## Illustration 15.3: Ideal and Viscous Fluid Flow

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Bernoulli's equation describes the conservation of energy in an ideal fluid system such as the one shown in this animation **(position is given in tenths of meters and pressure is given in pascals)**. Restart. The vertical tubes are open to the atmosphere. Notice that the water level is lower to the right, indicating a lower pressure. Why is the pressure lower in the narrower tube? Notice also that the pressure only changes when going from a wider to a narrower tube. The pressure indicator measures the gauge pressure, not the absolute pressure (gauge pressure is pressure above atmospheric pressure). When there is viscosity (that is the fluid sticks together a bit so there is some friction), but still a smooth (laminar) flow, the pressure drops along the length of a pipe. Try it. For viscous flow, notice that, to get the same volume per unit time (Av = volume/time, where A is the cross-sectional area and v is the speed of the fluid flow), the pressure drops more in the narrower tube than in the wider tube. The equation governing the flow is Poiseulle's equation, Av = πR^{4}ΔP/8ηL, where R is the radius of the tube, L is the length of the tube, ΔP is the pressure difference and η is the viscosity of the fluid.

*Note: The format of the pressure is written in shorthand. For example, atmospheric pressure, 1.01 x 10 ^{5} Pa, is written as 1.01e+005.*

Illustration authored by Anne J. Cox.