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Expressing the Navier-Stokes Equation in Cylindrical Coordinate Systems

Key Takeaways

  • To mathematically model physical problems involving fluid flow, the Navier-Stokes equations can be used. 

  • Depending on the shape of the domain, the Navier-Stokes equation is transformed to the appropriate coordinate system. 

  • Expressing the Navier-Stokes equation in cylindrical coordinates is ideal for fluid flow problems dealing with curved or cylindrical domain geometry.

 Navier-Stokes equation

Depending on the application domain, the Navier-Stokes 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 Navier-Stokes equations can be utilized to develop the numerical model of physical problems. Depending on the application domain, you can express the Navier-Stokes equations in cylindrical coordinates, spherical coordinates, or cartesian coordinates. This article will focus on how to express the Navier-Stokes equations in cylindrical coordinates.   

The Navier-Stokes Equations 

To mathematically model physical problems involving fluid flow, the Navier-Stokes equations are applied. The internal and external flow of compressible and incompressible fluids can be described using the Navier-Stokes equations. These equations address the changes in properties, such as speed, pressure, and density of the fluid, in fluid mechanics problems.

The Navier-Stokes equations can be rearranged to obtain the appropriate model according to the physical problem under consideration. For example, in problems involving turbulence or thermo-fluid interactions, the appropriate turbulent model or thermo-fluid model is developed to achieve credible solutions. In general, the Navier-Stokes 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 Navier-Stokes equations. 

Decomposing the Navier-Stokes 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 Navier-Stokes equations are the collection of three equations of conservation. The three equations of conservation are:

  1. 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.

  1. 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 three-dimensional fluid flow, the mathematical expression for  the conservation of momentum is referred to as the Navier-Stokes 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.

 Conservation of momentum equation

  1. 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.

Conservation of energy equation

Any fluid flow problem can be represented by the Navier-Stokes equations. However, the complexities and non-linear structure of the Navier-Stokes equations make it difficult to find exact solutions.  Usually, analytical methods and numerical methods are used to solve the Navier-Stokes equations.

Expressing the Navier-Stokes equations is important for obtaining credible solutions. The Navier-Stokes equations can be expressed in cartesian coordinates, cylindrical coordinates, or spherical coordinates. Depending on the shape of the domain, the Navier-Stokes equations are transformed to appropriate coordinate systems. Let’s discuss expressing the Navier-Stokes equations in cylindrical coordinates in the upcoming section.  

The Navier-Stokes Equations in Cylindrical Coordinates

The cylindrical polar coordinate system represents a point using ordered triples (r, θ, z). The cylindrical coordinates combine the two-dimensional polar coordinates (r, θ) with the cartesian z coordinate. Cylindrical coordinates are used to represent the physical problems in three-dimensional 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.

Transformation of coordinates

To solve fluid mechanics problems, the primary step is to choose the appropriate coordinate systems to express the Navier-Stokes equations. Using the continuity equation and thephysical details of the problem domain, the Navier-Stokes 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:

Continuity equation in cylindrical coordinates

The Navier-Stokes equations given in the second equation can be expressed in cylindrical coordinates as:

                        Cylindrical coordinates                                                                                               

Expressing the Navier-Stokes equations in cylindrical coordinates is ideal for fluid flow problems dealing with curved or cylindrical domain geometry. To solve Navier-Stokes 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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