Page 211 - Percolation Models for Transport in Porous Media With

12.4 THERMAL SLIDE 209

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Figure 75: Profile of velocities of the fluid in a capillary for the thermal slide

the developing temperature profile (93]. The given temperature profile is caused
by different degrees of compression of the medium by the propagating pressure
wave. Since the temperature of the medium is proportional to pressure (8.17), and
consequently the gradient oftemperature is proportional to the gradient of pressure
\lT "' \lp, it follows that the maximal velocity of thermal slide compensating
Poiseuille flow is directed as \lp, i.e., as k. Under the outlined assumptions,
the presence of thermal slide along the capillary surface results in the velocity
profile represented schematically in fig. 75. This profile can be described by the
relationship

v(ro) = -(1/4tt)\lp[(r- bt) 2 - r~] + "Yr\7T1J[ro - (r- bt)] (12.19)

where Ot is the thickness of the thermal flow layer, which has the order of the
diffusion layer thickness (93] (the latter is equal to several times the free path for
molecules of the fluid); "YT is the proportionality factor in the expression for the
velocity of thermal slide (93] vr = "Yr\lT; 17(·) is Heavyside's function. Using
(12.19), compute the flow of fluid in a capillary

r
q = 21fPJ j v(ro) ro dro
0
Having set it to zero, we obtain the correlation between the gradients of pressure
and temperature
(12.20)

Dissipation of energy due to the internal friction for Poiseuille flow (per unit length
of a capillary) equals
r
2
E1 = 27f/L J (8vf8ro) ro dro (12.21)
0
The corresponding heating of the fluid with specific heat cr by !::.T degrees absorbs
the energy equal to