Both methoas will be used in deriving the reynolds equation, and the assumptions inherent in reducing the navierstokes equations will be specified. He has combined navierstokes equations with continuity equation to generate a. 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. Each term in the transport equations is examined to see whether it complies with the einstein summation convention for curvilinear coordinates. The polar coordinate system is a twodimensional coordinate system in which each point p on a plane is determined by the length of its position vector r and the angle q between it and the positive direction of the xaxis, where 0 reynolds equation is a partial differential equation that describes the flow of a thin lubricant film between two surfaces. Navier stokes equation in curvilinear coordinate systems 1. However, entrainment velocities are known to vary with angle for some cases of pinondisk contact and chemical mechanical polishing cmp. Jim and bob are racing from the origin to the point 5, 10.
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. An alternative form of reynolds equation arises when using the angular coordinate whose origin is at the location of maximum film thickness. The details of the solution of stokess equation for the streamfunction. Bobs position at any time t is given by the parametric equations x 5t, y 10t. Find the slopes of the lines tangent to the cardioid r 1 sin. Let t be the number of seconds after the start of the race. 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. The graph of an equation in polar coordinates is the set of points which satisfy the equation.
The polar coordinate system describes the location of a point denoted as r. D rans equations of pipe flow as another example, we consider turbulent. We use cylindrical polar coordinates rather than cartesian and assume vanishing reynolds number. Our rst example focuses on the some of the more structurally simple polar equations. One of the fundamental results in low reynolds hydrodynamics is the stokes solution for steady. Solution of reynolds equation in polar coordinates. 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. 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.
Polar coordinates introduction and equations of lines. It should not be confused with osborne reynolds other namesakes, reynolds number and reynoldsaveraged navierstokes equations. 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. Coriolis force effects are included in the derivation of the model equation. A reynolds equation modelling chemical mechanical polishing in cylindrical polar coordinates is derived.
Partial differential equation switching to polar coordinates. The apllicatiuon range widely form the determination of electron charges to the physics of aerosols. Rans equations in curvilinear coordinates to begin with, the reynoldsaveraged navierstokes rans equations are presented in the familiar vector and cartesian tensor forms. Rans equations in curvilinear coordinates to begin with, the reynolds averaged navierstokes rans equations are presented in the familiar vector and cartesian tensor forms. Its graph is the circle of radius k, centered at the pole. Polar coordinates, parametric equations whitman college.
One of the fundamental results in low reynoldsnumber hydrodynamics is the stokes solution for steady. One of the fundamental results in low reynolds number hydrodynamics is the stokes solution for steady. Here are a set of practice problems for the parametric equations and polar coordinates chapter of the calculus ii notes. Jims position at any time t is given by the parametric equations x t, y 2t. Consider a solid body with surface z hx,y,t close to a solid plane at z 0. Lagrangian and euler coordinates kinematics is the description of motion without regard to forces. Figuring out where the equations for converting from rectangular to polar come from and then converting a bunch of lines. Corresponding expressions for classical reynolds equation follow. How long does it take each competitor to finish the race. Transformation between polar and rectangular coordinates of. 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.
The navierstokes equations in vector notation has the following form 8. For problems 5 and 6 convert the given equation into an equation in terms of polar coordinates. On solving the reynolds equation, the oil film force is always obtained in polar coordinates. We begin by considering the motion of a uid particle in lagrangian coordinates, the coordinates familiar from classical mechanics. It is derived from the navierstokes equations and is one of the fundamental equations of the classical lubrication theory. 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. Let us see how to convert the polar to cartesian coordinate and vice versa. But there is another way to specify the position of a point, and that is to use polar coordinates r. The equation defining an algebraic curve expressed in polar coordinates is known as a polar equation.
Advanced fluid dynamics 2017 navier stokes equation in. Calculus ii parametric equations and polar coordinates. 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. We would like to be able to compute slopes and areas for these curves using polar coordinates. Another key issue that was the focus of a great deal of research efforts on lubrication was the. 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. Solution of reynolds equation in polar coordinates applicable. The results are compared with the reynolds only model and navierstokes model using openfoam cfd code. Notes on the solution of stokess equation for axisymmetric. Homogeneous differential equation in polar coordinates. Integration in polar coordinates polar coordinates polar coordinates are a di. Derivation of reynolds equation in cylindrical coordinates. Hot network questions when did jack swigert realize that he forgot to file his taxes.
We consider an incompressible, isothermal newtonian flow density. 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. Polar coordinate system an overview sciencedirect topics. Notes on the solution of stokess equation for axisymmetric flow in spherical polar coordinates. 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. We will look at polar coordinates for points in the xyplane, using the origin 0.
How long will it take for the chain to slide off the. Determine a set of polar coordinates for the point. 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. 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. Department of chemical and biomolecular engineering. In cylindrical coordinates with the components of the velocity vector given by, the continuity equation is 18. Applications range widely from the determination of electron charges to the physics of aerosols. Polar coordinate system, equation of a line in polar form. Transformation between polar and rectangular coordinates. Lubrication theory flow in thin films it is an observed fact that thin layers of. Chapter 6 equations of motion and energy in cartesian. Pressure and body forces balance each other and at steady state the equation of. 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. Therefore, the transformation between the polar and rectangular coordinates.
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