# Contradiction Between Bernoulli's Equation and Fluid Flow

I read in my physics textbook that fluids flow from regions with higher mechanical energy to regions with lower mechanical energy (1).

Also, the Bernoulli's equation states that: $$PV + mgz + \frac{1}{2} mv^2 =constant$$

I find these two statements contradictory. If the total mechanical energy at any point in a fluid body is constant, as according to Bernoulli's equation, then there would be no flow in a fluid body at anytime. In other words, the statement (1) doesn't even apply in the first place because there is no variability in the mechanical energy in a fluid body, as according to Bernoulli's equation.

I know that I am missing something and appreciate any explanation.

• Can you provide an exact quote from your textbook?
– Gert
Dec 28 '20 at 18:02
• @Gert I uploaded the pictures. Dec 28 '20 at 18:08
• Thank you, will look into it.
– Gert
Dec 28 '20 at 18:16
• So you can't interconvert kinetic energy and potential energy without the total of the two changing? Dec 29 '20 at 13:23

Normally, in a instalation for water distribution, there is an water tank at a certain height $$h$$ over the floor.

If some tap is open, and the flow has a velocity $$v_1$$ in a pipe at the level of the water tank, and $$v_2$$ at a pipe in the floor: $$P_1 + \mu gh + \frac{1}{2}\mu v_1^2 = P_2 + 0 + \frac{1}{2}\mu v_2^2$$

If the pipes have the same diameter, the velocities are equal, and loss of gravitational potential results in elastic (pressure) potential. Otherwise there are also differences in the kinetic energy.

All that calculations neglect the loss due to friction between fluid and pipes. If they are taken in consideration (important for long pipe lengths), some energy is lost, and the pressure $$P_2$$in the example is smaller than calculated by the formula.

• How does your answer my question? Dec 29 '20 at 0:09
• There is always some loss of energy in the pipes, because real fluids have viscosity. So, it is right: the flow always have less energy downstream than upstream, even is the difference is small enough for short lengths of pipes, that we can use Bernoulli's equation with good precision. Dec 29 '20 at 0:30
• I apologize but it takes me longer than most to learn. I still don't really see how this answers my question. In the beginning of my question, I say it is in my book that says water flows from regions with higher mechanical energy to regions with lower mechanical energy, not regions with higher potential/kinetic energy/pressure to regions with lower ones. And mechanical energy is always constant in a fluid body, as accoding to Bernoulli's equation. This means there cannot even be any flow in the first place. This is where I am confused at. Dec 29 '20 at 1:12
• Mechanical energy, expressed by the sum of 3 terms of the Bernoulli's equation is almost constant. But some is lost by the friction with the pipes. The equation is a good approximation, but not exact in the real world. Dec 31 '20 at 1:25

There is no contradiction! Bernoulli's equation can be written as $$\frac{1}{2}\rho v^2+\rho gh+p=\mathrm{constant}$$

where the first term what sometimes called dynamic pressure. It can be thought of as the fluid's kinetic energy per unit volume. The second term denotes the potential energy of the fluid per unit volume.

If you rearrange these term a bit, $$\left(\frac{1}{2}\rho v_2^2+\rho gh_2\right)-\left(\frac{1}{2}\rho v_1^2+\rho gh_1\right)=p_1-p_2$$ Now everything get washed.

If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.

Edit: If you aren't able to see is what I meant is that on the left-hand side you have a difference in mechanical energy which is equivalent to saying in terms of pressure difference. Pressure difference implies the fluid flow.

• In my question, I say that water flows from regions with higher mechanical energy to regions with lower mechanical energy, not higher pressure to lower pressure. So I still don't see how this answers my question. Am I missing something? Dec 29 '20 at 0:12
• @ASlowLearner See the edit Dec 29 '20 at 5:02
• I think pressure is part of the total mechanical energy of a fluid. It is called elastic potential energy, which is from compressing a volume of fluid. Dec 29 '20 at 7:11