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pyroknife
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Can you derive the heat conduction equation from the navier stokes equations (particularly the energy eqn)?
Thanks. Question.boneh3ad said:The energy equation contains a heat conducting term already (the one containing ##\kappa \nabla T##), so then if you zeroed out all the other terms, then yes, you could get the heat equation. Since conduction is actually one of several parts of the energy equation, though, I don't know why you would want to do this. There are many other factors in energy transport.
pyroknife said:Thanks. Question.
Are equations like potential flow, Euler's, heat conduction equation formulated before the Navier Stokes equations? I see that Euler existed before Navier&Stokes.
If so, then were the Navier Stokes equations formulated on the basis of these simplified equations?
Wait I thought the Navier Stokes equations refers to the 3 conversation equations for mass, momentum, and energy? And hence the pluralness of "equationS?" If its just the momentum equation then why is it not called the Navier Stokes Equation?Andy Resnick said:A minor comment- the Navier Stokes equation(s) is/are concerned with momentum transport. Just as mass transport is handled by a different equation-the continuity equation- energy transport is handled by it's own equation. All three equations are, to some degree, independent:
http://www.ldeo.columbia.edu/~mspieg/mmm/Conserveq.pdf
Oh yeah I forget it is a vector equation. So the NS equations technically include the continuity and energy eqn?boneh3ad said:Momentum conservation results in one equation per spatial dimension, hence there are three momentum equations.
Really it's just a semantics debate, though.
No. It only addresses momentum. The continuity equation can be combined with the NS equation to obtain a different form of the equation, but its primary focus is still momentum. It is also possible to obtain the "mechanical energy balance equation" by dotting the NS equation with the velocity vector. This can be subtracted from the "overall energy balance equation " to yield the so-called "thermal energy balance equation".pyroknife said:Oh yeah I forget it is a vector equation. So the NS equations technically include the continuity and energy eqn?
The heat equation is a partial differential equation that describes the flow of heat in a given medium. It is a special case of the Navier-Stokes equations, which are a set of equations that describe the motion of fluids in various situations. In the heat equation, the velocity terms in the Navier-Stokes equations are assumed to be zero, as heat is not a fluid and does not have a velocity.
The main assumptions made in deriving the heat equation from Navier-Stokes equations are that the fluid is incompressible, the velocity terms are negligible, and that the thermal conductivity is constant. These assumptions simplify the Navier-Stokes equations and allow for a more specific description of heat flow.
The heat equation can be solved numerically using various methods, such as finite difference, finite element, or spectral methods. These methods discretize the equation and solve it iteratively, taking into account boundary conditions and initial conditions. The choice of method depends on the specific problem and the desired level of accuracy.
The heat equation has a wide range of applications in various fields, including physics, engineering, and mathematics. It is commonly used to model heat transfer in materials, such as metals, liquids, and gases. It is also used in the study of diffusion, electric circuits, and quantum mechanics.
One of the main limitations of the heat equation is that it assumes a steady-state condition, meaning that the temperature distribution does not change with time. Additionally, it does not take into account convection or radiation effects, which may be important in some situations. It also assumes a homogeneous medium, which may not be the case in real-world scenarios.