**Bernoulli’s Equation**

represents· a tube through which an ideal fluid is flowing at a steady rate. In a time interval At, suppose that a volume of fluid ·AV, colored purple in, enters the tube at the left (or input) end and an identical volume, colored green in emerges at the right (or output) end. The emerging volume must be the same as the entering volume because the fluid is incompressible, with an assumed constant density p. et YI’ “I’ and PI be the eleven ion, speed, and pressure of the fluid entering at the left, and Y2′ \’2′ and P2 be the corresponding quantities for the fluid emerging at the right. By applying the principle of conservation of energy to the fluid, we shall show that these quantities are related by

Put another way, where the streamlines are relatively close together (that is, where the velocity is relatively great), the pressure is relatively low, and conversely. The link between a change in speed and a change in pressure makes sense if you consider a fluid element. When the element nears a narrow region; the higher pressure behind it accelerates it so that it then has a greater speed in the narrow region. When it nears a wide region, the higher pressure ahead of it decelerates it so that it then has a lesser speed in the wide region. Bernoulli’s equation is strictly valid of to the extent that the fluid is ideal. If viscous forces are present. thermal energy will be involved. We take no account of this in the derivation that follows.

**Proof of Bernoulli’s Equation**

We shall apply the principle of conservation of energy to this system as it moves “For irrotational flow (which we assume), the constant in Eq. 15-29 has the same value for all pointswithin the tube of flow; the points do not have to lie along the same streamline. Similarly, the points I and 2 in Eq. 15-28 can lie anywhere within the tube of flow.

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