By Charles B Daniels; John Davison
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Extra resources for Ontology and Method in Wittgenstein's Tractatus
Example text
In the absence of motion, the contact line angle θ is fixed by the capillary properties of the substrate and the two fluids. Consider a liquid and a gas on a solid surface. The liquid–gas surface energy per unit area is the capillary tension σ . For the liquid–solid and the gas–solid interfaces we respectively have energies per unit area σls and σgs . Moving the contact point by a small distance δ x along the solid surface results in a gain of energy σgs δ x on one side and a loss (σls + σ cos θeq )δ x on the other side.
The representation of the interface as a contour with a specific value is used in level-set methods. The motion of the interface S is determined by the normal velocity V (u, v,t), for each point on S. This velocity may be that of the fluid itself, or may be different due to, for example, evaporation or condensation. In parametric form: ∂ x(u, v,t). 43) ∂t To update the location of S, we note that there is no need to specify a tangential velocity, although from a physical point of view the interface is made of fluid particles that travel at some tangential as well as normal velocity.
The equations of fluid motion in convective or non-conservative form. 20) states that the volume of any fluid element cannot be changed and these flows are therefore referred to as incompressible flows. 21) stating that inflow balances outflow. Notice that there is no requirement that the density is the same everywhere, for incompressible flows. The density of a material particle can vary from one particle to the next one, but the density of each particle must stay constant. When the density is not the same everywhere its value at any given point in space can change with time as material particles of different density are advected with the flow.