﻿Liquids under Capillary Pressure. 1153 



expression the rate of flow from the driving pressure made 

 up of three separate pressures, the unbalanced atmospheric 

 pressure, the hydrostatic pressure, and the capillary pressure. 

 In the case of horizontal tubes the first two pressures are 

 eliminated. The effective total driving pressure, however, 

 varies with the length of the column, since the frictional 

 resistance to the flow increases with the length of the 

 column in the capillary tube. "With this correction a some- 

 what different expression from that of Washburn is obtained 

 for the rate of penetration, which, however, reduces to the 

 form obtained by him on the neglect of terms which are 

 insignificant except for very small and very large values of 

 x the distance of penetration. A simple derivation of the 

 relationship may be obtained in the following manner : — 



Fig. 1. 



. J-T-Z— 



i i r 



i 



i . . i * 



i 

 i 



The forces acting on a column of liquid x cm. long in a 

 capillary tube r cm. radius are : 



(1) The surface tension forward, in magnitude 2iTry. 



(2) A retarding force due to the viscosity of the liquid in 

 the tube. 



According to Poiseuille's law, neglecting the slip factors, 

 this retarding force may be expressed in the following form, 



civ _ p 7rr 4 

 dt Srjx 



where -=- is the rate of flow ; hence, solving for P the 



pressure, we obtain 



_ Srjx dv Srix , „ . , 8-w.v x 



7T?' 4 dt 7T) A r 2 



The retarding force acting on the column is consequently 



F = irr 2 P = 8r)Xir&. 



The net force acting on the column thus varies not only with 

 the leugth of tube wetted, but also with the velocity of flow, 

 and is equal to 



27rry — Srjx tt&. 



