Quasi-Steady-State Approach to the Kinetics of Spreading

4 QUASI-STEADY-STATE APPROACH TO THE KINETICS OF SPREADING 

Let us first examine the problem of spreading of a low sloped drop of a viscous liquid on a horizontal surface covered with a layer of the same liquid with a thickness Hq — that is, the same problem as in Section 3.3. The same notations as in Section 3.3 are used here. Let h(r,t) be the equation of the drop profile r is the radial coordinate, and t is the time. We use the characteristic scales h, r , and t, respectively. The following relationships are satisfied h ho, r = ro(0)/2, where rdit) is the radius of drop spreading h r . Then, as shown in Section 3.3, in dimensionless variables h hlh, r - rln, t t/t., and ho ho/h, the spreading process is described by the differential equation (3.109), with conservation law (3.107) and boundary conditions (Equation 3.17 ) and (3.108). 

As was shown in Section 3.3, the condition e 1 is usually met i.e., viscous forces are small in comparison with capillary forces. Now, in Equation 3.109, we use a new quasi-steady-state variable  

Setting e = 0 in the equation, we obtain, in the same way as in Section 3.3, the outer solution of the problem  

In the vicinity of the moving apparent three-phase contact line, = 0, where the profile of the outer solution (3.141) intersects the surface of the liquid film, we introduce the inner variable as before  

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