Different phases of reynolds equation international journal of. The subject is mainly considered in the limit of incompressible flows with. Both equations 3 and 4 have the form of the general wave equation for a wave \, xt traveling in the x direction with speed v. Thus, study of hydrodynamic lubrication is from a mathematical. Finite element analysis of the reynolds lubrication. Derivation and definition of a linear aircraft model. A system is a quantity of matter of fixed identity.
Solution of reynolds equation for the short journal bearings rotation system 119 in totally hydrodynamic or fullfilm lubrication, the moving surface of the journal is totally separated from the bearing surface by a really thin film of lubricants. The uid is homogeneous, incompressible and newtonian. Download notes 2 derivation of the classical reynolds equation for. Very near the wall, the fluid motion is smooth and laminar, and molecular conduction and shear are important. The reynolds number can be defined for several different situations where a fluid is in relative motion to a surface. This dimension is a matter of convention for example radius and diameter are equally valid.
Solution of reynolds equation for the short journal. A generalized form of reynolds equation derived by dowson 6 can be deduced from eqn. The classical reynolds equation can be used to describe the pressure. Thin film fluid equation derivation for flat surfaces. This report documents the derivation and definition of a linear aircraft model for a rigid aircraft of constant mass flying over a fiat, nonrotating earth. A full derivation of the reynolds equation from the navierstokes equation can be found in numerous lubrication text books. Reynolds equation the uid ow between two solid surfaces is modeled with the reynolds equation for a incompressible uid, which is derived from the navierstokes equations using the orderofmagnitude analysis with the following assumptions. It should not be confused with osborne reynolds other namesakes, reynolds number and reynolds averaged navierstokes equations. Assuming that the continuum holds that is, the molecular mean free path of gas. The thin film fluid equation is derived from the navier stokes equation assuming uid motion on a at substrate. The general form of reynolds equation mit opencourseware. It is derived from the navierstokes equations and is one of the fundamental equations of the classical lubrication theory.
The treatment closely follows that in eckert and drake 1959. This value is obtained by comparing the inertial force with the viscous force. Reynolds transport theorem pennsylvania state university. School of aerospace engineering conservation equations. Reynolds equation is the governing equation that describes the pressure distribution in. The reynolds transport theorem rtt section 46 author.
The reynolds averaged navierstokes equations or rans equations are timeaveraged 1 equations of motion for fluid flow. Reynolds equation is a partial differential equation that describes the flow of a thin lubricant film between two surfaces. The derivation of this equation is somewhat simpler than equation 2. We prove that there exists a solution which converges, in the hydrodynamic limit, to a density maxwellian this density is obtained as the solution to the generalized reynolds equation. These definitions generally include the fluid properties of density and viscosity, plus a velocity and a characteristic length or characteristic dimension l in the above equation. This is a world with which we should be increasingly concerned.
Pdf derivation of the reynolds equation for lubrication of a. Reynolds equation an overview sciencedirect topics. Pdf rigorous derivation of the generalized reynolds. The derivation makes no assumptions of reference trajectory or vehicle symmetry. Fast methods to numerically integrate the reynolds equation for gas fluid films jt florin dimofte national aeronautics and space administration lewis research center cleveland, ohio 445 summary the alternating direction implicit adi method is adopted, modified, and applied to the reynolds equation for thin, gas fluid films. The reynolds number units in a dimensionless number. Three key assumptions are made to derive the reynolds equation from the navierstokes equations. Description and derivation of the navierstokes equations. The basic tool required for the derivation of the rans equations from the instantaneous navierstokes equations is the reynolds decomposition. Derivation and solution reynolds equation is a partial differential equation that describes the flow of a thin lubricant film between two surfaces. Reynolds number is a dimensionless value which is applied in fluid mechanics to represent whether the fluid flow in a duct or pat a body is steady or turbulent. It differs from elastohydrodynamic lubrication theory ehl due to negligible elastic deformation of the surfaces.
