![]() ![]() ![]() In other words, we assume one of the following conditions: 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. The standard treatment of inviscid flow begins with Euler’s equations, where incompressibility is generally assumed. If these forces are negligible, then we can reduce these equations to Euler’s equations. These equations generally treat any case where viscosity and the internal forces it produces must be considered in fluid flow. 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. 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:ĭissipation portion of the Navier-Stokes equationsįor an incompressible fluid, we apply the constant density continuity condition shown above. ![]() We also have a thermodynamic equation describing the flow:Įnthalpy portion of the Navier-Stokes equations 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. In this expression, μ and λ are proportionality constants used to describe linear stress-strain behavior for the fluid. (Alt text: Momentum portion of the Navier-Stokes equations) Momentum portion of the Navier-Stokes equations for compressible flows For compressible flows, we have the following equation describing conservation of momentum: 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. The momentum portion of the Navier-Stokes equations is derived from a separate equation from continuum mechanics, known as Cauchy’s momentum equation. Navier-Stokes Equationsįluid dynamics discussions generally start with the Navier-Stokes equations, which include the above continuity equation and an associated momentum equation. These are the Navier-Stokes equations and Euler’s equations. With these basic definitions, we can now examine the main equations of motion that govern inviscid and viscous flows. Assuming the density is constant in space and time (totally incompressible fluid), then the continuity equation reduces to:Ĭontinuity equation for incompressible flowsįinally, there is a particular vector operator that appears in fluid dynamics equations, known as the outer product: With this equation, we can immediately define the difference between compressible and incompressible flows. 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: In this definition, u is the fluid flow vector, which is generally expressed in Cartesian coordinates. Each of these relies on a particular differential operator, known as the material derivative: 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. Fundamental Fluid Dynamics Equationsįluid flow is largely described in four regimes: inviscid or viscous flow as well as compressible or incompressible flow. The equations used to describe fluid flow have rather simple meanings, even if their mathematical forms appear complex. In this article, we briefly explain fundamental fluid dynamics equations and their physical interpretations. 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. Important characteristics of fluid flow and compression/expansion are summarized in a few equations.Īlthough it isn’t obvious, these equations can be used to derive fluid behavior ranging from simple laminar flow to complex turbulence fields. The basic fluid dynamics equations are derived from Newton’s laws, with some assumptions on fluid behavior. ![]()
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