Essential Fluid Dynamics Equations

Key Takeaways

• The basic fluid dynamics equations are derived from Newton’s laws, with some assumptions on fluid behavior.

• Important characteristics of fluid flow and compression/expansion are summarized in a few equations.

• Although it isn’t obvious, these equations can be used to derive fluid behavior ranging from simple laminar flow to complex turbulence fields.

Just like in any other area of physics or engineering, fluid dynamics relies on several fundamental equations to describe fluid behavior, including turbulence, mass transport, and variable density in compressible fluids. In this article, we briefly explain fundamental fluid dynamics equations and their physical interpretations. The equations used to describe fluid flow have rather simple meanings, even if their mathematical forms appear complex.

Fundamental Fluid Dynamics Equations

Fluid flow is largely described in four regimes: inviscid or viscous flow as well as compressible or incompressible flow. In fluid dynamics classes, most treatments of fluid dynamics equations focus on inviscid incompressible flow as well as flow regimes where turbulence is not important. Each of these relies on a particular differential operator, known as the material derivative:

Material derivative operator

In this definition, u is the fluid flow vector, which is generally expressed in Cartesian coordinates. In addition, due to conservation of mass, we have a continuity equation that expresses the change in fluid density ⍴ as a function of flow variation in space:

Continuity equation for flow density

With this equation, we can immediately define the difference between compressible and incompressible flows. Assuming the density is constant in space and time (totally incompressible fluid), then the continuity equation reduces to:

Continuity equation for incompressible flows

Finally, there is a particular vector operator that appears in fluid dynamics equations, known as the outer product:

Definition of the outer product

With these basic definitions, we can now examine the main equations of motion that govern inviscid and viscous flows. These are the Navier-Stokes equations and Euler’s equations.

Navier-Stokes Equations

Fluid dynamics discussions generally start with the Navier-Stokes equations, which include the above continuity equation and an associated momentum equation. The momentum portion of the Navier-Stokes equations is derived from a separate equation from continuum mechanics, known as Cauchy’s momentum equation. The Navier-Stokes equations make combined statements that a flowing fluid must obey conservation of momentum as it undergoes motion and that mass is conserved during flow. For compressible flows, we have the following equation describing conservation of momentum:

Momentum portion of the Navier-Stokes equations for compressible flows

In this expression, μ and λ are proportionality constants used to describe linear stress-strain behavior for the fluid. Note that, in general, fluids do not undergo elastic deformation for every value of stress they experience; such a case is related to the treatment of non-Newtonian fluids.

We also have a thermodynamic equation describing the flow:

Enthalpy portion of the Navier-Stokes equations

In this equation, h is enthalpy, k is the fluid’s thermal conductivity, and the final term describes dissipation due to viscous effects and the stress-strain behavior of the fluid under compressive forces:

Dissipation portion of the Navier-Stokes equations

For an incompressible fluid, we apply the constant density continuity condition shown above. Note that there is ongoing controversy as to whether the stress-strain proportionality constant λ should also be set to 0 for nearly incompressible fluids, but it is often ignored in standard treatments. These equations generally treat any case where viscosity and the internal forces it produces must be considered in fluid flow. If these forces are negligible, then we can reduce these equations to Euler’s equations.

Euler’s Equations

The standard treatment of inviscid flow begins with Euler’s equations, where incompressibility is generally assumed. In the inviscid case, we have by definition μ = λ = 0; Euler’s equations are immediately derived by dropping any viscous terms from the Navier-Stokes equations. In other words, we assume one of the following conditions:

1. Any drag forces produced by viscosity are much smaller than any external forces.

2. The viscous term in the Navier-Stokes equations is very small compared to all other terms.

3. The system enforces vortical flow in a viscous fluid, which requires 𝛻2u = 0 and mimics inviscid flow.

In any of these cases, this reduces the Navier-Stokes equations to the following form for compressible fluids with no viscosity:

Momentum portion of Euler’s equations for compressible flows

For an incompressible fluid, we also apply the constant density continuity condition. For inviscid flows, it is also customary to set λ = 0, although, again, this remains a matter of controversy.

Finally, we have a different form for the enthalpy equation that includes a dissipation term proportional to λ:

Dissipation portion of Euler’s equations for compressible flows

Again, setting λ = 0 gets us back to the typical form of Euler’s equations found in textbooks on inviscid flow.

Bernoulli’s Equation

One useful relation for understanding incompressible steady flows is Bernoulli’s equation. This equation relates the energy (kinetic and potential) per unit mass of a fluid to its static pressure. For flows along a given streamline, the following equation is valid:

Bernoulli’s equation

This equation is system-specific; if you know the flow behavior for a given streamline at one point in the system, you can determine similar behavior at any other streamline in the system.