The Navier-Stokes Equation in Polar Coordinates for Circular Domain Physical Problems
The polar coordinates representing a point in 3-dimensional space are called 3D polar coordinates, otherwise known as spherical coordinates.
For describing engineering problems involving fluid dynamics, the Navier-Stokes equations are utilized in different coordinate systems.
The Navier-stokes equations in polar coordinates are beneficial in numerical simulations when the physical domain is circular in shape
Physical problems involving fluid flow are common in engineering systems. Usually, the properties of the fluid and the physical system influence the behavior of the fluid flow. Some properties include temperature, pressure, velocity, viscosity, and density. When developing a mathematical model for analysis, these properties have to be defined clearly.
In thermal-fluid problems, the numerical model is governed by the Navier-Stokes equation. The Navier-Stokes equation can be expressed in cartesian coordinates, cylindrical coordinates, or polar coordinates. The Navier-Stokes equation in polar coordinates can be expressed in a three-dimensional (3D) coordinate system. The 3D polar coordinate system is otherwise called a spherical coordinate system. In this article, we will explore polar coordinates, particularly the Navier-Stokes equation in 3D polar coordinates.
In a polar coordinate system, any point (P) can be represented using an angle (θ) and a distance (r). The distance is measured from the central point, otherwise called the pole, in polar coordinate systems. The pole is the same as the origin (0,0) in the cartesian coordinate system. The distance r is the length from the pole to the point P, and θ is the azimuth angle measured from the positive x-axis to the line connecting point P to the origin. The angle θ is measured in the anti-clockwise direction.
The polar coordinates can be expressed in two-dimension or three-dimension. Polar coordinates representing a point in 2-dimensional space are called 2D polar coordinates. They are the same as the ones mentioned above, expressed as (r, θ). The polar coordinates representing the point in 3-dimensional space are called 3D polar coordinates, otherwise known as spherical coordinates.
In a 3D polar coordinate system, a point is represented as (r,ɸ,θ), where r is the distance from the origin to the point, ɸ is the horizontal azimuth angle measured on the XY plane from the X-axis in the anti-clockwise direction, and θ is the azimuth angle measured from the Z-axis. 3D polar coordinates include one additional angle ɸ to describe the point compared to the 2D polar coordinate system.
It is easy to transform polar coordinates to Cartesian coordinates and vice versa. The relationship between 2D polar coordinates and cartesian coordinates (x, y) can be given as the following equation set:
Similarly, cartesian coordinates (x,y,z) can be transformed into the 3D polar coordinates (r,ɸ,θ) using the following equation set:
Polar coordinates are used in various fields such as the basic sciences, mathematics, and engineering to represent physical problems numerically. To describe engineering problems involving fluid dynamics, the Navier-Stokes equations are utilized in different coordinate systems. Let’s discuss the Navier-Stokes equation before expressing it in polar coordinates.
The Navier-Stokes Equation
To mathematically model physical problems involving thermo-fluid incidents such as convective heat transfer or conjugate heat transfer, the Navier-Stokes equations are applied. The Navier-Stokes equations address changes in properties such as speed, pressure, and density during thermo-fluid interactions. The Navier-Stokes equations are a collection of three fundamental equations:
- Continuity equation, which expresses the conservation of mass
- Newton’s second law, which expresses the conservation of momentum
- The first law of thermodynamics, which expresses the conservation of energy
The Navier-Stokes equations describe the conservation of mass, momentum, and energy and model fluid flow problems numerically. The expressions can be in cartesian, cylindrical, or polar coordinates in accordance with the domain of application.
Navier-Stokes Equations in Polar Coordinates
The fluid flow problems in engineering systems may occur in domains of complex geometries. These domains may be irregular or can be infinite. The Navier-Stokes equations representing the physical problems in such domains are difficult to solve. The transformation of the Navier-Stokes equations to a suitable coordinate system may help in making the problem-solving process easier.
Navier-Stokes equations in 3D polar coordinates or spherical coordinates are advantageous for domains with three-dimensional flow characteristics. The calculation of primitive variables or flow parameters becomes easier with the Navier-Stokes equation in 3D polar coordinates.
In a fluid flow problem, the conservation of mass is expressed as the continuity equation, and the same can be transformed into 3D polar coordinates as given in equation (3). Similarly, the conservation of momentum and energy can be expressed in polar coordinates.
The Navier-stokes equations in polar coordinates are useful in numerical modeling when the physical domain is circular in shape. To solve Navier-Stokes equations in thermo-fluid problems, you can rely on Cadence’s suite of CFD simulation tools to help you achieve fluid flow parameters.
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