Expressing the NavierStokes Equation in Cylindrical Coordinate Systems
Key Takeaways

To mathematically model physical problems involving fluid flow, the NavierStokes equations can be used.

Depending on the shape of the domain, the NavierStokes equation is transformed to the appropriate coordinate system.

Expressing the NavierStokes equation in cylindrical coordinates is ideal for fluid flow problems dealing with curved or cylindrical domain geometry.
Depending on the application domain, the NavierStokes equation is expressed in cylindrical coordinates, spherical coordinates, or cartesian coordinate
Physical problems such as combustion, turbulence, mass transport, and multiphase flow are influenced by the physical properties of fluids, including velocity, viscosity, pressure, temperature, and density. Properties can be categorized into thermodynamics, kinematic, or transport, depending on the type of physical problem. The NavierStokes equations can be utilized to develop the numerical model of physical problems. Depending on the application domain, you can express the NavierStokes equations in cylindrical coordinates, spherical coordinates, or cartesian coordinates. This article will focus on how to express the NavierStokes equations in cylindrical coordinates.
The NavierStokes Equations
To mathematically model physical problems involving fluid flow, the NavierStokes equations are applied. The internal and external flow of compressible and incompressible fluids can be described using the NavierStokes equations. These equations address the changes in properties, such as speed, pressure, and density of the fluid, in fluid mechanics problems.
The NavierStokes equations can be rearranged to obtain the appropriate model according to the physical problem under consideration. For example, in problems involving turbulence or thermofluid interactions, the appropriate turbulent model or thermofluid model is developed to achieve credible solutions. In general, the NavierStokes equations are the final word when it comes to the generation of numerical solutions for fluid flow problems. Next, we will look at the formulation of the NavierStokes equations.
Decomposing the NavierStokes Equations
In fluid flow problems, the movement of the fluid can be described using either the Langrarian method or the Eulerian method. In the Eulerian method, the fluid motion is defined by three equations based on the conservation of mass, momentum, and energy. Generally, the NavierStokes equations are the collection of three equations of conservation. The three equations of conservation are:

Continuity equation expressing the conservation of mass: According to the continuity equation, the mass flow difference between the system inlet and outlet is zero. The continuity equation can be given as the following equation, where is the density and V is the velocity of the fluid.

Newton’s second law expressing the conservation of momentum: The conservation of momentum in fluid flow is based on the expression Force, F=Mass (m) x Acceleration (a). In an incompressible threedimensional fluid flow, the mathematical expression for the conservation of momentum is referred to as the NavierStokes equation, given as the following equation, where g is the gravitational acceleration, is the viscosity coefficient and p is the static pressure of the fluid.

First law of thermodynamics or energy equation expressing the conservation of energy: According to the first law of thermodynamics, the summation of work done on the system and the heat added to it appears as an increase in the total energy of the system. The conservation of energy is commonly expressed in the following form, where h is the enthalpy, k is the thermal conductivity, and represents the heat dissipation term.
Any fluid flow problem can be represented by the NavierStokes equations. However, the complexities and nonlinear structure of the NavierStokes equations make it difficult to find exact solutions. Usually, analytical methods and numerical methods are used to solve the NavierStokes equations.
Expressing the NavierStokes equations is important for obtaining credible solutions. The NavierStokes equations can be expressed in cartesian coordinates, cylindrical coordinates, or spherical coordinates. Depending on the shape of the domain, the NavierStokes equations are transformed to appropriate coordinate systems. Let’s discuss expressing the NavierStokes equations in cylindrical coordinates in the upcoming section.
The NavierStokes Equations in Cylindrical Coordinates
The cylindrical polar coordinate system represents a point using ordered triples (r, θ, z). The cylindrical coordinates combine the twodimensional polar coordinates (r, θ) with the cartesian z coordinate. Cylindrical coordinates are used to represent the physical problems in threedimensional space in (r, θ, z). The transformation of cylindrical coordinates to cartesian coordinates (the first equation set) and vice versa (the second equation set) can be conducted as such.
To solve fluid mechanics problems, the primary step is to choose the appropriate coordinate systems to express the NavierStokes equations. Using the continuity equation and thephysical details of the problem domain, the NavierStokes equations are simplified and solved for parameters such as velocity and pressure.
The continuity equation given in the very first equation can be expressed in cylindrical coordinates as:
The NavierStokes equations given in the second equation can be expressed in cylindrical coordinates as:
Expressing the NavierStokes equations in cylindrical coordinates is ideal for fluid flow problems dealing with curved or cylindrical domain geometry. To solve NavierStokes equations in fluid flow problems, you can rely on the CFD simulation tools available in Cadence’s suite of CFD software.
Subscribe to our newsletter for the latest CFD updates or browse Cadence’s suite of CFD software, including Fidelity and Fidelity Pointwise, to learn more about how Cadence has the solution for you.