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86 Chap. 7 <br />A solution satisfying the differential equation and the boundary conditions <br />is: <br />2 <br />h=fit ( x ~ eudu ( 1) <br />7- <br />2~KD ~ x u2 <br />(_.~ <br />Values of this integral may be obtained from Table 9. At x.= 0 this becomes: <br />~. ~ 7 2 <br />~)o = 2aKD ( ) <br />It will be of interest to compute the flow f passing between planes a unit <br />distance apart. This flow is, to a first approximation: <br />x2 ~ 2 <br />8h _~ - ~e~t+~ a-u du <br />f =-Kpax 2n x 2 <br />u <br />x <br />To obtain this result the procedure for differentiating an integral has been <br />followed. This was described previously. In the present case, however, she <br />variation is in the lower limit which introduces a negative sign. An evalua- <br />tion of the integral is needed. To obtain this integrate by parts with <br />2 <br />ul = e-u dvl ' ~ <br />u <br />2 <br />dul = -2ue-u du vl u <br />Then since <br />• <br />Ir <br />T <br />c <br /> <br />~uldvl = ulvl -1vldul <br />