Navier-Stokes-Fourier Exact Model
Let ๐ = ๐(t) denote an arbitrary volume that is moving with the fluid, and let F(t, x) be a function of position vector x and time t.
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1.1 The Transport Theorem Let QJ = QJ(t) denote an arbitrary volume that is moving with the fluid, and let F(t, x) be a function of position vector x and time t. The volume integral
J
F(t,x)dtJ
m(t)
is then a well-defined function of time. Its derivation is given by the following formula (1.1 ) where the velocity vector of a particle of fluid is u = dxl dt = u(t, x). Equation (1.1) can be expressed in an alternate way that brings out clearly its kinematical significance (independently of any meaning attached to function F). By virtue of dFI dt = รF Iรt + u . grad F, the integrand on the right side of (1.1) can be written 8FIรt + div(Fu), and then by application of the divergent classical theorem, we find that :t
J
F(t,x)dtJ
m(t)
=
:t J
FdtJ +
m(t)
J
Fuยท nds.
(1.2)
(5
Here, ร I รt denotes differentiation with volume QJ( t) held fixed, and n is the outer normal to the surface 6. Equation (1.2) equates the rate of change of total F across a material volume QJ(t) to the rate of change of total F across the fixed volume that instantaneously coincides with QJ(t) plus the flux of F out of the bounding surface 6. Of course, the function F is continuously derivable in terms ofvolume QJ(t) and the surface 6 of this volume is assumed suitably smooth. R. K. Zeytounian, Theory and Applications of Viscous Fluid Flows ยฉ Springer-Verlag Berlin Heidelberg 2004
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1 Navier-Stokes-Fourier Exact Model
1.2 The Equation of Continuity We suppose that the fluid posesses a density function p = p(t, x), which serves by means of the formula
I
9J1 =
pdtJ
'!!(t)
to determine the mass 9J1 of fluid occupying a region m(t). We naturally assume that p > 0 and assign "mass per unit volume" to the physical dimension. According to the principle of conservation of mass, the mass 0/ fluid in a material volume m(t) does not change as m(t) moves with the fluid, and consequently,
I
:t
pdtJ = O.
'!!(t)
From (1.1),
I [~+PdiVU]dtJ=O,
(1.3a)
'!!( t)
or dp dt
+ P d'IVU = 0 ,
(1.3b)
because m(t) is arbitrary [this derivation is substantially attributable to Euler (1755)J. As a consequence of (1.1) and (1.3b), we conclude with the formula that is valid for an arbitrary function F = F(t, x), d dt
I
I
pF dtJ =
'!!(t)
dF dtJ . Pdi
(1.4)
'!!(t)
1.3 The Cauchy Equation of Motion According to the principle of conservation of linear momentum, the rate of change of linear moment um of a material volume m(t) equals the resultant force on the volume [see Serrin (1959a, p.134)J:
:t III '!!(t)
pudtJ =
III '!!(t)
pfdtJ
+
II l5
tdร,
(1.5)
1.4 The Constitutive Equations of a Viscous Fluid
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where i is the extraneous force per unit mass and t is the stress vector. The stress vector may therefore be expressed as a linear function of the components of n, that is, (1.6) We observe that, according to Cauchy's stress principle (1828), "upon any imagined closed surface (5 there exists a distribution of stress vector t whose resultant and moment are equivalent to those of the actual forces of material continuity exerted by the
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