Reynolds equation wikipedia the free encyclopedia pdf the reynolds equation is a partial differential equation governing the pressure distribution of thin viscous fluid films in lubrication theory. This is the note prepared for the kadanoff center journal club. This is important, because increased mixing and shearing occur in. Learn more about reynolds number formula and solved examples. The subject is mainly considered in the limit of incompressible flows with constant properties. A solution method for the reynolds averaged navierstokes equation t. When the surfaces are deformable, as in rolling contact. Derivation of the navier stokes equations i here, we outline an approach for obtaining the navier stokes equations that builds on the methods used in earlier years of applying m ass conservation and forcemomentum principles to a control vo lume. Reynolds equation article about reynolds equation by the. Apr 25, 2016 advantages of navier stoke equation equation is based on simple newtons second law so therefore it is not a empirical equation.
While deriving this equation reynolds neglected fluid inertia and gravitational. Reynolds equation wikipedia the free encyclopedia pdf. The following is not a rigorous derivation of reynolds number, for that would defeat the purpose for the paper. Fast methods to numerically integrate the reynolds equation. Derivation of the continuity equation the reynolds transport theorem states that. This value is obtained by comparing the inertial force with the viscous force. Derivation of twodimensional nonnewtonian reynolds equation. Lee mechanical and aerospace engineering, semte, arizona state university, tempe, az, 85287 abstract using the lagrangian transport of momentum, the reynolds stress can be expressed in terms of basic turbulence parameters. Some interesting nonlocal effects appear due to the nonlinearity. Pdf derivation of the reynolds equation for lubrication of. 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. Bernoulli equation is also useful in the preliminary design stage.
Fluid property which is proportional to amount of mass extensive property. The uid is homogeneous, incompressible and newtonian. Derivation of the stokes drag formula in a remarkable 1851 scientific paper, g. Chapter 1 derivation of the navierstokes equations 1. At a location downstream from the junction before the two streams completely merge, the. Then, we study the asymptotic behavior of the solution in case of periodic roughness via homogenization method. The reynolds averaged navierstokes equations or rans equations are timeaveraged 1 equations of motion for fluid flow. Very near the wall, the fluid motion is smooth and laminar, and molecular conduction and shear are important. This, together with condition of mass conservation, i. Pressureviscosity behaviour at the highest pressure within the contact is necessary for the calculation of traction.
Pdf on aug 19, 2019, eduard marusicpaloka and others published analysis of the reynolds equation for. 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. Consider a stream tube in an onedimensional flow as sketched. The terms in the equation are velocity,pressure and body force which can be easily determine for a fluid so therefore the equation are fairly simple to use. The interpretation of inertial to viscous forces comes from looking at the nondimensional equation. In deriving this equation the thermal, compressibility, viscosity variation, slip at the surface, inertia and surface roughness effects were ignored. Finite difference method to solve reynolds equation youtube.
Formulation of the reynolds equation on a timedependent. Derive reynold s equation apply reynolds equation to bearing 1. 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. It should not be confused with osborne reynolds other namesakes, reynolds number and reynoldsaveraged navierstokes equations. Description and derivation of the navierstokes equations. Another version of the reynolds number uses velocity, pipe id in inches and kinematic viscosity. Box 86 carlisle, pennsylvania 170086 august 16, 2001 preface and dedication this paper is a form of plagiarism for it contains few new thoughts. There is one way to do that, which ends up being the reynolds number.
The instataneous quantities of u, v, w and p are split up in an averaged part and a fluctuating part. The derivation makes no assumptions of reference trajectory or vehicle symmetry. Hello, reading a text book about the derivation of the reynolds equation i did not understand the following point. Reynolds number formula definition, formula and solved examples. In the next lecture we shall nd an example which has within it a di usion equation. The equations of motion and navierstokes equations are derived and explained conceptually using newtons second law f ma. The fluid pressure does not vary across the film thickness and fluid inertia effects are ignored. The general form of reynolds equation mit opencourseware. Analysis of the reynolds equation for lubrication in case of. Stokes first derived the basic formula for the drag of a sphere of radius ra moving with speed uo through a viscous fluid of density. Consider a liquid flowing through a thin film region separated by two closely spaced moving surfaces. A solution method for the reynoldsaveraged navierstokes. On an inconsistency in the derivation of the equations of.
U correlation between system controlvolume for comprehensive. Solution of reynolds equation for the short journal. 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. Solving the equations how the fluid moves is determined by the initial and boundary conditions. The navierstokes equations in vector notation has the following form 8.
First solutions of lubrication theory were obtained by reynolds himself and can be found in the original work. The derivation of this equation is somewhat simpler than equation 2. Deformable control volumes and we now apply the reynolds transport theorem eqn. In 1886, within the wake of a classical experiment by beauchamp tower 7, he developed an equation celebrated as. Hydrodynamic lubrication of rough surfaces diva portal.
We study the reynolds equation, describing the ow of a lubricant, in case of pressuredependent viscosity. Reynolds number formula definition, formula and solved. Thus, study of hydrodynamic lubrication is from a mathematical. According to the classic lubrication theory, the pressure distribution in the annular region of a squeeze film damper is determined by using reynolds equation, where it is assumed that the inertial forces are negligible relative to viscous forces i. Since then many works on the solutions of the equation have been published. The traditional approach is to derive teh nse by applying newtons law to a nite volume of uid. 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. An important parameter that indicates the relative importance of viscous and inertial forces in a given situation is the reynolds number. The reynolds equation can be derived either from the navierstokes and continuity equations or from first principles, provided of course that the samebasic assumptions are adopted in each case. 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. Derivation and definition of a linear aircraft model. It involves mathematical analysis, it involves a merger or we say marriage of. The reynolds transport theorem u correlation between system lagrangian concept controlvolume eulerian concept for comprehensive understanding of fluid motion. Pdf analysis of the reynolds equation for lubrication in case of.
