Expressing the Navier-Stokes Equations in Cartesian Coordinates

Key Takeaways

• In a two-dimensional Cartesian coordinate system, the location of a point can be given by Cartesian coordinates (x, y), where x and y are the perpendicular distances to the point from the y-axis and x-axis, respectively.

• The Navier-Stokes equations in the Cartesian coordinate system are compact in representation compared to cylindrical and spherical coordinates.

• The  Navier-Stokes equations in the Cartesian coordinate system can be converted into cylindrical and spherical coordinate systems easily.

Fluid flow is one of the main physical problems encountered in engineering, and engineers use modeling to help solve fluid flow issues. Using the Navier-Stokes equations is the best way to develop a numerical model for fluid flow, as fluid motion is governed by these equations. Depending on the application domain, the Navier-Stokes equations can be expressed in Cartesian coordinates, cylindrical coordinates, or spherical coordinates. In this article, we will focus on expressing the Navier-Stokes equations in Cartesian coordinates, but first, let’s look at the importance of the Navier-Stokes equations.

The Importance of the Navier-Stokes Equations in Fluid Dynamics

The Navier-Stokes equations form the base of fluid flow modeling. Solving the Navier-Stokes equations along with a set of boundary conditions provides a clear picture of flow parameters such as fluid velocity and pressure. Numerical modeling using the Navier-Stokes equations helps engineers understand physical problems such as water flow in a pipe, airflow around an aircraft wing, weather behavior, and ocean currents. Simplified or complete forms of the Navier-Stokes equations are employed in the design of trains, cars, aircraft, power stations, and blood flow.

The Navier-Stokes equations are a collection of three fundamental equations:

1. The continuity equation governs the conservation of mass in a physical problem.
2. The momentum equation, or Newton's second law, governs the conservation of momentum.
3. The first law of thermodynamics governs the conservation of energy in a physical problem.

Let’s take a look at the Cartesian coordinate system to better understand how the Navier-Stokes equations can be expressed in Cartesian coordinates.

The Cartesian Coordinate System

Cartesian coordinates are used to label points on a plane using distances from two perpendicular lines. The two perpendicular lines are called the x-axis and y-axis, collectively called the coordinate axes, representing the two-dimensional Cartesian coordinate system. The Cartesian coordinate system is also known as the rectangular coordinate system. The plane formed by the coordinate axes is called the Cartesian plane. The coordinate axes intersect at a point called the origin or zero of the Cartesian system. The origin is represented as (0, 0), which indicates the point is at zero distance from the x-axis and y-axis, respectively.

In a two-dimensional Cartesian coordinate system, the location of a point can be given by Cartesian coordinates (x, y), where x and y are the perpendicular distances to the point from the y-axis and x-axis. The coordinate numbers representing the distance from the axes are signed numbers; they can be positive, negative, or zero. If the location of the point is on the right side of the y-axis, the x-coordinate is positive. If the location of the point is on the left side of the y-axis, the x-coordinate is negative. For a point existing above the x-axis, the y-coordinate is positive, if the point is below the x-axis, the coordinate is negative.

The Cartesian coordinate system can represent the location of points in one-dimensional, two-dimensional, or three-dimensional space. In a three-dimensional Cartesian coordinate system, the location of the point on the Cartesian plane is given by a triplet of numbers, represented as (x, y, z). These three numbers represent the distance in the direction along the x-axis, y-axis, and z-axis to the point, respectively.

Let’s take a look at expressing the Navier-Stokes equations in Cartesian coordinates.

The Navier-Stokes Equations in Cartesian Coordinates

The Navier-Stokes equations in the Cartesian coordinate system are compact in representation compared to cylindrical and spherical coordinates. The Navier-Stokes equations in Cartesian coordinates give a set of non-linear partial differential equations.  The velocity components in the direction of the x, y, and z axes are described as u, v, and w in equations represented in the Cartesian coordinate system.

The Navier-Stokes equations in the Cartesian coordinate system can be converted into cylindrical and spherical coordinate systems easily. The transformation of cylindrical coordinates (r, θ, z) to Cartesian coordinates (the first equation set) and vice versa (the second equation set) can be conducted as such:

The relationship between Cartesian and spherical coordinate systems  (r, ɸ, θ) can be represented as:

The Navier-Stokes equations for an incompressible fluid can be expressed in Cartesian coordinates as:

The first equation gives the continuity equation and the last three equations represent the velocity components of the incompressible Navier-Stokes equations in the x, y, and z directions, respectively.

Expressing the Navier-Stokes equations in Cartesian coordinates is appropriate for physical problems involving non-curvilinear geometric application domains. To solve the Navier-Stokes equations in fluid flow problems, you can rely on the CFD simulation tools available in Cadence’s suite of CFD software.