On the derivation of reynolds equation emerald insight. It should not be confused with osborne reynolds other namesakes, reynolds number and reynoldsaveraged navierstokes equations. In deriving this equation the thermal, compressibility, viscosity variation, slip at the surface, inertia and surface roughness effects were ignored. Consider a stream tube in an onedimensional flow as sketched. 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. The navierstokes equations this equation is to be satis. In what follows, we describe several approaches turbulence models for solving the rans equations. Finite element analysis of the reynolds lubrication.
It should not be confused with osborne reynolds other namesakes, reynolds number and reynolds averaged navierstokes equations. Contrary to 4 where, starting from the full navierstokes equations, we derived rigorously the classical reynolds equation for lubrication of a. At low reynolds numbers, flows tend to be dominated by laminar sheetlike flow, while at high reynolds numbers turbulence results from differences in the fluids speed and direction, which may sometimes intersect or even move counter to the. 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. Since the term only appears due to the reynolds average, it is called reynolds stress or apparent turbulent shear stress. The uid is homogeneous, incompressible and newtonian. Rigorous derivation of the generalized reynolds equation from.
This is the continuity or mass conservation equation, stating that the sum of the rate of local density variation and the rate of mass loss by convective. Derivation of twodimensional nonnewtonian reynolds equation. Reynolds equation and applications applied tribology. The load number was found to be inversely proportional to n.
Article information, pdf download for analytical solution of reynolds equation. Derivation of the reynolds equation for lubrication of a rotating shaft. We study the reynolds equation, describing the ow of a lubricant, in case of pressuredependent viscosity. Lee mechanical and aerospace engineering, arizona state university, tempe, az, 85287 abstract navierstokes equations are difficult to solve, and also the reynolds stress that arises in the reynoldsaveraged navierstokes rans equations is associated with the closure. A rigorous derivation of the stationary compressible.
Reynolds equation is the governing equation that describes the pressure distribution in a hydrodynamic film. Derivation of reynolds equation cfd online discussion forums. The mechanics of a vanishingly thin fluid film at the interface between two moving surfaces has been described by the reynolds equation in a broad range of lubrication problems with indisputable success 2,3. Analysis of the reynolds equation for lubrication in case of.
The derivation of the classical reynolds equations for a flow between two plain surfaces was. The equations of motion and navierstokes equations are derived and explained conceptually using newtons second law f ma. In developing a mathematical theory of boundary layers, the rst step is to show the existence, as the reynolds number rtends to in nity, or the kinematic viscosity tends to zero, of a limiting form of the equations of motion, di. 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.
It then reflects tremendous insight into fluid behavior in bearing lubricant films, which then gave birth to the science of hydrodynamic lubrication. Subsequently the incompressible equations are derived and. Using the asymptotic expansion in the stokes equations, we show the existence of singular perturbation phenomena whenever the two surfaces are in relative motion. Reynolds number formula derivation reynolds number for. A solution method for the reynoldsaveraged navierstokes equation t. If you want to learn differential equations, have a look at differential equations for engineers if your interests are matrices and elementary linear algebra, try matrix algebra for engineers if you want to learn vector calculus also known as multivariable calculus, or calculus three, you can sign up for vector calculus for engineers. Mathematics, numerics, derivations and openfoam v5. Based on the fact that engineering applications are mostly turbulent, the reynolds averaging methods are presented and explained. Derivation of the reynolds averaged navierstokes equations part 2.
Analysis of the reynolds equation for lubrication in case. Ordinary and partial differential equations by john w. Solution of reynolds equation for the short journal bearings. Then, the equation was solved with regular perturbation method, obtaining the analytical solution. A full derivation of the reynolds equation from the navierstokes equation can be found in.
Since the twodimensional nonnewtonian reynolds equation of sliding bearings is important in investigating the lubrication performances of hydrodynamic bearings, a further derivation is useful for engineering. Pdf derivation of the reynolds equation for lubrication of. A generalized form of reynolds equation derived by dowson 6 can be deduced from eqn. Analysis of lubricant flow through reynolds equation comsol. Analytical solution of reynolds equation under dynamic. 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. So, first this slide shows reynolds equation, which was derived in previous lecture, left hand side shows pressure term. Jun 05, 2009 a form of reynolds equation is derived in this manuscript from the navierstokes equations without entrainment velocity assumptions. In this paper, unsteady form of reynolds equation was simplified and nondimensionalized at first, based on certain assumptions. Then, we study the asymptotic behavior of the solution in case of periodic roughness via homogenization method. Since then many works on the solutions of the equation have been published.
