Chapter 1: Foundations of mathematical modeling 9
Equation 1.2 is called the continuity equation and is a statement of conserva-
tion of mass. Equation 1.3 is the vector form of conservation of linear momen-
tum. The dependent variables in Equation 1.2 and Equation 1.3 are ρ, the
mass density of the uid, v, the three-dimensional velocity vector, and p, the
pressure. The body forces are represented by F
b
, and μ is the dynamic coef-
cient of viscosity. There are ve dependent variables, but Equation 1.2 and
Equation 1.3 provide only four scalar equations. If the ow is incompressible,
then conservation of mass implies that ∇ ⋅ v = 0, and Equation 1.2 implies that
the density is constant. In other cases, an equation of state relating pressure
and density is used along with Equation 1.2 and Equation 1.3. Equations of
state often involve temperature as a variable. If temperature varies, then an
energy equation is used in conjunction with Equation 1.2 and Equation 1.3.
A second method of modeling is to apply basic forms of the conservation
laws directly to the system. If time is an independent variable in the problem
formulation, the conservation laws are applied at an arbitrary instant. In all
cases, the conservation laws are applied for arbitrary values of the depen-
dent variables. Two of the basic laws are conservation of momentum and
conservation of energy.
Conservation of momentum is often applied to a free-body diagram of an
in nitesimal particle drawn at an arbitrary instant. Conservation of momen-
tum, Newton’s second law as applied to a particle, is formulated as:
Fa
∑
= m (1.4)
where ∑ F represents the resultant force acting on a free-body diagram of the
particle drawn at an arbitrary instant and a is the acceleration vector. Conser-
vation of angular momentum is an appropriate form of the moment equation.
Conservation of energy as applied to a control volume takes the form that
the rate of energy accumulation within the control volume is equal to the rate at
which energy is transferred into the control volume through its boundaries.
A complete mathematical formulation often requires that the basic conser-
vation laws be supplemented by laws speci c to the system being modeled.
For example, conservation of energy can be supplemented by Newton’s law
of cooling and Fourier’s conduction law. The Navier-Stokes equations are
written for a viscous uid which satis es Newton’s viscosity law. Constitutive
equations are used to represent stress-strain relations for many solids.
The method of application of mathematical modeling is different for differ-
ent types of systems, yet the concepts are the same for all systems. The prob-
lem is rst identi ed. Basic questions are answered. Why is a mathematical
model necessary? What level of abstraction is required? What assumptions
are necessary for successful modeling? What are the independent and depen-
dent variables? What parameters are used in the modeling? Are numerical
values of these parameters necessary? Based on answers to these questions,
the problem is re ned. Basic laws of physics, in some form, are applied to the
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