# Bernoulli's Principle

Figure represents a tube of flow (or an actual pipe, for that matter) through which an ideal fluid is flowing at a steady rate. In a time interval Î” t, suppose that a volume of fluid Î” V, coloured purple in Figure (a), enters the tube at the left (or input) end and an identical volume, coloured green in Figure (b), emerges at the right (or output) end. The emerging volume must be the same as the entering volume because the fluid in incompressible, with an assumed constant density p.

Let y

_{1}, v

_{1}and p

_{1}be the elevation, speed, and pressure of the fluid entering at the left, and y

_{2}, v

_{2}and p

_{2}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

-----(i)

We can also write this equation as

--------(ii)

Like the equation of continuity, BernoulliÃ¢â‚¬â„¢s equation is not a new principle but simply the reformulation of a familiar principle in a form more suitable to fluid mechanics. As a check, let us apply BernoulliÃ¢â‚¬â„¢s equation to fluids at rest, by putting v

_{1}= v

_{2}= 0 in the equation (i). The result is

P

_{2}= p

_{1}+ pg(y

_{1}- y

_{2})

A major prediction of BernoulliÃ¢â‚¬â„¢s equation emerges if we take y to be a constant (y = 0, say) so that the fluid does not change elevation as it flows. Equation then becomes

Which tells us that:

If the speed of a fluid particle increases as it travels along a horizontal streamline, the pressure of the fluid must decrease, and conversely.

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# Proof of Bernoulli's Equation

Let us take as our system the entire volume of the (ideal) fluid shown in the figure. We shall apply the principle of conservation of energy to this system as it moves from its initial state (a) to its final state (b). The fluid lying between the two vertical planes separated by a distance L in the figure does not change its properties during this process; we need to be concerned only with changes that take place at the input and the output ends.

We apply energy conservation in the form of the work-kinetic energy theorem.

W = Î” K

Which tells us that the change in the kinetic energy of our system must equal the net work done on the system. The change in kinetic energy results from the change in speed between the ends of the pipe and is

Î” K = Î” m v - Î” m v

= p Î” V (v - v)

The work done on the system arises from two sources. The work W

_{g}done by the weight (Î” m g) of mass Î” m during the vertical lift of the mass from the input level to the output level is

W

_{g}= -Î” m g (y

_{2}- y

_{1})

= -pg Î”(y - y

_{1})

This work is negative because the upward displacement and the downward weight point in opposite directions.

Work W

_{p}must also be done on the system (at the input end) to push the entering fluid into the tube and by the system (at the output end) to push forward the fluid ahead of the emerging fluid. In general, the work done by a force of magnitude F, acting on a fluid sample contained in a tube of area A to move the fluid through a distance Î” x, is

FÎ” x = (pA) (Î” x) = p(AÎ” x) = pÎ” V

The work done on the system is then p

_{1}Î” V, and the work done by the system is -p

_{2}Î” V. Their sum W

_{p}is

W

_{p}= -p

_{2}Î”V + p

_{1}Î” V

= -(p

_{2}- p

_{1}) Î” V

The work-kinetic energy theorem now becomes

W = W

_{g}+ W

_{p}= Î” K.

Substituting for the values for W

_{g}, W

_{p}and Î”K we get,

=pgÎ” V(y

_{2}- y

_{1}) - Î” V(p

_{2}- p

_{1}) = pÎ” V(v - v)

This, after a slight rearrangement, matches the equation of continuity, which we set out to prove.