Flow velocity
Vector field describing fluid motion at each point and time.
Flow velocity, also known as the velocity field, is a vector field used in continuum mechanics to mathematically describe the motion of a fluid. It gives the velocity of an element of fluid at a given position and time, and its length is the scalar flow speed.
- field
- Continuum mechanics, fluid dynamics, statistical mechanics, electromagnetism
- also_called
- Velocity field, macroscopic velocity, drift velocity
- scalar_quantity
- Flow speed
- vector_definition
- u = u(x, t)
- bulk_velocity_formula
- u̅ = V̇ / A
Lore & Background
In continuum mechanics, flow velocity is defined as a vector field u = u(x, t), representing the velocity of a fluid element at position x and time t. Its magnitude, the flow speed q = ‖u‖, is a scalar field. The concept is also referred to as macroscopic velocity in statistical mechanics and drift velocity in electromagnetism. When evaluated along a line, it is called a velocity profile, as in the law of the wall.
Reader's Guide
Flow velocity is fundamental to describing fluid motion. Many physical properties of a fluid are expressed mathematically in terms of this vector field. For steady flow, the velocity does not vary with time (∂u/∂t = 0). For incompressible flow, the divergence of u is zero (∇·u = 0), making u a solenoidal vector field. Irrotational flow has zero curl (∇×u = 0), and in a simply-connected domain can be described by a velocity potential Φ such that u = ∇Φ. If both irrotational and incompressible, the Laplacian of the velocity potential is zero (ΔΦ = 0). Vorticity ω is defined as ω = ∇×u; zero vorticity indicates irrotational flow. In engineering, when local velocity is unknown, the bulk velocity u̅ is used, defined as the volume flow rate V̇ divided by cross-sectional area A.
Did You Know?
- Flow velocity is also called macroscopic velocity in statistical mechanics and drift velocity in electromagnetism.
- The length of the flow velocity vector is the scalar flow speed.
- A flow is steady if the flow velocity does not vary with time.
- If a flow is both irrotational and incompressible, the Laplacian of the velocity potential must be zero.
Frequently Asked Questions
What is Flow velocity in the Mechanics And Fluid Dynamics series?
Flow velocity is a vector field that assigns a specific velocity vector to every point in a fluid at a given instant. It encodes both the direction and the magnitude of how a tiny fluid element is moving at that location and time.
What is Flow velocity's mathematical form?
It is written as u = u(x, t), meaning the velocity is a function of spatial position x and time t. This single expression maps each point in space and each moment in time to a unique velocity vector.
How does Flow velocity differ from Flow speed?
Flow speed is only the magnitude (length) of the velocity vector, making it a pure scalar. Flow velocity, by contrast, carries directional information in addition to that magnitude, which is why it is a vector quantity.
In which fields does Flow velocity appear and what are its alternate names?
It is central to continuum mechanics, fluid dynamics, statistical mechanics, and electromagnetism. Depending on the subfield, it is also called the velocity field, macroscopic velocity, or drift velocity.
What is the bulk velocity formula and what does it represent?
The bulk velocity is given by u̅ = V̇ / A, where V̇ is the volumetric flow rate and A is the cross-sectional area. It yields the average velocity of the fluid across an entire pipe or channel cross-section.
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