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jostpuur

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In summary, the conversation discusses the Navier Stokes equations and their role in defining the time evolution of velocity in fluids. The pressure in the fluid depends on the model being used, such as Newtonian fluid or incompressible flow. The pressure gradient can act as a driving force in certain types of flows, but may not play a significant role in others. Ultimately, the pressure can be seen as a Lagrange Multiplier to the incompressibility constraint of the velocity. However, there may be additional conditions that need to be satisfied for the pressure in some cases.

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jostpuur

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gammamcc

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Depends on the model: Newtonian fluid, ideal gas, certain types of fluids, incompressible flow, etc.

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jostpuur

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The description of the millennium problem (http://www.claymath.org/millennium/Navier-Stokes_Equations/ ) says, that there we can restrict to incompressible fluids, so I'll stick with it. If have difficulty believing, that the PDEs given in the problem description are the whole truth about the problem, because the PDEs don't even contain the time derivative of the pressure anywhere. Would any pair (u,p) that satisfies the given Navier-Stokes equations really suffice? My instinct says that there must be more conditions to be satisfied for the pressure.

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dreamboy

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momentum_waves

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In higher-speed flows, the compressibility effect of the fluid can affect the pressure gradient. In low-speed momentum-driven flows, the pressure gradient term often plays little role in in the dynamics.

The Navier-Stokes equation is a set of partial differential equations that describe the motion of fluid substances. It is important in studying fluid dynamics because it allows us to understand and predict the behavior of fluids in various situations, such as in pipes, pumps, and other engineering applications.

The pressure term in Navier-Stokes equation is a result of the fluid's resistance to changes in velocity. In other words, as the fluid moves, it creates pressure that opposes any changes in its motion. This pressure evolves as the fluid experiences external forces, such as gravity or applied forces, and as it interacts with its surroundings.

The evolution of pressure in Navier-Stokes equation is affected by various factors, including the fluid properties (such as density and viscosity), the geometry of the system, the boundary conditions, and the external forces acting on the fluid.

The evolution of pressure in Navier-Stokes equation plays a crucial role in determining the behavior of fluids. It affects the flow velocity, direction, and patterns, and can lead to phenomena such as turbulence, vortices, and boundary layer separation. Understanding pressure evolution is essential in designing efficient and stable fluid systems.

The evolution of pressure in Navier-Stokes equation is a complex phenomenon, and there are still many challenges in fully understanding and modeling it. Some of these challenges include accurately capturing the boundary conditions, simulating highly turbulent flows, and dealing with the non-linearity of the equations. Additionally, the Navier-Stokes equation is a continuum model and may not accurately describe certain fluid behaviors at a molecular level.

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