NUMERICAL SOLUTION OF REYNOLDS EQUATION US ING DIFFERENTIAL TRANSFORM METHOD

Ahmed f. Koura 1 , m. Elhady 2 and m.s.metwally 3 . 1. Department of Basic Science,Al-Safwa High Institute of Engineering. 2. Egyptian Space Program,National Authority for Remote Sensing & Space Science. 3. Department of Mathemat ics,Faculty of Science, Suez University . ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History Received: 09 September 2018 Final Accepted: 11 October 2018 Published: November 2018 Reynolds equation is a partial differential equation, derived from the Navier-Stokes equations.Reynolds equation is the fundamental equations of the hydrodynamic lubrication theory.SolutionofReynolds equation describes the pressure distribution of the lubricant in a journal bearing with finite length. The parameters involved in the Reynolds equation are viscosity, density and film thickness of lubricant. However, an accurate analysis of the fluid film hydrodynamics obtained using many numerical solution of the Reynolds equation. Differential Transform Method (DTM) is one of the powerful numerical methods applied to solve linear and nonlinear part ial differential equations. This study aims to apply DTM to solve Reynolds equation in partial differential form to get pressure distribut ion of journal bearing. Results obtained from the DTM compared with available solutions obtained using other numerical methods and show good agreement. The obtained results reveal that the technique used here is good, effective and convenient for such kind of problems.


