Transcript
Page 1: Water Infiltration into Soil in Response to Ponded-Water Head

SOIL SCIENCE SOCIETY OF AMERICAJOURNALVOL. 55 NOVEMBER-DECEMBER 1991 No. 6

DIVISION S-l-SOIL PHYSICS

Water Infiltration into Soil in Response to Ponded-Water HeadD. Swartzendruber* and W. L. Hogarth

ABSTRACTThe pressure head of water ponded on the soil surface can increase

the infiltration of water into soil, but the effect has often been com-plicated to describe mathematically. This study was conducted todevise a simpler mathematical description without undue sacrificeof accuracy. A new three-parameter infiltration equation was ex-amined for its capability in describing the effect of soil-surface-pond-ed water head, h, on the cumulative quantity of water infiltrated intothe soil with time. The infiltration equation, as reduced to two-pa-rameter dimensionless form, was fitted by nonlinear least squaresto dimensionless data generated from mathematical descriptions ofinfiltration that included the effect of h in somewhat complicatedparametric form. The fitted two-parameter equation gave an excel-lent description of all the generated data, in terms of both goodnessof fit and in recovery of the dimensionless ponded head p used asan input into the generated data. From an overall perspective, re-covery of h was achieved within a relative error of ± 1.4% acrossthe complete range of the .generated data, thus validating the newand relatively simple equation in its description of the general effectof ponded head on the cumulative infiltration process.

THE EFFECT OF A NONZERO HEAD of ponded wateron the infiltration of water into soil has not only

been included in the first theoretically derived infil-tration equation (Green and Ampt, 1911), but has alsobeen recurrently handled in more general treatmentssince then. Philip (I958a) provided a rigorous and gen-eral analysis within the framework of his solution bypower series involving time exponents in multiples ofone-half. The effects, however, were not simple; thatis, for increased ponded-water head, the first two coef-ficients in the series were both increased, but the thirdcoefficient was decreased.D. Swartzendruber, Dep. of Agronomy, Univ. of Nebraska, Lincoln,NE 68583; and W.L. Hogarth, Division of Australian Environmen-tal Studies, Griffith Univ., Nathan, Queensland, 4111, Australia.Contribution from Division of Australian Environmental Studies,Griffith Univ. Received 15 Oct. 1990. 'Corresponding author.

Published in Soil Sci. Soc. Am. J. 55:1511-1515 (1991).

In the recent work of Hogarth et al. (1989), the effectof ponded head was assessed by using the parametricequations of Parlange et al. (1985). By their very na-ture, however, these parametric equations are morecomplicated to use than a single equation. Our objec-tive in this study was to provide a simplified mathe-matical form that would retain acceptable generalityand accuracy. To this end, we examined and evaluateda recently derived infiltration equation.

THEORETICAL CONSIDERATIONSThe new infiltration equation, obtained as an exact inte-

gration of the quasi solution (Swartzendruber, 1987b) ofRichards' (1931) equation, is expressible most simply by thethree-parameter form

= (S/A0)[\ - [1]where / is the cumulative quantity (volume of water per unitcross-sectional bulk area of soil) of water infiltrated into thesoil in time t after the initial and instantaneous ponding ofwater on the soil surface. The constants S, KQ, and A0 are thesorptivity, the sated (near-saturated) hydraulic conductivity,and a time-decay constant arising from the quasi solution,respectively. Furthermore, on the basis of the quasi solutiongiving rise to Eq. [1], S and A0 in principle do include theeffect of the constant ponded head, h, of water covering thesoil surface, whereas KQ is generally construed to be indepen-dent not only of h but of all other components of hydraulichead as well. Note also that the small-time form (gravity ef-fects negligible) of Eq. [1], obtained by allowing t —> 0, be-comes

/ = St1'2, [2]as would be expected.

For investigating the relationship of 5 to h, we proceedfrom the work of Hogarth et al. (1989), beginning with

S2 = SI + 2K0h(60 - 6n), [3]where we explicitly recognize 5 as the total sorptivity andSs as the sorptivity of the soil alone (with h = 0 for aninfinitesimally thin ponding of water), 00 is the sated (near-saturated) constant volumetric water content of the soil due

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1512 SOIL SCI. SOC. AM. J., VOL. 55, NOVEMBER-DECEMBER 1991

to surface ponding, and 0n is the constant initial soil watercontent prior to surface ponding. For economy of symbols,we define 5W as the component of sorptivity due to h, or

