## U

In (8.26), the denominator of the fraction is the gamma function and the numerator is the complementary gamma function as defined by Tricomi (1954). From this expression, another constraint for k is obtained. In fact, if k > 1, the solution diverges for u ^ 0. That is, Equation (8.26) is finite everywhere only if (0 < k < 1).

In Figure 8.5, the behaviour of the dimensionless diffusivity X(&), according to Equations (8.24) and (8.26), is presented for three different values of the parameter k . It is easy to verify that, in the aforementioned range for the parameter k(0 < k < 1), the dimensionless

Figure 8.5 Behaviour of the dimensionless diffusivity X(\$), according to Equations (8.24) and (8.26), for three different values of the parameter k . It is easy to verify that all the three curves have to be monotonic increasing function since 0 < k < 1. Moreover, only the functions with 0 < k < 1/2 have an inflection point (curve with k = 0.1 in the figure)

Figure 8.5 Behaviour of the dimensionless diffusivity X(\$), according to Equations (8.24) and (8.26), for three different values of the parameter k . It is easy to verify that all the three curves have to be monotonic increasing function since 0 < k < 1. Moreover, only the functions with 0 < k < 1/2 have an inflection point (curve with k = 0.1 in the figure)

diffusivity X(&) increases from zero to infinity. Moreover, if 0 < k < 1/2,X(&) has an inflection

^2 — k/ \2 /// \2 — k/ be seen in Figure 8.5 where only the curve (k = 0.1) has the inflection point.

Example: the function &(u) is known

Experimental vertical profiles of the soil water content, which are initially convex in their upper part, during evaporation process, may sometimes become concave (Hillel 1980b; Menziani etal. 1999). This is more probable during strong drying processes. The inflection point appearing in the soil moisture profile allows discerning the development of a drying front, which moves downward.

Directly from Equation (8.11), d2û Jz2

Introducing the transformation u = [— log(1 — &)](1/a) in (8.29), obtained from (8.28), the dimension-less diffusivity function X(&) is:

 ri i r -, -log(l - Û) a The behaviour of the dimensionless diffusivity X(&), according to Equation (8.30), for three different values of the parameter a (0.9, 1.0, 1.1) is presented in Figure 8.6. dX dX /d# Since (8.29), it is obvious that X(û) is everywhere increasing if a < l while it has a minimum if l < a < 2.
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