Continuity Equation: Fluid Mechanics Explained

In the final lesson of Module 3 we derived a single, generic equation — the Reynolds Transport Theorem (RTT) — and then made a promise: set its intensive property b to 1, and out drops the continuity equation. This lesson collects on that promise. We will not re-derive RTT; we will simply specialize it to mass and read off the most-used equation in fluid mechanics.

Recall the move. The extensive property B = mass m has intensive property b = dm/dm = 1. Plugging b = 1 into RTT, the left side becomes dm/dt of a system — and a system, by definition, always contains the same matter, so dm/dt = 0. Setting the right-hand side to zero gives the integral continuity equation: the mass inside any control volume can change only because mass flows across its boundary. Nothing else. Mass is neither created nor destroyed; it is only transported.

That one line — the accumulation of mass inside the control volume balances the net mass flowing across its surface — is the whole of mass conservation written for a region of space. Every special case we use, from a contracting nozzle to a branching pipe network, is this same equation with some terms dropped. The rest of this lesson is about which terms to drop, and when.

Real hardware — pumps, valves, lengths of pipe — connects to the rest of a system through a small number of openings, or ports. When the flow through each port is steady and roughly uniform across the opening, the surface integral collapses into a sum over ports, and each port contributes a single number: the mass flow rate

where ρ is the density, A the port's cross-sectional area, and V the average speed normal to that area. This port form was introduced in the RTT lesson; here it becomes our working currency. Pair it with the volume flow rate Q = A V (cubic metres per second), and note that ṁ = ρ Q — mass flow is just density times volume flow.

A useful bookkeeping habit concerns sign. At an outlet the velocity points outward (V·n̂ > 0) so the port's ρAV is positive; at an inlet the velocity points inward (V·n̂ < 0) so the same product enters with a minus sign. Keeping those signs straight is what turns a jumble of connected pipes into a solvable mass balance.

The integral form is completely general but is rarely used as-is, because two simplifications apply to almost every engineering flow.

The first is steady flow. 'Steady' means nothing changes with time at any fixed point, so the accumulation term ∂/∂t ∫_CV ρ dV vanishes. What remains says: the net mass flowing out across the control surface is zero — equivalently, the total mass flow rate in equals the total mass flow rate out. In port form, Σṁ_out = Σṁ_in. This steady balance is the form used for the great majority of design calculations.

The second is steady plus incompressible. A liquid such as water is, for practical purposes, incompressible — its density ρ is constant and cancels out of the balance. The mass balance then becomes a volume balance: the total volume flow in equals the total volume flow out, ΣQ_out = ΣQ_in. For the commonest geometry of all — a single inlet and a single outlet — this collapses to the beautifully compact

which says that the product of area and velocity is the same at every cross-section. That single equation is the continuity equation as most engineers actually use it. Squeeze the area (a nozzle) and the velocity rises to keep A·V constant; widen it (a diffuser) and the velocity falls. The duct simulator below lets you feel that trade-off directly.