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64 CHAPTER 3. GENERAL EQUATIONS
3.3 Equations of motion
The equations of motion are obtained by applying Newton’s Second Law of Mechanics,
to the material element in Fig. 3.1.
Let V be the volume of said element. Then ρV will be the mass and ρV ¯v the
momentum, for which the time variation when following the element is
D (ρV ¯v)
Dt
= ρV D¯v
Dt , (3.12)
since ρV does not vary. Therefore, the variation of momentum of the mass contained
in the unit volume is ρ (D¯v/Dt). Such variation originates from:
1) The surface forces acting upon the unit volume.
2) The mass forces acting upon the unit volume.
Let us calculate both, separately.
Let ¯f be the force that acts upon the surface element dσ at a point of Σ, Fig. 3.1.
It has been shown in chapter 1 that ¯f is determined by the stress tensor τ e of components
τ ij and that as a function of the tensor it can be expressed as follows
¯f = ¯n · τ e dσ, (3.13)
where ¯n is the outward normal to the surface element dσ at the point. As it can easily
be proved, the resultant of all forces ¯f per unit volume is ∇ · τ e . 6
Let ¯F be the field intensity of the mass forces, that is to say, the force per unit
mass. The force per unit volume is, evidently, ρ ¯F . 7
Now, by expressing Newton’s Second Law of Mechanics, we obtain
ρ D¯v
Dt = ∇ · τ e + ρ ¯F . (3.14)
This vectorial equation is equivalent to three scalar equations corresponding to
the three components of the reference system. For example, in rectangular cartesian
coordinates, the component of this equation parallel to the coordinate axis x i (i =
1, 2, 3) is
ρ ∂v i
∂t + ρv ∂v i
j = ∂τ ij
+ ρF i , (3.15)
∂x j ∂x j
6 In fact, the resultant of the forces acting upon Σ is RR Σ ¯n · τ e dσ. Now, by applying Ostrogradsky’s
theorem we obtain RR Σ ¯n · τe dσ = RRR V ∇ · τe dV. Therefore, the force per unit volume is ∇ · τe.
7 If the force per unit mass depends on the species and ¯F i is the intensity for species A i , ρ ¯F must be
substituted by P ρ i ¯Fi = ρ P Y i ¯Fi .
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