Model incompressible fluids

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Iadda89] Ladd, A.J.C. and D.Frenkel, Dynamics of colloidal dispersions via lattice-gas models of an incompressible fluid, pages 242-245 in [mann89]. 

As will be outlined below, the computation of compressible flow is significantly more challenging than the corresponding problem for incompressible flow. In order to reduce the computational effort, within a CED model a fluid medium should be treated as incompressible whenever possible. A rule of thumb often found in the literature and used as a criterion for the incompressibility assumption to be valid is based on the Mach number of the flow. The Mach number is defined as the ratio of the local flow velocity and the speed of sound. The rule states that if the Mach number is below 0.3 in the whole flow domain, the flow may be treated as incompressible [84], In practice, this rule has to be supplemented by a few additional criteria [3], Especially for micro flows it is important to consider also the total pressure drop as a criterion for incompressibility. In a long micro channel the Mach number may be well below 0.3, but owing to the small hydraulic diameter of the channel a large pressure drop may be obtained. A pressure drop of a few atmospheres for a gas flow clearly indicates that compressibility effects should be taken into account. 

Two common types of one-dimensional flow regimes examined in interfacial studies Poiseuille and Couette flow [37]. Poiseuille flow is a pressure-driven process commonly used to model flow through pipes. It involves the flow of an incompressible fluid between two infinite stationary plates, where the pressure gradient, Sp/Sx, is constant. At steady state, ignoring gravitational effects, we have... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

Consider a straight tubular runner of length L. A melt following the power-law model is injected at constant pressure into the runner. The melt front progresses along the runner until it reaches the gate located at its end. Calculate the melt front position, Z(f), and the instantaneous flow rate, Q t), as a function of time. Assume an incompressible fluid and an isothermal and fully developed flow, and make use of the pseudo-steady-state approximation. For a polymer melt with K = 2.18 x 10 N s"/m and n = 0.39, calculate Z(t) and Q(t)... 

We consider a single polymer molecule in solution. The solvent is treated as a viscous incompressible fluid. The polymer chain contains N monomer units or beads, and each individual bead is regarded as a point source of friction in the solvent. This is the standard model. 

Abstract A general theoretical and finite element model (FEM) for soft tissue structures is described including arbitrary constitutive laws based upon a continuum view of the material as a mixture or porous medium saturated by an incompressible fluid and containing charged mobile species. Example problems demonstrate coupled electro-mechano-chemical transport and deformations in FEMs of layered materials subjected to mechanical, electrical and chemical loading while undergoing small or large strains. 

Equations (8-111) to (8-115) are restricted to incompressible fluids. For gases and vapors, the fluid density is dependent on pressure. For convenience, compressible fluids are often assumed to follow the ideal gas law model. Deviations from ideal behavior are corrected for, to first order, with nonunity values of compressibility factor Z (see Sec. 2, Physical and Chemical Data, for definitions and data for common fluids). For compressible fluids... 

The elastic membrane model assumes that the cell is a thin-walled sphere filled with incompressible fluid. Because the wall is thin, it may be treated as a mechanical membrane. It can be presumed that the wall cannot support out-of-plane shear stresses or bending moments. This situation is described as plane stress, as the only non-zero stresses are in the plane of the cell wall. Furthermore, the stresses can be expressed as... 

P. R. Peck, M. S. Jhon, R. F. Simmons, and T. J. Janstrom, Mathematical modeling of lubrication for the head-disk interface using incompressible fluids, J. Appl. Phys. 75(10), 5747-5749 (1994). 

This model is useful strictly for incompressible fluids only, but for small pressure changes it may be used for gas networks as well. The equations for general network simulation are based on eqn (1) and eqn (2) and a variety of models analogous to eqn (3). It goes beyond the scope of this paper to elaborate on networks with loops, this is dealt with in details by for example Mach (1), Gay and Preece (2,3) and Carnahan and Wilkes (4). A variety of... 

Let us write the model of nonstationary flow distribution as applied to the problem of search for the maximum pressure rise at a given node of the hydraulic circuit at a fast cut off of the flow in one of its branches (or the largest drop at pipe break) provided that there is an isothermal motion of viscous incompressible fluid subjected to the action of the pressure, friction, and inertia forces (Gorban et al., 2006). find... 

The isotherms for the liquid phase on the left side of Fig. 3.2 are very steep and closely spaced. Thus both (dV/dP)T and dV/dT)P, and hence both /3 and k, are small. This characteristic behavior of liquids (outside the region of the critical point) suggests an idealization, commonly employed in fluid mechanics and known as the incompressible fluid, for which /3 and k are both zero. No real fluid is in fact incompressible, but the idealization is nevertheless useful, because it often provides a sufficiently realistic model of liquid behavior for practical purposes. The incompressible fluid cannot be described by an equation of state relating V to T and P, because V is constant. 

Here we explain in brief the extended continuum model and its application to the calculation of C. We start from the Navier-Stokes equation with a position-dependent viscosity Tj(r) for a slow steady flow of an incompressible fluid given as ... 

It is known that incompressible fluids represent a useful model for real fluids in fluid mechanics and heat and mass transfer. Their thermal equation of state is v = v0 = const. For pure substances and also for mixtures, isobaric and isochoric specific heat capacities agree with each other, cp = cv = c. 

It is apparent from equations 3.2.4-3.2.7 that the determination of the concentration field is dependent on the values of the Gaussian dispersion parameters a, (or Oy in the fully coupled puff model). Drawing on the fundamental result provided by Taylor (1923), it would be expected that these parameters would relate directly to the statistics of the components of the fluctuating element of the flow velocity. In a neutral atmosphere, the factors affecting these components can be explored by considering the fundamental equations of fluid motion in an incompressible fluid (for airflows less than 70% of the speed of sound, airflows can reasonably be modeled as incompressible) when the temperature of the atmosphere varies with elevation, the fluid must be modeled as compressible (in other words, the density is treated as a variable). The set of equations governing the flow of an incompressible Newtonian fluid at any point at any instant is as follows ... 

The original VOF model designed for free surface flow simulations constitutes the mass and momentum conservation equations for incompressible fluids in the jump condition form [155, 108[. 

Obviously, if the Newtonian constitutive model for an incompressible fluid is to be consistent with the second law of thermodynamics, we require that the viscosity be nonnegative, that is,... 

The new integral term corresponds to the pressure availability, ap, of equation [3]. In the incompressible fluid model this term is readily evaluated as... 

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