Partial differential equations nondimensionalization

Although the partial differential equation Eq. 25-10 is linear and looks rather simple, explicit analytical solutions can be derived only for special cases. They are characterized by the size of certain nondimensional numbers that completely determine the shape of the solutions in space and time. A reference distance x0 and a reference time f0 are chosen that are linked by ... 

When dealing with simple equations (as in the previous three models), the dimensional equations are solved without recourse to the process of nondimen-sionalisation. Now, we must deal with partial differential equations, and to simplify the notation during the analysis and also to deduce the proper dimensionless parameters, it is necessary to reduce the equations to nondimensional form. To achieve this, we introduce the following nondimensional variables and parameters ... 

Step 1. Setting up the model equations on the particle scale. These equations are generally nonlinear partial differential equations (PDEs). For analysis in the frequency domain, it is most convenient to use nondimensional concentrations and temperatures, dehned as relative deviations from their steady-state values. 

The system equations (eqs. 8.4-1 and 8.4-5) are coupled nonlinear partial differential equations and must be solved numerically. To facilitate with the numerical analysis, we define the following nondimensional variables... 

The answer to this question is the subject oiscaling and dimensional analysis. In general, scaling involves the nondimensionalization of the conservation equations where the characteristic variables used for nondimensionalization are selected as their maximum values, e.g., the maximum values of velocity, temperature, length, and the like, in a particular problem. Let s see specifically how this method works and why it can often lead to a simplification of partial differential equations. 

The first step in scaling involves nondimensionalization of the partial differential equation (PDE). To accomplish this we introduce dimensionless variables (denoted by an asterisk) for every dependent and independent variable of the PDE, such as... 

---