Reynolds equation approximation

In 1959, Burgdorfer [39] first introduced a concept of the kinetic theory to the field of gas film lubrication. This was to derive an approximation equation, called the modified Reynolds equation, using a slip flow velocity boundary con-... 

The flow configuration of building block 3, of two non-parallel plates in relative motion, shown in Fig. 6.19, was analyzed in detail in Example 2.8 using the lubrication approximation and the Reynolds equation. This flow configuration is not only relevant to knife coating and calendering, but to SSEs as well, because the screw channel normally has constant-tapered sections. As shown in Fig. 6.19, the gap between the plates of length L is Ho and II at the entrance and exit, respectively, and the upper plate moves at constant velocity Vo-... 

R. M. Crone, M. S. Jhon, T. E. Karis, and B. Bhushan, The behavior of a magnetic sliders over a rigid-disk surface A comparison of several approximations of the modified Reynolds equations, Adv. Inform. Storage Syst. 4, 105-121 (1992). 

However, the flow through the brush layer may be ignored in a first approximation [240], whereby the thickness h, appearing in Eq. (3.85), should be identified with the aqueous core thickness, /i2 (rather than hw) [241], The aqueous core thickness is plotted in Fig. 3.36, ( ). The dramatic influence on the interpretation is better seen in Fig. 3.37, ( ). The dependence is linear down to about /itot 90 nm. Thinner films drain faster initially and later on slower than predicted by the linear dependence, i.e. by Reynolds equation. The disjoining pressure isotherm (Fig. 3.38) is no more monotonous. 

In this subsection, we begin by considering the limit of the thin-film equations in which Re < 1. In this case, the problem reduces to solving a classical lubrication problem, and a slightly modified version of the Reynolds equation, (5-79), can be used to obtain the leading-order approximation to the pressure distribution in the thin gap. 

When the dimensionless form of the thermal energy equation is compared with the dimensionless Navier-Stokes equation, it is clear that the Peclet number plays a role for heat transfer that is analogous to the Reynolds number for fluid motion. Thus it is natural to seek approximate solutions for asymptotically small values of the Peclet number, analogous to the low-Reynolds-number approximation of Chaps. 7 and 8. 

The low Reynolds number approximation of the Navier-Stokes equations (also known as Stokes equations) is an acceptable model for a number of interfacial flow problems. For instance, the typical example of drop coalescence belongs to this case. A BI method [3] arises from a reformulation of the Stokes equations in terms of BI expressions and the subsequent numerical solution of the integral equations. This technique is further described in chapter 18. 

The Reynolds equations, the continuity equation, which is turned into an equation for pressure correction [10], and the transport equations for the turbulence quantities k and e, are integrated over the respective finite volume elements resulting from the discretization of the stirred tank domain. The convection and diffusion terms in the transport equations are approximated using the hybrid-scheme of Patankar [10]. The resulting algebraic equations are then solved with the aid of the commercial CFD software PHOENICS (Version 2.1). 

In order to approximate the implicit performance function g(X), the load capacity is deduced by the integration (over the fluid film surface 9f ) of the pressure distribution (obtained by the Reynolds equation) as follows ... 

The data were obtained by a proper solution of the Reynolds equation in finite difference form, as described in Reference 3, with the thermal effects accounted for via the Couette approximation [7]. By ignoring the effect of the pressure gradients on temperature variation, the latter permits a decoupling of the energy equation from the Reynolds equation. 

In case of a linear differential equation, the so-called Correction Scheme is applicable [5]. Since the Reynolds equation is non linear, the Full Approximation Scheme (FAS) had to be used in the calculations. 

In the above studies, the squeeze film term has been neglected mainly due to the small influence of load on film thickness in the case of steady state loads. The effect of suddenly applied or step load on gear film thickness is an area in which little has been published. Vichard (1) included the squeeze film terms in the Reynolds equation, and made use of Grubin s approximation. This is a comprehensive paper which included... 

