Jatkuvuusyhtälöllä

Jatkuvuusyhtälöllä, known in English as the Continuity Equation, is a fundamental principle in fluid dynamics which describes the conservation of mass in a flowing fluid. It states that, for an incompressible or compressible fluid, the mass flow rate entering a control volume must equal the mass flow rate leaving it, assuming no mass sources or sinks are present within the volume.

Mathematically, the continuity equation can be expressed as the differential form: ∂ρ/∂t + ∇·(ρv) = 0, where ρ represents fluid density, t is time, v is the flow velocity, and ∇·(ρv) denotes the divergence of mass flux. For incompressible fluids, where density remains constant, the equation simplifies to the more familiar form: ∇·v = 0, indicating that the divergence of velocity is zero, and fluid flow is divergence-free.

The continuity equation is vital for analyzing and designing systems involving fluid flow, such as pipelines, air ducts, and cardiovascular systems. It ensures that mass is conserved across different sections of a system, facilitating the calculation of flow velocities and pressure distributions.

This principle, derived from the law of conservation of mass, was first formalized in the 18th century, building on work by mathematicians and physicists like Daniel Bernoulli. Its applications extend across engineering, physics, and environmental sciences, underpinning the understanding of fluid behavior in various natural and industrial processes.