# Tag Info

21

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 describe the system. Any new equation, whether it's the Navier-Stokes equations, Einstein's equations, the Schrodinger equation, or whatever, must be ...

18

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 volume and the result is the Navier Stokes equations. The Navier Stokes equations, in the usual form, apply to Newtonian fluids, that is fluids whose stress ...

17

Consider a standard volume of $1\textrm{ m}^3$ of air. This contains on the order of $10^{25}$ molecules of O2 and N2. If you needed to simulate or explain the physics occurring in that volume of air, would you want to model $10^{25}$ molecules and all the interactions between them or, say, 100x100x100 cells based on the Navier-Stokes equations? ...

12

Frank White's Viscous Fluid Flow book contains a good list of these "exact" solutions. I am not sure if it is complete though. I've provided links to a few of the solutions. Steady flow between a fixed and moving plate Axially moving concentric cylinders Flow between rotating concentric cylinders Hagan-Poiseuille flow Combined Couette-Poiseuille flow ...

12

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 correlation functions, see, for example, here and here. These fractional powers are seen in molecular dynamics (they are more difficult to see experimentally), ...

11

Dear Ondřeji, a good question but a part of the answer is that your equation for the fluid is underdetermined. It treats $p,\rho$ as independent variables. But the physical system only knows how to behave if you also substitute some equation of state, i.e. a function $p=p(\rho)$ or $p=p(\rho,\vec v)$. Note that your Ansatz for the stress-energy tensor ...

11

There is a simple general argument for why you get small-scale motion from large-scale motion in any nonlinear nonintegrable continuous mechanical system, whether it is fluids, or electromagnetic waves interacting with charged plasmas, or surface waves on water, or anything nonlinear at all. This argument must break down for those special cases where the ...

10

Actually, there are two different viscosity coefficients. You can see this from the stress tensor $$\sigma_{ij} = -p_0 \delta_{ij} + \eta \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} - \frac{2}{3} \delta_{ij} \frac{\partial v_k}{\partial x_k} \right) + \zeta \delta_{ij} \frac{\partial v_k}{\partial x_k}$$ which has the two ...

10

This is difficult to answer, because the answer depends on whether the answer is positive or negative. Although most people, including myself, expect a positive answer, a negative answer is actually possible, unlike say, for the question of P!=NP, or the well-definedness/mass-gap of gauge theory where we are 100% sure that we know the answer already (in the ...

10

Turbulence is indeed an unsolved problem both in physics and mathematics. Whether it is the "greatest" might be argued but for lack of good metrics probably for a long time. Why it is an unsolved problem from a mathematical point of view read Terry Tao (Fields medal) here : ...

7

The equation is correct--- the (laminar) flow at small Reynolds number is given by making the flow be along the pipe, and substituting into the Navier Stokes equations, which reduces to your thing. The one issue is the sign--- $\Delta P$ is negative if you mean the flow is going to be in the positive z direction. I will absorb the constants and consider the ...

5

Strictly speaking, turbulence doesn't exist in two dimensions. The energy cascade required for turbulence to develop (transfer energy from large scales to small scales) is due to the (incompressible for illustration) vorticity equation: $\frac{D\vec{\omega}}{Dt} = \left(\vec{\omega}\cdot\nabla\right)\vec{v} + \nu\nabla^2\vec{\omega}$ specifically the ...

5

I belive you have it pretty much settled already. If I was to change anything, I would shrink instead of adding more items: Identify the relevant quantities of your system: Energy, Momentum, entropy, electric charge, mass ... Which may or may not be conserved. If you have boundary conditions, most probably you don't have energy and/or momentum ...

5

Turbulence is not one of the great unsolved problems in physics. Physics tells us exactly how turbulence emerges as a direct consequence of local mass and momentum conservation. We can create multiparticle computer models such as lattice gas automata that generate turbulence at large length and time scales. We can write down the equations that govern ...

5

For a single-component fluid, the conservation of mass follows $$\left(\begin{array}{c}\text{mass of fluid } \\ \text{in volume }\Delta V\end{array}\right)=\left(\begin{array}{c}\text{flux of fluid } \\ \text{in/out of volume }\Delta V\end{array}\right)+\left(\begin{array}{c}\text{sources or} \\ \text{sinks in }\Delta V\end{array}\right)$$ In terms of a ...

