Surface, equations unit normal vector

Macroscopic Equations An arbitraiy control volume of finite size is bounded by a surface of area with an outwardly directed unit normal vector n. The control volume is not necessarily fixed in space. Its boundary moves with velocity w. The fluid velocity is v. Figure 6-3 shows the arbitraiy control volume. 

Interphase momentum transfer is the focus of this section. Macroscopic correlations are based on dynamic forces due to momentum flux that act across the fluid-solid interface, similar to terms of type 2, 3, and 4 in the equation of motion. Gravity enters into this discussion via the hydrostatic contribution to fluid pressure, because volumetric body forces are not operative across an interface. The outward-directed unit normal vector from the solid surface into the fluid is n. As discussed earlier, forces due to total momentum flux, transmitted in the —n direction from the flnid to the solid across the interface at r = / , are (i.e., see equation 8-20) ... 

In these equations, m is the droplet mass and is the mass of species i, is the density of the vapor at the drop surface, Mj = Uj n is the normal component of the vapor velocity, where n denotes the droplet outward unit normal vector, and Sd is the droplet surface. Further, Cpi and Ti denote the mass-averaged liquid heat... 

In this section, we begin by reviewing the implicit representation of an interface as the level surface of a function defined over all space, together with appropriate definitions of the unit normal vector and the curvature at that interface. We then present a key result from vector calculus that is needed in deriving the proper boundary conditions at an interface. This sets the stage for the formulation of two-phase flow problems including the governing equations and boundary conditions, as well as a whole-domain formulation that combines the two. 

In this equation, d2u) represents the angle of the radiated SH light with respect to the surface normal, 7(co) is the pump intensity, and e(2co) is the polarization at the SH frequency. The vectors e(co) and e(2co) are related to the unit polarization vectors e(co) and e(2co) in medium 2 by Fresnel coefficients. The effective surface nonlinear susceptibility incorporates the surface nonlinear susceptibility x( and the bulk magnetic dipole contributions to the nonlinearity. The result simplifies since, for isotropic media, there are only three nonzero independent elements of xf These are x%, X% = X%.> and XsfL where 1 =... 

Figure 1.16 Mappiitg of a minimal surface from real >ace to the complex plane. A point P on the surface, whose normal vector at P is n, is transformed to a point P by the Gauss map, given by the intersection of n (placed at the origin of the unit sphere O) with the sphere. P is mapped into a point P" on die complex plane (real and imaginary axes aand rresp.) by stereographic projection from the north pole of die sphere, N, onto the complex plane, which intersects the sphere in its equator.

In either formulation, an expression for VFR is required, which can be obtained from simple geometrical considerations. Let I represent the entire ellipsoidal surface area in Equation 18-6 and dS denote the differential surface area on S. Then, if n is the local unit normal drawn perpendicular to dS, and q is the Darcy velocity vector, we simply have... 

E is the vector electric field and the integral is taken over a closed surface S da is an element of this surface, n is a vector of unit length and direction normal to the surface and the sum is taken over all charges. Putting equation... 

A transformation from the Cartesian coordinate system to the local (r, u, v) frame can then be made. It can be shown that this transformed coordinate system is orthogonal with unit basis vectors r, u, v where the r-direction is parallel to the local layer normal of the Dupin cyclide. Moreover, it can be shown [210] that Vr = r is the unit layer normal to the cyclide surface, so that a = f automatically fulfils the necessary requirements a a = 1 and V x a = 0, stated at equations (6.3)i and (6.4). It consequently follows that for a fixed value of r, the inner part of a Dupin cyclide surface is obtained by varying the values of u and v in the above expressions for the Xy y and z coordinates of the surface. An alternative parametrisation for the full Dupin cyclide surfaces is available in the book by Forsyth [90, p.326], but care needs to be exercised in the context of parallel layers for liquid crystals because complicated restrictions on the range of the parameters must be introduced. 

Here n is the unit vector perpendicular to the surface S and directed outward, and Tn the normal component of an arbitrary vector X, which is a continuous function within volume V. As was pointed out, Equation (1.67) has an infinite number of solutions let us choose any pair of them, U p) and U2(p), and form their difference ... 

---