ABSTRACT
Many phenomena in hydrodynamics are described by partial differential equations (PDEs) that cannot be solved analytically. As a result, solutions for those equations are feasible only by means of numerical algorithms. The general equation for linear PDEs of the second-order in two independent variables reads as follows:
A
f
x B
f x y
C f
y D
f x
E f y
Ff G ∂ ∂
+ ∂
∂ ∂ + ∂ ∂
+ ∂ ∂ +
∂ ∂ + + =
2 0 (3.1)
where A to G are constant coefficients. Depending only on the coefficients of the second-order derivatives, a classification of those PDEs is accomplished based on the value of the discriminant:
Δ = B2 − 4AC (3.2)
Thus, Δ < 0 equations are classified as elliptic, Δ = 0 as parabolic and Δ > 0 as hyperbolic. This classification, in addition to being important from a mathematical point of view, has great significance for the hydrodynamic phenomena represented by the different classification groups. Under steadystate conditions (time-independent problems) the variables x and y are the spatial coordinates. Under unsteady conditions (time-dependent problems) the variable x is the spatial variable, while the variable y is replaced by the time variable t (Mitchell and Griffiths 1980; Lapidus and Pinder 1999).
