Solution of Reynolds Equation in Polar Coordinates Applicable to Nonsymmetric Entrainment Velocities

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
Kurt Beschorner ◽  
C. Fred Higgs ◽  
Michael Lovell
Keyword(s):  
Reynolds Equation ◽  
Stokes Equations ◽  
Angular Position ◽  
Technical Note ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Polar Form ◽  
Fluid Films ◽  
Navier Stokes Equations ◽  
Cylindrical Polar Coordinates

Reynolds equation in polar cylindrical (polar) coordinates is used for numerous tribological applications that feature thin fluid films in sliding contacts, such as chemical mechanical polishing and pin-on-disk testing. Although unstated, tribology textbooks and literary resources that present Reynolds equation in polar coordinates often make assumptions that the radial and tangential entrainment velocities are independent of the radial and tangential directions, respectively. The form of polar Reynolds equation is thus typically presented, while neglecting additional terms crucial to obtaining accurate solutions when these assumptions are not met. In the present investigation, the polar Reynolds equation is derived from the cylindrical Navier–Stokes equations without the aforementioned assumptions, and the resulting form is compared with results obtained from more traditionally used forms of the polar Reynolds equation. The polar form of Reynolds equation derived in this manuscript yields results that agree with the commonly used Cartesian form of Reynolds equation but are drastically different from the typically published form of the polar Reynolds equation. It is therefore suggested that the polar form of Reynolds equation proposed in this technical note be utilized when entrainment velocities are known to vary with either radial or angular position.

On the energy dissipative spatial discretization of the barotropic quasi-gasdynamic and compressible Navier–Stokes equations in polar coordinates

2018 ◽  
Vol 33 (3) ◽  
pp. 199-210 ◽  
Author(s):  
Alexander Zlotnik
Keyword(s):  
Stokes System ◽  
Stokes Equations ◽  
Physical Property ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Uniform Mesh ◽  
Viscous Stress ◽  
Spatial Discretization ◽  
Navier Stokes Equations ◽  
Viscous Stress Tensor

Abstract The barotropic quasi-gasdynamic system of equations in polar coordinates is treated. It can be considered a kinetically motivated parabolic regularization of the compressible Navier–Stokes system involving additional 2nd order terms with a regularizing parameter τ > 0. A potential body force is taken into account. The energy equality is proved ensuring that the total energy is non-increasing in time. This is the crucial physical property. The main result is the construction of symmetric spatial discretization on a non-uniform mesh in a ring such that the property is preserved. The unknown density and velocity are defined on the same mesh whereas the mass flux and the viscous stress tensor are defined on the staggered meshes. Additional difficulties in comparison with the Cartesian coordinates are overcome, and a number of novel elements are implemented to this end, in particular, a self-adjoint and positive definite discretization for the Navier–Stokes viscous stress, special discretizations of the pressure gradient and regularizing terms using enthalpy, non-standard mesh averages for various products of functions, etc. The discretization is also well-balanced. The main results are valid for τ = 0 as well, i.e., for the barotropic compressible Navier–Stokes system.

scholarly journals A class of exact solutions of equations for plane steady motion of incompressible fluids of variable viscosity in presence of body force

2018 ◽  
Vol 7 (3) ◽  
pp. 77
Author(s):  
Mushtaq Ahmed ◽  
Waseem Ahmed Khan ◽  
S M. Shad Ahsen
Keyword(s):  
Exact Solutions ◽  
Body Force ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Steady Motion ◽  
Incompressible Fluids ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
First Case

This paper determines a class of exact solutions for plane steady motion of incompressible fluids of variable viscosity with body force term in the Navier-Stokes equations. The class consists of stream function  characterized by equation , in polar coordinates ,  where ,  and  are continuously differentiable functions, derivative of  is non-zero but double derivative of  is zero. We find exact solutions, for a suitable component of body force, considering two cases based on velocity profile. The first case fixes both the functions ,  and provides viscosity as function of temperature. Where as the second case fixes the function , leaves  arbitrary and provides viscosity and temperature for the arbitrary function . In both the cases, we can create infinite set of expressions for streamlines, viscosity function, generalized energy function and temperature distribution in the presence of body force.  

INFLUENCE OF FLAT WALL DIFFUSER WALL OSCILLATION ON HYDRODYNAMIC PARAMETERS OF VISCOUS FLUID FLOW

2019 ◽  
Vol 9 (1) ◽  
pp. 119-125
Author(s):  
Evgeny A. KRESTIN
Keyword(s):  
Fluid Flow ◽  
Viscous Fluid ◽  
Stokes Equations ◽  
Hydraulic Drive ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Viscous Fluid Flow ◽  
Moving Wall ◽  
Hydrodynamic Parameters

In order to reduce the energy consumption, increase the reliability of the hydraulic drive of construction machines and mechanisms, studies of the hydrodynamic parameters of the viscous fluid flow in a flat diffuser during the oscillation of one of the walls of the channel are carried out. Navier-Stokes equations together with the continuity equation are used to construct velocity and pressure fields. The problem is solved in polar coordinates with boundary conditions. The General solution of the problem, which corresponds to the self-similar boundary condition on the moving wall, is obtained. The radial velocity profile has sections of forward and reverse currents and is a standing wave along the angular coordinate. The forces acting on the movable and stationary walls of the diffuser are determined.