5W = [2K0h(00 - 0n)]'/2

so that Eq. [3] becomesS2 = 5? + SI

[4]

[5]A dimensionless measure of the ponded head, 7, was em-ployed by Hogarth et al. (1989), namely

7 = S2JS\ [6]but we utilize here the more direct alternative

p = S*/Sl [7]through which, in conjunction with Eq. [5], S is expressibleby

s = [8]Combining Eq. [6] and [7] enables the conversion of 7 to pby

7 = p/(l + p). [9]The particular dimensionless^formulations by Hogarth et

al. (1989) of their cumulative quantity of infiltration 7* andtime f* caused 7 to be embedded in both I* and f. To avoidthis, we define the dimensionless cumulative quantity of in-filtration Y and time T by

Y =

T=[10]

[11]and recast the two parametric equations used previously(Parlange et al., 1985; Hogarth et al., 1989). The latter au-thors' /*, <*, Sg, Ks, 6S, 0j, 7 (of Eq. [9]), and a are convertedto or replaced by the symbols used here: Y, T, Ss, K0, 0p, 0n>p, and 5, respectively, but with initial hydraulic conductivityK(6n) = K{ = 0 retained, to obtain finallyY = P/(2(dY/dT - 1)]

+ (1/25) ln[l + d/(dY/dT - 1)] [12]andT = [1/25(1 - 8)] ln[l + 5/(dY/dT- 1)]+ p/[2(dY/dT - 1)] - {[1 + (1 - S)p]/2(l - 8)}

l/(dY/dT - 1)]. [13]The soil-characterizing parameter 8 ranges from a maximumof one to a minimum of zero, with o — > 0 reducing Eq. [12]and [13] to the Green and Ampt (1911) equation

7 - [(p + 1)12} ln[l + 2Yf(p + 1)] = T. [14]; As shown in the Appendix, the I(f) relationship from Eq.[12] and [13] at very small times (gravity effects negligible)is the same for all 5 (0 < 8 < 1), and thus includes the Greenand Ampt Eq. [14] (6 — » 0). This therefore generalizes to all5 the customary Green and Ampt (1911) small-time I(t),which is

7 = [2K0(h [15]where P is the constant suction head associated with thewetting front, not only as originally conceived (Green andAmpt, 1911) but also as reiterated since then (Philip, 1954,1958b). Comparing Eq. [15] with Eq. [A4] (Appendix) gives

Ss(l + p)1'2 = [2K0(h + P)(60 - 0n)]1/2- [16]Appropriate combination of Eq. [4], [7], and [16] yields

P = h/PSI - 0n)

and from Eq. [8] and [16], we note that5 = [2K0(h + P)(B0 - »

[17][18]

[19]which is the desired relationship between S and h for a given«o, P, and (00 - «J.

Although P has been construed as fictitious by two author s(Philip, 1958b, p. 333; Panikar and Nanjappa, 1977, p. 13),a recent rigorous derivation (Swartzendruber, 1987a) showssuch a contention to be superfluous. That is, the expression

TI>

= A51 J.K(r)dr, [20]

where T = r(6) is the suction-head function, K(r) is the hy-draulic conductivity as a function of T, and rn = r(0J. Thisis quite general and not limited to the Green and Amptassumption, and does confer on P the meaning of a weightedmean value of r across the interval (0, rn) with weightingfactor K(T)/KQ. Philip (1958b) found, for what he designatedas Yolo light clay, that the S calculated from Eq. [19] wasalways within the range —0.5 to —1.2% of his exact valuesof S, for h ranging from 0.10 to 2.00 m. Such remarkableaccuracy is supported by our present finding that Eq. [19] isimplied by the small-time I(f) of Eq. [12] and [13] indepen-dently of the soil as characterized by 5. Finally, to give someidea of the value of P, we note from Philip (1958b) that 5S= 125.38 nm/s1'2, f0 = 123.0 nm/s, and 00 - 0n = 0.2574,with Eq. [18] then giving P =- 248.3 mm for Yolo light clay.

The use of Eq. [8], [10], and [11] in Eq. [1] yieldsY = «-'[! - exp(-«r/2)] + T, [21]

wheren = «/(! + p)1/2

a[22][23]

As the dimensionless counterpart of Eq. [1], Eq. [21] is atwo-parameter form with n and a as the dimensionless pa-rameters. Because Ss and K0 are constants for a given soil,a is a dimensionless form of A0 by Eq. [23]. Also, p is adimensionless form of h by Eq. [17], S/SS = (1 + p?n is adimensionless form of S by Eq. [8], and n'1 = (1 + p)l/2/afrom Eq. [22] is a dimensionless form of the quantity S/A0appearing in Eq. [1].

