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And that (126) 44 and recalling that (V, - u)·v, = 0 on 8 a (J (=a material surface with respect to the solid). ){-'C T 'V' + OH'} 8t 'I' p, " , J, 1 -- Uti 1 St. oH 8£';' J, ·v, dS - T,, (1 - f)'1-. (126) 8t Equations (124) and (126) are two macroscopic differential balance equations. one for the fluid (having an averaged temperature T/) and one for the solid (at an averaged temperature of T, '). By expressing tile dispersive and diffusive fluxes also in terms of T/ and T,'. we obtain two equations that have to be solved simultaneously for T/ and T,'.

31) 23 Another example of interest is the internal energy. pecific inte~al energy, u", by U" = 8U I. (32) 8m" p,T,m'"~,, Then I _ 8U m" U LU"P" = L = -U =pu. ('l') ('l') 8m" p,T,m','n 8U (33) Equation (33) relates the internal energy density ofa phase to the partial specific energies of the components. This relationship should be compared with that expressing the relationship between the energy density of a phase and that of the components, given above. With the definition of partial specific values, new weighted velocities may be defined.

Of the latter expresses the drag at the fluid-solid interface, Le . 1 Uo Sf. , 11"•• " . J dS -- _Til CI a"q . r A2 'J rJ' (139) / The remaining term, 'ji"'lJ 2 qri/{}Zi{}zi, represents the resistance to flow due to the internal friction in the fluid. By assembling terms, Bear and Bachmat (1990) obtain the approximate form of the averaged momentum balance equation _/{ {)qi P Tt + (140) 50 where Tti and Qii(= iii - Vi",,;/) are two tensorial properties of the configuration of the S/I-surface in saturated, single phase flow.

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