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- Thread starter ramzerimar
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The term viscoelastic relates to the mechanical behavior of the fluid, not the heat transfer behavior.In summary, the Navier-Stokes equations can be applied to incompressible flows, but not to compressible flows. They also have limitations in terms of the type of fluid they can describe, such as non-Newtonian fluids. Some examples of when the equations may not apply include when dealing with compressible flow or when the fluid behavior is not adequately modeled by a continuum. Additionally, the Prantdl number, which relates to heat transfer behavior, is not relevant when considering the mechanical behavior of viscoelastic fluids.

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One example it can't be applied in is when the system is not Galilean invariant. This occurs for a fluid which is subject to a nonlinear vector potential which depends on the density ##\rho## of the fluid. In this situation, the kinetic energy density of the fluid is nonlinear in ##\rho## and the pressure of the fluid depends explicitly on the flow.ramzerimar said:I would like to know in which situations I cannot apply the NS equations, and what is the common procedure when something like that happens (like, there are any other set of PDEs that describe fluid flow in situations where the NS equations are invalid?).

The way forward in these kind of situations is to start from first principles without assuming anything in particular..In your case Cauchy's equation would do.

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boneh3ad said:

I've read that one of the assumptions is that the NS equations are for a incompressible flow. So, for dealing with compressible flow problems, the NS equations cannot be applied?

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The NS equations can be applied to incompressible flows.ramzerimar said:I've read that one of the assumptions is that the NS equations are for a incompressible flow. So, for dealing with compressible flow problems, the NS equations cannot be applied?

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To analyze compressible flow I would need to rewrite the NS equations or just add the energy equation to the mix? The difference that I see is that, in compressible flow, the density also changes and so becomes a variable, and I would need another equation to describe the flow.Chestermiller said:The NS equations can be applied to incompressible flows.

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Yes, and you would also have to use the compressible form of the continuity equation. And you might have to allow for temperature-dependent viscosity.ramzerimar said:To analyze compressible flow I would need to rewrite the NS equations or just add the energy equation to the mix? The difference that I see is that, in compressible flow, the density also changes and so becomes a variable, and I would need another equation to describe the flow.

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There are the basic ##u##, ##v##, ##w##, ##p##, ##\rho##, and ##T## variables. Additionally, you could have ##\mu## (dynamic viscosity), ##\lambda## (second coefficient of viscosity), ##c_p##, ##c_v##, ##\kappa## (thermal conductivity), or even ##X_i## (mass fractions of various chemical species). The Navier-Stokes equations can technically apply to problems involving all of those variables, both compressible and incompressible.

The two most important limitations on the Navier-Stokes equations is that they only apply to (a) fluids that can adequately be modeled by a continuum and (b) Newtonian fluids. Some examples of when the continuum breaks down are in the upper atmosphere where density is so low that the mean free path is comparable to the flow scales or in microfluidics when the flow scales are so small that they are comparable to the mean free path.

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Respected professorChestermiller said:

i just want to know the examples of viscoelastic fluids with the values of Prandtl number . I saw a lot of articles , where viscoelastic fluids are used with Prandtl number 0.7 , 1 and 3 etc . which are shocking for me because these Prandtl numbers are for air and water and air ,water are Newtonian fluids .

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The Prantdl number relates to the heat transfer behavior of the fluid, not the mechanical behavior. The term viscoelastic relates to the mechanical behavior of the fluid, not the heat transfer behavior.Idrees Afridi said:Respected professor

i just want to know the examples of viscoelastic fluids with the values of Prandtl number . I saw a lot of articles , where viscoelastic fluids are used with Prandtl number 0.7 , 1 and 3 etc . which are shocking for me because these Prandtl numbers are for air and water and air ,water are Newtonian fluids .

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Professor thanks... but by taking Pr =0.7 means v r talking about air ... is ve take a air as viscoelastic fluid...Chestermiller said:The Prantdl number relates to the heat transfer behavior of the fluid, not the mechanical behavior. The term viscoelastic relates to the mechanical behavior of the fluid, not the heat transfer behavior.

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This is too far removed from the original theme of the present thread to continue along these lines. Please start a new thread that addresses the topic of how to analyze heat transfer to viscoelastic fluids.Idrees Afridi said:

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I have recently solved the Navier-Stokes d.e. for air and water and other Madelung's fluids.ramzerimar said:

Those fluids are characterized by having a (well-behaved and eventually time-dependent) density. Please do see my publication cited as

R. Meulens , "A note on N-soliton solutions for the viscid incompressible Navier–Stokes differential equation", AIP Advances 12, 015308 (2022) https://doi.org/10.1063/5.0074083

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