By James N. Pitts, etc.
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Extra info for Advances in Photochemistry
In the heat equation we shall neglect the contribution of frictional viscous heating. , (d=L) ! 1, and the thermal gradients not too high (a , 10À3 À 10À4 , Introduction 21 DT ¼ 10 K, aDT ! 1, rgxT d ! 1, agd=Cp ! 1, where Cp is the speciﬁc heat and xT the isothermal compressibility). For illustration, let us now particularize the thermohydrodynamic equations and boundary conditions for various illustrative cases to be studied at length in this book. A. Thermal Convection Take, for instance, a liquid layer heated by its solid support.
The complete expression of the mode in Eq. (13) is i A þ kB exp(ikx þ lt), j¼À (21) l m according to the above indicated kinematic boundary condition at z ¼ 0. Equation (13) provides the real part of Eq. (21). In the earlier high frequency approximation the boundary condition involving the Marangoni effect in Eq. (11) yields B¼ pﬃﬃﬃ A 2(3 þ i) 1=2 þ 0(vÀ1=2 ) v (22) which shows indeed that for transverse oscillations the rotational part is negligible in comparison with the potential one in the Fourier normal mode solutions.
Assuming that the vibration period is much smaller than all characteristic hydrodynamic time scales and the sound velocity is practically unattainable, then the BO approximation can be applied . Using the method of averaging over fast time we obtain a closed system of equations for a slow ﬂow component [72,73]. Substituting the superposition of the average and pulsational components into Eqs (26) and (40) and selecting the leading fast periodic terms we obtain div v0 ¼ 0 (45) @v0 1 ¼ À rp0 þ nT abV2 cos Vt @t r (46) @T 0 þ v0 Á rT ¼ 0 @t (47) In Eq (46) the potential part of the inertial force is equilibrated by the pressure gradient, while the solenoidal part generates a pulsational ﬂow, w.
Advances in Photochemistry by James N. Pitts, etc.