Flow velocity

In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar. It is also called velocity field; when evaluated along a line, it is called a velocity profile (as in, e.g., law of the wall). The flow velocity u of a fluid is a vector field which gives the velocity of an element of fluid at a position and time The flow speed q is the length of the flow velocity vector and is a scalar field. The flow velocity of a fluid effectively describes everything about the motion of a fluid. Many physical properties of a fluid can be expressed mathematically in terms of the flow velocity. Some common examples follow: Steady flow The flow of a fluid is said to be steady if does not vary with time. That is if Incompressible flow If a fluid is incompressible the divergence of is zero: That is, if is a solenoidal vector field. Irrotational flow A flow is irrotational if the curl of is zero: That is, if is an irrotational vector field. A flow in a simply-connected domain which is irrotational can be described as a potential flow, through the use of a velocity potential with If the flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero: Vorticity The vorticity, , of a flow can be defined in terms of its flow velocity by If the vorticity is zero, the flow is irrotational. Potential flow If an irrotational flow occupies a simply-connected fluid region then there exists a scalar field such that The scalar field is called the velocity potential for the flow. (See Irrotational vector field.

Anomalous dissipation and other non-smooth phenomena in fluids

Anomalous dissipation and other non-smooth phenomena in fluids