Continuity equation

One of the basic equations in hydrodynamics, expressing the law of conservation of mass for any volume of a moving fluid (or gas). In Euler variables the continuity equation has the form

\[ \frac{\partial \rho}{\partial t } + \text{div}(\rho \mathbf{v}) \equiv \frac{\partial \rho}{\partial t} + \frac{\partial (\rho v_x)}{\partial x} + \frac{\partial (\rho v_y)}{\partial y} + \frac{\partial (\rho v_z)}{\partial z} = 0, \]

where $ \rho $ is the density of the fluid, $ \mathbf{v} $ is its velocity at a given point, and $ v_x, v_y, v_z $ are the projections of the velocity on the coordinate axes. If the fluid is incompressible $ (\rho = \text{const}) $, then the continuity equation takes the form

\[ \text{div } \mathbf{v} = 0 \quad \text{or} \quad \frac{\partial v_x}{\partial x} + \frac{\partial v_y}{\partial y} + \frac{\partial v_z}{\partial z} = 0. \]

For a stationary one-dimensional flow in a tube, canal, etc., with cross-sectional area $ S $, the continuity equation gives the law $\rho S\mathbf v=\text{const}$ for the flow.