The details of the solution of stokess equation for the streamfunction. A reynolds equation modelling chemical mechanical polishing in cylindrical polar coordinates is derived. Another key issue that was the focus of a great deal of research efforts on lubrication was the. We consider an incompressible, isothermal newtonian flow density. In many cases, such an equation can simply be specified by defining r as a function of the resulting curve then consists of points of the form r. There are several relationships between polar and rectangular coordinates, diagrammed in figure 1.
Rans equations in curvilinear coordinates to begin with, the reynoldsaveraged navierstokes rans equations are presented in the familiar vector and cartesian tensor forms. One of the fundamental results in low reynolds hydrodynamics is the stokes solution for steady. Consider a solid body with surface z hx,y,t close to a solid plane at z 0. The polar coordinate system describes the location of a point denoted as r. We use cylindrical polar coordinates rather than cartesian and assume vanishing reynolds number. Determine a set of polar coordinates for the point. The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation. One of the fundamental results in low reynolds number hydrodynamics is the stokes solution for steady. Transformation between polar and rectangular coordinates of.
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. Chapter 6 equations of motion and energy in cartesian coordinates equations of motion of a newtonian fluid the reynolds number dissipation of energy by viscous forces the energy equation the effect of compressibility resume of the development of the equations special cases of the equations restrictions on types of motion isochoric motion. It is derived from the navierstokes equations and is one of the fundamental equations of the classical lubrication theory. Partial differential equation switching to polar coordinates. Equation of motion of a chain a uniform chain of length a is placed on a horizontal frictionless table, so that a length b of the chain dangles over the side. A coordinate system with radial and tangential r, t axes is conveniently defined. Hot network questions when did jack swigert realize that he forgot to file his taxes. He has combined navierstokes equations with continuity equation to generate a. Lubrication theory flow in thin films it is an observed fact that thin layers of. Notes on the solution of stokess equation for axisymmetric. Polar coordinate system an overview sciencedirect topics. However, entrainment velocities are known to vary with angle for some cases of pinondisk contact and chemical mechanical polishing cmp.
Polar coordinates introduction and equations of lines. Let t be the number of seconds after the start of the race. Reynolds 1886 derived and employed an equation for the analysis of fluid film lubrication which has by now become a basic governing equation and is named after him as reynolds equation. The navierstokes equations in vector notation has the following form 8.
Chapter 6 equations of motion and energy in cartesian. However, if the reynolds number is small enough that secondary flows do not occur, then the velocity at a given value of the radius may be approximated as a. The results are compared with the reynolds only model and navierstokes model using openfoam cfd code. Polar coordinate system, equation of a line in polar form. Department of chemical and biomolecular engineering. We begin by considering the motion of a uid particle in lagrangian coordinates, the coordinates familiar from classical mechanics. 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.
If youd like a pdf document containing the solutions the download tab above contains links to pdf s containing the solutions for the full book, chapter and section. On solving the reynolds equation, the oil film force is always obtained in polar coordinates. An alternative form of reynolds equation arises when using the angular coordinate whose origin is at the location of maximum film thickness. D rans equations of pipe flow as another example, we consider turbulent. Its graph is the circle of radius k, centered at the pole. Navier stokes equation in curvilinear coordinate systems 1.
Integration in polar coordinates polar coordinates polar coordinates are a di. 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. But there is another way to specify the position of a point, and that is to use polar coordinates r. It should not be confused with osborne reynolds other namesakes, reynolds number and reynoldsaveraged navierstokes equations. Find the slopes of the lines tangent to the cardioid r 1 sin. We would like to be able to compute slopes and areas for these curves using polar coordinates. In cylindrical coordinates with the components of the velocity vector given by, the continuity equation is 18. Polar coordinates, parametric equations whitman college. We will look at polar coordinates for points in the xyplane, using the origin 0. Jims position at any time t is given by the parametric equations x t, y 2t. Elasticity problems in polar coordinates 10 i main topics a motivation b cartesian approach c transformation of coordinates d equilibrium equations in polar coordinates e biharmonic equation in polar coordinates f stresses in polar coordinates ii motivation a many key problems in geomechanics e. Jun 05, 2009 traditional tribology references typically provide the cylindrical or polar reynolds equation, which may not be applicable when entrainment velocities vary with radius andor angle. Bobs position at any time t is given by the parametric equations x 5t, y 10t.
Transformation between polar and rectangular coordinates. Here are a set of practice problems for the parametric equations and polar coordinates chapter of the calculus ii notes. The apllicatiuon range widely form the determination of electron charges to the physics of aerosols. For problems 5 and 6 convert the given equation into an equation in terms of polar coordinates. Pressure and body forces balance each other and at steady state the equation of. Polar coordinatespolar to cartesian coordinatescartesian to polar coordinatesexample 3graphing equations in polar coordinatesexample 5example 5example 5example 6example 6using symmetryusing symmetryusing symmetryexample symmetrycirclestangents to polar curvestangents to polar curvesexample 9. Applications range widely from the determination of electron charges to the physics of aerosols.
How long does it take each competitor to finish the race. Both methoas will be used in deriving the reynolds equation, and the assumptions inherent in reducing the navierstokes equations will be specified. Calculus ii parametric equations and 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 pinondisk testing. Each term in the transport equations is examined to see whether it complies with the einstein summation convention for curvilinear coordinates. This is a constant coe cient equation and we recall from odes that there are three possibilities for the solutions depending on the roots of the characteristic equation. 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. Solution of reynolds equation in polar coordinates. Lagrangian and euler coordinates kinematics is the description of motion without regard to forces. The reynolds equation is a partial differential equation governing the pressure distribution of thin viscous fluid films in lubrication theory. Traditional tribology references typically provide the cylindrical or polar reynolds equation, which may not be applicable when entrainment velocities vary with radius andor angle.
Advanced fluid dynamics 2017 navier stokes equation in. Therefore, the transformation between the polar and rectangular coordinates. Notes on the solution of stokess equation for axisymmetric flow in spherical polar coordinates. Our rst example focuses on the some of the more structurally simple polar equations. Homogeneous differential equation in polar coordinates.
How long will it take for the chain to slide off the. One of the fundamental results in low reynoldsnumber hydrodynamics is the stokes solution for steady. Corresponding expressions for classical reynolds equation follow. Coriolis force effects are included in the derivation of the model equation. Derivation of reynolds equation in cylindrical coordinates. Figuring out where the equations for converting from rectangular to polar come from and then converting a bunch of lines.
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