Tanners Law

We estimate the effect of the velocity fluctuations on the capillary pressure, using the Hoffman-Voinov-Tanner law which is valid at 9d < 135° and Ca < 0 (0.1)... 

A spherical cap [Z(r) > b. Here normal viscous flow takes place, the slip is negligible, and the kinetics follows the Tanner laws.(55)... 

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 ... 

Contrary to the hydrodynamic approach, the molecular model describes the experimental results in partially wetting (and liquid—liquid) systems (see as example, the fitting in Fig. 15) with physically reasonable adjustable parameters. However, it fails to predict Tanners law for complete wetting situation. 

They observed that while the central dry zone increases, a bump is built up between the receding contact line and the liquid film the latter remains static and the receding contact line moves at constant dewetting velocity, V. They investigate the dependence of this velocity with the diffetent parameters of the system. The main results are that V4 does not depend on the film thickness (for h he) and that for viscous and non-polar liquids and small static contact angles (up to 50°) the dewetting capillary number Caj = i VdlV scales as the cube of (P while the prefactor varies weakly with the studied system. This result, that resembles Tanners law, was explainedby means of a simple hydrodynamic model that assumes a circular cross section for the bump and symmetrical dissipation at both of it ends. Following later observations of asymmetries in the bump s profile this last assumption was modified. 

If the liquid completely wets the wall, the front angle 0d is the dynamic contact angle given by Tanner law (10) the back angle vanishes and the liquid leaves a small film behind of thickness d given by equation (13), the upper meniscus curvature is thus smaller than b these two contributions lead to a pressure gradient... 

This shows that in general the one dimensional profile is unstable to perturbations of the contact line. A more detailed stability analysis has been performed in reference (37) which shows that the wavelength of the fastest growing mode is of the order of the size of the bump 1 and also varies slightly with the microscopic cutoff k of Tanner law (the thickness of the precursor film). 

Schelling, T. C. (1983) Ethics, law, and the exercise of self-command, in S. McMurrin (ed.). The Tanner Lectures on Human Values IV, Salt Lake City University of Utah Press, 43-79. Cited after the reprint in Schelling (1984). 

A review by Bird and Wiest [6] gives a more complete list of existing viscoelastic models. The upper convective model and the White-Metzner model are very similar with the exception that the White-Metzner model incorporates the strain rate effects of the relaxation time and the viscosity. Both models provide a first order approximation to flows, in which shear rate dependence and memory effects are important. However, both models predict zero second normal stress coefficients. The Giesekus model is molecular-based, non-linear in nature and describes thepower law region for viscosity andboth normal stress coefficients. The Phan-Thien Tanner models are based on network theory and give non-linear stresses. Both the Giesekus and Phan-Thien Tanner models have been successfully used to model complex flows. 

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