A non-viscous liquid of constant density `1000kg//m^3` flows in a streamline motion along a tube of variable cross section. The tube is kept inclined in the vertical plane as shown in Figure. The area of cross section of the tube two point P and Q at heights of 2 metres and 5 metres are respectively `4xx10^-3m^2` and `8xx10^-3m^2`. The velocity of the liquid at point P is `1m//s`. Find the work done per unit volume by the pressure and the gravity forces as the fluid flows from point P to Q.

Answer

B, C, D

Explanation

Given `A_(1)=4xx10^(-3)m^(2), A^(2)=8xx10^(-3)m^(2)`
`h_(1)=2 m, h_(2)=5m, v_(1)=1m//s` and `rho=10^(3) kg//m^(3)`

From continutity equation, we have
`A_(1)v_(1)=A_(2)v_(2)`
or, `v_(2)=(A_(1)/(A_(2))) v_(1)` or `v_(2)=((4xx10^(-3))/(8xx10^(-3)))(1m//s)`
`v_(2)=(1)/(2) m//s`
Applying Bernoulli's equation at sections 1 and 2
`p_(1)+(1)/(2)rho v_(1)^(2)+rho gh_(1)=p_(2)+(1)/(2)rho v_(2)^(2) +rho gh_(2)`
or, `p_(1)-p_(2)=rhog(h_(2)-h_(1))+(1)/(2) rho (v_(2)^(2)-v_(1)^(2))` ....(i)
(i) Work done per unit volume by the pressure as the fluid flows from P to Q.
`w_(1)=p_(1)-p_(2)`
`=rho g(h_(2)-h_(1))+(1)/(2)rho (v_(2)^(2)-h_(1)^(1))` [From Eq. (i)]
`={(10^(3))(9.8)(5-2)+(1)/(2) (10)^(3)((1)/(4)-1)}J//m^(3)`
`=[29400=375]J//m^(3)=29025 J//m^(3)`.
(ii) Work done per unit volume by the gravity as the fluid flows from P to Q.
`W_(2)=rho g(h_(2)-h_(1))={(10)^(3)(9.8)(5-2)}J//m^(3)`
or, `W_(2)=29400 J//m^(3)`