4

It's not that the random motion decreases when the flow rate increases. It is only that the random motion stays the same but the coherent motion dominates. If the diffusion velocity in a gas is 1 and the convective velocity of the flow is 1000 (units don't matter), then the diffusive action can be pretty safely ignored. The important thing to remember is ...

4

My answer will focus just on the mathematical parts pertaining to partial differential equations. Scale invariance is the fact that some partial differential equations stay the same if you appropriately scale the variables. For example the heat equation (where $\boldsymbol{x}$ is the position vector in 1, 2 or 3D, doesn't matter) $$\partial_t ... 4 Solutions of the form$$ cos(x_i)e^{-x_j}$$are common specific solutions of the Navier-Stokes equations in simplified (not simple) problems. These are however problems where inertia is ignored, which you include. (Please note that I am using index notation, with i,j\in\{1,2,3\}). x_j is then the wall normal direction. This is actually quite well ... 4 I don't know a good answer to your first question (I'd be interested in a good text for that myself), but I can answer the second. It's easier to explain if we temporarily imagine \phi represents the concentration of some dye made up of little particles suspended in the fluid. The convective term (aka advective term) is transport of \phi due to the ... 4 I don't think that such a computation of a theoretical limit of accuracy is possible. There are several sources of uncertainty in weather models: initial and boundary data, parameterizations, numerical instability, rounding and approximation errors of the numerical scheme employed to solve the Navier-Stokes equations for the atmosphere. The term ... 3 I am not sure how useful this "back of the envelope" calculation of reliability of Numerical Weather Prediction is going to be. Several of the assumptions in the question are not correct, and there are other factors to consider. Here are some correcting points: The Weather is 3 dimensional and resides on the surface of the planet up to a height of at ... 3 Let me here just derive the equation (6.11) that follows the sentence, you mention. The Navier-Stokes equation (6.6a) reads$$\partial_t v_i + v_j\partial_j v_i = -\partial_i p + f_i + \nu~\partial_j\partial_j v_i.$$The incompressibility condition (6.6b) reads$$\partial_jv_j=0. $$Hence we have in the unprimed and the primed point that$$\partial_t ...

3

I'm far from expert and my advice is basically to read Terry Tao's piece linked in John Rennie's comment. (Read Terry Tao is always good advice.) But I would like to make a couple of points. Singularities are not impossible or unphysical Mainly, I wanted to clear up the idea that, if we find smooth solutions that became singular in finite time that would ...

3

First, Navier-Stokes governs the fluid in your setup. So, anything apart from the fluid will be an external force in N-S equation. Body-force means an external force that applies in the bulk of the fluid, like gravity or a magnetic force. Interaction with a "body", as a wing, which is external to the fluid domain, is done through boundary conditions : the ...

3

You almost give the answer in your question: With increasing flow rates, the inertial forces become larger than the viscous forces We can state it more precisely: The inertial forces scale quadratically with the flow speed $U$: $\rho \mathbf u\nabla \mathbf u\sim \rho U^2/L$, while the viscous forces scale linearly: $\mu \nabla^2\mathbf u \sim \mu ... 3 For Newtonian fluids (such as water and air), the viscous stress tensor,$T_{ij}$, is proportional to the rate of deformation tensor,$D_{ij}$: $$D_{ij} = \frac{1}{2}\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right)$$ $$T_{ij} = \lambda\Delta\delta_{ij} + 2\mu D_{ij}$$ where$\Delta \equiv D_{11} + D_{22} + D_{33}$. The ... 3 Most importantly, the Navier-Stokes equations are based on a continuum assumption. This means that you should be able to view the fluid as having properties like density and velocity at infinitely small points. If you look at e.g. liquid flows in nanochannels or gas flows in microchannels you could be in a regime where this assumption breaks down. As far as ... 3 IANAFD but I'll stick my neck out and say this: resolving the Clay problem one way or another won't cause people doing CFD to lose any more sleep than they already do. First of all, Jean Leray proved the existence of weak solutions to Navier Stokes in$R^3$way back in the 1930s, and that is pretty much what matters for the task of getting numerical ... 3 There is a normalized form, though it's properly called the dimensionless Euler equations. The way to do it is define: scale time$t_0$scale density$\rho_0$scale length$L_0\$ and then derive the scales from these: $$v_0 = \frac{L_0}{t_0},\quad p_0=\rho_0v_0^2$$ NB: it is possible to use other combinations, but I find that these are often the ...

3

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 solutions regard simple situations with particular boundary conditions; a general solution that covers all possible cases is not known in either case. One should not ...

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