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?).
boneh3ad said:Presumably you made some assumptions when studying the derivation of those equations. Can you list what those assumptions are? That will give you a pretty big hint about what the equations' limitations are.
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?
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.
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.
Respected professorChestermiller said:The Navier Stokes equations apply to fluids described by Newton's law of viscosity (i.e., Newtonian Fluids). If the fluid exhibits more complicated behavior than that of a Newton fluid (a Newtonian fluid is one for which the stress tensor in linearly proportional to the rate of deformation tensor), the Navier Stokes equations will not apply. Such fluids are called non-Newtonian fluids, and include viscoelastic fluids and purely viscous non-Newtonian fluids. Examples of such fluids are polymer melts and solutions, and suspensions. And, of course, the Navier Stokes equations don't apply to solids.
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 .
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.
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:If i want to discuss the thermal behaviour of viscoelastic fluid then kindly suggest me a suitable Prandtl number . For which i have a justification that why i am taking this Prandtl number ?
I have recently solved the Navier-Stokes d.e. for air and water and other Madelung's fluids.ramzerimar said:I'm studying the Navier Stokes equations right now, and I've heard that those set of equations are invalid in some situations (like almost any mathematical formulation for a physics problem). 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 Navier Stokes equations are a set of partial differential equations that describe the motion of fluids. They are used to model a wide range of phenomena, from the flow of air over an airplane wing to the circulation of blood in the human body.
The Navier Stokes equations are used in many fields, including fluid mechanics, aerodynamics, and weather forecasting. They are also fundamental to the study of turbulence, which is present in many natural and man-made processes.
The Navier Stokes equations are only valid for certain types of fluids, such as incompressible, Newtonian fluids. They also assume that the flow is laminar and that the fluid is continuous, which may not always be the case in real-world situations.
The Navier Stokes equations become invalid when certain assumptions are not met, such as when the fluid is compressible, or when the flow becomes turbulent. They also break down at very small length scales, where the effects of viscosity become dominant.
Scientists use various techniques to deal with the limitations of the Navier Stokes equations. These include simplifying the equations or using numerical methods to solve them in complex situations. They also use experimental data and computational simulations to validate and improve the equations.