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Governing Equations
Navier-Stokes equations are essential in describing the motion of fluids because they apply Newton’s second law to fluid motion. The equations use the viscous forces acting on a fluid, such as friction or gravity, to exhibit rates of change present in fluid motion. These equations give the fluid’s velocity, but not the position of particular fluid particles, in a solution called a velocity field. More often than not, there are not solutions to these equations because of their nonlinearity. This nonlinearity allows any type of flow, whether turbulent or laminar, to be modeled using the differential equations. In particular, it is believed by many scientists that the Navier-Stokes equations are able to properly model turbulent flows. The particular equation most closely associated with Newton’s second law is:
The left side of the equation describes acceleration of the flow. It is the pressure gradient that occurs from normal stresses within a particular fluid force, as well as a gradient surface force that describes viscosity for the incompressible force. "f" represents other forces acting on the fluid such as gravity. The following equation describes the incompressible flow assumption for Newtonian fluids, such as water:
The Navier-Stokes equations can be used in three different coordinate systems to describe the motion of fluids. These equations will prove to be useful because of the role they play in analyzing the many forces acting on a fluid. In particular, Navier-Stokes equations can be used to describe chaotic mixing in groundwater remediation.
The left side of the equation describes acceleration of the flow. It is the pressure gradient that occurs from normal stresses within a particular fluid force, as well as a gradient surface force that describes viscosity for the incompressible force. "f" represents other forces acting on the fluid such as gravity. The following equation describes the incompressible flow assumption for Newtonian fluids, such as water:
The Navier-Stokes equations can be used in three different coordinate systems to describe the motion of fluids. These equations will prove to be useful because of the role they play in analyzing the many forces acting on a fluid. In particular, Navier-Stokes equations can be used to describe chaotic mixing in groundwater remediation.
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