Inviscid fluid

The inviscid fluid, though an idealized system, yields a suitable reference frame in which real fluids can easily be taken up. This model is in use since the 18th century. It has achieved the first step for the extension of classical mechanics to deformable 

In the particular case of incompressible fluids the continuity equation, given by the relation 

This equation means that, as the magnetic field in electrodynamics, the velocity vector has no sources in incompressible fluids. 

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Emery, A.F., An Evaluation of Several Differencing Methods for Inviscid Fluid-Flow Problems, Sandia Corporation, Livermore Laboratory Monograph No. SCL-DC-66-78, Livermore, CA, 57 pp., March 1967. 

X = 10 4 r0 = initial bubble radius tcou = collapse time scale, for an inviscid fluid, collapse occurs at t = 0.915 xco ... 

Nonlinear viscous fluid (non-Newtonian) Linear viscous fluid (Newtonian) Inviscid fluid (Pascalian)... 

Euler expressed Newton s second law of motion for a frictionless (inviscid) fluid along a streamline as... 

Pressure is the normal stress for an inviscid fluid. The shear stress for a Newtonian fluid is given as... 

A flowing fluid is required to do work to overcome viscous frictional forces so that in practice the quantity W0 is always positive. It is zero only for the theoretical case of an inviscid fluid or ideal fluid having zero viscosity. The work W, may be done on the fluid by a pump situated between points 1 and 2. 

For an inviscid fluid, ie frictionless flow, and no pump, equation (1.10) becomes... 

The force due to the movement of the liquid surrounding the bubble is m (dt>ldt). For a sphere moving in an infinite medium of an inviscid fluid, the mass of the liquid m is equal to half the mass of the displaced liquid. The authors, however, assumed merely a direct proportionality between m and the mass of the displaced fluid, instead of the above relationship, because they considered their flow not to be irrotational. 

The equations for the constant flow rate have already been derived in Section IV for various situations. For inviscid fluids with surface tension, the equation is reproduced below. 

Equations (89)-(92) have been verified by the authors, both for the final overall values and for the influence of individual variables. Their typical results for inviscid fluids are presented in Fig. 15, which shows a good agreement between the theoretical and the experimental values. The authors have obtained higher bubble volumes than those of Davidson and Schuler (D9) under otherwise similar conditions. This is probably due to absence of liquid circulation in the present condition. 

The above discussion dealt with only that particular situation where the continuous phase approximated to an inviscid fluid. However, the equations thus derived can be easily modified to include the effects of viscosity of the of the continuous phase. Under constant pressure conditions also, viscosity of the continuous phase tends to increase the bubble volume by increasing the drag during both the expansion and detachment stages. 

If the viscosity term is removed, then these equations reduce to the ones given earlier for inviscid fluids. Thus, these general equations are applicable to inviscid fluids, both with and without the surface tension at the orifice tip. 

From the above discussion, it is seen that the bed acts as an inviscid fluid without surface tension, and that the bubble formation takes place under constant flow conditions. Thus, the theories which are applicable to the above conditions should also be applicable to bubble formation in fluidized beds. 

This model is a modification of the model developed by Kumar and Kuloor (K18) for bubble formation in inviscid fluids in the absence of surface-tension effects. The need for modification arises because the bubble forming nozzles actually used to collect data on bubble formation in fluidized beds differ from the orifice plates in that they do not have a flat base. Under such conditions the bubble must be assumed to be moving in an infinite medium and the value of 1/2 is more justified than the value 11/16. 

The model proposed for constant flow conditions is again used here to explain the data. The modified equations for inviscid fluids are ... 

In practice all real fluids have nonzero viscosity so that the concept of an inviscid fluid is an idealization. However, the development of hydrodynamics proceeded for centuries neglecting the effects of viscosity. Moreover, many features (but by no means all) of certain high Reynolds number flows can be treated in a satisfactory manner ignoring viscous effects. 

The components of this equation are sometimes referred to as the Navier-Stokes equations when the viscosity is set equal to zero (inviscid fluid), then these equations reduce to the Euler equations. 

Figure 6.22 illustrates the solution to this problem for several Reynolds numbers. The boundary layer forms near the inlet boundary, owing to the axial no-slip condition. The inner portions of the flow (i.e., near the centerline) tend to behave as an inviscid fluid, as evidenced by the linear v profile. As expected, the boundary layer thins as the Reynolds number increases. 

A SECOND GRADIENT MODEL FOR DEFORMABLE POROUS MATRICES FILLED WITH AN INVISCID FLUID... 

We analyze, within a linearized second gradient theory, the static infinitesimal deformations of an annular porous cylinder filled with an inviscid fluid and with the inner and the outer surfaces subjected to uniform external pressures pj xt and iff1 respectively. We assume that surface tractions on the inner and the outer surfaces of the cylinder, in the reference configuration, equal -po and postulate that... 

For solids discharge of the fluidized spout, the interconnecting aperture represents the main resistance to solids flow. By treating the solids emulsion as an inviscid fluid, as proposed by Jones and Davidson (1965) and recommended by Leung (1980), the solids discharge is likened to the efflux... 

The Beverloo relation for sohds discharge may be contrasted with the mass flow rate of an inviscid fluid from an opening of area A, or... 

Wallis GB (1989) Inertial Coupling in Two-Phase Flow Macroscopic Properties of Suspensions in an Inviscid Fluid. Multiphase Science and Technology 5 239-... 

As a bubble rises, the particles move aside, as would an incompressible inviscid fluid having the same bulk density as the whole bed at incipient fluidization Ps(l - Smf) + PgSmf-... 

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