the local velocity magnitudes and directions from the macroscopic values of these quantities separately, a general macroscopic momentum equation for fluid flow in a porous medium is obtained after averaging over a REV. From the general equation, most of the established relations for nonDarcy flow can be recovered as special cases. Explicit analytic expressions are obtained for the involved inertial coefficients from which the origin and nature of nonlinear (inertial) effects for high velocity flow in a porous medium is clearly demonstrated. It is also shown that the coefficient associated with the quadratic term for nonDarcy flow is not material and is a function of the macroscopic flow. Finally, some previous results are discussed and an extension of the derived equation to include higher-order nonlinear effects, with regard to the resistivity force, is proposed.

Viscothermal theory of sound wave propagation in chemically reacting mixtures of nondiffusive fluids

Measurements of the absorption and dispersion of forced acoustic waves propagating in fluids in equilibrium have been used to determine thermostatic properties and the kinetics of reactions. Here is presented a theoretical treatment of this problem using the linearized balance equations of mass, linear momentum, and energy. The reacting fluid mixture is considered to be linearly viscous (Newtonian), Fourier heat-conducting, and nondiffusive. General results are obtained in the form of a biquadratic characteristic equation for the complex propagation variable χ = −(α+iω/c), where α is the attenuation coefficient, c is the phase speed of the progressive wave, and ω = 2πf is the angular frequency. This represents a modified form of the Kirchhoff–Langevin equation which includes the effects of viscosity and thermal conduction and is applicable to a system of coupled reactions. The reactions were treated using ’’normal modes’’ for a set of extent-of-reaction variables and the equations were necessarily symmetrized so that results for the reacting, but inviscid and nonconducting case, could be expressed in simple forms and compared to previous work. For the single reaction case, the comparison is exact; however, for the multiple reaction case, the forms derived here differ somewhat from those used by other investigators and should be noted.

Gaseous diffusion in a stressed-thermoelastic solid. Part I: The thermomechanical formulation

A phenomenological theory is proposed for the diffusion of a dilute solution of a gas in a thermoelastic solid. It is assumed that the thermomechanical behavior of the solid is unaffected by the presence of the gas. On the other hand the presence of the solid is recognized by the gas by letting certain thermomechanical variables of the solid to enter into the constitutive equations of the gas. The constitutive functionals of the gas are restricted by the principles of continuum physics. These principles are currently referred to as equipresence, material objectivity, entropy production inequality, as well as the balance equations of mass, linear and angular momentum and internal energy. By this approach, axiomatic statements on the existence of equations of state are avoided and the classical results of linear irreversible thermodynamics are obtained by further specialization of the proposed theory.