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The equations of the lubricant liquid

Let's consider the movement of thin layer of lubricant between surfaces and (fig. 1.1).

Fig. 1.1

Let's enter motionless orthogonal system of co-ordinates, Cartesian or curvilinear, arranging a plane between surfaces and. Let the equations of the deformed surfaces in system of co-ordinates looks like

, (1.1)

Let bodies are pressed one to another by loading and make movement owing to which in some domain filled with lubricant, there is the elevated pressure. The movement of the surface, we will describe a two-dimensional vector and a variable, where are components of the vector of movement of the point with co-ordinates along axes and accordingly. It is obvious that


We will neglect the influence of movement of the surfaces caused by their deformations, on speeds of surfaces. It means that speeds of surfaces are set by movement of bodies as rigid bodies and they are known. As a result we receive a problem about movement of a layer of lubricant between the surfaces at the known movement of surfaces.

Prominent feature of the given problem is the small size of the ratio, where is the characteristic size of a layer of liquid in the co-ordinate direction, are the characteristic sizes of domain in the directions and accordingly. Taking into account this ratio the equations of movement of liquid become [1]

,,                                           (1.3)

, , , .                                                     (1.4)

Here are components of stress tensor in the liquid, is pressure in the liquid. Approximate equalities in (1.3), (1.4) differs from exact equalities that in them neglected the members having an order and above in comparison with remained members. It follows from (1.3) that in liquid it is possible to consider pressure is not depend on co-ordinate, in other words.

Last two equations in (1.4) can be written down more compactly as follows


where vector, , are basic vectors of axes and accordingly.

The density of liquid and components of speed of liquid particle should satisfy to the equation of continuity [1]

.                                       (1.6)

On surfaces owing to lubricant adhesion to surfaces the speed of movement of liquid should coincide with speed of movement of surfaces, that is conditions should be satisfied

, (1.7)


Let's integrate the equation of continuity (1.6) on the variable from to at the fixed values. Considering (1.8) and (1.2), we receive


where .

The lubricant density changes both with pressure change and with temperature change. Most often in the theory of lubrication this dependence accept in the form of [1]


where is temperature, is lubricant density at atmospheric pressure and temperature, is temperature expansion factor, are constants. Typical values of constants are:

, , .

Temperature change on thickness of lubricant layer (on co-ordinate) has an order . Therefore relative change of density on thickness of lubricant layer equals to. Therefore the change of density of liquid on film thickness can be neglected and the equation of continuity (1.9) can be written down in a kind


where is the two-dimensional vector, which components are equal to speeds of movement of the liquid on the axes and accordingly.

In sliding bearing the temperature change in sliding direction can reach several tens of degrees Celsius. It also leads to minor alteration of density of lubricant therefore in most cases it is possible to use the formula


This formula considers only change of density of lubricant with pressure change.

In most cases the thermal emission from shear is the principal source of heat. Neglecting a thermal emission from compression of liquid and neglecting in the equation of energy members having an order and above, we receive [1]:


where is liquid temperature,are thermal capacity and heat conductivity factor of lubricant, accordingly.

The equations (1.5), (1.11), (1.12), (1.13) are fair for all types of liquid lubricants. If functions defining the thickness of lubricant layer are set, these five scalar equations include seven unknown scalar functions:

, , , .

The missing equations are the equations defining interrelation between stress tensor and strain rate tensor. In the case under consideration these dependences are reduced to dependences between vectors and. In the simplest case of Newtonian liquids this dependence looks like


whereis the viscosity of lubricant depending on pressure and temperature.

Differentiating both members of equation (1.14) on a variable, and taking into account the dependence (1.5), we receive


The right member of equation (1.15) does not depend on variable. It allows integrating it. Thus it is necessary to consider that viscosity can change on the thickness of lubricant layer owing to temperature change. Integrating the equation (1.15) and substituting the received result in the equation (1.11), we receive after simple transformations


where. The equation (1.16) is called as Reynolds's equation.

In an isothermal case the given equation becomes

.               (1.17)

The viscosity of lubricant entering into the Reynolds's equation depends both on pressure, and on temperature. For the account of dependence of viscosity from pressure two models are most often used. The most simple is model of Barus [2] according to which viscosity exponential grows with pressure growth, that is


where is the viscosity of lubricant at atmospheric pressure, is pressure coefficient of viscosity. More exact is the model of Roelands [3]

, (1.19)

where is a constant (usually).

In order to the equation (1.16) or (1.17) has the unique decision it is necessary to set initial and boundary conditions. The initial conditions are uniquely defined at specific target statement therefore we will not consider here these conditions. Concerning boundary conditions for Reynolds's equation the considerable quantity theoretical and experimental researches [4-6] are spent. Results of these researches say that at the high pressures developed in lubricant layer, the condition can be used


where is the domain wall of domainoccupied with lubricant.

The equation (1.16) (or (1.17)) together with a condition (1.20) unequivocally define function if entry conditions are set, that is and functions are set, and also the domain wall of domain changing with time is set. The domain wall can be divided into 2 parts. Through one part of domain wall lubricant arrive in a gap, and this part of domain wall, as a rule, is known. It is defined by construction of oil-circulation system and conditions of oil supply. Through the second part of domain wall lubricant follows from a gap. This part of domain wall which we designate is unknown and the additional condition is necessary for its definition. At high pressures developed in the lubricant layer, the condition (1.21) can be used.


Here is a unit normal vector to border.