Flows

Flow is the movement of a quantity per unit time through a region or across a boundary. It is typically described by a flow rate, which specifies how much of the quantity crosses a given surface per unit time. Commonly studied flows include fluids, traffic, and information.

In fluid dynamics, flow describes the movement of liquids and gases. The volumetric flow rate Q equals cross-sectional area A times average velocity v for steady, incompressible flow (Q = Av). The mass flow rate m-dot equals ρQ. The continuity equation ∂ρ/∂t + ∇·(ρu) = 0 expresses conservation of mass; for incompressible fluids where density is constant, ∇·u = 0. The motion is governed by Euler or Navier–Stokes equations, and can be laminar or turbulent. Flow can be steady or unsteady, compressible or incompressible, and may involve boundary layers and viscosity.

In networks and operations research, flow refers to the amount of a commodity moving through a system of edges and nodes per unit time. Each edge has a capacity, and node balances require what enters a node to exit, in order to maximize overall throughput. The max-flow min-cut theorem is a foundational result. Applications include transportation, logistics, and communication networks.

Flow concepts also apply to information, ideas, or influence moving through social or technical networks. Information flow quantifies data rate and the spread of signals, while social flows study how attention or opinions propagate. Pedestrian flow models describe how people move in crowds, linking density and speed to predict congestion and safety.

Common units include volumetric flow rate (cubic meters per second) and mass flow rate (kilograms per second). Flow analysis uses conservation laws, dimensional analysis, and computational methods to simulate and optimize systems.