Boundary conditions Young-Laplace equation

Equations II-12 and 11-13 illustrate that the shape of a liquid surface obeying the Young-Laplace equation with a body force is governed by differential equations requiring boundary conditions. It is through these boundary conditions describing the interaction between the liquid and solid wall that the contact angle enters. 

The first approach developed by Hsu (1962) is widely used to determine ONE in conventional size channels and in micro-channels (Sato and Matsumura 1964 Davis and Anderson 1966 Celata et al. 1997 Qu and Mudawar 2002 Ghiaasiaan and Chedester 2002 Li and Cheng 2004 Liu et al. 2005). These models consider the behavior of a single bubble by solving the one-dimensional heat conduction equation with constant wall temperature as a boundary condition. The temperature distribution inside the surrounding liquid is the same as in the undisturbed near-wall flow, and the temperature of the embryo tip corresponds to the saturation temperature in the bubble 7s,b- The vapor temperature in the bubble can be determined from the Young-Laplace equation and the Clausius-Clapeyron equation (assuming a spherical bubble) ... 

Static capillary phenomena lead to precisely determined geometrical shapes like sessile menisci, pendent menisci, minimal surfaces, which can be used for the physical determination and measurement of the surface tension or the interfacial tensions between fluids. In addition to the simple forms considered herein, more complex forms (e.g., sessile lenticular drops) can be studied. Mathematical resolution of these shapes is a combination of the (numerical) solution of the highly nonlinear Young-Laplace equation together with an appropriate set of boundary conditions. For practical purposes, only axisymmetric forms are readily amenable to mathematical analysis. 

The linearized dynamic boundary condition is physically the same as given by Eq. (10.4.7) for the plane surface wave, which with allowance for the cylindrical symmetry of the jet problem may, from the Young-Laplace equation, be written... 

For the pendant drop in figure I, the equations above are subjected to the no-slip boundary conditions at solid surfaces and the kinematic condition on the free surfaces. The kinematic condition implies that there is no liquid crossing the boundary into the gas phase, or in other words forms a definite boundary between the phases. For creeping flows into an atmosphere of gas with minimal velocities there will be no interfacial shear stress tangential to the surface and the normal stress inside the fluid is balanced by the surface tension as described by the famous Young-Laplace equation. 

FIGURE 5.7 The original Young-Laplace equation was concerned with narrow cylindrical capillary tubes of circular and constant cross section (left). For noncircular cross sections, the capillary pressure depends on the two principal radii of curvature (middle). For irregular cross sections under the influence of a vertical gravitational field, a second-order differential equation and the related boundary conditions must be satisfied (right). 

With the help of the Young-Laplace equation, it is possible to calculate the equilibrium shape of a liquid surface. If we know the pressure difference and some boundary conditions (such as the volume of the liquid and its contact line), we can calculate the geometry of the liquid surface. 

We have thus shown that the profile r(z) that minimizes the free energy in Eq. (22) has to obey the Laplace equation with Young s law as a boundary condition. 

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