Conservation of Mass Equation — Lesson 3

Conservation of Mass in Fluid Dynamics

Introduction to the Continuity Equation

• The conservation of mass, also known as the continuity equation, states that mass cannot be created or destroyed. The net mass crossing a system must balance with accumulation or depletion within that system.

• An example illustrating this principle is water flowing through a hose: what enters one end must exit from the other.

Compressible Fluids and Conservation of Mass

• In compressible fluids, mass can vary within a control volume. The discussion begins with the Lagrangian form where mass variation equals zero.

• Applying Reynolds transport theorem allows transformation into an equation suitable for analysis in fluid dynamics (O114).

Mathematical Derivation

• Utilizing the divergence theorem enables substitution of surface integrals with volume integrals, leading to a general form independent of control volume.

• For arbitrary volumes, if the integral equals zero, it leads to the final expression for conservation of mass in O114.

Forms of Conservation of Mass

• The conservation of mass can be expressed in various forms; here, we focus on Cartesian coordinates for three-dimensional fields.

• For incompressible flows where density remains constant, simplification occurs: divergence of velocity equals zero. This indicates that fluid entering a domain equals fluid exiting it.

Implications for Incompressible Flows