# Why incompressible flow does not satisfy energy equation?

• I
I encountered this statement on my lecture notes today, I dont understand why compressible flow needs to have another constraint of energy equation while incompressible flow only satisfies continuity and momentum equation. And how is this energy equation related to the speed of sound?

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Chestermiller
Mentor
I encountered this statement on my lecture notes today,
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I dont understand why compressible flow needs to have another constraint of energy equation while incompressible flow only satisfies continuity and momentum equation. And how is this energy equation related to the speed of sound?
I don't know what the speaker is referring to. Incompressible flow of a Newtonian Fluid certainly does satisfy the thermal energy balance equation as well as the mechanical energy balance equation.

Gold Member
My guess is that whoever made the slides was simply being inexact with their language. Incompressible flows absolutely do satisfy the energy equation. However, for an incompressible flow, the energy equation is entirely decoupled from the mass and momentum conservation equations, so you can solve the entire velocity and pressure fields without ever touching the energy equation. That's one of the major advantages of treating a flow as incompressible. So, for an incompressible flow, while it does still satisfy the energy equation, we can entirely ignore it unless we specifically care about the temperature field.

Chestermiller
Mentor
My guess is that whoever made the slides was simply being inexact with their language. Incompressible flows absolutely do satisfy the energy equation. However, for an incompressible flow, the energy equation is entirely decoupled from the mass and momentum conservation equations, so you can solve the entire velocity and pressure fields without ever touching the energy equation. That's one of the major advantages of treating a flow as incompressible. So, for an incompressible flow, while it does still satisfy the energy equation, we can entirely ignore it unless we specifically care about the temperature field.
Yes. As you say, if there is significant viscous heating and/or direct external heat transfer to the fluid, the viscosity of the fluid can change, and this then couples to the equation of motion. So, under these circumstances, the energy balance equation can have a significant effect on the fluid mechanics, even for an incompressible fluid.

Gold Member
Yes. As you say, if there is significant viscous heating and/or direct external heat transfer to the fluid, the viscosity of the fluid can change, and this then couples to the equation of motion. So, under these circumstances, the energy balance equation can have a significant effect on the fluid mechanics, even for an incompressible fluid.
There was a fair bit of discussion about this a few months back and whether this truly qualifies as being incompressible or not. Different authors will treat the question differently. Personally, for exactly this reason, I'd argue for there being essentially three flow regimes: incompressible, variable-density, and compressible (or I guess four regimes if you split compressible into subsonic and supersonic). Not everyone agrees with me, though.

Chestermiller
Mentor
There was a fair bit of discussion about this a few months back and whether this truly qualifies as being incompressible or not. Different authors will treat the question differently. Personally, for exactly this reason, I'd argue for there being essentially three flow regimes: incompressible, variable-density, and compressible (or I guess four regimes if you split compressible into subsonic and supersonic). Not everyone agrees with me, though.
I'm confused. Viscous dissipation is a completely different phenomenon than compressibility/incompressibility (which I'm sure you know).

Gold Member
I'm confused. Viscous dissipation is a completely different phenomenon than compressibility/incompressibility (which I'm sure you know).
Yes, I was referring to your comment about how if you have large temperature gradients (for example) then the equations can remain coupled. That's a situation where the kinematics can remain incompressible in a certain sense but the density can still vary in space (or viscosity). Often those are called "variable-density flows" to distinguish them from truly incompressible flows. Depending on your definition of "incompressible", they may or may not still fall within that umbrella.

Chestermiller
Mentor
Yes, I was referring to your comment about how if you have large temperature gradients (for example) then the equations can remain coupled. That's a situation where the kinematics can remain incompressible in a certain sense but the density can still vary in space (or viscosity). Often those are called "variable-density flows" to distinguish them from truly incompressible flows. Depending on your definition of "incompressible", they may or may not still fall within that umbrella.
I wasn't referring to the effect of temperature on density. I was referring to the effect of temperature variations on viscosity variations, which, in turn, couples to the fluid dynamics.