Incompressible fluid, assumption

Flow of incompressible liquids through pipes is described by the mechanical energy balance (Equation 4-1) combined with the incompressible fluid assumption (Equation 4-2). The net result is... 

If we ignore inertia force and follow the conventional assumptions in liquid lubrication, motion equation (here we consider the incompressible fluids in the absence of volume force and volume momentum) is ... 

Flow through abrupt expansion Using the one-dimensional flow assumption for a single-phase incompressible fluid, the energy equation becomes... 

The separation process shown in Fignre 2.35a consists of the formation of two new interfaces, each of nnit cross-sectional area, at a location where no interface previously existed. The free energy change associated with the separation process comes directly from the definition of surface energy [Eq. (2.61)] where two snrfaces of unit surface area are formed. With appropriate assumptions regarding constant temperature, pressure, and incompressible fluids, we can equate this free energy change with the... 

Unfortunately, there is no concrete evidence that this is a correct approximation, and the dilemma remains. Nevertheless, it is a commonly made assumption and is used in most formulations of the Navier-Stokes equations. In the case of an incompressible fluid,... 

Solution This flow is z-axisymmetric. We, thus, select a cylindrical coordinate system, and make the following simplifying assumptions Newtonian and incompressible fluid with constant thermophysical properties no slip at the wall of the orifice die steady-state fully developed laminar flow adiabatic boundaries and negligible of heat conduction. 

Equation E15.2-21 can also he derived by a controlled volume approach. Consider the a element confining node m in Fig. 15.6(b) (shaded area). For an incompressible fluid and under the same assumptions as earlier we can make the following flow rate balance... 

The boundary layer equations may be obtained from the equations provided in Tables 6.1-6.3, with simplification and by an order-of-magnitude study of each term in the equations. It is assumed that the main flow is in the x direction. The terms that are too small are neglected. Consider the momentum and energy equations for the two-dimensional, steady flow of an incompressible fluid with constant properties. The dimensionless equations are given by Eqs. (6.46) to (6.48). The principal assumption made in the boundary layer is that the hydrodynamic boundary layer thickness 8 and the thermal boundaiy layer thickness 8t are small compared to a characteristic dimension L of the body. In mathematical terms,... 

This is a different order of approximation than the assumption of an incompressible fluid. 

Starting with the open system balance equation, derive the steady-state mechanical energy balance equation (Equation 7.7-2) for an incompressible fluid and simplify the equation further to derive the Bernoulli equation. List all the assumptions made in the derivation of the latter equation. 

This method has been approved as a ASTM procedure [120-122] and used in the commercial computer controlled Coulter Porometer (Coulter Electronic Ltd) for pore sizes much larger than 0.44 pm. However the theoretical basis used for the evaluation of the accumulated data neglects the specificity of gas flow in pores and incorrectly considers the flowing gas as an incompressible fluid. The assumption of gas flow dependency only on AP distorts the resulting... 

Equations 3-27 and 3-28 provide the most compact and general form of the equation of conservation of mass. These expressions are limited only by the assumption that the fluid is incompressible this assumption is reasonable for all of the problems discussed in this text. 

In the case of compressible fluids the necessary vent area is determined using the same assumptions described for incompressible fluids, except that the resulting equation includes an additional flow function / ... 

It is possible to model the deformation of film bubbles with a system of dimensionless equations that is derived according to the following assumptions [13] steady-state and axisymmetrical flow (z-axis) of an incompressible fluid thin and flat film external forces on the bubble are neglected Newtonian, pseudoplastic, or viscoelastic fluids and linear temperature profiles between die exit and freezeline position. The system of dimensionless fundamental equations can be represented, irrespective of the rheological constitutive equation used, as shown in the following equations ... 

Therefore with such (or similar) additional assumptions, the constant density for incompressible fluid mixture should be achieved and also other properties are obtained (properties from Rem. 3 based on barycentric velocity are often used [58])... 

