continuity equation

Meaning: In steady flow with no leaks, the mass entering a section of pipe each second equals the mass leaving. If the fluid is incompressible, its density is constant, so the volume passing each point per second (the flow rate) is also constant. A smaller cross-sectional area therefore requires a greater speed. Where a vessel branches, the flow into the branch point equals the sum of the flows out of the branches, so the total cross-sectional area of the branches is what matters.

Equation: A₁v₁ = A₂v₂, where A is cross-sectional area (m²) and v is average fluid speed (m/s). The product Q = Av is the volume flow rate (m³/s). For a compressible gas, ρ₁A₁v₁ = ρ₂A₂v₂, where ρ is density (kg/m³).

On the MCAT: A passage describes a pipe or blood vessel that narrows, widens or branches, and asks how speed changes. Since A = πr², halving the radius cuts the area to one fourth and makes the speed four times larger. Blood flow is the usual setting: the aorta is a single large vessel, but the capillaries have the greatest total area and the slowest flow. The question is often paired with Bernoulli's equation, where higher speed goes with lower pressure.

Common confusion: Narrowing a pipe does not reduce the flow rate. Speed rises and Q stays the same. In branched systems, compare the total area of all branches, not the area of one small vessel.