# Bernoulli's equation's contradiction

Using Bernoulli's equation P Pressure p density V velocity of fluid

$$P_1+ \rho gy_1+\frac{1}{2}\rho V_1^2 = P_2+\rho gy_2+\frac{1}{2}\rho V_2^2$$ $$V_1^2-V_2^2 =\left(2g(y_2-y_1) +\frac{2(P_2-P_1)}{p}\right)$$ $$V_1^2-V_2^2 =K$$ (1) Where K is constant

Using equation of continuity $$V_1^2(\frac{ A_1}{A_2})^2 = V_2^2$$

Substitution in (1) gives $$V_1^2(1-(\frac{A_1}{A_2})^2)=K$$ Here as A1 increase V1 increase which is opposite to equation of continuity in which as A1 increase V1 decrease.

Help(・へ・)

• Hi and welcome to physics SE. Please, use laTex notation for formulae. It's about writing them in between of dollar symbols, and laTex commands inside. See here: math.meta.stackexchange.com/questions/5020/… Commented Jun 10, 2019 at 15:14
• FGSUZ done. THANKS FOR SUGESTION :) Commented Jun 11, 2019 at 2:14
• The equations are not all rendering correctly and the some of the equations that did render appear to be wrong. Please check carefully your LaTex and equations are correct.
– hft
Commented Jun 11, 2019 at 2:27

## 1 Answer

The relation is true for A1=A2 only, in essence a varying area must lead in varying a pressure, there you assumed that the change in all external pressures is zero whereas area has changed, which is wrong.

Addendum: the relation is also true for V1,V2=0 where no change in external pressure causes no flow, any fluid moves from high pressure to low pressure.

• But pressure at a point is pgh .Where is area in this expression. Commented Jun 11, 2019 at 1:40
• The pressure is not just given by $\rho g h$. It also depends on the velocity, per the Bernoulli equation.
– hft
Commented Jun 11, 2019 at 2:32
• Oh yes I completely forgot when the equation was in front of me. Thanks :-) Commented Jun 11, 2019 at 3:00
• Well the fact that the pressure at a point is p.+pgh is only true for static fluids or those moving with constant velocity/area (you can make sure of this yourself using the equation) otherwise in hydraulics/fluid dynamics the relation is not true at all and you only find pressure using bernoulli's Commented Jun 16, 2019 at 9:46