Cylindrical coordinates physical components

In order to understand the physics behind the observed super-high sensitivity, we investigated the optical field distribution in the microtube using cylindrical coordinates z, r, and time-independent field distribution for the resonant mode can be separated into a radial-dependent mode component and an azimuthal-dependent phase term, T lYjexpfiw J, where / is the amplitude of the axial magnetic (TE) or electric (TM) modal field and m is the azimuthal quantization... 

The nonstationary motion of the medium was modeled by difference analogs of the Euler gasdynamics equations. For plane-parallel or axisymmetrical flows with a cylindrical or spherical system of coordinates, it can be assumed that d/d = 3/3 = 0. The physical components of the velocity vector W will be presented in the form of projections onto the directions and g e, and we shall also introduce the variables T] = 1 - RJr and 0 If all the linear dimensions are scaled with R, then in the dimensionless form the equations take the form ... 

We assume isothermal flow and a Newtonian fluid. (Isothermality is the more serious of the two assumptions. Even if the physical properties of the melt are independent of temperature, we must deal with the possibility of solidification of the melt near the cold mold faces during filhng.) For radial flow we have a single velocity component in cylindrical coordinates, Vr, and the continuity equation from Table 2.1 is... 

Firstly, notice that the expressions for Ui, n, Ay, Wij, Ni and i are exactly the same as those derived earlier in equations (5.243) to (5.247), the only difference being that 0 will turn out not to be constant. Prom formulae (C.8) to (C.IO) in Appendix C and the form for the director given by (5.279), the non-zero physical components of [Vn] = riij in cylindrical coordinates are... 

Equations (5.329) and (5.330) will be expressed in the relevant physical components in cylindrical polar coordinates. As in the Couette flow examples above, the velocity and director are independent of time t and therefore v = (v V)v and h = (v V)n, and by appropriate applications of the formula (C.4) in Appendix C we have... 

The physical components in cylindrical polar coordinates of the gradient of any scalar function p are then given by [115, p.l35]... 

Appendix C Physical Components in Cylindrical Polar Coordinates... 

The divergence of the stress tensor T with Cartesian components T j is defined in this text by [V -T] = Tijj. The physical components of this divergence in cylindrical polar coordinates are [115, p.l44]... 

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