Expressing the NavierStokes Equations in Cartesian Coordinates
Key Takeaways

In a twodimensional 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 yaxis and xaxis, respectively.

The NavierStokes equations in the Cartesian coordinate system are compact in representation compared to cylindrical and spherical coordinates.

The NavierStokes 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 NavierStokes 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 NavierStokes equations can be expressed in Cartesian coordinates, cylindrical coordinates, or spherical coordinates. In this article, we will focus on expressing the NavierStokes equations in Cartesian coordinates, but first, let’s look at the importance of the NavierStokes equations.
The Importance of the NavierStokes Equations in Fluid Dynamics
The NavierStokes equations form the base of fluid flow modeling. Solving the NavierStokes 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 NavierStokes 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 NavierStokes equations are employed in the design of trains, cars, aircraft, power stations, and blood flow.
The NavierStokes equations are a collection of three fundamental equations:
 The continuity equation governs the conservation of mass in a physical problem.
 The momentum equation, or Newton's second law, governs the conservation of momentum.
 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 NavierStokes 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 xaxis and yaxis, collectively called the coordinate axes, representing the twodimensional 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 xaxis and yaxis, respectively.
In a twodimensional 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 yaxis and xaxis. 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 yaxis, the xcoordinate is positive. If the location of the point is on the left side of the yaxis, the xcoordinate is negative. For a point existing above the xaxis, the ycoordinate is positive, if the point is below the xaxis, the coordinate is negative.
The Cartesian coordinate system can represent the location of points in onedimensional, twodimensional, or threedimensional space. In a threedimensional 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 xaxis, yaxis, and zaxis to the point, respectively.
Let’s take a look at expressing the NavierStokes equations in Cartesian coordinates.
The NavierStokes Equations in Cartesian Coordinates
The NavierStokes equations in the Cartesian coordinate system are compact in representation compared to cylindrical and spherical coordinates. The NavierStokes equations in Cartesian coordinates give a set of nonlinear 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 NavierStokes 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 NavierStokes 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 NavierStokes equations in the x, y, and z directions, respectively.
Expressing the NavierStokes equations in Cartesian coordinates is appropriate for physical problems involving noncurvilinear geometric application domains. To solve the NavierStokes equations in fluid flow problems, you can rely on the CFD simulation tools available in Cadence’s suite of CFD software.
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