If fitted to a dimensionless data set (Y,T) generated insome fashion for given p from such equations as Eq. [12],[13], or [14], Eq. [21] could then be assessed for its validityand applicability. In addition to the goodness of fit as ex-pressed by the residual sum of squares, we could also ex-amine how well the given p is recovered. Recognizing thatthe p embedded in the fitted Eq. [21] by Eq. [22] would notrecover the given p exactly, we designate the p in Eq. [22]as pc and solve for it, obtaining

pc = (a/n)2 - 1. [24]Using the fitted values of a and n in Eq. [24] thus yields pcfor comparison with the given p used to generate the dataset to which Eq. [21] was fitted.

METHODSTo generate (Y, T) data for fitting by Eq. [21 ], we employed

the two parametric equations of Hogarth et al. (1989) recasthere as Eq. [12] and [13]. For given 5 and/?, 51 pairs of(Y,T)were determined from Eq. [12] and [13] by computerizedNewton-Raphson and bisection methods, with common T

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SWARTZENDRUBER & HOGARTH: INFILTRATION IN RESPONSE TO PONDED WATER 1513

values varying from 10~5 to 105. The 51 (Y,T) pairs wereobtained at each of 13 values of p ranging from 0 to 100,and carried out for 5 = 0.850 and 0.425. For 5 = 0, the 51(Y,T) pairs were obtained at each of the 13/7 values by usingEq. [14] since, even for 5 as small as 0.425, the calculationsbecame very time consuming. The value of 5 = 0.850 wassuggested by Parlange et al. (1982). By taking 5 = 0 and0.850, we are considering the extremes of soil types. Thevalue of 5 = 0.425 gives an intermediate value between theextremes.

The fitting of Eq. [21] to a given set of 51 (Y,T) pairs wasactually carried out in terms of natural logarithms, namelythe 51 data pairs (In Y, In T). Logarithms were employed togive equal weighing to small and large values of Y.and T.Transforming Eq. [21] logarithmically by v = In Y and u =In T yieldsv = ln{l + n exp u — exp[—a exp(«/2)]} — In «.[25]Equation [25] was then fitted to each (In Y, In T) data setby nonlinear least squares, using the NLIN package program(Marquardt searching option) of the Statistical Analysis Sys-tem (SAS Institute, 1990), with derivatives dv/dn and dv/dafrom Eq. [25] as inputs. Beginning at T = 10~5, the 51 com-mon T values were selected to progress by increments of InT very nearly evenly spaced. Specifically, for any given dec-ade range of T, the values were 1.0, 1.6, 2.5, 4.0, and 6.4,each of these being multiplied by 10% where N took on thesuccessive values —5, —4, . . . , 4, and 5 for each decaderange. This is the same scheme used by Swartzendruber andClague (1989, p. 621, second column), but wherein it is er-roneously stated that the number of T values was 30 insteadof the correct number 51.

As a check on the correctness of the overall computationalprocess, Eq. [25] was modified by setting n = a and thenfitting the resulting one-parameter equation to two data sets,for 8 = 0.850 and p = 0 in Eq. [12] and [13], and for p =0 in Eq. [14] (5 = 0). The resulting fitted a and the residualsum of squares (RSS) agreed exactly with the correspondingfindings of Swartzendruber and Clague (1989) for their datagenerated from the equations of Parlange et al. (1982) andGreen and Ampt (1911), respectively.

To compare the pc of Eq. [24] with the input p of Eq. [12]and [13], or [14], the customary percentage error of pc>termed Ep, is

Ep = 100(pc — p)/p, [26]but Eq. [26] does fail for p = 0. Of as much or even moreinterest is an examination of how well the term (h + P) of

Table 1. Results of fitting Eq. [25] (parameters n and a) to datagenerated from Eq. [12] and [13] with soil-characterizing param-eter 5 = 0.850, using different input p (dimensionless pondedhead); pc of Eq. [24] is the recovered p, while E, and Eh+, are thepercentage errors in pc and (Ac + P), respectively.

Eq. [19] is recovered by (/zc + P), as assessed by the (hc +P) percentage error = Eh+P = 100[(/zc + P) - (h + P)]/(h+ P), or

Eh+P = 100(/zc - h)/(h + P). [27]If h = hc when p = pc in Eq. [17], then the resulting /zc =pj> and the h = pP from Eq. [17] itself allow Eq. [27] tobecome

Eh+P = 100(/>c - p)l(p + 1), [28]with Eq. [28] being useable at p = 0.