The various other forms of reynolds equations can also be derived from the generalized eqn. The governing equation of hydrodynamic lubrication, the reynolds equation, was established in 1886 1. The bernoullis equation for incompressible fluids can be derived from the eulers equations of motion under rather severe restrictions the velocity must be derivable from a velocity potential external forces must be conservative. Pdf in this paper, using the asymptotic expansion, we prove that the. Reynoldsaveraged navier stokes equations thermal fluids. The applied load causes the center line of the journal to be displaced from the center line. Simple derivation of electromagnetic waves from maxwells. We study the stationary boltzmann equation in a thin slab for a rarefied gas for which the molecular mean free path is comparable to the film thickness.
So, first this slide shows reynolds equation, which was derived in previous lecture, left hand side shows pressure term. Being dimensionless, the reynolds number re is a natural parameter when it comes to the navierstokes equations. Hirschberg eindhoven university of technology 28 nov 2019 this is an extended and revised edition of iwde 9206. A control volume is a region in space chosen for study. The lecture presents the derivation of the reynolds equation of classical lubrication theory.
Another version of the reynolds number uses velocity, pipe id in inches and kinematic viscosity. The load number was found to be inversely proportional to n. Absence of the elastic deformations simplifies the problem as compared to ehl theory, but allows one to construct some important analytical solutions as. Consider a liquid flowing through a thin film region separated by two closely spaced moving surfaces.
A formal mathematical expression which allows the time rate of change of an extensive property for a given quantity of mass, a system, to be expressed in terms of quantities related to a specific region of space, a control volume. Reynolds transport theorem and continuity equation 9. Read online notes 2 derivation of the classical reynolds equation for. The idea behind the equations is reynolds decomposition, whereby an instantaneous quantity is decomposed into its timeaveraged and fluctuating quantities, an idea first proposed by osborne reynolds. The convergence is proved using a truncated hilbert. Analysis of the reynolds equation for lubrication in case. While deriving this equation reynolds neglected fluid inertia and gravitational effects in. Solving the equations how the fluid moves is determined by the initial and boundary conditions. Equating the speed with the coefficients on 3 and 4 we derive the speed of electric and magnetic waves, which is a constant that we symbolize with c. Reynolds number formula definition, formula and solved. Derivation of the reynolds equation for lubrication of a rotating shaft. Lubrication approximation for fluids with sheardependent viscosity. The reynolds equation is a partial differential equation governing the pressure distribution of.
Since then many works on the solutions of the equation have been published. A generalized form of reynolds equation for fluid lubrication has been derived considering the. In the derivation of the load number, n 1 the substitution u. If mass in v is conserved, the rate of change of mass in v must be equal to. First, the boundary conditions that will be used in the derivation will be listed. Fast methods to numerically integrate the reynolds. If the thickness of the domain is small, the solution can be fairly approximated by the solution of the reynolds equations 4, 14 indeed, if we derive a formal asymptotic expansion of the solution to the system in powers of, then the solution of the reynolds equation makes the first term of the expansion see, e. The accuracy of the compressible reynolds equation for. Similarly, when the reynolds averaging is applied to the conservation equation of water vapor mixing ratio q, we obtain the reynolds averaged equations for q dq u q v q w q s dt x y z. Here and in the sequel the differential operators div and are taken only. The theory of lubrication is concerned with the solution of the reynolds equation, sometimes in combination with the equation of energy, under various lubricating conditions.
The reynolds equation is a partial differential equation governing the pressure distribution of thin viscous fluid films in lubrication theory. All books are in clear copy here, and all files are secure so dont worry about it. We present dimensionally reduced reynolds type equations for steady lubricating flows of incompressible nonnewtonian fluids with sheardependent viscosity. Reynolds number the nondimensional velocity can be defined as the ratio. Pdf derivation of the reynolds equation for lubrication. Likewise, consistent with these effects and the requirement of the particular bearing problems, it may become necessary to relax few of the assumptions used for derivation of the reynolds equation. Derivation of reynolds equation and possible solution methods are given here. When the surfaces are deformable, as in rolling contact bearings or human and animal joints, the equations of elasticity and the pressure dependence of. This nonlinear reynolds stress term requires additional modeling to close the rans equation for solving, and has led to the creation of many different turbulence models. Nevertheless, it is often a good approximation to adopt a solution to the equations of. A precise study of asymptotic behavior of the newtonian flow in a thin domain. The fluid pressure does not vary across the film thickness and fluid inertia effects are ignored. The equations of motion and navierstokes equations are derived and explained conceptually using newtons second law f ma.