Numerical Procedure. In the computer program one main loop was employed to solve the Reynolds equation In Its approximate form. Two convergency criteria were adopted, one for the maximum difference on any nodal pressure from the previous cycle and the other for a global difference check. Several thousand nodal points were adopted for each solution and careful trials established that a local criterion of 0.001 and a global one of 0.01 resulted In excellent accuracy whilst minimizing the computer run time. 

Suppose we have an approximation of the free boundary. If we use the Dirichlet b.c. on the free boundary, we can calculate the oil stream over that boundary. We can use that oil stream to improve the initial approximation. To do that we split up the free boundary into line pieces of equal length X. Now we can discretize O in fig.2 with rhombs of side length X and split up the rhombs into two triangles (linear-conformal element) to get a discretization of the Reynolds equation. This procedure will give a very regular M matrix and the solution of the linear equations will fulfil the maximum-minimum principle. 

This paper deals with the elTect of the finite width of a foilbearing. The film is described by the 2-D Reynolds equation, the foil by a 1-D membrane equation. These equations are then put in nondimensional form. First an analytical approximation is given in the case of a large width and a large wrap angle. Then an approximative 1-D Reynolds equation is derived, which accounts for side leakage. This equation is solved numerically. Finally the 2-D Reynolds equation is solved numerically and comparisons are made. 

As a matter of fact the parabolical approximation for P can be regarded as the first 2 symmetrical terms of a Taylor series expansion, whereas the Reynolds equation is weighted with a... 

For comparing the approximated solution with the real solution the full 2-D Reynolds equation is solved also numerically. For approximation in the Y-diiection a series expansion is used. 

The pressure P can be expanded with respect to Y in several ways and also different sets of weighting functions for the Reynolds equation can be used. The choice as indicated in 4.2 seems a natural one. But with this set of basis and weighting functions, and also with other sets of polynomial functions, the resulting matrix appears to be very ill-conditioned for larger sets (N>5). For this reason Fourier series are used, so P is approximated by... 

For a large width the 2-dimensional Reynolds equation is approximated very well by a 1-dimensional one which accounts for side leakage. 

In order to generate the dimensionless variables required for the thermal network analysis, a dimensionless form of the Reynolds equation was solved using a finite difference approximation technique (5). An Isovlscous lubricant was assumed. 

These calculations concerning total time for antifoam action rely essentially on the relevance of the Reynolds equation [193] and require knowledge of pf , which is given approximately by the hydrostatic head for a foam only after drainage has ceased (see Section 1.4.1). We have already noted the limitations of the applicability of the Reynolds equation in this context. At most we can state that arguments about the overall time for antifoam action based on the use of that equation are likely to be overestimates except in the exceptional cases where the air-water rheology is consistent with rigid immobile film behavior and film sizes are so small that dimple formation is not possible (i.e., <100-micron diameter) [194],... 

There is more to formation invasion than Darcy s law q = - (k/ u) 9p/9x. Equations 16-1 to 16-6 are derived using Darcy s law, a low Reynolds number approximation to the Navier-Stokes momentum equations, in conjunetion with a requirement for mass eonservation. It is almost never correct to approach simulation by setting, say q = - (k/ p) 9p/9x = constant, to solve a problem, since this does not account for the underlying lineal, radial, or spherical geometry, or for pressure boundary conditions. Yet, this is often done Darcy fluid mechanics requires the solution of pressure boundary value problems. 

The first theoretical description of elastohydrodynamic lubrication for the fine contact problem relevant to gears that combined the Reynolds equation. Bams law, and a Hertzian contact mechanics was developed by Ertel [961] and pubhshed 10 years later by Gmbin and Vinogradova [962]. The approach was to assume a Hertzian contact to calculate the pressure in the gap and let the pressure in the lubricant before the entrance increase exponentially to match the Hertzian contact pressure curve. The pressure distribution for such a line contact between parallel cylinders is shown in Figure 9.12. With this approximation, Ertel derived an expression for the average thickness ho of the lubricant film within the gap, which was later refined by Dowson [963] ... 

Most screw channels and flight lands have shell-like geom cal configmations in which its thickness is much smaller than the other physical dimensions. Under the lubrication approximation, the generalized Reynolds equation and energy equation are described as follows ... 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

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