45
votes
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How were the Navier-Stokes equations found in the first place if we can't solve them?
I just wanted to give a more concrete idea of how we know these equations even though we have trouble proving analytical theorems about them.
Stuff moving in space
Consider any stuff (as in, any ...
- 35.2k
32
votes
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Why can't the Navier Stokes equations be derived from first principle physics?
None of the interesting equations in physics can be derived from simpler principles, because if they could they wouldn't give any new information. That is, those simpler principles would already fully ...
- 334k
24
votes
Why can't the Navier Stokes equations be derived from first principle physics?
They are derivable from classical mechanics using either the continuum or molecular points of view.
Starting with a continuum view, one applies conservation of mass, momentum, and energy to a control ...
- 3,007
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votes
Why can't the Navier Stokes equations be derived from first principle physics?
I once asked Putterman after a similar colloquium what he meant by this statement, and his answer was "long time tails". Long time tails are fractional powers that appear in the long time behavior of ...
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20
votes
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What does it mean that a substance can be smelled from far away?
You are not missing anything. Rather I think you are placing too much emphasis on the scientific accuracy of something said for effect in a very chatty presentation.
The spoken words almost ...
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20
votes
How were the Navier-Stokes equations found in the first place if we can't solve them?
As @QMechanic mentioned in a comment, the Navier-Stokes equations are just $F = ma$, but they look much scarier. Assuming an incompressible fluid, you have:
$$ \rho \frac{D u_i}{D t} = -\frac{\...
- 16k
17
votes
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How was Reynolds number derived?
There's no magic behind it. It was done by non-dimensionalizing the momentum equation in the Navier-Stokes equations.
Starting with:
$$\frac{\partial u_i}{\partial t} + u_j\frac{\partial u_i}{\partial ...
- 16k
16
votes
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Why don't the Navier-Stokes equations simplified for hydrodynamics contain gravitational acceleration?
If you go through the process of non-dimensionalizing the equations, the math becomes more clear. If you start with the momentum equation (ignoring viscous forces because they aren't important for the ...
- 16k
13
votes
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Why is the pressure gradient zero at a wall?
This usually only applies to a wall bounded flow and is normally restricted to incompressible fluids. This result usually manifests in boundary layer theory and can be obtained through order of ...
- 878
13
votes
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Navier-Stokes Derivation
The traditional derivation of the Navier-Stokes equations starts by looking at a fluid parcel and the different fluxes over the surface in the integral form. The integral form is preferred as it is ...
- 1,529
12
votes
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Life at low Reynolds number (E. M. Purcell)
1. Why is force linear in the velocity?
On a conceptual level, the idea behind stokes flow is that it is a very overdamped system because the viscosity is very high. When I say "overdamped" I am ...
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12
votes
How were the Navier-Stokes equations found in the first place if we can't solve them?
The Navier Stokes equations are a combination of Newton's 2nd law of motion (differential form) with the 3D version of Newton's law of viscosity (i.e., the mechanical constitutive equation for a ...
- 28.4k
12
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Why don't the Navier-Stokes equations simplified for hydrodynamics contain gravitational acceleration?
The NS equations do include the gravity term, see the Wikipedia entry where it's included as a body force term.
Under some conditions, it can be neglected: large Froude number (as tpg2114 showed) or ...
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11
votes
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Is it possible to derive Navier-Stokes equations of fluid mechanics from the Standard Model?
One way to derive fluid dynamics is to start from the equations of motion for $N$ particles, and use these to compute the evolution of average quantities (like the density) of the distribution of ...
- 35.5k
9
votes
Why does moving air have low pressure?
It's all about conservation of momentum, $F=ma$. Fluid can only change velocity by experiencing a force, and the only force it can feel is a pressure difference. So if there's a velocity difference, ...
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9
votes
Why is the pressure gradient zero at a wall?
It comes from the notion of the boundary layer and whether it stays attached to the wall or not. If you consider the momentum equation normal to the wall, the only way there can be a pressure gradient ...
