Governing conservation equations

In order to study polymer flows in processing equipment, it is essential to consider first the governing flow equations. The flow of incompressible fluids (such as polymer solutions and melts, at least in situations where they are considered as incompressible for pressures below 100 MPa) is governed by the conservation equations of mass, momentum, and energy [2, 6, 21], that is. 

The above system of conservation equations is usually called the Navier-Stokes equations in fluid mechanics. 

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Equations (88), (89), and (91) constitute a useful set of governing conservation equations for characteristic analyses of one-dimensional, unsteady flows. 

The governing conservation equations and the boundary conditions in non-isothermal laminar tube flow can be formulated as follows ... 

The governing conservation equations in the inertial frame of reference were solved with the boundary conditions of temperature and velocity ... 

The conservation equations (2.202), (2.207) and (2.213) are rigorous (i.e., for mono-atomic gases) consequences of the Boltzmann equation (2.185). It is important to note that we have derived the governing conservation equations without knowing the exact form of the collision term, the only requirement is that we are considering summation invariant properties of mono-atomic gases. That is, we are considering properties that are conserved in molecular collisions. 

The preceding sections in this chapter deal with the kinetic theory of dilute gases summarizing the statistical modeling concepts, deriving the governing conservation equations and fairly accurate relations determining the transport coefficients from first principles. 

The governing conservation equations developed here lead to a well-posed system of partial differential equations, on specification of the appropriate boundary conditions. It can be noted here that boundary conditions corresponding to DNA hybridization are already incorporated through specification of the source term / , (Eq. 21) for control volumes adjacent to the channel-fluid interface and need not be duplicated in prescription of boundary conditions. Other pertinent boundary conditions are summarized in T able 1. 

Substitution of expansions (4, 5, and 6) into the governing conservation equations and the EOS and retention of leading terms results in the equations of linear acoustics in the form... 

Equations (16.14)-(16.16) are solved using COMSOL Multiphysics [32]. This simulation environment solves the governing conservation equations using the flnite element method [33]. It also readily accommodates introduction of the appropriate form for the permeation flux and its dependence on gas partial pressure. 

List governing (conservation) equations, which apply to the physical problem—mass, momentum, energy, charge, species, etc. 

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