Singular treatment of viscous flow near the corner by using matched eigenfunctions

2018 ◽  
Vol 233 (5) ◽  
pp. 1660-1676 ◽  
Author(s):  
Ali Deliceoğlu ◽  
Ebutalib Çelik ◽  
Fuat Gürcan
Keyword(s):  
Stokes Equations ◽  
Careful Analysis ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Flow Topology ◽  
Navier Stokes Equations ◽  
Upper Lid ◽  
Eigenfunction Method ◽  
Numerical And Analytical Methods ◽  
The Stokes Equation

In this paper, the local singular behavior of Stokes flow is solved near the salient and re-entrant corners by the matching eigenfunction method. The flow in a rectangular and an L-shaped cavity are considered as a model for the flow generated by the motion of the upper lid. The solutions of the Stokes equation in polar coordinates are matched with a velocity vector components obtained by analytic or numerical solution for the streamfunction developed for any values of the heights of the rectangular and an L-shaped cavity. Streamline patterns near the corner are simulated for a different aspect ratio A. The techniques are tested on a flow problem undergoing Stokes or Navier–Stokes equations in a square cavity. It is seen that the method appears to be cheaper and more accurate than the numerical and analytical methods. It is expected that the study will lead to useful insights into the understanding of the flow topology near a re-entrant corner from a combined analytical-numerical method. Attention is then focused on the topological behavior near the re-entrant corner of the L-shaped cavity. Careful analysis of the streamlines of streamfunction near the re-entrant corner by using wall shear stress allows us to give a possible flow bifurcation of dividing streamline.

scholarly journals A NOTE ON NAVIER–STOKES EQUATIONS WITH NONORTHOGONAL COORDINATES

2018 ◽  
Vol 59 (3) ◽  
pp. 335-348 ◽  
Author(s):  
J. M. HILL ◽  
Y. M. STOKES
Keyword(s):  
Coordinate System ◽  
Stokes Equations ◽  
Flat Space ◽  
Curvilinear Coordinates ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Curvilinear Coordinate System ◽  
Navier Stokes Equations ◽  
Curvilinear Coordinate ◽  
Stress Boundary Conditions

There are many fluid flow problems involving geometries for which a nonorthogonal curvilinear coordinate system may be the most suitable. To the authors’ knowledge, the Navier–Stokes equations for an incompressible fluid formulated in terms of an arbitrary nonorthogonal curvilinear coordinate system have not been given explicitly in the literature in the simplified form obtained herein. The specific novelty in the equations derived here is the use of the general Laplacian in arbitrary nonorthogonal curvilinear coordinates and the simplification arising from a Ricci identity for Christoffel symbols of the second kind for flat space. Evidently, however, the derived equations must be consistent with the various general forms given previously by others. The general equations derived here admit the well-known formulae for cylindrical and spherical polars, and for the purposes of illustration, the procedure is presented for spherical polar coordinates. Further, the procedure is illustrated for a nonorthogonal helical coordinate system. For a slow flow for which the inertial terms may be neglected, we give the harmonic equation for the pressure function, and the corresponding equation if the inertial effects are included. We also note the general stress boundary conditions for a free surface with surface tension. For completeness, the equations for a compressible flow are derived in an appendix.

Effects of stator splitter blades on aerodynamic performance of a single-stage transonic axial compressor

2020 ◽  
Vol 14 (4) ◽  
pp. 7369-7378
Author(s):  
Ky-Quang Pham ◽  
Xuan-Truong Le ◽  
Cong-Truong Dinh
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Aerodynamic Performance ◽  
Numerical Results ◽  
Single Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Operating Stability ◽  
Length Coverage

Splitter blades located between stator blades in a single-stage axial compressor were proposed and investigated in this work to find their effects on aerodynamic performance and operating stability. Aerodynamic performance of the compressor was evaluated using three-dimensional Reynolds-averaged Navier-Stokes equations using the k-e turbulence model with a scalable wall function. The numerical results for the typical performance parameters without stator splitter blades were validated in comparison with experimental data. The numerical results of a parametric study using four geometric parameters (chord length, coverage angle, height and position) of the stator splitter blades showed that the operational stability of the single-stage axial compressor enhances remarkably using the stator splitter blades. The splitters were effective in suppressing flow separation in the stator domain of the compressor at near-stall condition which affects considerably the aerodynamic performance of the compressor.

Sign in / Sign up

Export Citation Format

Share Document