The NavierStokes Equation in Polar Coordinates for Circular Domain Physical Problems
Key Takeaways

The polar coordinates representing a point in 3dimensional space are called 3D polar coordinates, otherwise known as spherical coordinates.

For describing engineering problems involving fluid dynamics, the NavierStokes equations are utilized in different coordinate systems.

The Navierstokes 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 thermalfluid problems, the numerical model is governed by the NavierStokes equation. The NavierStokes equation can be expressed in cartesian coordinates, cylindrical coordinates, or polar coordinates. The NavierStokes equation in polar coordinates can be expressed in a threedimensional (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 NavierStokes equation in 3D polar coordinates.
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 xaxis to the line connecting point P to the origin. The angle θ is measured in the anticlockwise direction.
The polar coordinates can be expressed in twodimension or threedimension. Polar coordinates representing a point in 2dimensional 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 3dimensional 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 Xaxis in the anticlockwise direction, and θ is the azimuth angle measured from the Zaxis. 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 NavierStokes equations are utilized in different coordinate systems. Let’s discuss the NavierStokes equation before expressing it in polar coordinates.
The NavierStokes Equation
To mathematically model physical problems involving thermofluid incidents such as convective heat transfer or conjugate heat transfer, the NavierStokes equations are applied. The NavierStokes equations address changes in properties such as speed, pressure, and density during thermofluid interactions. The NavierStokes 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 NavierStokes 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.
NavierStokes 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 NavierStokes equations representing the physical problems in such domains are difficult to solve. The transformation of the NavierStokes equations to a suitable coordinate system may help in making the problemsolving process easier.
NavierStokes equations in 3D polar coordinates or spherical coordinates are advantageous for domains with threedimensional flow characteristics. The calculation of primitive variables or flow parameters becomes easier with the NavierStokes 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 Navierstokes equations in polar coordinates are useful in numerical modeling when the physical domain is circular in shape. To solve NavierStokes equations in thermofluid problems, you can rely on Cadence’s suite of CFD simulation tools to help you achieve fluid flow parameters.
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