- 16k
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Is the Navier-Stokes equation valid in $d=2$ spatial dimensions?
There has actually been a fair amount of activity in this area recently, see, for example, this set of lecture notes.
I would argue that the answer to your question is, "yes", if properly ...
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9
votes
Is it possible to derive Navier-Stokes equations of fluid mechanics from the Standard Model?
From your comment :
So is it possible to prove the consistence of fluid mechanics with the Standard Model?
The standard model is consistent with special relativity and quantum theory. We know those ...
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votes
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Why is the Reynolds number "the way it is?" Why is its order the way it is?
The Reynolds number, with $\rho$ the density, $u$ the velocity magnitude, $\mu$ the viscosity and $L$ some characteristic length scale (e.g. channel height or pipe diameter) is given by
$$\text{Re}=\...
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8
votes
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Deriving Stokes' law ($f_v=6\pi\eta Rv$) in a simple way
As already stated I am not familiar with a simpler way than the standard derivation. I don’t even think it is possible to derive it in a easier manner: For Stokes’ formula it is necessary to find ...
- 1,529
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votes
Why hasn't an exact solution to the Navier-Stokes equations been found?
There are known solutions to the Navier-Stokes equations. A simple example would be laminar shear-driven flow between two moving plates. Just as in the case of Einstein's equations, the known ...
- 32.6k
7
votes
Accepted
No diffusion term in conservation of mass in Navier-Stokes equations?
For a single-component fluid, the conservation of mass follows
$$
\left(\begin{array}{c}\text{mass of fluid } \\ \text{in volume }\Delta V\end{array}\right)=\left(\begin{array}{c}\text{flux of fluid }...
- 24.7k
7
votes
How were the Navier-Stokes equations found in the first place if we can't solve them?
To add to tpg124's answer and to answer the implict question in your statetent:
How were the equations discovered in the first place if we can't solve them?
simplicity and clarity of an equation's ...
- 85.4k
7
votes
Is Navier-Stokes a turbulence model?
The Navier-Stokes Equations are not a 'turbulence model', they are more fundamental than that: they are the fundamental equations that govern all of fluid dynamics (assuming the continuum assumption ...
- 3,954
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votes
Accepted
How does hot air rise?
The reason for this is actually the temperature-dependent distribution of the absolute value of particle velocities (higher temperature results in higher average velocities as well as greater variance ...
- 1,529
7
votes
What does $ \mu \nabla^{2} \vec V$ mean in the Navier-Stokes equations?
Since you are asking for some intuition, forget about the positive coefficient $\mu>0$ and about the other terms in the Navier-Stokes apart from:
$$
\partial_t \bf v = \mu \nabla^2v
$$
All the ...
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6
votes
How was Reynolds number derived?
The way it was explained to me: you start by thinking of all the possible factors that could play in drag (size, velocity, density, viscosity, ...); then you do dimensional analysis and find ...
- 117k
6
votes
Accepted
Original Derivations of Euler Equations or Navier-Stokes Equations
According to the book Worlds of Flow: A History of Hydrodynamics from
the Bernoullis to Prandtl by Olivier Darrigol, the derivation of Euler's equation by Euler and of Navier-Stokes by Stokes are ...
6
votes
Accepted
What is the meaning of pressure in the Navier-Stokes equation?
There are two pressures: Thermodynamic pressure $p_\text{thermo}$, and Mechanical pressure $p_\text{mech}$. Thermodynamic pressure, a concept from equilibrium thermodynamics and therefore applicable ...
- 6,197
5
votes
Is there a nice way to write Navier-Stokes equations in exterior calculus
MS Mohamed et al begin with a "standard vector calculus formulation of the NS equations",
$$
\frac{∂ \bf u}{∂ t} - \mu ∆ {\bf u} + ({\bf u} \cdot \nabla) {\bf u} + \nabla p = 0
$$
$$
\nabla \cdot {\...
- 865
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