When the surfaces are deformable, as in rolling contact bearings or human and animal joints, the equations of elasticity and the pressure dependence of. Finite difference method to solve reynolds equation duration. 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. Two rivers merge to form a larger river as shown in figure 2. The reynoldsaveraged navierstokes equations or rans equations are timeaveraged equations of motion for fluid flow. Recalling the definition of the local pressureviscosity coefficient. 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 governing equation of hydrodynamic lubrication, the reynolds equation, was established in 1886 1. In fluid mechanics, the reynolds number re is a dimensionless quantity that is. Since the publication of reynolds seminal work, the reynolds equation has been the central, driving formulation behind theoretical and computational lubrication analysis.
Sommerfeld 1 solved this equation by neglecting the axial flow the hypothesis of an infinitely long bearing. Navier stoke equation and reynolds transport theorem. In everyday practice, the name also covers the continuity equation 1. The reynolds equation is an equation with partial derivatives of second order and of elliptical type that does not possess analytical solutions, except for a few simple cases. Reynolds equation is a partial differential equation that describes the flow of a thin lubricant film between two surfaces. One of the most important analytical solutions of the hydrodynamic theory of the most interest in tribology field was obtained by martin in 1916. 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 various other forms of reynolds equations can also be derived from the generalized eqn. Oct 11, 20 control volume and reynolds transport theorem. Because the mean operator is a reynolds operator, it has a.
We combine with this with an integrated analysis of the flow in the vertical direction, and the use of what is commonly called the kinematic boundary condition at. Close form solution of reynolds equation can not be obtained therefore finite elements method is used to solve it. On the ground of the rabinowitsch fluid model, a nonlinear twodimensional nonnewtonian reynolds equation of slider bearings will be derived for engineering application. Derivation and solution reynolds equation is a partial differential equation that describes the flow of a thin lubricant film between two surfaces. Reynolds transport theorem and continuity equation 9. Peters 1 and loren sumner 2 1 physics and 2 mechanical engineering departments. V and let the net force given by the right hand side of 1. 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. As stated in the introduction to the rans approach, to lead to the closure of the equation system, an approximation for the reynolds. If you inspect the magnitude of the terms, namely the convective or inertial term and the viscous term, the role of the number should be obvious. The reynolds number units in a dimensionless number.
The accuracy of the compressible reynolds equation for. The derivative describes the change in velocity as the element moves in space. An example of onedimensional powerlaw curved film slider bearings lubricated with dilatant and pseudoplastic fluids is guided to illustrate the application of the nonlinear. Pdf a generalized reynolds equation based on nonnewtonian. Reynolds equation an overview sciencedirect topics. Reynoldsaveraged navierstokes equations for turbulence modeling. 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. The equation is solved with the assumption that there is no lubricant flow in the axial direction infinitely long bearing assumption.
Derivation of reynolds equation cfd online discussion forums. A precise study of asymptotic behavior of the newtonian flow in a thin domain. The navierstokes equations september 9, 2015 1 goal in this lecture we present the navierstokes equations nse of continuum uid mechanics. Derivation of reynolds equation 2 derivation of reynolds equation 3 2 2 dy d u dx dp k 7. The momentum equation the derivation of the momentum equation in di. The lecture presents the derivation of the reynolds equation of classical lubrication theory. Indeed, if we derive a formal asymptotic expansion of the.
Lubrication theory, reynolds equation, rotation number. Harish hirani, department of mechanical engineering, iit delhi. The ratio, called the reynolds number 3 can be thought of as a comparative measure of inertial and viscous diffusive effects within the flow field. Pdf derivation of the reynolds equation for lubrication. Euler equation and navierstokes equation weihan hsiaoa adepartment of physics, the university of chicago email. Cope 2 modified the reynolds equation by including viscosity and density variation along the fluid film. The reynolds numbers equation is the same in terms of the ratio of inertial force to viscous force, but the inertial force equation will be different. Both methoas will be used in deriving the reynolds equation, and the assumptions inherent in reducing the navierstokes equations will be specified. Apr 01, 2014 three key assumptions are made to derive the reynolds equation from the navierstokes equations. In what follows, we describe several approaches turbulence models for solving the rans equations.
Different phases of reynolds equation international journal of. Since the term only appears due to the reynolds average, it is called reynolds stress or apparent turbulent shear stress. 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. 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. 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. Salih department of aerospace engineering indian institute of space science and technology, thiruvananthapuram february 2011 this is a summary of conservation equations continuity, navierstokes, and energy that govern the ow of a newtonian uid. Solution of reynolds equation for the short journal bearings. The author is extremely indebted to the following two groups of engineers for developing its concepts fully. The reynolds equation is a partial differential equation governing the pressure distribution of thin viscous fluid films in lubrication theory. First we prove the existence and uniqueness of the solution.
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. Analytical solutions of reynolds equation exist only for certain assumptions. Reynolds equation of slider bearings will be derived for engineering application. A generalized form of reynolds equation derived by dowson 6 can be deduced from eqn.
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