Laplacian cylindrical coordinates

In these equations the Laplacian operator V2 is given in cylindrical coordinates as... 

The solution of these equations by means of standard eigenfunction expansions can be carried out for any curvilinear, orthogonal coordinate system for which the Laplacian operator V2 is separable. Of course, the most appropriate coordinate system for a particular application will depend on the boundary geometry. In this section we briefly consider the most common cases for 2D flows of Cartesian and circular cylindrical coordinates. 

Clearly, this equation needs to be completed with boundary conditions corresponding to the cylindrical shape of the TAP reactor. This requires the use of cylindrical coordinates (r,0,z) instead of the Cartesian coordinates (x,y,z). Note that the cylindrical z coordinate is longitudinal and corresponds to the Cartesian x coordinate used so far. Substituting the coordinate transformation equations and applying the chain rule we obtain the following expression for the Laplacian in cylindrical coordinates ... 

If we consider long enough pores to ignore end effects in the interior of the pore, there is a cylindrical symmetry about the axis of the pore. For this two-dimensional problem, the above Poisson-Boltzmann equation is solved in the cylindrical coordinate system of Figure 3.13, where r is the radial distance from the pore axis. Rewriting the Laplacian in the cylindrical coordinate system,... 

Notice finally that the appearance of the logarithm in the inflection point criterion is related to ln(r) being the two-dimensional Coulomb potential, i.e., the Green function of the cylindrically symmetric Laplacian. In the corresponding three-dimensional (spherical) problem of charged colloids the Green function 1/r would be the appropriate choice for plotting the radial coordinate [31,32], More details can be found in Ref. 4. 

Write the expression for the Laplacian in cylindrical polar coordinates... 

We can proceed in this way to obtain higher order derivatives, but only first and second derivatives are needed to handle most engineering problems. For cylindrical and spherical coordinates, the shifted position procedure in Problem 12.8 shows how to deal with the Laplacian operators. 

Table 1 7.1 For a scalar / or a vector u, this table gives the gradient cif, the divergence V u, and the Laplacian V-/ in Cartesian, cylindrical, and spherical coordinates. 

Now, we are in cylindrical polar coordinates and so must use the appropriate form of the Laplacian ... 

Equation (2) represents the heat flow into the volume V and can be derived from Eq. (11) in Fig. 1.2. The symbols have the standard meanings p is the density and Cp, the specific heat capacity. Standard techniques of vector analysis now dlow the heat flow into the volume V to be equated with the heat flow across its surface. This operation leads to the Foimer differential equation of heat flow, given as Eq. (3). The letter k represents the thermal diffusivity, which is equal to the thermal conductivity k divided by the density and specific heat capacity. Its dimension is m /s. The Laplacian operator, v2, is 32/3j2 + 2/ 2 + a2/aj2 where x,y and z are the space coordinates. In the present example of cylindrical symmetry, the Laplacian operator, operating on temperature T, can be represented as — i.e., the... 

Example 8.25. Write the expression for the Laplacian in cylindrical polar coordinates For cylindrical polar coordinates, hr =, h = r, and = 1. 

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