RESULTS AND DISCUSSIONFor the data generated from Eq. [12] and [13] with

5 = 0.850, the results are presented in Table 1. Asregards goodness of fit, the largest RSS is 2.985 X 10'3at p = 0. This RSS is less than that of the correspond-ing case of the one-parameter equation of Swartzen-druber and Clague (1989) fitted to the data setgenerated from the equation of Parlange et al. (1982)with 6 = 0.850, which gave an RSS of 3.65 X id'3and a maximum relative error of ±2.35%. The 18%reduction in RSS is attributable to Eq. [25] and [21]having two fitted parameters, n and a, rather than hav-ing n = a as in the single-parameter fit (Swartzen-druber and Clague, 1989). Hence, the goodness of fitof Eq. [25], and thus of Eq. [21], is deemed excellentfor all values of p in Table 1.

For input p > 0.2 (Table 1), recovery of the inputp by the pc of Eq. [24] is within an error on the orderof 2%, but the error increases distinctly as p decreasesbelow 0.2, apparently because of the small magnitudeof p. For, examining the error Eh+P with which (hc +P) recovers (h + P), we again find it maximal at p =0 but its value is now the very acceptably small1.356%. Hence, with the error Eh+P within ± 1.4% forall values ofp in Table 1, the recovery of (h + P) andthus of h, by Eq. [25] and thus of Eq. [21], is thereforedeemed excellent for 6 = 0.850.

Results are presented in Table 2 for the data gen-erated from Eq. [12] and [13] with 5 = 0.425. Withthe largest RSS of 1.576 X 10~3 being even less thanthe largest RSS of Table 1, the goodness of fit of Eq.[25] is even better than it was for the case 8 — 0.850.

Table 2. Results of fitting Eq. [25] (parameters n and a) to datagenerated from Eq. [12] and [13] with soil-characterizing param-eter S = 0.425, using different input p (dimensionless pondedhead); pc of Eq. [24] is the recovered p, while Ep and Ettf are thepercentage errors in pc and (Ac + P), respectively.

Input p,Eq. [12]and [13]

Fitted parameters

n a

0.00.10.20.40.60.81.02.06.0

10.014.020.0

100.0

150.9524 151.9726121.8770102.626677.752262.320851.846044.296525.36499.18455.58664.01212.81940.5681

128.2704112.614491.975278.730569.429262.504943.807124.229818.477915.497712.88735.6968

Sum ofsquaresin 10~3

2.9850.5460.2190.4310.7450.985.156.501.581.555.537

1.5221.491

PcEq. [24] Eq. [26]

En,Eq. [28]

Input p,Eq. [12]and [13]

0.0135630.1076670.2041150.3993170.5959530.7933040.9910831.982795.95979.9398

13.92119.89399.56

1.3567.6672.058

-0.171-0.675-0.837-0.892-0.860-0.672-0.602-0.566-0.533-0.44

0.6970.343

-0.049-0.253-0.372-0.446-0.574-0.576-0.547-0.528-0.507-0.44

0.00.10.20.40.60.81.02.06.0

10.014.020.0

100.0

Fitted parameters

n a

——— inlO-2 ———

110.3987 110.880092.866080.405263.375452.226244.362138.525223.13388.83065.44863.93912.78270.5665

97.646288.208274.998666.021959.453654.405839.983123.304818.025615.218312.72085.6813

Sum ofsquaresin 10-'

1.5760.3880.1350.1580.3160.4700.6000.9711.3001.3691.3991.4221.470

Eq. [24] Eq. [26]Eittr

Eq. [28]

0.0087380.1055980.2035100.4004410.5980820.7961070.9943461.987165.96489.9448

13.92619.89899.58

0.8745.5981.7550.110

-0.320-0.487-0.565-0.642-0.586-0.552-0.530-0.512-0.42

0.5090.2920.032

-0.120-0.216-0.283-0.428-0.502-0.501-0.495-0.488-0.42

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1514 SOIL SCI. SOC. AM. J., VOL. 55, NOVEMBER-DECEMBER 1991

Table 3. Results of fitting Eq. [25] (parameters n and a) to datagenerated from the Green and Ampt (1911) Eq. [14] as yieldedfrom soil-characterizing parameter S —> 0 in Eq. [12] and [13],using different input p (dimensionless ponded head); pc of Eq. [24]is the recovered p, while E. and E/, f P are the percentage errorsin pe and (hc + P), respectively.

