Equations for drainage under pressure drop

The linear flow rate in a border averaged by cross-section is expressed by Eq. (5.2). From the condition of flow continuity it follows that the volumetric flow rate in an arbitrary cross-section qJJ) is proportional to the volume of the border part Vir from a given cross-section to the border mouth 

The comparison of two flow rates for two arbitrary cross-sections (for example, for r = r and r = rmm) gives the following expression for the pressure gradient 

From Eq. (5.28) it is seen that in order to determine the pressure gradient at a random border cross-section it is necessary to know the quantity dVi dVL,max which can be determined from the border profile Vt,r(0 or K0- Therefore, the border profile cannot be estimated directly from Eq. (5.27) and Laplace s equation, in contrast to the case of a steady- 

The simplest model of a border profile is a cylinder, the bottom of which is a slit between three adjoining cylinders (cylindrical model of a Plateau border). The cylindrical border profile is realised at the initial and final stage of drainage. 

Hi is the shortest distance from the border outset to a border cross-section with radius r L is the total length of the border, from the outset to a cross-section of radius r, accounting also for the influx , L = pH. Bearing in mind the constancy of the border cross-section along its length, from Eq. (5.28) it follows 

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