Chapter 4
Mathematical Properties of the Fluid
Dynamic Equations
J.D. Anderson, Jr.
4.1 Introduction
The governing equations of fluid dynamics derived in Chap. 2 are either integral
forms (such as Eq. (2.23) obtained directly from a finite control volume) or partial
differential equations (such as Eqs (2.36a–c) obtained directly from an infinitesimal
fluid element). The governing equations in the form of partial differential equations
are by far the most prevalent form used in computational fluid dynamics. Therefore,
before taking up a study of numerical methods for the solution of these equations,
it is useful to examine some mathematical properties of partial differential equations themselves. Any valid numerical solution of the equations should exhibit the
property of obeying the general mathematical properties of the governing equations.
Examine the governing equations of fluid dynamics as derived in Chap. 2. Note
that in all cases the highest order derivatives occur linearly, i.e. there are no products
or exponentials of the highest order derivatives—they appear by themselves, multiplied by coefficients which are functions of the dependent variables themselves.
Such a system of equations is called a quasilinear system. For example, for inviscid
flows, examining the equations in Sect. 2.8.2 we find that the highest order derivatives are first order, and all of them appear linearly. For viscous flows, examining the
equations in Sect. 2.8.1 we find the highest order derivatives are second order, and
they always occur linearly. For this reason, in the next section, let us examine some
properties of a system of quasilinear partial differential equations. In the process,
we will establish a classification of three types of partial differential equations—all
three of which are encountered in fluid dynamics.
4.2 Classification of Partial Differential Equations
For simplicity, let us consider a fairly simple system of quasilinear equations. They
will not be the flow equations, but they are similar in some respects. Therefore, this
section serves as a simplified example.
J.D. Anderson, Jr.
National Air and Space Museum, Smithsonian Institution, Washington, DC
e-mail: AndersonJA@si.edu
J.F. Wendt (ed.), Computational Fluid Dynamics, 3rd ed.,
c Springer-Verlag Berlin Heidelberg 2009
77
78
J.D. Anderson, Jr.
Consider the system of quasilinear equations given below.
a1
∂u
∂u
∂v
∂v
+ b1 + c1 + d1 = f1
∂x
∂y
∂x
∂y
(4.1a)
a2
∂u
∂u
∂v
∂v
+ b2 + c2 + d2 = f2
∂x
∂y
∂x
∂y
(4.1b)
where u and v are the dependent variables, functions of x and y, and the coefficients
a1 , a2 , b1 , b2 , c1 , c2 , d1 , d2 , f1 and f2 can be functions of x, y, u and v.
Consider any point in the xy-plane. Let us seek the lines (or directions) through
this point (if any exist) along which the derivatives of u and v are indeterminant,
and across which may be discontinuous. Such lines are called characteristic lines.
To find such lines, we assume that u and v are continuous, and hence
since u = u(x, y) : du =
∂u
∂u
dx + dy
∂x
∂y
(4.2a)
since v = v(x, y) : dv =
∂v
∂v
dx + dy
∂x
∂y
(4.2b)
Equations (4.1a and b) and (4.2a and b) constitute a system of four linear equations with four unknowns (∂u/∂x, ∂u/∂y, ∂v/∂x, and ∂v/∂y). These equations can be
written in matrix form as
⎤⎡
⎡
⎤ ⎡ ⎤
⎢⎢⎢ a1 b1 c1 d1 ⎥⎥⎥ ⎢⎢⎢∂u/∂x⎥⎥⎥ ⎢⎢⎢ f1 ⎥⎥⎥
⎢⎢⎢⎢ a2 b2 c2 d2 ⎥⎥⎥⎥ ⎢⎢⎢⎢∂u/∂y⎥⎥⎥⎥ ⎢⎢⎢⎢ f2 ⎥⎥⎥⎥
⎥⎢
⎢⎢⎢
⎥=⎢ ⎥
(4.3)
⎢⎢⎣ dx dy 0 0 ⎥⎥⎥⎥⎦ ⎢⎢⎢⎢⎣∂v/∂x ⎥⎥⎥⎥⎦ ⎢⎢⎢⎢⎣du⎥⎥⎥⎥⎦
0 0 dx dy ∂v/∂y
dv
Let [A] denote the coefficient matrix.
⎡
⎢⎢⎢ a1
⎢⎢⎢ a
[A] = ⎢⎢⎢⎢ 2
⎢⎢⎣ dx
0
b1
b2
dy
0
c1
c2
0
dx
⎤
d1 ⎥⎥
⎥
d2 ⎥⎥⎥⎥
⎥
0 ⎥⎥⎥⎥⎦
dy
Moreover, let ;|A| be the determinant of [A]. From Cramer’s rule, if |A| 0,
then unique solutions can be obtained for ∂u/∂x, ∂u/∂y, ∂v/∂x, and ∂v/∂y. On
the other hand, if |A| = 0, then ∂u/∂x, ∂u/∂y, ∂v/∂x and ∂v/∂y are, at best, indeterminant. We are seeking the particular directions in the xy-plane along which these
derivatives of u and v are indeterminant. Therefore, let us set |A| = 0, and see what
happens.
&
&&
&& a1 b1 c1 d1 &&&
&& a2 b2 c2 d2 &&
&=0
&&
&& dx dy 0 0 &&&
& 0 0 dx dy &