Final answer:
The continuity equation in fluid dynamics relates the flow rate and velocity of an incompressible fluid. It can be derived using an infinitesimal control volume of rectangular shape. The equation states that the mass flow rate into a volume is equal to the mass flow rate out of the volume.
Step-by-step explanation:
The continuity equation in fluid dynamics relates the flow rate and velocity of an incompressible fluid. It states that the mass flow rate into a volume is equal to the mass flow rate out of the volume. The equation can be derived from first principles using an infinitesimal control volume of rectangular shape.
To derive the continuity equation, we can consider a fluid flowing through a pipe with two different cross-sectional areas A1 and A2. The mass flowing into the pipe (Pin) multiplied by the cross-sectional area (A1) and velocity (V1) must equal the mass flowing out of the pipe (Pout) multiplied by the cross-sectional area (A2) and velocity (V2). This can be written as A1V1 = A2V2.