The velocity of liquid flowing through a tube at certain distance from the axis of tube

Explanation

Construction: It consists of two wider tubes A and A' (with cross sectional area A) connected by a narrow tube B (with cross sectional area a). A manometer in the form of U-tube is also attached between the wide and narrow tubes as shown in Figure. The manometer contains a liquid of density `'rho_(m)'`.

Theory: Let `P_(1)` be the pressure of the fluid at the wider region of the tube A. Let us assume that the fluid of density `'rho'` flows from the pipe with speed `'v_(1)'` and into the narrow region, its speed increases to `'v_(2)'`.
According to the Bernoulli's equation, this increases in speed is accompanied by a decrease in the fluid pressure `P_(2)` at the narrow region of the tube B. Hence, the pressure difference between the tubes A and B is noted by measuring the height difference `(DeltaP=P_(1)-P_(2))` between the surfaces of the manometer liquid.
From the equation of continuity, we can say that `Av_(1)=av_(2)` which means that
`v_(2)=(A)/(a)v_(1)`
Using Bernoulli's equation
`P_(1)+rho(v_(1)^(2))/(2)=P_(2)+rho(v_(2)^(2))/(2)=P_(2)+rho(1)/(2)((A)/(a)v_(1))^(2)`
From the above equation, the pressure difference
`DeltaP=P_(1)-P_(2)=rho(v_(1)^(2))/(2)((A^(2)-a^(2)))/(a^2)`
Thus, the speed of flow of fluid at the wide end of the tube A
`v_(1)^(2)=(2(DeltaP)a^2)/(rho(a^(2)-A^(2)))`
`rArrv_(1)=sqrt((2(DeltaP)a^(2))/(rho(a^(2)-A^(2))))`