The continuity equation is a statement of the conservation of mass in a system. Consider a pipe that is uniform in diameter at both ends but has a constriction between the ends, called a Venturi tube. Furthermore, assume that fluid is flowing through the pipe from one end through the narrow throat of the tube with cross-sectional areas A1 and A2, respectively. Let V1 and V2 be the average flow speeds at these cross sections. Assume also that there are no leaks in the pipe nor is fluid being pumped in through the sides. The continuity equation states that the fluid “mass flow rate”—the amount of fluid per unit time—must be the same at any cross section of the pipe or else there is an accumulation of mass—"mass creation"—and the steady flow assumption is violated. Simply stated,
(Mass rate)1 = (Mass rate)2
where
Mass rate = Density x Area x Velocity
This equation reduces to
p1A1V1 = p2A2V2
Since the fluid is assumed to be incompressible, p is a constant and equation (3) reduces to
A1V1 = A2V2
This is the simple continuity equation for inviscid, incompressible, steady, one-dimensional flow with no leaks. If the flow were viscous, the statement would still be valid as long as average values of V1 and V2 across the cross section were used.
By rearranging, one obtains
V2 = (A1/A2)V1
Since cross-section A1 is greater than cross-section A2, it can be concluded that V2 is greater than V1. This is a most important result. It states that the flow speed increases where the area decreases and the flow speed decreases where the area increases.
In fact, by the continuity equation, the highest speed is reached where the area is the smallest. This is at the narrowest part of the constriction, commonly called the throat of the Venturi tube.
It would be nice if this up and down crap would cease.
Back online