Continuum assumption

We begin our discus.sion with the top-down approach. Let F be a two or three dimensional region filled with a fluid, and let v x,t) be the velocity of a particle of fluid moving through the point x = ( r, y, z) at time t. Note that v x, t) is a vector-valued field on F, and is to be identified with a macroscopic fluid cell. The fact that we can make this so-called continuum assumption - namely that we can simultaneously speak of a velocity of a particle of fluid and think of a particle of fluid as a macroscopic cell - is not at all obvious, of course, and deserves some attention. 

The vapor-layer model developed in Section IV.A.2 is based on the continuum assumption of the vapor flow. This assumption, however, needs to be modified by considering the kinetic slip at the boundary when the Knudsen number of the vapor is larger than 0.01 (Bird, 1976). With the assumption that the thickness of the vapor layer is much smaller than the radius of the droplet, the reduced continuity and momentum equations for incompressible vapor flows in the symmetrical coordinates ( ,2) are given as Eqs. (43) and (47). When the Knudsen number of the vapor flow is between 0.01 and 0.1, the flow is in the slip regime. In this regime, the flow can still be considered as a continuum at several mean free paths distance from the boundary, but an effective slip velocity needs to be used to describe the molecular interaction between the gas molecules and the boundary. Based on the simple kinetic analysis of vapor molecules near the interface (Harvie and Fletcher, 2001c), the boundary conditions of the vapor flow at the solid surface can be given by... 

Then under the simplifying assumptions that pwi l (6=l/kT, in which T is the absolute temperature and k Boltzmarm s constant), so that multiple links parallel to the x axis are negligible, and a continuum assumption stated below, the probability p+(h) tha at any given y and time t, an elongation exceeding h (subsequently called an h exceedence) is observed can be shown to be... 

Fig. 2.1 Processes like chemical vapor deposition must sometimes consider the effects of submicron features at the deposition surfaces. When the features sizes are on the order of the mean-free-path length, then continuum assumptions can be questionable.

Detailed CFD models of fuel cells (see Chapters 3 and 4), on the other hand, use continuum assumption to predict the 3-D distributions of the physical quantities inside the fuel cells. These models are more complex and computationally expensive compared to reduced order models especially due to the disparity between the smallest and largest length scales in a fuel cell. The thickness of the electrodes and electrolyte is usually tens of microns whereas the overall dimensions of a fuel cell or stack could be tens of centimeters. Though some authors used detailed 3-D models for cell or stack level modeling, they are mostly confined to component level modeling. In what follows, we present the governing equations for some of these models. 

The Eulerian continuum approach, based on a continuum assumption of phases, provides a field description of the dynamics of each phase. The Lagrangian trajectory approach, from the study of motions of individual particles, is able to yield historical trajectories of the particles. The kinetic theory modeling for interparticle collisions, extended from the kinetic theory of gases, can be applied to dense suspension systems where the transport in the particle phase is dominated by interparticle collisions. The Ergun equation provides important flow relationships, which are useful not only for packed bed systems, but also for some situations in fluidized bed systems. 

As mentioned, the pseudocontinuum approach can be used for the multiphase flow analysis only when the control volume contains enough particles that the volume average is of statistical significance, and at the same time, the control volume is small enough to ensure the validity of the continuum assumption. Thus, in order to have a statistically meaningful phase average, there exists a minimum averaging volume. 

An instability analysis of a multiphase flow is usually based on the continuum assumption and the equations of motion of the phases. Two different approaches are common. One approach is to study the wave propagation speed from the equations of motion using an analogy to the surface wave situation. The other approach is to study the growth rate of the perturbation wave when a small perturbation wave is introduced to the system. The criterion for stability can then be derived. A perturbation wave can be expressed by... 

The basic equation (34) involves the continuum assumption and is, therefore, valid only if h is large compared to in addition,Gf should have a... 

Each of these different types of flows is governed by a set of equations having special features. It is essential to understand these features to select an appropriate numerical method for each of these types of equations. It must be remembered that the results of the CFD simulations can only be as good as the underlying mathematical model. Navier-Stokes equations rigorously represent the behavior of an incompressible Newtonian fluid as long as the continuum assumption is valid. As the complexity increases (such as turbulence or the existence of additional phases), the number of phenomena in a flow problem and the possible number of interactions between them increases at least quadratically. Each of these interactions needs to be represented and resolved numerically, which may put strain on (or may exceed) the available computational resources. One way to deal with the resolution limits and... 

The basic governing equations (2.1 to 2.10) along with appropriate constitutive equations and boundary conditions govern the flow of fluids, provided the continuum assumption is valid. To obtain analytical solutions, the governing equations are often simplified by assuming constant physical properties and by discarding unimportant... 

By use of the typical numerical value for the mean free path given above for a gas at temperature iOOK and pressure 101325Pa, this condition indicates that the continuum assumption is valid provided that the characteristic dimension of the apparatus is L > 0.001cm. Nevertheless, it is noted that at low gas pressures, say lOPa, the mean free path is increased and might be comparable with the characteristic dimensions of the apparatus. In particular, for low pressures and small characteristic dimensions we might enter a flow regime where the continuum assumption cannot be justifled. 

It follows from the continuum assumption that the integrands in (3.415) are continuous and differentiable functions, so the integral theorems of Leibnitz and Gauss (see app. A) can be applied transforming the system description into an Eulerian control volume formulation. The governing mixture... 

Jointed rock mass can be regarded as equivalent continuum media if the distribution of fractures are in uniform style, a RVE (representative volume element) can be established, and the fracture flow controls the seepage of rock masses. A hydro-mechanical model of jointed rock mass coupled with damage is developed in this paper, based on the equivalent continuum assumption. From eqations (l) (i4), a FEM formulation can be established as below... 

Consider a spherical point dipole solute molecule with a dipole moment, p, at infinite dilution in a spherical container of supercritical fluid. With a continuum assumption, the fluid s electrical properties may be represented by a homogeneous dielectric constant, e. The inhomogeneous field of the dipole polarizes the fluid which reacts and gives rise to a field, R, at the dipole. R will be proportional to p as long as no saturation effects occur. 

Applying the continuum assumption, microfluidic flows can be modeled by the continuity equation... 

Represents a correction to Stokes drag and the friction between objects due to the breakdown of the continuum assumption for very small objects. 

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