Equation of continuity

To discuss the mass flow rate of a fluid through a pipe, the flow is first assumed to be STEADY: at any given point in the pipe, the velocity of every fluid particle passing through it remains constant with respect to time (under this condition, every fluid particle's path is a streamline). Consider a pipe running from A to B whose cross-sectional area varies from $a_1$ (a wider section) to $a_2$ (a narrower section, with $a_1>a_2$); a non-viscous, incompressible liquid flows steadily through it with speed $v_1$ where the area is $a_1$ and speed $v_2$ where the area is $a_2$. In a small time interval $\Delta t$, the mass of fluid crossing the wide section is $m_1=(a_1v_1\Delta t)\rho$, and the mass crossing the narrow section in the same time is $m_2=(a_2v_2\Delta t)\rho$. Because the liquid is incompressible, mass must be conserved along the pipe: $m_1=m_2$, i.e. $a_1v_1\Delta t\,\rho=a_2v_2\Delta t\,\rho$, which, cancelling the common factors, gives $a_1v_1=a_2v_2$, or simply $av=\text{constant}$ -- the EQUATION OF CONTINUITY, a direct statement of the conservation of mass in fluid flow. Since $av$ stays constant, the flow speed v must be LARGER wherever the pipe's cross-sectional area a is SMALLER, and vice versa -- the smaller the cross section, the greater the fluid velocity there. This equation of continuity is exactly why, for instance, blood flowing from a s …