State and prove Bernoulli's theorem for a non-viscous liquid.
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Answer:
Bernoulli’s theorem: According to Bernoulli’s theorem, the sum of pressure energy, kinetic energy, and potential energy per unit mass of an incompressible, nonviscous fluid in a streamlined flow remains a constant. Mathematically, PρPρ + 1212v2 + gh = Constant This is known as Bernoulli’s equation. Proof: Let us consider a flow of liquid through a pipe AB. Let V be the volume of the liquid when it enters A in a time t which is equal to the volume of the liquid leaving B in the same time. Let aA , vA and PA be the area of cross section of the tube, velocity of the liquid and pressure exerted by the liquid at A respectively. Let the force exerted by the liquid at A is FA = PA aA Distance travelled by the liquid in time t is d = va t Therefore, the work done is W = FA d = PA aA vA t But a v t = aA d = V, volume of the liquid entering at A. Thus, the work done is the pressure energy (at A), W = PA d = PA V Pressure energy per unit volume at Pressure energy per unit mass Since m is the mass of the liquid entering at A in a given time, therefore, pressure energy of the liquid at A is EPA = PA V = PA V × ( mmmm) = mPAρPAρ Potential energy of the liquid at A, PEA = mghA Due to the flow of liquid, the kinetic energy of the liquid at A, KEA = 1212 mvA2A2 Therefore, the total energy due to the flow of liquid at A, EA = EPA + KEA + PEA EA = mmPAρmPAρ + 1212mv2AvA2 + mghA Similarly, let aB , vB and PB be the area of cross section of the tube, velocity of the liquid and pressure exerted by the liquid at B. Calculating the total energy at FB, we get EB = mmPBρmPBρ + 1212mv2BvB2 + mghB From the law of conservation of energy, EA + EB Thus, the above equation can be written as PρgPρg + 1212v2gv2g + h = Constant The above equation is the consequence of the conservation of energy which is true until there is no loss of energy due to friction. But in practice, some energy is lost due to friction. This arises due to the fact that in a fluid flow, the layers flowing with different velocities exert frictional forces on each other. This loss of energy is generally converted into heat energy. Therefore, Bernoulli’s relation is strictly valid for fluids with zero viscosity or non-viscous liquids. Notice that when the liquid flows through a horizontal pipe, then h = 0 ⇒ PρgPρg + 1212 v2gv2g = Constant
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