Tanner’s law

By the definition of the capillary number, we therefore find that 9 oc Uq slow velocities imply small dynamic contact angles. This expression has been termed Tanner s law. 

Calculate the time dependence of the growth of a three-dimensional droplet using Tanner s law and consen/ation of matter. 

Duffy, B.R. and Wilson, S.K., A third-order differential equation arising in thin-film flows and relevant to Tanner s law, Appl. Math. Lett., 10, 63, 1997. 

The results can be expressed in terms of a contact angle Oq which depends on the spreading time t. When surfaces are smooth and clean, and for non-volatile liquids, the law is remarkably universal. It is known as Tanner s law ... 

For non-polymeric or low-molecular-weight molten polymers, the dynamic contact angle is related to the spreading velocity by Tanner s law, given by ... 

Due to surface roughness, the relationship between the spreading velocity and the dynamic contact angle changes (equation (7.38)). In the case of parallel V-shaped surface channels, measurements of the spreading of polydimethylsiloxane parallel to the surface channels showed that Tanner s law is still approximately valid at short times, shifted by a constant, ki, to higher spreading velocities. The data followed a modified Tanner law, as follows ... 

The macroscopic contact angle 0 of a simple, pure, nonvolatile wetting liquid wedge advancing with a small constant velocity U on a smooth solid substrate (Fig. 5.1) is given by the so-called Tanner s law. A simple form reads ... 

The length a belongs to the molecular scale. Usually, the ratio x/xi is of the order 10 therefore Kj in Tanner s law is around 120. The liquid-solid interaction enters in the interface profile... 

Consequences are remarkable, both on the shape and on the dynamics of a macroscopic nematic droplet spreading spontaneously on a solid substrate. When spreading proceeds, the initial spherical cap profile evolves progressively toward a characteristic bell shape, while the dynamics is accelerated with respect to the one of a simple spreading liquid. Tanner s law is no longer obeyed. 

The exponents are independent of the spreading parameter, and in very good agreement with Tanner s law. ... 

Figure 5.7 The dynamic contact angle data of silicon oil inside a 2 imn diameter capillary as a fnnction of capillary number. The solid line represents the Tanner s law profile for different constants...

This is quite different from Tanner s law. For a real drop of volume Q, the radius increases with time during the spreading as... 

As in Tanner s law, this gives a relation between the local contact angle... 

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