The fact that frictional losses represent a small contribution to the overall energy balance also justifies the initial assumptions of the calculation. Since pressure drop is small, steam may indeed be treated as an incompressible fluid. The amount of work that is dissipated is correspondingly small (3.319 kJ/kg) therefore, the temperature rise that would be expected (recall that the pipe is insulated) must also be small. This temperature rise maybe estimated by assuming that the enthalpy of steam at the exit has increased by the amount of frictional losses. Interpolation in the steam tables shows that the expected temperature rise over the entire pipe length is about 4 °C. 

The following assumptions are considered in the mathematical models (1) steady-state and isothermal conditions, (2) fully developed conditions at inlet, (3) laminar flow, and (4) incompressible fluid with constant density p, viscosity p, and diffusivity D. Based on these assumptions, the governing equations can be expressed as... 

As a basic system to be studied, we consider a suspension of identical spherical particles of radius a and density p, in an incompressible fluid of density and viscosity ij,. From the very beginning, we presume that the spherical particles are involved in a chaotic fluctuating motion, and that their collisions dominate in the interparticle exchange of fluctuation energy and momentum. In particular, the last assumption implies that ... 

At elevated pressures, the Poynting correction [in Eq. (22.48)] can be integrated for incompressible fluids. This assumption cannot be applied near the critical point... 

At moderate pressures, Eq. (22.53) gives a good approximation for incompressible fluids. At high pressures and especially in the region around the critical point, this assumption no longer holds. The integral in Eq. (22.48) and Eq. (22.53) can be solved for the EOS by RKS [33] [see Eq. (22.48)], resulting in Eq. (22.55). AH basic thermodynamic data were taken from [18]. Equation (22.55) was applied for aU calculations for... 

The material flow in a thin cavity can be modeled by the Hele-Shaw model. Also a lubrication approximation can be used with assumptions such as a negligible inertial force, dominant shear stress, a thin cavity (compared with the planar dimensions), incompressible fluid, a small capillary number and a generalized Newtonian fluid (Hieber and Shen, 1980 Ilicker and Folgar,1983). 

According to G.G.Kirs [12] assumptions, for Incompressible fluid and rough surfaces possessing narrow-grooves, the following statements are reasonable ... 

In general the net macroscopic pressure tensor is determined by two different molecular effects One pressure tensor component associated with the pressure and a second one associated with the viscous stresses. For a fluid at rest, the system is in an equilibrium static state containing no velocity or pressure gradients so the average pressure equals the static pressure everywhere in the system. The static pressure is thus always acting normal to any control volume surface area in the fluid independent of its orientation. For a compressible fluid at rest, the static pressure may be identified with the pressure of classical thermodynamics as may be derived from the diagonal elements of the pressure tensor expression (2.189) when the equilibrium distribution function is known. On the assumption that there is local thermodynamic equilibrium even when the fluid is in motion this concept of stress is retained at the macroscopic level. For an incompressible fluid the thermodynamic, or more correctly thermostatic, pressure cannot be deflned except as the limit of pressure in a sequence of compressible fluids. In this case the pressure has to be taken as an independent dynamical variable [2] (Sects. 5.13-5.24). 

For equilibrium viscometric flows of incompressible fluids (the latter is a reasonable assumption for polymer melts and solutions in most engineering circumstances), the situation is brighter. With the coordinate directions assigned as described for viscometric flows in the last chapter, the shear stresses T12 and T21 are equal (or there would be a rotational flow component), and ti3 = t3i — T32 = T23 = 0, Because ofthe arbitrary nature of the static pressure definition, two independent diflerences of the deviatoric normal stresses are commonly defined ... 

The complex flow was simplified by the assumption that the screw chaimel is fully filled with a steady isothermal flow of an incompressible fluid. The Reynolds number of the flow is very small. We ignored the mass force and inertia force, since they are not to be compared with the big viscous force. 

Bearing in mind these estimates and the assumptions that [Pg.406]

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. 

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