ISSN: 2320-5407
Int. J. Adv. R es. 6 (11), 729-737 730 Finally, three PDE problems with constant and variable coefficients are solved by the present method . The calculated results are compared very well with those obtained by other analytical or appro ximate methods.
Jang et al. [3] present the definition and operation of the two -dimensional differential transform. A distinctive feature of the differential transform is its ability to solve linear and nonlinear differential equations. Partial differential equation of parabolic, hyperbolic, elliptic and nonlinear types can be solved by the differential transform. Hassan [4] studied the differential transformat ion technique which is applied to solve eigenvalue problems and to solve partial d ifferential equations. First,usingthe one-dimensional differential transformation to construct the eigenvalues and the normalized eigenfunctions for the differential equation of the second-and the fourth-order. Second,using the two-dimensional differential transformat ion to solve P.D.E. of the first -and secondorder with constant coefficients. In both cases, a set of difference equations is derived and the calculated results are compared closely with the results obtained by other analytical methods.
Ayaz [5] studied two-dimensional d ifferential transform method of solution of the initial value p roblem for part ial differential equations. New theorems have been added and some linear and nonlinear PDEs solved by using this method. The method can be easily applied to linear or nonlinear problems and is capable of reducing the size of computational work. Ayaz [6] introduced three-dimensional differential transform method and fund amental theorems have been defined for the first time. Moreover, as an application of two and three -dimensional differential transform, exact solutions of linear and non-linear systems of partial d ifferential equations have been investigated. The results of the present method are compared very well with those obtained by decomposition method. Differential transform method can easily be applied to linear or non -linear problems and reduces the size of computational work.
Kurnaz et al. [7] solved partial differential equations (PDEs) using the generalization of the differential transformation method to n-dimensional case. A distinctive practical feature of this method is its ability to solve especially nonlinear d ifferential equations efficiently. The results applied to a few init ial boundary-value problems to illustrate the proposed method. Hassan [8] compared the differential transformation method DTM and adomian decomposition method ADM to solve partial differential equations (PDEs). A distinctive practical feature of the differential transformation method DTM is ability to solve linear or nonlinear differential equations. Higher-order dimensional differential t ransformat ions are applied to a few some initial value problems to show that the solutions obtained by the proposed method DTM coincide with the approximate solution ADM and the analytic solutions.
Murat DUZ and UgurILTER. [9] givedifferential transforms of first, second and third derivatives of a complex function. Later, third order co mp lex equations were solved using two dimensional differential transform.Kangalg il and Ayaz [10] present a reliable algorithm in o rder to obtain exact and appro ximate solutions for the nonlinear dispersive KdV and mKd V equations with initial p rofile. The approach rest ma inly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be ach ieved by the known forms of the series solutions.
Elrod-Adams model [11][12] is not straightforwardly accomplished with the FEM formulat ion. Essentially, the main difficult ies arise in the discretization of the convective term of the modified equation, as well as in the enforcement of the flo w conservation on the cavitation boundaries throughout the lubricated contact.
Elrod and Brewe [13] developed a numerical reduction approach to solve the Reynolds equation coupled with the 2D energy equation with Dirich let boundary conditions. Temperature and flu idity (inverse of viscosity) are approximated by third order Legendre polynomials across the fluid film thickness. Elrod used Lobatto point quadrature method to discretize and calcu late the integral quantities across the film thickness. The pressure and temperature are discretized using the classical fin ite difference methods in the other directions. The method showed good agreement with classical approaches.
Elrod [14] imp roved the precision of the method by approximat ing the temperature and the fluid ity using arbitrary orders Legendre polynomials. In 2005, Moraru [15] extends the approach presented by Elrod [14] to co mpressible flu ids and takes also into account a temperature-dependent density. In his work, a 2D formu lation of the energy equation neglecting the axial heat conduction is used. In contrast to [13] and [14], the density is also approximated 731 by Legendre polynomials across the fluid film thickness. The governing partial differential equations are so lved by fin ite difference methods with upwind scheme for nu merical stability.
In 2009, Feng and Kaneko [16] used the same approach as Moraru to calculate the temperature and the pressure distributions in a multi-wound foil bearing while taking into account foil deflections. Unlike Moraru, Feng and Kaneko solved the energy equation on a 3D co mputational do main using finite d ifference methods. In 2015, Mahner et al. [17] used the reduction approach to analyze steady state performances of thrust and slider b earings operating with a compressible fluid. The authors used the Quadrature Method, the Modified Quadrature Method, Lobatto Point Collocation Method and the Galerkin Method in order to reduce number of unknowns of the discretized equations. According to the authors, all these methods yielded a significant time reduction compared to the classical methods.
Silun Zhang et al. [18] present numerical solution of the Reynoldsequation coupled with the energy transport equation. A Spectral approach named Lobatto Point Collocation Method (LPCM) is studied. The combination of LPCM with two different film rupture/reformat ion models is validated using numerical results published in theliterature in the cases of 1D slider.Sfyris and Chasalevris [19] solve the Reynolds equation for the pressure distribution of the lubricant in a journal bearing with finite length analytically. Using the method of separation of variables and compare the results with past numerical solutions.

Mathematical modeling:
The problem of the lubrication of journal bearings with fin ite length is defined in this work as the calculation of the pressure distribution of the Newtonian lubricant that is assumed to flow under laminar, isoviscous, and isothermal conditions in between the rotating journal and the static bearing. The journal o f radius R and length L b is assumed to be rotating with a constant rotational speed and to be constantly located in a point of eccentricity e with respect to the geometric center of the bearing of radius R+C r and length L b afteran application of a virtual vertical load W as shown in Fig. 1. . The dynamic v iscosity of the lubricant is assumed to be constant and equal to  through the entire control volume (notified with shadow in Fig.   1) that is defined fro m the bearing and the journal surfaces. The attitude angle of the journal is defined as 0  with respect to the vertical coordinate axis (see Fig. 1). The starting point is the equation of Reynolds which is expressed as 3) The Boundary conditions are 3) is the one that we are going to work with.
In a real application, and when The fundamental mathematical operations performed by two-dimensional differential transform method are listed in Table 1.   Results of the proposed numerical technique applied to Reynolds equation of lubrication are shown inFigs. 2and 3.The DTM numerical method results presentedshow good agreement with exact solution in [19]. There are some differences in the maximu m values of pressure but also in the domain of maximu m values.Co mparing now the DTM results with the exact analytical result. In (Fig. 2a)