Equations Reynolds

As of this writing, the only practical approach to solving turbulent flow problems is to use statistically averaged equations governing mean flow quantities. These equations, which are usually referred to as the Reynolds equations of motion, are derived by Reynold s decomposition of the Navier-Stokes equations (18). The randomly changing variables are represented by a time mean and a fluctuating part ... 

These extra turbulent stresses are termed the Reynolds stresses. In turbulent flows, the normal stresses -pu, -pv, and -pw are always non-zero beeause they eontain squared veloeity fluetuations. The shear stresses -pu v, -pu w, -pv w and are assoeiated with eorrelations between different veloeity eomponents. If, for instanee, u and v were statistieally independent fluetuations, the time average of their produet u v would be zero. However, the turbulent stresses are also non-zero and are usually large eompared to the viseous stresses in a turbulent flow. Equations 10-22 to 10-24 are known as the Reynolds equations. 

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 failure of TFL only means a loss of mobility here, but monolayers can stay on solid surfaces to separate the solid surfaces in relative motion, and subsequently sustain a feasible boundary lubrication state [10]. Because the film thickness of TFL is of the nano scale or molecular order, from a mechanical point of view, TFL is the last one of the lubrication regimes where the Reynolds equation can be applied. 

Being compared to conventional Reynolds equations, /12 can be regarded as a modification coefficient of the micropolar effects on viscosity, and its effects are shown in Fig. 8. This shows that the microstructure and microrotation will add an increase in lubricant viscosity. When the ratio hH increases, the viscosity enhancement decreases further increasing the ratio, the modiflcation approaches unit. Because I is related to the molecular size, and h is the film gap, this means that if the problem scale is much larger than the molecular dimension, microrotation and the microstructure of particles will contribute msignrhcantly to the macroscopic properties. The larger N is, the more the increase is, as also evidenced by Fig. 8. 

Singh, C., and Suiha, P., The Three-Dimensional Reynolds Equation for Micropolar-Fluid Lubricated Bearings, Wear, Vol. 76, No. 2,1982,pp. 199-209. 

Modified Reynolds Equations for Ultra-Thin Film Gas Lubrication... 

In spite of the different assumptions adopted in derivation, all of the modified Reynolds equations presented in the literature up to now has the same form as below... 

Among these models, the Fukui-Kaneko model is regarded as the most accurate one because it is derived from the linearized Boltzmaim equation. However, the flow rate coefficient, Qp, in the Fukui-Kaneko model, is not a unified expression, as defined in Eqs (6)-(8). This would cause some inconveniences in a practical numerical solution of the modified Reynolds equation. Recently, Huang and Hu [17] proposed a representation of Qp, as shown in Eq (10), for the whole range of D from 0.01 to 100, by a data-fitting method, to replace the segmented expressions of the Fukui-Kaneko model... 

For steady state lubrication, the general modified Reynolds equation can be written as... 

The main task of head-disk gas lubrication analysis is to calculate the air pressure distributions for various known slider surface structures and given working conditions. In the following calculation examples, the modified Reynolds equation derived by Fukui and Kaneko was used, and the flow rate coefficient Qp was calculated by using the Eq (10). 

In order to analyze the dynamic behavior of a magnetic head, time-dependent pressure should be calculated. This can be attained by solving the modified Reynolds equation including a squeezing term, i.e.. 

Mitsuya, Y., Modified Reynolds Equation for Ultra-thin Film Gas Lubrication Using 1.5-order Slip-Flow Model and Considering Surface Accommodation Coefficient," ASME J. Tri- [25] fcoZ.,Vol. 115,1993, pp. 289-294. 

Huang, P., Niu, R. J., and Hu, H. H., A New Numerical Method to Solve Modified Reynolds Equation for Magnetic [31] Head/Disk Working in Ultra Thin Gas Films, Sci. China,... 

Li, W. L. and Weng, C. I., "Modified Average Reynolds Equation for Ultra-Thin Film Gas Lubrication Considering Roughness Orientations at Arbitrary Knudsen Numbers, Wear, Vol. 209,1997,pp. 292-230. 

Instead of dividing the computation domain into lubrication regions and asperity contact areas, the mixed lubrication model proposed by the present authors assumes that the pressure distribution over the entire domain follows the Reynolds equation ... 

In this model, there will be no asperity contacts in the traditional sense, but as the film thickness between the interacting asperities decreases below a certain level, the right-hand terms in Eq (9), which represent the lubricant flow caused by pressure gradient, become so insigniheant that the pressure can be predicted by a reduced Reynolds equation [16,17] ... 

An effective viscosity rp has been introduced in the Reynolds equation to describe the non-Newtonian lubricant properties. Ignoring the variation of viscosity across the film thickness, one may evaluate the effective viscosity via the following rheological model that considers a possible shearthinning effect [19],... 

The approach has proven to be advantageous in comparison with the conventional" method. Because the reduced equation is a special form of the Reynolds equation, a full numerical solution over the entire computation domain, including both the hydrodynamic and the contact areas, thus obtained through a unified algorithm for solving one equation system. In this way, both hydrod5mamic and con-... 

This chapter describes a DML model proposed by the authors, based on the expectation that the Reynolds equation at the ultra-thin film limit would yield the same solutions as those from the elastic contact analysis. A unified equation system is therefore applied to the entire domain, which gives rise to a stable and robust numerical procedure, capable of predicting the tribological performance of the system through the entire process of transition from full-film to boundary lubrication. 

Yang, P. and Wen, S., A Generalized Reynolds Equation for Non-Newtonian Thermal Elastohydrodynamic Lubrication," ASMEJ. Tribol,MoX.Wl, 1990,pp.631-636. 

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