Navier-Stokes equations corrections

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

Turbulent inlet conditions for LES are difficult to obtain since a time-resolved flow description is required. The best solution is to use periodic boundary conditions when it is possible. For the remaining cases, there are algorithms for simulation of turbulent eddies that fit the theoretical turbulent energy distribution. These simulated eddies are not a solution of the Navier-Stokes equations, and the inlet boundary must be located outside the region of interest to allow the flow to adjust to the correct physical properties. 

A number of authors from Ladenburg (LI) to Happel and Byrne (H4) have derived such correction factors for the movement of a fluid past a rigid sphere held on the axis of symmetry of the cylindrical container. In a recent article, Brenner (B8) has generalized the usual method of reflections. The Navier-Stokes equations of motion around a rigid sphere, with use of an added reflection flow, gives an approximate solution for the ratio of sphere velocity in an infinite space to that in a tower of diameter Dr ... 

Clearly the assumption of a flat velocity profile is not correct. For a film in steady, laminar motion one may obtain an expression for the velocity distribution from the Navier-Stokes equations of motion [Eq. (9)]. For this case the Navier-Stokes equations simplify to... 

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,... 

Harlow and Nakayama (H2), noting that the length scale will be used to determine the dissipation, proposed a closure model for the exact differential equation for S> derivable from the Navier-Stokes equations. They experimented with the use of this equation in MTEN closures. The Los Alamos group (private communication) has now abandoned the MTE closure in favor of MRS closures, which also use the SD equation for inference of length scales [see Eq. (61) ]. They refer to the SD equation as a dissipation equation, which as we have noted is not strictly correct. 

Kelkar, K.M. and Patankar, S.V. (1989), Development of generalized block correction procedure for the solution of discretized Navier-Stokes Equation, Comput. Phys. Commun., 53, 329-336. 

As we have seen in the previous sections, the friction term in the Navier-Stokes equation (3.98) may not be neglected for large Reynolds numbers Re — oo, if we wish to correctly describe the flow close to the wall, and satisfy the no-slip condition. The region in which the friction forces may not be neglected compared to the inertia forces is generally bounded by a very thin zone close to the wall, as... 

The Enskog [24] expansion method for the solution of the Boltzmann equation provides a series approximation to the distribution function. In the zero order approximation the distribution function is locally Maxwellian giving rise to the Euler equations of change. The first order perturbation results in the Navier-Stokes equations, while the second order expansion gives the so-called Burnett equations. The higher order approximations provide corrections for the larger gradients in the physical properties like p, T and v. 

Another consequence of the integral theorem (8-111) is that we can calculate inertial and non-Newtonian corrections to the force on a body directly from the creeping-flow solution. Let us begin by considering inertial corrections for a Newtonian fluid. In particular, let us recall that the creeping-flow equations are an approximation to the full Navier-Stokes equations we obtained by taking the limit Re -> 0. Thus, if we start with the ftdl equations of motion for a steady flow in the form... 

By removing the inconsistency, Oseen thus placed Stokes result on a firmer mathematical footing. In effect, Oseen s solution provides a uniformly valid zero-order approximation to the solution of the Navier-Stokes equations throughout all space. Unfortunately, a stronger interpretation than this was ascribed to Oseen s solution. Prevailing opinion at the time, and indeed for many years after, held that Oseen s solution was, in fact, an asymptotically valid solution of the Navier-Stokes equations to 0(R). This led to the well-known Oseen correction of Stokes law, ... 

Proudman and Pearson obtained Oseen s result, Eq. (200). That they obtained the same result as did Oseen is, however, fortuitous. For Oseen s original velocity field does not agree to 0(R) with the correct asymptotic solution of the Navier-Stokes equations. Rather it is correct only to 0(R°). 

Stokes law is an analytic solution of the Navier-Stokes equation for the simplified flow case with solid particles and creeping flow. If the particles are fluid and in the absence of surface-active components, internal circulation inside the particle will reduce the drag. (Note that this is not necessarily valid for small fluid particles, but these are irrelevant in gravity separation.) The viscosity correction term for this case is given in Eq. (9). From this equation it can be seen that, for large viscosity differences between the dispersed and continuous phases, the settling will approach the Stokes velocity or 3/2 Stokes velocity (the two limiting... 

This corrected second-order boundary cmidition was used to the Navier-Stokes equations for confined fluids at the micro- and nanoscale. 

It is clear that the general procedure used to derive the Navier-Stokes equations can be used to obtain the corrections to the hydrodynamic equations to higher order in the uniformity parameter. The order equations— the... 

These are the so-called stick boundary conditions that are usually imposed when the Navier-Stokes equations are solved. For this reflection mechanism, the normal solution breaks down near the walls already at order . As a result the kinetic boundary layer in the stick case is of order /u., whereas it is of order /i, in the slip case. The existence of such a boundary layer leads to corrections of order /i in the boundary conditions (124). The boundary conditions for the tangential velocity and the temperature at the wall actually become of the form ... 

The normal solution method just outlined leads to two principal results, both of which can be tested experimentally. These are (i) explicit expressions for the coefficients of viscosity 17 and thermal conductivity A for dilute monatomic gases, in terms of the intermolecular potential < (r), and (ii) an explicit form for the Burnett and higher-order corrections to the Navier-Stokes equation, together with expressions for the associated (higher-order) transport coefficients in terms of the intermolecular potential. 

It is clear from the above discussion that surface properties are extremely important in microscale systems and their importance grows as the characteristic channel dimension decreases. However, there is no straightforward way to take these effects into account, with the models developed to describe this phenomenon being problem specific. Generally they are based on a combination of classical solutions of the Navier-Stokes equations, coupled with ad hoc models of molecular slip flow. Therefore, in the simulation of microchannel flows, it is important to keep in mind that the use of the no-slip boundary condition may not be appropriate and that additional physics may need to be included in the modeling to capture the correct behavior. 

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