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Suppose we have an incompressible, viscous sessile drop subject to a time-dependent pressure field ##p## on a substrate. Let ##\mu## be dynamic viscosity, ##u## be the fluid velocity field, ##\kappa_{1/2}## curvatures of the fluid surface, ##\sigma## surface tension, ##\hat n## normals to the equilibrium surface, and ##\eta## the disturbed interface.

Disturbances to the equilibrium surface generate pressure gradients, and thereby flows. A pressure balance at the interfacial surface yields $$p-\mu \hat n \cdot(\nabla \otimes u) \cdot \hat n = - \sigma( \Delta_\Gamma \eta + (\kappa_1^2+\kappa_2^2)\eta)$$

The RHS is flow from the capillary pressure (Young-Laplace equation). The LHS is inertial pressure (first term) and viscous pressure (second term). I do not understand where the viscous pressure entered the pressure balance.

After googling I found this site: http://web.mit.edu/1.63/www/Lec-notes/Surfacetension/Lecture2.pdf

where equation (3) looks like the LHS, and if we look at their definition of ##T## we see there is a transpose velocity component (not shown in the pressure balance above, and the implication ##\hat n \cdot -p I \cdot \hat n = -p##)? Can someone help me understand this? Thanks!

Disturbances to the equilibrium surface generate pressure gradients, and thereby flows. A pressure balance at the interfacial surface yields $$p-\mu \hat n \cdot(\nabla \otimes u) \cdot \hat n = - \sigma( \Delta_\Gamma \eta + (\kappa_1^2+\kappa_2^2)\eta)$$

The RHS is flow from the capillary pressure (Young-Laplace equation). The LHS is inertial pressure (first term) and viscous pressure (second term). I do not understand where the viscous pressure entered the pressure balance.

After googling I found this site: http://web.mit.edu/1.63/www/Lec-notes/Surfacetension/Lecture2.pdf

where equation (3) looks like the LHS, and if we look at their definition of ##T## we see there is a transpose velocity component (not shown in the pressure balance above, and the implication ##\hat n \cdot -p I \cdot \hat n = -p##)? Can someone help me understand this? Thanks!

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