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Hari Seldon
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Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?
boneh3ad said:Did you try Googling this question first? It seems to turn up several hits.
the Navier-Stokes is a system with energy dissipation. The variational principle for the Euler equations is contained in M. Taylor's PDE vol 3Hari Seldon said:Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?
It seems to some that those equations could be approached with such methods:Hari Seldon said:Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?
The Navier-Stokes equation is a set of partial differential equations that describe the motion of fluid particles in a continuous medium. It is important in fluid dynamics because it allows scientists to predict the behavior of fluids in various scenarios, such as in pipes, around objects, and in turbulent flows.
The Lagrangian method is a mathematical approach that describes the fluid motion in terms of individual fluid particles, while the Hamiltonian method describes the fluid motion in terms of the overall energy of the system. The Lagrangian method is often used for problems involving small-scale fluid motion, while the Hamiltonian method is more suitable for large-scale fluid motion.
The Navier-Stokes equations incorporate the effects of viscosity and turbulence through the inclusion of the viscous stress tensor and the turbulent stress tensor, respectively. These terms account for the friction and random fluctuations within the fluid that affect its motion.
No, the Navier-Stokes equations cannot be solved analytically for most real-world scenarios. This is due to their nonlinear nature and the complex boundary conditions that are often present in fluid dynamics problems. Instead, numerical methods and approximations are used to solve these equations.
The Navier-Stokes equations have numerous applications in various fields, including aerodynamics, weather prediction, oceanography, and engineering. They are used to design efficient aircraft and cars, understand weather patterns, and optimize the performance of turbines and pumps, among others.