1.6

Input p,Eq. [14]

Fitted parameters71 a »

urn ofquares

t,Eq. [24]

:_ *A-I :_ IA-I

0.00.10.20.40.60.81.02.06.0

10.014.020.0

100.0

56.5082 56.310951.381747.108640.391735.352231.431528.294018.87668.10095.15843.78432.70420.5632

53.703651.428847.632144.570142.032739.885432.595121.3731

I IV

.408

.410

.412

.416

.419

.421

.424

.432

.44717.0627 1.45414.6187 1.45812.3612 1.4635.6477 1.478

-0.0069710.0924210.1918250.3906420.5894770.7883160.9871891.981655.96099.9412

13.92319.89599.56

Eq. [26]£,.,

Eq. [28]

-0.697-7.579-4.088-2.340-1.754-1.460-1.281-0.917-0.651-0.588-0.552-0.525-0.44

-0.689-0.681-0.668-0.658-0.649-0.641-0.612-0.558-0.534-0.516-0.500-0.44

The error Ep ofpc is also less than it was in Table 1,as is likewise true of error Eh+P of (hc + P), althoughthe maximal value ofEh+P = 0.874% again occurs atp-0.

For d = 0, results for the data generated from Eq.[14] are given in Table 3. Since the RSS all fall withinthe narrow band 1.408 X 1Q-3 to 1.478 X 10"3, all ofwhich are less than the maximum value 1.576 X 10'3in Table 2, the goodness of fit of Eq. [25] is generallyat least as good as for d = 0.425 (Table 2). The errorEp of pc (Table 3) at small values of p is similar inmagnitude to that in Table 1 (5 = 0.850) but of re-versed sign. Error EH+P of (hc + P) is again maximalin magnitude (—0.697%) at p = 0, but continues thetrend of reduction by being of even smaller magnitudethan the 0.874% of Table 2.

Since the results in Tables 2 and 3 are generallysomewhat improved over those of Table 1, the be-havior exhibited in Table 1 can be taken as setting theinclusive limits for all the data of this investigation;namely, 0 < 6 < 0.850 and 0 < p < 100. The goodnessof fit of Eq. [21], as manifested by the logarithmictransformation via Eq. [25], is generally excellent.Also, the recovery of (h + P) by (hc + P) is accuratewithin ± 1.4%, which is only inconsequentially largerin magnitude than the —1.2% found originally by Phil-ip (1958b) as the maximum magnitude of error insorptivity S as a function of ponded depth h, as ex-pressed by Eq. [19].

To illustrate the effect of ponded-water head on thedimensionless cumulative infiltration Y vs. T, we em-ploy a first set of curves given by the sum of the firstthree terms of the tl/2 series expression of Philip(1958a), after using Eq. [10] and [11] to express histhree-term equation in the dimensionless variables Yand T. The Philip data for Yolo light clay with h =0, 250, and 1000 mm of ponded water were selected,with the corresponding p from Eq. [17] being 0, 1.007,and 4.027, respectively, because P = 248.3 mm asalready noted. For a second set of curves, we tookadvantage of an earlier comparison (Swartzendruberand Clague, 1989) that had shown for h = 0 that theY(T) curve for Yolo light clay and the Y(T) equation

LJ

I-

oo Q

1.2

0.8

LU

1 0.4

• Parlange et cil. (1985), i = 0.850in Eq. [12] and [13].

i Swartzendruber(1987b), Eq. [21].

Phillip (1958a).

YOLO LIGHT CLAY

0 0.1 0.2 0.3 0.4DIMENSIONLESS TIME (T)

Fig. 1. Influence of three different ponded-water heads (p) on cu-mulative infiltration curves in dimensionless form, for three math-ematical descriptions.

of Parlange et al. (1982) with 6 = 0.850 were alwayswithin ±0.58% of each other. Hence, 5 = 0.850 wasused in Eq. [12] and [13] at p values of 0, 1.007, and4.027, this second set of Y(T) curves thus representingthe basic description of Parlange et al. (1985) and Ho-garth et al. (1989). A third set of Y(T) curves was cal-culated from Eq. [21], using for each p value (0, 1.007,and 4.027) the corresponding pair (n, a) as least-squares-fitted by Eq. [25] to the data generated fromEq. [12] and [13] with 5 == 0.850.

