Dissipative function

We now need to add the dissipation term. A Rayleigh dissipation function will suffice for this purpose, since the hydrodynamic interactions in the.elastomer should be well screened. Let F be a matrix such that XF has elements of the form... 

Here we no longer have molecular diffusion operating exclusively ( ) and the diffusion coefficient, frictional factor and chemical potential are no longer Interrelated. Also, the energy dissipation function is probably no longer quadratic. For simplicity, at unit rejection, for the steady state... 

According to the second law, the dissipation function must be positive if not zero, which of course is to be expected here, since we are dealing with a spontaneously occurring passive process. The thermodynamic force A/x+, which contains both a concentration-dependent component and an electrical component, is the sole cause of the flow J+. In a system in which more than one process occurs, each process gives rise to a term in the dissipation function consisting of the product of an appropriate force and its conjugate flow. In the case of active transport of the cation, as found, for example, in certain epithelial tissues, the cation flux is coupled to a metabolic reaction. If we represent the flow or velocity of the reaction per unit area of membrane by Jr, the appropriate force driving the reaction is... 

The kinetic constants of the system enter into the phenomenological L-coefficients, which are parameters of state. According to the reciprocity theorem of Onsager, the cross-coefficients L+r and Lr+ are identical. Now the definition of the efficiency 17 emerges directly from the dissipation function... 

The thermal energy equation now has a single term that involves the viscosity it is called the dissipation function... 

The dissipation function, also called viscous dissipation, represents the irreversible conversion of kinetic energy into thermal energy. Since the dynamic viscosity p, is positive and all the terms are squared, the first two terms of the dissipation must be always positive. The bulk viscosity can be negative the Stokes hypothesis (Section 2.11) says that k = —2p/3. It turns out that the necessary condition for the dissipation function to be positive is that... 

In this equation it still remains to write out the components of the mass-flux vector (e.g., jkZ) in terms of the appropriate composition (and possibly temperature) gradients, Section 3.5.2. Moreover the dissipation function contributes a lot of terms that must be written out in cylindrical coordinates, Eq. 3.201. 

It remains to compare Cerf s results with the one derived be Budtov and Gotlib (183). These authors made use of the following dissipation function, which is valid anly for a two-dimensional description ... 

Biot and Daughaday (B6) have improved an earlier application by Citron (C5) of the variational formulation given originally by Biot for the heat conduction problem which is exactly analogous to the classical dynamical scheme. In particular, a thermal potential V, a dissipation function D, and generalized thermal force Qi are defined which satisfy the Lagrangian heat flow equation... 

Using all these variables the relations, which form the starting point for the further calculations, can be constructed. These relations are the energy density , the dissipation function R, the Gibbs-relation and the Gibbs-Duhem relation. To illustrate the idea of our model we split up e and R into several parts according to the different origin of the variables ... 

By a similar construction we write down the dissipation function as (see Table 2 for a list of the thermodynamic variables and their conjugates)... 

The set of macroscopic hydrodynamic equations we now deal with, (16), (18)-(20), (27), and (28), follows directly from the initial input in the energy density and the dissipation function without any further assumptions. 

From the various versions of this method we will choose only one. Let V < 0 and, only at the rest point under study c, V - 0. Then let Vhave its minimum, V(c) = at the point c and for some e > Vmin the set specified by the inequality V(c0) < e is finite. Therefor any initial conditions c0 from this set the solution of eqn. (73) is c(t, k, c0) - c at t - oo. V(c) is called a Lyapunov function. The arbitrary function whose derivative is negative because of the system is called a Chetaev or sometimes a dissipative function. Physical examples are free energy, negative entropy, mechanical energy in systems with friction, etc. Studies of the dissipative functions can often provide useful information about a given system. A modern representation for the second Lyapunov method, including a method of Lyapunov vector functions, can be found in ref. 20. 

Consequently, if the law of mass/surface action is suggested from the existence of at least one PDE, then it follows that there exists a dissipation function of the composition G whose derivative equals zero only at PDEs. The product RTG has the dimensions of energy. 

We have proved that any positive PDE N is asymptotically stable in the polyhedron D (it is even a "node ). In this point constructed above the G dissipation function, a minimum of free energy is achieved and the point of minimum is unique. Whence we obtain that TV is a point of minimum G and a unique positive PDE in D. 

In the previous section we introduced the Lyapunov functions for chemical kinetic equations that are the dissipative functions G. The function RTG is treated as free energy. Since G < 0 and the equality is obtained only at PDE, and for the construction of G it suffices to know only the position of equilibrium N, there exist limitations on the non-steady-state behaviour of a closed system that are independent of the reaction mechanism. If in the initial composition N = N, the other composition N can be realized during the reaction only in the case when... 

Table 5.6 Viscous Dissipation Function for Incompressible Newtonian Fluids...

The last term is the rate of viscous energy dissipation to internal energy, Ev = jv <5 dV, also called the rate of viscous losses. These losses are the origin of frictional pressure drop in fluid flow. Whitaker and Bird, Stewart, and Lightfoot provide expressions for the dissipation function <5 for Newtonian fluids in terms of the local velocity gradients. However, when using macroscopic balance equations the local velocity field within the control volume is usually unknown. For such... 

The term fa) is the volume-averaged dissipation function for the energy dissipated by the viscous force, which is irreversible dissipation of mechanical work into thermal energy or heat. For the solid-particle phase, the kinetic energy loss by attrition or inelastic collision may be included in this term. 

What are the forces conjugate to the flux of heat, electric current, diffusion and chemical reaction if the dissipation function is used to define the forces ... 

---