# Can a fluid approach the speed of light according to the equation of continuity?

We all are well aware with the equation of continuity (I guess) which is given by:

$$A_1V_1= A_2V_2$$

Where $$A_1$$ and $$A_2$$ are any two cross sections of a pipe and $$V_1$$ and $$V_2$$ are the speeds of the fluid passing through those cross sections.

Suppose $$A_1$$ is $$1\ m^2$$ and $$V_1$$ is $$2\ m/s$$ and $$A_2$$ is $$10^{-7}\ m^2$$. This will mean that $$V_2$$ will be $$2×10^{7}\ m/s$$ which is much closer to the speed of light.

But my teacher just said it is not possible. He didn't give a reason.

Is he saying this because density of the liquid will change (given mass increases and length contracts with higher speed)?

Why can't we use this equation to push fluids to a higher speed?

If there is a limit on the maximum speed we can get to with this equation, what is it?

For simplicity of calculations, you may take water.

• "For simplicity, you can take water"... I hate to break this to you... – jacob1729 Dec 3 '20 at 13:10
• If you're wondering what Jacob's getting at, water is weird, with an unusually high boiling point (at standard atmospheric pressure) for such a light molecule. Also, take a look at en.wikipedia.org/wiki/Water_(data_page)#Phase_diagram & other stuff on that page. – PM 2Ring Dec 3 '20 at 14:05
• I'm surprised no one has yet mentioned this: what-if.xkcd.com/147 – Vorbis Dec 4 '20 at 8:20
• "We all are well aware " what can I make of myself if I'm not aware of that equation? – infinitezero Dec 5 '20 at 9:39
• Pressure drop scales as the square of velocity. There is no physical device that can tolerate the pressures required to accelerate a small stream of any fluid to relativistic velocities. – David White Jun 7 at 0:55

That simplified form of continuity equation assumes that the fluid is incompressible. That is only a valid assumption at low Mach numbers. I think a typical “rule of thumb” is that a Mach number less than 0.3 is required for the assumption to hold. So for the continuity equation to hold in that form requires a speed which is much less than the speed of sound which in turn is much less than the speed of light. You cannot use the continuity equation to achieve supersonic flow, let alone relativistic flow.

Note, that is not to say that supersonic flow is impossible, but rather that it is not possible simply by application of that form of the continuity equation. You need a form that accounts for compressibility. Superluminal flow is fundamentally impossible as no massive particle can reach c with finite energy.

• The OP's asking about relativistic subluminal flow, not superluminal. – PM 2Ring Dec 3 '20 at 14:07
• @Ankit yes, it becomes compressible. The speed of sound is the speed at which pressure waves travel. So at high Mach number you cannot assume that the pressure is uniform. Hence the flow is compressible. – Dale Dec 3 '20 at 14:27
• @Ankit yes $\partial \rho/\partial t + \nabla \cdot (\rho \vec v)=0$ is the general form – Dale Dec 3 '20 at 14:50
• @Ankit All real fluids are compressible, but at low Mach number you may assume incompressibility as a simplifying approximation. As for relativistic flow, well, plasma is trickier stuff than gas or liquid. – John Doty Dec 3 '20 at 21:20
• @Ankit you really should go to physicsforums.com or a similar discussion site. Stack exchange is a Q&A site. I know you are aware that comments here are not for discussion. But it is clear that you really want discussions. That is fine, but not here. I think you would be happier at a site that encourages discussion rather than suppresses it. – Dale Dec 4 '20 at 4:06

You can check your hypothesis by closing the mouth of a pipe (from which water is flowing out) to different degrees. You will find that the flow velocity increases initially as the outlet area is decreased, reaches a maximum and then falls off to zero as you proceed to completely close the outlet. This is because, when the outlet area is very small, viscous forces and surface tension effects become dominant and retard the flow velocity.

That is one version of the continuity equation for sure. But that is not the most general one. As a matter of fact, that condition can be deduced from the incompressibility of a fluid as in a limit of the Navier-Stokes equation:

$$\rho(\partial_t + v \cdot \nabla)v + \nabla p = 0,$$

where $$\rho$$ is the density, $$p$$ is the pressure and $$v$$ is the velocity field in the fluid, assumed to satisfy $$\nabla\cdot v = 0$$. But even if you take everything under consideration in this description, the formalism will allow for fluids to have a speed higher than the speed of light.

The reason for this is that even the Navier-Stokes equation is a limit of its relativistic counterpart, the conservation of the stress-energy tensor:

$$\nabla^\mu T_{\mu\nu} = 0.$$

And from this equation, it is easy to show that your fluid will not propagate faster than light under the assumption that your $$T_{\mu\nu}$$ is physical.