The applied load causes the center line of the journal to be displaced from the center line. Peters 1 and loren sumner 2 1 physics and 2 mechanical engineering departments. The validity and accuracy of the compressible reynolds equation in predicting the local pressure in a gaslubricated, textured parallel slider bearing is studied by comparing the local bearing pressure obtained with the reynolds equation and the navierstokes equations for a single dimple textured bearing model. 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. Since the publication of reynolds seminal work, the reynolds equation has been the central, driving formulation behind theoretical and computational lubrication analysis. Keywords reynolds equation, journal bearings, perturbation, analytical. The general form of reynolds equation mit opencourseware. Modified reynolds equation for steady flow through a curved pipe. A new extended reynolds equation for gas bearing lubrication. Derivation of reynolds averaged navier stokes youtube. Mathematics, numerics, derivations and openfoam v5 holzmann cfd. Some interesting nonlocal effects appear due to the nonlinearity. Solving the equations how the fluid moves is determined by the initial and boundary conditions.
Reynolds equation is the basis of lubrication theory. This is important, because increased mixing and shearing occur in. Reynolds equation an overview sciencedirect topics. Pdf derivation of the reynolds equation for lubrication of a. A general finite volume method for the solution of the. We prove that the reynolds equation is an approximation of the stokes equations and that the kind of. As stated in the introduction to the rans approach, to lead to the closure of the equation system, an approximation for the reynolds. Derivation and solution reynolds equation is a partial differential equation that describes the flow of a thin lubricant film between two surfaces. Derivation of the reynoldsaveraged navierstokes equations.
The transition between the stokes equations and the. Derivation of twodimensional nonnewtonian reynolds equation and application to powerlaw film slider bearings. Description and derivation of the navierstokes equations. This equation describes reynolds transport theorem rtt. Their derivation focused on the poiseuille flow rate and no analytical. The unique book of holzmann cfd namely mathematics, numerics, derivations and openfoam build for the openfoam community. The fluid pressure does not vary across the film thickness and fluid inertia effects are ignored. The instataneous quantities of u, derivation of reynolds equation cfd online discussion forums. Pdf a rigorous derivation of the stationary compressible.
Here and in the sequel the differential operators div and are taken only. The navierstokes equations classical mechanics classical mechanics, the father of physics and perhaps of scienti c thought, was initially developed in the 1600s by the famous natural philosophers the codename for physicists of the 17th century such as isaac newton. Derivation of reynolds equation in three dimensions. Derivation of reynolds transport theorem enggnotes. Solution of reynolds equation for the short journal. A rigorous derivation of the stationary compressible reynolds equation via the navierstokes equations article pdf available in mathematical models and methods in. Another version of the reynolds number uses velocity, pipe id in inches and kinematic viscosity. Nov 27, 2014 derivation of the reynolds averaged navierstokes equations part 2.
Rigorous derivation of the generalized reynolds equation. Consider a liquid flowing through a thin film region separated by two closely spaced moving surfaces. A rigorous derivation of the stationary compressible reynolds equation via the navierstokes equations article pdf available in mathematical models and methods in applied sciences may 2017. Formulation of the reynolds equation on a timedependent. The following is not a rigorous derivation of reynolds number, for that would defeat the purpose for the paper. A survey of the compressible navierstokes equations desjardins, benot and lin, chikun, taiwanese journal of mathematics, 1999. Pressures separate surfaces o normal loads on bodies o convergent profile between surfaces o tangential motion between surfaces o viscous effects generate shear stresses o pressures equilibrate shear stresses o surfaces lift apart. The lecture presents the derivation of the reynolds equation of classical lubrication theory. 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.
Although the most rigorous derivation of the conservation of momentum equations also stems from the general form continuity equation formed above, a quicker and nearly as rigorous derivation can be done using newtons laws and an application of the chain rule. The reynolds and energy equations fundamentals of tribology. So putting all that together, the extensive property of the closed system can be expressed in terms of a control volume as shown below. As known, the pressure distribution in bearing clearance can be described with reynolds equation. This value is obtained by comparing the inertial force with the viscous force. A solution method for the reynolds averaged navier stokes. Overall, the proposed formulation essentially addresses lubrication interfaces undergoing large deformations and a priori satisfies all objectivity requirements, neither of which are features of the classical reynolds equation. The blowup criterion for the compressible navierstokes system with a yukawapotential in the critical besov space chikami, noboru, differential and integral equations, 2014. Differential equations hong kong university of science. The governing equation of hydrodynamic lubrication, the reynolds equation, was established in 1886 1. Analytical derivation and experimental evaluation of short bearing. Hello, reading a text book about the derivation of the reynolds equation i did not understand the following point.
