By L.D. Landau, J.B. Sykes
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Extra resources for Fluid Mechanics: Vol 6 (Course of Theoretical Physics)
U~ = 0 on r . 44a) with 8>" 0 0 8n = n . 44b) then compute V = u~ + v>.. , and replace u~ by V in u(O, x) = u~ in n. 44c) In general, the data and constant are assumed to be "sufficiently smooth," which we avoid trying to define too carefully, except to insist that n·ur(t, P) be continuous in time, so that 8/8t[n . ur(t, P)] exists, and to require that n· u~ is a continuous function of the position space vector x as x --+ r, where the boundary unit normal n is imagined to be translated (in the -n direction) to form n· u~.
26) CvP ~~ + pdivu = ifJ(u) + div(kgradT) , with ifJ(u) = 2 Trace [(D(U))2] + A(V· U)2 . If A = A0 , J1 = J1 0 , and k = kO, are constant and if we assume that AO = -32 J1 0 . 6 The Navier-Stokes-Fourier Equations d l~; p + V' . u = 0, du Pdt = pI - 21 p = RpT , 1 V'p + 3)L°V'(V' . 28) dT CvPdt + pV'. 29) for the Laplacian of u. 2 Dimensionless NSF Equations Below, we work mainly with dimensionless equations. From (1. 25a-c), the dimensionless full unsteady-state NSF equations, written with the same notations for the dimensionless veloeity u and thermodynamic functions p, p, and T, are the following: dp S dt + pV' .
38) It gives the possibility of deriving, in the framework of low Mach number asymptotics of NSF equations, the full viscous and heat-conducting Boussinesq equations. 41 ) plays a significant role in the derivation of OB equations. 42) Ra is the Rayleigh number. 43) is the Peclet number and for Pr « 1 with Re » 1 but Pe = 0(1), the fluid motion is quasi-nonviscous but thermally conducting, and the so-called "high thermal conductivity" model equations are valid, at least when the Mach number, M 2 « Re, is not very high and far from the wall.
Fluid Mechanics: Vol 6 (Course of Theoretical Physics) by L.D. Landau, J.B. Sykes