Variational geometric approach to the thermodynamics of porous media

Abstract

Many applications of porous media research involves high pressures and, correspondingly, exchange of thermal energy between the fluid and the matrix. While the system is relatively well understood for the case of nonmoving porous media, the situation when the elastic matrix can move and deform, is much more complex. In this paper, we derive the equations of motion for the dynamics of deformable porous media, which includes the effects of friction forces, stresses, and heat exchanges between the media, by using the new methodology of variational approach to thermodynamics. This theory extends the recently developed variational derivation of the mechanics of deformable porous media to include thermodynamic processes and can easily include incompressibility constraints. The model for the combined fluid-matrix system, written in the spatial frame, is developed by introducing mechanical and additional variables describing the thermal energy part of the system, writing the action principle for the system, and using a nonlinear, nonholonomic constraint on the system deduced from the second law of thermodynamics. The resulting equations give us the general version of possible friction forces incorporating thermodynamics, Darcy-like forces, and friction forces similar to those used in the Navier-Stokes equations. The equations of motion are valid for arbitrary dependence of the kinetic and potential energies on the state variables. The results of our work are relevant for geophysical applications, industrial applications involving high pressures and temperatures, food processing industry, and other situations when both thermodynamical and mechanical considerations are important.

Variational and thermodynamically consistent finite element discretization for heat conducting viscous fluids

“…We briefly explain how the variational formulation ( 9)-( 22)-( 23)-( 12) yields the equations ( 14) together with the conditions in (15), with the last one replaced by (24). We proceed similarly as in §B.2 before.…”

“…The last condition in (15) is the insulated boundary condition. We refer to [23,16,24] for the use of this type of variational formulation for modelling purposes in nonequilibrium thermodynamics.…”

“…The first equation in (14) follows exactly as before. Now the kinematic constraint ( 22), together with (89) and the first condition in (91) give both the second equation in (14) and the boundary condition (24).…”

Variational Methods for Fluid-Structure Interactions

“…In particular, our approach includes the solid phase as the porous matrix, takes into account the dynamics of the fluid in the variational principle, and incorporates thermodynamic effects to derive the consistent expressions for transitions between solid and broken components. Our work will be based on the variational approach to the thermodynamics porous media without damage [18], which, in turn, is based on the variational derivation of porous media motion taking into account purely mechanical effects [19,20]. It is unrealistic to provide a detailed description of the different models of porous media here, thus, we refer the reader to the reviews [21,22] and in particular to the fundamental treatises [23,24] for historical introduction and the background of different approaches to porous media modeling.…”

“…That description in [19] was based on the classical Arnold description of incompressible fluid as geodesic motion [41]. The thermodynamics effects were included in the variational principle in [18], which allowed for the conservation of total energy and the use of second law of thermodynamics to derive thermodynamically consistent laws of motion generalizing the Darcy-Brinkman model of porous media [42][43][44][45]. The present paper builds on these considerations to model the breaking of the porous media using a variational approach incorporating both mechanical and thermodynamics effects.…”