Viscous pressure

Notation-. T is the temperature, Vi the fluid velocity, II,j the viscous pressure tensor, Jg the heat current density, p its chemical potential, the current density of molecular species a, v J the stoichiometric coefficient (13), and Wp the speed of reaction p. 

Viscous Pressure Loss. The viscous pressure loss as described by the measured permeability of cellophane (15) is... 

Since x77 =f(z) for tapered dies, dx7,Jdz is nonzero. For reasonable taper values, they find these elastic forces to contribute less than 10% of the viscous pressure drop. They have also calculated the drag and pressure forces on the mandrel. 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

Polymer solutions may have the memory effects observed in viscoelastic phenomena. This requires additional relaxation terms in the constitutive equations for the viscous pressure tensor, which may be affected by the changes in the velocity gradient. Besides this, the orientation and stretching of the macromolecules may have an influence on the flow. 

The following summary is from Jou and Casas-Vazquez (2001). In the extended nonequilibrium thermodynamics for a binary liquid mixture, the viscous pressure tensor Pv and the diffusion flux J are considered as additional independent variables. The viscous pressure tensor, Pv, by the simplest Maxwell model, is defined by the following constitutive equation ... 

In extended nonequilibrium thermodynamics of polymer solutions, the generalized extended Gibbs equation for a fluid characterized by internal energy U and viscous pressure Pv is... 

The respective relaxation times of diffusion flux and viscous pressure tensor are... 

If Pv is the sum of various viscous pressure contributions P/, each of them with its own relaxation time t2 Eq. (14.61) becomes... 

Here, the first equation is the usual Fourier law, the second relates the viscous pressure tensor to the internal variable W, and the last is the evolution of the internal variable. The matrix of the transport coefficients /.(/ is positive definite... 

The viscous pressure drop in the continuous (wetting) phase provides the force necessary to sustain motion of a ganglion in spite of the capillary forces. Darcy s Law for the wetting phase may be written as ... 

Viscous Pressure Drop. For continuous bubble trains with perfectly mobile interfaces moving through a given Dm channel, the dynamic pressure drop in the gas (foam) phase over a single tube segment D follows from Equation 16 ... 

A qualitative prediction of the role of the capillary number in drop deformation can be obtained from the normal-stress balance. Now, if the shape is spherical, the capillary term on the right-hand side is a constant, and thus the viscous pressure and stress contributions... 

Now, if the shape is spherical, the capillary term on the right-hand side is a constant, and thus the viscous pressure and stress contributions on the left must also produce a constant value that corresponds to the jump in pressure across the interface that is due to surface tension. In general, however, the pressure and stress differences will not reduce to this very simple form, but will instead vary as a function of position on the surface of the drop. Such a nonuniform distribution of pressure and stress will tend to deform the drop. In fact, in this case, the normal-stress balance can be satisfied only if the drop deforms to a shape where the interface curvature (V n) varies in precisely the same way as the surface pressure and stress difference. 

Notice that the accumulation rate process and convective forces scale as pV /L, whereas viscous, pressure, and gravity forces scale as pV/L . If one takes the ratio of these two dimensional scaling factors, an important dimensionless number is obtained ... 

In the limit of vanishingly small Reynolds numbers, forces due to convective momentum flux are negligible relative to viscous, pressure, and gravity forces. Equation (12-4) is simplified considerably by neglecting the left-hand side in the creeping flow regime. For fluids with constant /r and p, the dimensionless constitutive relation between viscons stress and symmetric linear combinations of velocity gradients is... 

Initial lymphatics in skeletal muscle have intraluminal valves that consist of bileaflets and a funnel structure [Mazzoni et al., 1987]. The leaflets are flexible structures and are opened and closed by a viscous pressure drop along the valve funnel. In closed position, these leaflets can support considerable pressures [Eisenhoffer et al., 1995 Ikomi et aL, 1997]. This arrangement preserves normal valve function even in initial lymphatics with irregularly shaped lumen cross sections. 

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