By Charles R. Doering
The Navier-Stokes equations are a collection of nonlinear partial differential equations that describe the elemental dynamics of fluid movement. they're utilized regularly to difficulties in engineering, geophysics, astrophysics, and atmospheric technology. This ebook is an introductory actual and mathematical presentation of the Navier-Stokes equations, targeting unresolved questions of the regularity of ideas in 3 spatial dimensions, and the relation of those concerns to the actual phenomenon of turbulent fluid movement. The aim of the ebook is to give a mathematically rigorous research of the Navier-Stokes equations that's obtainable to a broader viewers than simply the subfields of arithmetic to which it has commonly been limited. as a result, effects and strategies from nonlinear useful research are brought as wanted with a watch towards speaking the fundamental principles in the back of the rigorous analyses. This publication is acceptable for graduate scholars in lots of parts of arithmetic, physics, and engineering.
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Additional info for Applied Analysis of the Navier-Stokes Equations
12) a d-dimensional set. 13) where U(x) is specified for x E 852. Then there are again four parameters in the problem: p, v, a length scale set by the system size, and a velocity scale set by the boundary conditions. (There is also a velocity scale set by the initial conditions which may dominate the flow at early times, an observation which applies to the body-forced example as well. ) There are a number of possibilities for the system size length scale. A natural choice may be a length scale L based on the system volume, say, L- (fddx) 1/d.
The fitting parameter x does not enter into this limit of the model. 30) yields U . - x logUR, (3 . 2 . 33) Asymptotically at high Reynolds numbers, this theory predicts a turbulent drag force proportional to the square of the speed U with logarithmic 48 Turbulence z/h Fig. 3. Reynolds stress across the layer. Note that u. is the scale of the velocity fluctuations. corrections. 4. Although this example has been developed for illustrative purposes with a minimum of fitting parameters (one), it does reproduce some essential features of the conventional wisdom concerning the structure of both the mean and Reynolds stress profiles.
N"n = Rl'n = Rr < oo implies a supercritical bifurcation precisely at the critical Reynolds number R. ) The nature and stability of the bifurcating solution require their own analysis, for example, via the asymptotic methods of amplitude equations. 3 References and further reading Linear and nonlinear stability theory is discussed in detail in Drazin and Reid . The energy method for nonlinear stability is developed for a variety of problems in Straughan . Linear stability can, in some cases, be elevated to "local" nonlinear stability, and these methods have been used to identify stable solutions of the Navier-Stokes equations from solutions of approximate problems (see Titi ).