Balance laws

We shall always assume isothermal conditions and therefore ignore thermal effects. In these circumstances, as in any classically based continuum theory, conservation laws for mass, linear momentum and angular momentum must hold. The balance law for linear momentum, given below, is basically similar to that for an isotropic fluid, except that the resulting stress tensor (to be derived later) need not be symmetric. The balance law for angular momentum is also suitably augmented to include explicit external body and surface moments. 

For a volume V of nematic liquid crystal bounded by the surface S the three conservation laws for mass, linear momentum and angular momentum are, respectively. 

We can apply Reynolds transport theorem, as stated in Appendix B, to the mass conservation law (4.29). If we set 5 = p in equation (B.7) then the equation for mass conservation leads to the relation 

Since is an arbitrary volume, we can write this result in point form as 

Prom the definition of the material derivative (4.5), this result can be recast into the more familiar form known as the equation of continuity [159, p.2] 

The Maxwell transport equation can be derived from (4.5) requiring that V (c) is a function of c only. With (4.20), the Maxwell transport equation can be written as  

Taking = m in (4.59) yields the local form of the continuity equation for the granular material  

It is customary to introduce a total pressure tensor defined as the sum of the kinetic and collisional pressure tensors  

The second last term on the RHS denotes kinetic pressure and the last term the convective rnomenmm flux. From (4.28) it is immediately concluded that 2 (me) = 0 because the me is a summational invariant (i.e., in a particle collision the momentum is conserved). 

---

Herrman, W., Balance Laws for Deformable Continua, Sandia National Laboratories Report No. SAND81-1757, Albuquerque, NM, 43 pp., August 1981. 

In the late 1970 s, the country s mood changed to accept not just some risk, but to include in the decision equation, specific consideration for economic factors, social benefits and impact on jobs. And so, the Clean Air Act was amended in 1977, changing the thrust of the law to include economic and practical considerations in carrying out the mandate of that law. And so too, the TSCA law that finally passed in late 1976, was a "balancing law," the first really true balancing law to be passed by Congress. It con-... 

The derivation of the mixture-balance laws has been given by Chapman and Cowling for a binary mixture. Its generalization to multicomponent mixtures, as in Equation 5-1, uses a determination of the invariance of the Boltzmann equation. This development has been detailed by Hirschfelderet These derivations were summarized in the notes of Theodore von Karmin s Sorbonne lectures given in 1951-1952, and the results of his summaries were stated in Pinner s monograph. For turbulent flow, the species-balance equation can be represented in the Boussinesq approximation as ... 

The balance laws (4)-(5) are to be supplemented by constitutive relations we express these in terms of the internal energy. We presume that the mixture is... 

For each compartment in the configuration, apply the mass-balance law to obtain the differential equation expressing the variation of amount per unit of time. In these expressions, constant or variable fractional flow rates k can be used. 

Jenkins, J.T, and Mancini, F. (1987), Balance laws and constitutive relations for plane flows of a dense binary mixture of smooth, nearly elastic circular disks, J. Appl. Meek, 54, 27. 

General physical laws often state that quantities like mass, energy, and momentum are conserved. In computational mechanics, the most important of these balance laws pertains to linear momentum (when reckoned per unit volume, linear momentum may be expressed as the material density p times velocity v). The balance equation for linear momentum may be considered as a generalization of Newton s second law, which states that mass times acceleration equals total force. As we saw in the previous section, stresses in a material produce tractions, which may be considered as internal forces. In addition, external forces such as gravity may contribute to the total force. These are commonly reckoned per unit mass and are usually referred to as body forces to distinguish them from tractions, which may be considered as surface forces. For a one-dimensional motion, balance of linear momentum requires that (37,38)... 

In addition to the momentum balance equation (6), one generally needs an equation that expresses conservation of mass, but no other balance laws are required for so-called purely mechanical theories, in which temperature plays no role (as mentioned, balance of angular momentum has already been included in the definition of stress). If thermal effects are included, one also needs an equation for the balance of energy (that expresses the first law of thermodynamics energy is conserved) and an entropy inequality (that follows from the second law of thermodynamics the entropy of a closed system cannot decrease). The entropy inequality is, strictly speaking, not a balance law but rather imposes restrictions on the material models. 

The balance laws described in the previous section are assumed to be valid for all materials. They do not, however, completely specify the mechanical response in the sense... 

The equations expressing these balance laws are, by themselves, insufficient to uniquely define the system, and statements on the material behavior are also required. Such statement are termed constitutive relations or constitutive laws. 

We have shown that all the balance laws for Eulerian CVs can be cast in the same standard form, applicable in any coordinate system. The coordinate system is chosen to proceed with the solution for the problem in question in a convenient way. 

For each balance law, the values of -0, J and 4> defines the transported quantity, the diffusion flux and the source term, respectively, v denotes the velocity vector, T the total stress tensor, gc the net external body force per unit of mass, e the internal energy per unit of mass, q the heat flux, s the entropy per unit mass, h the enthalpy per unit mass, u>s the mass fraction of species s, and T the temperature. 

Jenkins JT, Mancini F (1987) Balance Laws and Constitutive Relations for Plane Flows of a Dense, Binary Mixture of Smooth, Nearly Elastic, Circular Disks. Journal of Applied Mechanics 54 27-34... 

Forces and Balance Laws 2.3.1 Forces Within Continua Stress Tensors... 

The free—energy function (Equation i) is related to the mass action laws, and the constraints (Equation ii) are simply the mass balance laws. See Ref. [5],... 

The computation of chemical composition involves two types of problems. The first is to find an initial feasible solution—that is, one that satisfies the mass balance laws with all x s positive. The second is to find an optimal solution that satisfies both mass balance and mass action laws, using the initial solution as a starting point. Control card SIMPLE creates an initial feasible solution, and control card SOLVE finds an optimal solution. However, it is not necessary for the control card user to use the SIMPLE card for a solution, since the SOLVE instruction automatically does this if it is needed. The only time the SIMPLE card would be used would be when one wanted a feasible solution but not an optimal one, which seldom happens. 

Balance laws (3.67) are known to exhibit shock waves, i.e. solutions w with discontinuities of the (negative) tangent angle w at shock positions s t). See Figure 3.12 and, for some qualitative remarks, also [62]. The Rankine-Hugoniot condition specifies the shock speed s t) to satisfy... 

Fig. 3.12. Collision of spirals and shocks in solutions w of balance laws (3.67).

The balance laws of linear momentum in local form for constituents are... 

---