Coordinate system normal/tangential

It is often convenient to use some other coordinate system besides the Cartesian system. In the normal/tangential system (Figure 2-10), the point of reference is not fixed in space but is located on the particle and moves as the particle moves. There is no position vector and the velocity and acceleration vectors are written in terms of... 

In addition to the Cartesian and normal/tangential coordinate systems, the cylindrical (Figure 2-11) and spherical (Figure 2-12) coordinate systems are often used. 

To calculate the tangential stress pT = p, it is convenient to define a rectangular coordinate system (x, y, z) with the (xt y) plane parallel to the transition layer and the 2-axis normal to the surface and directed from liquid to vapor. The tangential stress is the sum of the momentum transport and the force transmitted across a strip of unit width, perpendicular to the transition layers and extending from 2 — — 1/2 in the interior of the liquid to 2 = 1/2 in the gas phase. Kirkwood and Buff expressed this stress in terms of the intermolecular force and the pair distribution function. Thus they obtained... 

Consider creeping viscous flow of an incompressible Newtonian fluid past a stationary gas bubble that is located at the origin of a spherical coordinate system. Do not derive, but write an expression for the tangential velocity component (i.e., vg) and then linearize this function with respect to the normal coordinate r within a Ihin mass transfer boundary layer in the liquid phase adjacent to the gas-liquid interface. Hint Consider the r-9 component of the rate-of-strain tensor ... 

Figure 9. A schematic of the sliding microindentation event with the point normal and tangential loads as well as a moving coordinate system.

Two key features are exploited by the tracking. When the orientation and position of a front is known, one can locally rotate into a coordinate system that is aligned with the wave front so that the normal and tangential directions to the front are grid lines in the local coordinate system. Such a grid substantially improves the quality of the solution near the front. 

T is often referred to it as traction and clearly it represents a stress vector acting parallel to the surface. From the Pythagorean relation shown in Fig. 1.12, the sum of the squares of the normal and tangential stresses on any face of an elementary cube under arbitrary stress is equal to their sum. Furthermore, the shear component, T, is usually not parallel to any of the axes of a chosen coordinate system, as indicated in Fig. 1.12. It is, however, common to resolve this shear stress into two components, each of which is parallel to a chosen reference coordinate system. In Fig. 1.13, the stresses are indicated on the z plane. 

In the intrinsic coordinate system, the Euler equations for a steady flow are written along the tangential and normal directions, respectively, as ... 

When calculating adsorption-desorption probabilities one has to use models, that allow one to couple the energetic variables in the gas phase with those in the adsorbed layer. Let us divide the translational motion of an adsorbed molecule into normal and tangential components. We shall denote the energies of the molecular motion along x, y, z eixes as Ey, E respectively. The momentum of the projectile molecule in the same coordinate system can be presented as... 

At a point in the filament boundary the unit outward normal vector is n and the tangential vector is t. The appropriate coordinate system for the problem is cylindrical with the axis of symmetry coinciding with the z axis (Fig. 9.3). The velocity vector is v and the total stress tensor is rr. The components... 

The unit vectors in this system are er, eg and e. The unit vector er is directed outwards from the coordinate origin O to point P, the unit vector eunit vector is tangential to the circle shown in Figure 0.2 and points in the direction of increasing [Pg.5]

Recently, Wiggins et al. [15] provided a firm mathematical foundation of the robust persistence of the invariant of motion associated with the phase-space reaction coordinate in a sea of chaos. The central component in RIT that is, unstable periodic orbits, are naturally generalized in many DOFs systems in terms of so-called normally hyperbolic invariant manifold (NHIM). The fundamental theorem on NHIMs, denoted here by M, ensures [21,53] that NHIMs, if they exist, survive under arbitrary perturbation with the property that the stretching and contraction rates under the linearized dynamics transverse to jM dominate those tangent to M. Note that NHIM only requires that instability in either a forward or backward direction in time transverse to M is much stronger than those tangential directions of M, and hence the concept of NHIM can be applied to any class of continuous dynamical systems. In the case of the vicinity of saddles for Hamiltonian problems with many DOFs, the NHIM is expressed by a set of all (p, q) satisfying both q = p = Q and o(Jb) + En=i (Jb, b) = E, that is. 

Consider a drop of phase a, immersed in phase and described by polar coordinates r,d,4f. As in 2.4 the system under discussion is bounded by two concentric spheres, one of radius R", which lies in phase a, and one of radius in phase. An arbitrary dividing surfi of radius R separates the two phases, R planar surface, the symmetry of the system requires that the pressure tensor has only two independent components, normal and tangential ... 

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