Incompressible Flow

The general conservation of mass equations can be further simplified for incompressible flows, a condition typically applicable to liquids. Since density ($\rho$ρ) is constant, it can be canceled from both sides of the general steady-flow relation, yielding:

Steady, incompressible flow:$$\sum _{\text{in}}\dot{V}=\sum _{\text{out}}\dot{V}$$where $\dot{V}$˙ is the volume flow rate.

For a system with a single inlet and a single outlet, this simplifies to:

Steady, incompressible flow (single stream):$${\dot{V}}_1={\dot{V}}_2\text{or}{A}_1{V}_1=A_2{V}_2$$A critical point must be emphasized: there is no fundamental “conservation of volume” principle in physics. For a steady-flow device, volume flow rates at the inlet and outlet can differ even when the mass flow rate is constant. A common example is an air compressor: the outlet volume flow rate is significantly lower than the inlet flow rate due to the increased density of the compressed air, despite a constant mass flow rate.

For steady flows of liquids, however, the constant-density assumption means that volume flow rates remain constant along with mass flow rates. The flow of water through a garden hose nozzle illustrates this case.

The conservation of mass principle is a fundamental law grounded in experimental observation—it requires that all mass entering, leaving, and accumulating within a system must be precisely accounted for. Conceptually, it is similar to balancing a checkbook, where you must track all deposits (inlets), withdrawals (outlets), and changes in the account balance (accumulation). If you can manage that, you are well-prepared to apply the conservation of mass to engineering systems.