Conservation of Mass

The conservation of mass is a fundamental principle in physics, and its mathematical formulation within continuum mechanics is known as the continuity equation. This equation provides a precise statement about how the density of a material changes in space and time. The equation \(\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0\) applies at every point within the continuum. The term \(\frac{\partial \rho}{\partial t}\) represents the rate of change of density at a fixed point (the local or unsteady term), while the term \(\nabla \cdot (\rho \mathbf{u})\) is the divergence of the mass flux (\(\rho \mathbf{u}\)), representing the net rate of mass flowing out of an infinitesimal volume around that point.

The equation essentially states that if the density at a point is increasing, it must be because more mass is flowing into the infinitesimal volume than is flowing out, and vice versa. For a special case known as an incompressible flow, the density \(\rho\) of a fluid parcel is assumed to be constant as it moves. In this case, the continuity equation simplifies significantly to \(\nabla \cdot \mathbf{u} = 0\). This simplified form is widely used in modeling liquids like water and in low-speed aerodynamics. The continuity equation is one of the core governing equations, alongside the conservation of momentum and energy, used in virtually all analyses in fluid dynamics and solid mechanics.