Bernoulli's theorem explain
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вernoυllι'ѕ тнeoreм. вernoυllι'ѕ тнeoreм, ιn ғlυιd dynaмιcѕ, relaтιon aмong тнe preѕѕυre, velocιтy, and elevaтιon ιn a мovιng ғlυιd (lιqυιd or gaѕ), тнe coмpreѕѕιвιlιтy and vιѕcoѕιтy (ιnтernal ғrιcтιon) oғ wнιcн are neglιgιвle and тнe ғlow oғ wнιcн ιѕ ѕтeady, or laмιnar.
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According to it for an stream flow of an liquid, the total energy [sum of kinetic energy, potential energy and pressure energy] per unit mass remains constant.. At every cross section throughout the liquid flow.
Consider a streamline flow of ideal liquid across a pipe AB as shown in figure....
For an ideal liquid = constant
= 0
a1, a2 be the area of cross section at A and B
v1, v2 be the volume of liquid at A and B.
P1, p2 the pressure of liquid at A and B.
= density of liquid.
If m is mass entering per sec.
Then;
a1v1 = a2v2 = m
a1v1 = a2v2 = = v
F = P1a1
W1 = P1 a1 v1 = P1 V [Work done per pressure at A]
W2 = P2 a2 v2 = P2 V [Work done per pressure at B]
Net work done = P1 V - P2 V
Kinetic Energy at A = m
Kinetic Energy at B = m
Potential Energy at A = mg
Potential Energy at B = mg
According to work energy principle...
= +
+ + = + + ....(1)
Divide eq. (1) by m
+ + = + +
+ + = + +
+ + = Constant...
Hence, proved...
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