The foregoing results are shown graphically in Fig.1, in which the upper limit T = 0.385 is dictated bythe accuracy of the three-term Philip (1958a) series. Itis evident that the values from Eq. [12] and [13] andfrom Eq. [21] fall so closely on the solid-line Philipcurves that the differences can hardly be displayed onthe graph. More specifically, all of the points desig-nated by circular or square symbols in Fig. 1 werewithin ±1.5% of their respective Philip curves. Forthe much larger time range 0 < T < 105, the Y valuesfrom Eq. [21] were always within the following errorbands centered on the rvalues from Eq. [12] and [13],namely: ±2.1, ±1.3, and ± 1.4% for the p values 0,1.007, and 4.027, respectively. Hence, it is concludedthat Eq. [21] provides a highly acceptable descriptionof Eq. [12] and [13] across the range 0 < p < 4.027,for all times T > 0.

The nature of a (the dimensionless counterpart ofA0, Eq. [23]) as a function of the dimensionless pondedhead p is displayed in Fig. 2 for the three values of d.At p = 100 (not shown in Fig. 2), the a are the verynearly equal values 0.056968,0.056813, and 0.056477,corresponding to 5 of 0.850, 0.425, and 0, respectively.Because the graph in Fig. 2 is semilogarithmic, theobvious absence of any straight-line relationshipsmakes it clear that a(p) is not simply the commonexponential function with —p in the argument. Weplan to give further consideration to the character of

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SWARTZENDRUBER & HOGARTH: INFILTRATION IN RESPONSE TO PONDED WATER 1515

1.5 -

ct:

0,15 -

15 20 25

RELATIVE PONDED HEAD (p)Fig. 2. Behavior of dimensionless parameter a (Eq. [23]) with di-

mensionless ponded-water head p, for three values of the soil-characterizing parameter {. Note logarithmic scale on ordinate.

CONCLUDING COMMENTSThe validity of Eq. [21] has thus been established,

in terms of the generated data of this study. The natureof the validation process, i.e., by including the ex-tremes 5 = 0 and 0.850, ensures a more stringent testof the new infiltration equation than testing on ex-perimental data with inherent noise. Dimensionallyexpressed, Eq. [21] becomes7 = [2K£h + F)(60 - 0n)]'/2

[1 - exp(-.V"2)]A4o + K0t, [29]which we therefore propose as a relatively simple butaccurate mathematical description of water infiltrationinto a wide variety of uniform soils under the influenceof a nonzero head h of water ponded on the soil sur-face. The equation holds for all times t > 0. Thissimple form contrasts with the more complicated onesof Philip (1958a) and Parlange et al. (1985). The avail-ability of S as a function of h, as expressed in Eq. [19],has enabled the derivation and testing of Eq. [21] with-out knowing the form of a = a(p) or of A0 = A0(h).

APPENDIXTo determine the behavior of Y from Eq. [12] and [13] at

very small T, we form the difference Y — T from the twoequations and set U = Y - T, from which dU/dT = dY/dT — 1. The overall relationship can ultimately be written

2(1 - 6)17 = [1 + (1 - «)p] ln[l + l/(dU/dT)]- ln[l + 8f(dU/dT)]. [Al]

Next, although Y _ O and U -> O as T -> O, d£//d T becomeslarge without limit, so that l/(dU/dT) and 5/(dU/dT) in turn

become very small. Hence, for T sufficiently small, the log-arithmic terms in Eq. [Al] can be replaced by the leadingterms of their respective series expansions, so that Eq. [Al]becomes, after manipulation and simplification,

2UdU/dT = 1 + p. [A2]Separating variables, integrating and observing that 7 = 0and U = O when T = O, yields from Eq. [A2] that U = (1+ p)l/2'P/2. Substituting Y — T for f/and rearranging yieldsY = (1 + pyi2T'2 + T, but again, for T sufficiently small,it is negligible in comparison with (1 + pY/2T/2, so that therelationship between Y and T becomes simply

Y = (1 + pyW2, [A3]which, employing Eq. [10] and [11], yields in dimensionalterms

7 = S.U + P)l'2t1'2. [A4]Hence, for very small T and t, note that Eq. [A3]

and [A4] thus apply for Eq. [12] and [13] indepen-dently of the value of 5, because 5 has disappearedalready at the stage of Eq. [A2]. By Eq. [8], the Ss(l+ p)112 of Eq. [A4] is simply S, thus showing the iden-tity of Eq. [A4] and [2].

ACKNOWLEDGMENTSD. Swartzendruber expresses thanks to the University of

Nebraska, Lincoln, for a faculty development leave at Grif-fith University, where this work was done.


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