When an object is placed in an initially uniform flow, a portion of the fluid approximate volume l 3 is forced to redirect around the object, causing the flow to accelerate upstream and. Reynolds equation article about reynolds equation by the. The derivation of this equation is somewhat simpler than equation 2. The brinkmanextended darcy equations and stokes equations are utilized to model the flow in the porous region and fluid film region, respectively. Apr 01, 2014 the validity and accuracy of the compressible reynolds equation in predicting the local pressure in a gaslubricated, textured parallel slider bearing is studied by comparing the local bearing pressure obtained with the reynolds equation and the navierstokes equations for a single dimple textured bearing model. Article pdf available january 2000 with 1,204 reads.
Analytical solution of reynolds equation under dynamic conditions. 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. Rigorous derivation of the generalized reynolds equation from the boltzmann equation. A complete summary of the basic equations needed for the computational fluid dynamics and much more. The convergence is proved using a truncated hilbert. Reynolds equation is a partial differential equation that describes the flow of a thin lubricant film between two surfaces. Derivation of the reynolds averaged navierstokes equations part 1 duration. The goal of this work is to formulate the reynolds equation with respect to such a continuously evolving surface.
Safa, in tribology and dynamics of engine and powertrain, 2010. On the incompressible euler equations and the blowup problem chae, dongho, 2007. The various other forms of reynolds equations can also be derived from the generalized eqn. V d q m v d a m a v v d m d the reynolds re number is a quantity which engineers use to estimate if a fluid flow is laminar or turbulent. Pdf derivation of the reynolds equation for lubrication. The reynolds number re is an important dimensionless quantity in fluid mechanics used to help predict flow patterns in different fluid flow situations. The reynolds equation is a partial differential equation governing the pressure distribution of. The accuracy of the compressible reynolds equation for. May 09, 2017 derivation of reynolds averaged navier stokes qiqi wang.
The stress jump boundary condition at the porous mediafluid film interface and effects of viscous shear are included in deriving the modified reynolds equation. Derivation of reynolds equation 2 derivation of reynolds equation 3 2 2 dy d u dx dp k 7. All equations are summed up with a one page summary at the end. It is derived from the navierstokes equations and is one of the fundamental equations of the classical lubrication theory. The fundamentals of the ebfvm and their applications to the modi. Although such a derivation has been carried out for dilute gases, a corresponding exercise for liquids remains an open problem.
The reynolds equation is used to calculate the pressure distribution in a thin layer of lubricant film between two surfaces. Two case studies, related to pinondisk and cmp, are presented and results from the derived form of reynolds equation are compared with results from the traditionally used form. When the surfaces are deformable, as in rolling contact bearings or human and animal joints, the equations of elasticity and the pressure dependence of. In additional, prandtl separately derived through experiment the following equation of an exponential function as the velocity distribution of a turbulent flow in a. In the derivation of the load number, n 1 the substitution u. 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 reynolds equation is a partial differential equation governing the pressure distribution of thin viscous fluid films in lubrication theory. This chapter begins by presenting the derivation of the classical reynolds equation based on the work of osborne reynolds. First we prove the existence and uniqueness of the solution. The intent of this article is to highlight the important points of the derivation of msi k8n neo4 manual pdf the navierstokes equations as well as the application and formulation for different. We provide a rigorous derivation of the compressible reynolds system as a singular limit of the compressible barotropic navierstokes system on a thin domain.
Reynolds transport theorem and continuity equation 9. The classical reynolds equation can be used to describe the pressure. Derivation of modified reynolds equationa porous media. Cope 2 modified the reynolds equation by including viscosity and density variation along the fluid film. 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. This equation is considered applicable only in the neighbourhood of the wall from the viewpoint of its derivation. Thus, study of hydrodynamic lubrication is from a mathematical. In particular, the existence of solutions to the navierstokes system with nonhomogeneous boundary conditions is shown that may be of independent interest. Derivation of twodimensional nonnewtonian reynolds. Derivation of reynolds equation in cylindrical coordinates. Derivation of modified reynolds equationa porous media model.
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