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Streamline fluid flow and varying tube diameter. Continuity equation.

Fluid flow. Suppose fluid is flowing in a tube in which the cross-sectional area is greater in one section than in another section. Then the velocity of the fluid will be higher in the section of the tube with the smaller cross-section. See Fig. 1. Why is this so and what is the mathematical law relating the velocities and the cross-sections?

Examples of this idea

- fluid flow in a tube where the diameter reduces down

- fluid flow in a river that is wide and deep in one place and narrow and shallow in another

- vehicles moving on a highway in which several lanes are funneled into a single lane (perhaps due to road construction)

- marbles flowing in a channel that reduces down in diameter

If a pipe has a cross-sectional area of 1 sq. ft. in one section and 1/3 sq. ft. in another section, then water flowing in the 1/3 sq. ft. section will flow at three times the speed of the water in the 1 sq. ft. section. Give an intuitive explanation of why this must be true.

Consider the example in Fig. 2 in which marbles in three channels (or lanes) are fed into a single channel. Let “d” be the diameter of a marble. Then it is intuitively obvious that when a marble in the single channel portion has moved a distance of 3d the marbles in the 3-channel section will have moved a distance of only 1d. Thus if the marbles in the 1 channel section are moving at a rate of 3d per second, then the marbles in the 3 channel section will be flowing at a rate of only 1d per second.

Consider now the fluid flow shown in Fig. 3 where the cross-sectional area at point 1 is A1 and at point 2 is A2. In time t, the fluid at point 1 will advance a distance of d1 = V1t and a volume of liquid equal to A1d1 = A1V1t will pass point 1. Similarly, the volume of fluid passing point 2 in time t will be A2V2t. The volume of fluid passing points 1 and 2 in time t must be equal so

A1V1t = A2V2t

and

A1V1 = A2V2

which is the equation of continuity for the steady flow of an incompressible liquid.