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The dimensionless temperature can be represented as a product of two functions of 1 variable each. 56) V is a function of dimensionless time only and <>| is a function of dimensionless space only. Eq. 57) where Xn2 is a constant. The choice of the negative sign for the constant is to stay consistent with the decaying exponential in time of transient temperature. Solving the two ordinary differential equations shown in Eq. 59) It can be expected that the transient temperature profile will be symmetrical wrt to the x = 0 axis in the finite slab.
The Boltzmann equation that can be used to study any transport phenomenon, is given as a function of an arbitrary distribution, P. 88) where, a is the acceleration of the particle or phonon or item of interest, P is the distribution function which depends on the particle position, r, velocity, v, and time t. The speed of phonons is nearly that of sound and does not change significantly over a wide frequency range. As a result, the term, -dP/dv, can be neglected. The v VP represents the advection or drift of the distribution.
Joseph and Preziosi [1989, 1990] provide a review of constitutive equations for thermal wave behavior. Constitutions between the heat flux vector and the temperature gradient including the Jeffreys type and the Guyer and Krumhansl model for second sound propagation in dielectric materials are discussed. Coupling of the thermal relaxation behaviour with the mass and momentum transfer in fluidlike structures have been given by Choi and Wilhelm [1976]. Mass transport was discussed by Roetman [1975] and Sieniutycz [1981].