Simulations via Micropolar Theory

From experimental results, the variation of film thickness with rolling velocity is continuous, which validates a continuum mechanism, to some extent in TFL. Because TFL is described as a state in which the film thickness is at the molecular scale of the lubricants, i.e., of nanometre size, common lubricants may exhibit microstructure in thin films. A possible way to use continuum theory is to consider the effect of a spinning molecular confined by the solid-liquid interface. The micropolar theory will account for this behavior. 

When the length scale approaches molecular dimensions, the inner spinning of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and 

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Size-dependent effects will be seen in fluids with micro-structure, and the thinner the gap is, the greater the effect will be. During lubrication when the him approaches molecular dimensions, the microstructure will play a dominant role. In the present paper, micropolar theory is incorporated into the theory accounting for very thin him EHL. The effects of coupling number and characteristic length are analyzed for specihc conditions. 

The conservation laws of the hydrodynamics of isotropic polar fluids (conservation of mass, momentum, angular momentum, and energy, respectively) are written as follows  

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