Illustration 15.1: The Continuity Equation

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A continuity equation is simply a way to express that "what goes in must come out." This simply means that as a fluid goes from a pipe of one diameter to another, the fluid flow changes. Restart. Assume an ideal fluid (position is given in meters and pressure is given in pascals). The dark blue in the animation is a section of water as it flows through the pipes from left to right (assume they are cylindrical, that is, the vertical distances in the animation correspond to the diameter of a circular cross section). Notice that as the water enters the narrower pipe, it goes faster. How long does it take the dark blue region to cross a line in the wide region and a line in the narrow region? The volume of the dark blue region divided by this time is the volume flow rate for each region. It should be the same for both regions because whatever goes in (per second) must come out (per second). We express this with the continuity equation Av = constant, where A is the cross-sectional area and v is the speed of the fluid flow (What are the units of Av? They should be volume/time). When we couple this with Bernoulli's equation (conservation of energy),

P + (1/2)ρv2 + ρgy = constant,

where P is the pressure, ρ is the density of the fluid, v is the speed of the fluid flow, and y is the height of the fluid (you can, of course, pick any point to be y = 0 m), we find a change in pressure as well. In this case, because the pipe is horizontal, y is the same, so we simply use P + (1/2)ρv2 = constant, so as the speed increases, the pressure decreases. Note the pressure readings by sliding the pressure indicator along the center of the pipe.

Note: The format of the pressure is written in shorthand. For example, atmospheric pressure, 1.01 x 105 Pa, is written as 1.01e+005.

Illustration authored by Anne J. Cox.

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