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Conservation of Mass

Mass is neither created nor destroyed in a closed system. This principle is crucial for analyzing fluid flow and conservation.
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The statement of the theorem

The conservation of mass principle, applied to a continuous fluid flow within a control volume Ω(t)⊂R3\Omega(t) \subset \mathbb{R}^3, is mathematically expressed by the continuity equation. Assuming the fluid is described by a density function ρ(x⃗,t)\rho(\vec{x}, t) and a velocity field u⃗(x⃗,t)\vec{u}(\vec{x}, t), the statement is:\n\n∂ρ∂t+∇⋅(ρu⃗)=0in Ω(t)\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0 \quad \text{in } \Omega(t)\n\nThis equation asserts that the rate of change of density within the control volume (the first term) must be exactly balanced by the net flux of mass out of the volume (the divergence of the mass flux, ∇⋅(ρu⃗)\nabla \cdot (\rho \vec{u})). Alternatively, using the Reynolds Transport Theorem (RTT) for a fixed control volume VV, the conservation law states that the rate of change of mass within VV equals the net mass flux across the boundary surface S=∂VS = \partial V:\n\nddt∫Vρ dV=−∮Sρu⃗⋅n⃗ dS\frac{d}{dt} \int_{V} \rho \, dV = - \oint_{S} \rho \vec{u} \cdot \vec{n} \, dS\n\nHere, n⃗\vec{n} is the outward-pointing unit normal vector to the surface SS. The differential form, ∂ρ∂t+∇⋅(ρu⃗)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) = 0, is the most fundamental statement for local conservation.