# Where is the limitation within the assumptions leading to Bernoulli's law?

Lets say there is a horizontal pipe $P_1$ of diameter $d_1$, connected to another horizontal pipe $P_2$ with diameter $D_2$.

The fluid inside shall be considered as "ideal" without viscosity and flows with $v_2$ at the outlet.

The pressure at the outlet of $P_2$, $p_2$ shall be normal pressure.

Applying Bernoulli's law I get a reduced pressure $p_1$ as compared to $p_2$. By reducing $D_1$ more and more, $p_1$ is will decrease too. But then there is a point, where it would be zero or even negative. Since negative pressures are nonphysical, there must be a limit, where the assumptions behind the derivation of Bernoulli's law are not met anymore.

However, I cannot identify that point: I know, that cavitation effects will start to occur, but those are more a property of the particular fluid and in principle I could think (?) about a hypothetical fluid not subject to cavitation. So I think there must be a pitfall in the derivation of Euler's equations, from which Bernoulli's law is to be derived by integration.

• The limitation is that the fluid will capitate (basically boil) if the pressure drops below its equilibrium vapor pressure at the fluid temperature. – Chet Miller May 24 '18 at 11:01

I believe that your question is aimed at the fact that if the pipe diameter is kept on decreasing, then the velocity of fluid increases tremendously, and the pressure at the inlet point (on application of Bernoulli's theorem) becomes negative.

The explanation to this fact lies with the physical laws. When we deal with pipe flows, we usually consider Reynolds' number as a governing non-dimensional parameter, which is a ratio of Inertial forces experienced by the fluid element to the viscous forces. This parameter plays an important role upto a certain pipe diameter.

Once the pipe diameter is reduced to a value 'd'<< D (characteristic diameter for normal pipe flow), then the predominant force becomes SURFACE TENSION force, and not just the VISCOUS forces. Weber's Number becomes a governing non-dimensional parameter. Hence, it is not true that the velocity of the fluid keeps on increasing till infinity. Accordingly, the Bernoulli's theorem needs to be modified to account for surface tension forces.

Also, as you have stated above, a negative pressure value basically indicates a reversal in flow direction. This can be visualized assuming that the fluid particles do not possess sufficient energy to overcome the adverse pressure gradient, and hence, they tend to reverse their path. This is very much plausible, and in case calculations lead us to a negative pressure flow, the same should be investigated in the laboratory or maybe, simulated using commercial solvers.

• I understand the limitation of the model with regard to reality. But still I do not understand, why the assumption of an in-viscid fluid (which is wrong in reality) leads to negative pressures. Where is the point, where Euler's equations fail? – michael May 24 '18 at 11:30
• This is because Euler's equation(which is basically conservation of momentum) does not take into account the Surface tension forces, and thus, there is a missing component which is not being integrated to give Bernoulli's equation. So, of course, the results are bound to differ and as seen in this case, produce erroneous results. – Subhas Nandy Jul 19 '18 at 18:03

An ideal inviscid fluid is the assumption which breaks at smaller $D_1$; as you decrease $D_1$ the viscosity start playing a more dominant role in any real fluid.

Mathematically speaking, the Bernoulli equation $$p_1+0.5\rho v_1^2=p_2+0.5\rho v_2^2=\textrm{constant}$$ for an ideal incompressible fluid does not preclude the possibility of either pressure becoming negative to compensate for high velocity, or velocity becoming imaginary to compensate for high pressure. That (absolute) pressure and velocity are both real and positive is an additional physical stipulation, based on the known physical fact that energy cannot be negative.

• Right - in fact I never thought about why pressure is always positive in fluids... – michael May 25 '18 at 11:43