• I appreciate your efforts but i am a high schooler and what you wrote are just going above my head :). Sorry if it hurts you. – Ankit Dec 3 '20 at 14:20
• Ohh, I'm sorry if I hurt you too. Than the short answer is: In Newtonian physics and everything you do without mention to relativity it will always be possible to "go faster than light". For example, if you uniformly accelerate anything for long enough in Newtonian mechanics, the theory tells you that it is possible to reach the speed of light and even surpass it. The only way to really handle these things properly is by using relativistic mechanics, which will recover the Newtonian theory in what is called the nonrelativistic limit, which is only valid for speeds much smaller than $c$ – Tales Rick Perche Dec 3 '20 at 15:22
• @Ankit Perhaps the problem is that you are wishing to ask difficult questions without wanting difficult answers. I mean yeah, I think it is reasonable to think that perhaps there are simple explanations behind everything, but probably this is a point to re-evaluate beliefs. – Buraian Dec 8 '20 at 9:13

One must look at all relevant equations when discussing what's possible or what's doable. In this case, you must look also at the momentum equation, which is Newton's Law, and when you do, you see that the pressure required for high Mach number in fluids becomes impossible to achieve in practice.

In talking about incompressible flow, from a fundamental point of view, it's not possible to achieve even Mach 1 in an "incompressible" fluid. The criterion by which a fluid can be considered incompressible is that the Mach number squared be much less than unity. That means the Mach number itself should be less than about 0.3, and above that, the fluid is compressible; i.e. changes in it's density become important. Thus, one can argue that it's impossible for an incompressible fluid to have a velocity close to the speed of sound in that fluid, simply because it must experience compressibility for Mach number above about 0.3.

Also, when you go through the numbers, you will find that it takes very large pressures to accelerate say, water to high Mach number. It's thus pure fantasy to even imagine speeds approaching the speed of light.

• is there a physical reason why it becomes incompressible ? – Ankit Dec 5 '20 at 4:42
• @Ankit Of course. The pressure forces and the difference in pressure forces between adjacent packets of fluid are great enough to significantly change local density. The criterion M^2 << 1 is only saying that, because M = v/c where v is velocity and c is sound speed (fluid property). v^2 represents the dynamic pressure, so when dynamic pressures are large enough, local density changes. This also means that local temperature changes, which then requires another equation, the Energy Equation, which accounts for temperature changes. – ttonon Dec 6 '20 at 17:03

One thing not pointed out in the above answers is that if a fluid was truly "incompressible", it would violate special relativity. This is because the speed of sound in a fluid can be derived from its equation of state $$\rho(P,T)$$ which gives the density as a function of the pressure and temperature. Specifically, we have $$\frac{1}{v^2} = \left( \frac{\partial \rho}{\partial P} \right)_s,$$ where $$v$$ is the speed of sound in the material and the derivative is taken while holding entropy constant. If $$\rho$$ were really & truly "incompressible", then it would be constant with respect to $$P$$, and the above derivative would vanish. Thus, $$v^2 \to \infty$$, which is faster than light.[citation needed]

The resolution to this is that "incompressible" doesn't literally mean incompressible. It just means "the range of pressures we're looking at is so small that the compression of the fluid is negligible."

• +1 thanks for the answer. Does density increase or decrease with speed ? Why ! – Ankit Dec 5 '20 at 14:21
• @Ankit: The equation of state generally is only assumed to hold in the rest frame of the fluid, which is why $v$ doesn't appear in it. (Note that the $v$ in my equation above is the speed of waves in the fluid, not the speed of the fluid itself.) – Michael Seifert Dec 6 '20 at 14:15

I recieved a lot of great answers.

But I came up with an idea this morning (which I would like to share) which is not much different from the answers I recieved (I guess).

When Sir Isaac Newton was trying to estimate the speed of sound, he assumed that the process of transmission is isothermal stating that the molecules get enough time to be in thermal equilibrium with the surrounding. But with this he got a different (more specifically: smaller) value for the speed of transmission.

But then came Laplace and he corrected Newton by saying that the process is too fast to be isothermal and thus the process is definitely adiabatic.

So if we take the above paragraph in consideration, we can similarly come to an estimate (not a certain reason though).

If the speed of water is something like $$2150-200$$ $$m/sec$$ , the molecules have some time to interact with its surrounding and so pressure inside the tube doesn't drop much. But as it approaches the speed of sound or any higher speed the molecules have very less time to interact and the pressure on the walls of the tube decreases too a great extent and thus this situation becomes similar to that of an imploding can.

We can't even exceed the speed of sound with this equation. So thinking to reach the speed of light with this equation is completely stupid 😌.