Page 443 - Introduction to Continuum Mechanics

7

Integral Formulation of General Principles

In Sections 3.15,4.4,4.7,4.14, the field equations expressing the principles of conservation
of mass, of linear momentum, of moment of momentum, and of energy were derived by the
consideration of differential elements in the continuum. In the form of differential equations,
the principles are sometimes referred to as local principles. In this chapter, we shall formulate
the principles in terms of an arbitrary fixed part of the continuum. The principles are then in
integral form, which is sometimes referred to as the global principles. Under the assumption
of smoothness of functions involved, the two forms are completely equivalent and in fact the
requirement that the global theorem be valid for each and every part of the continuum results
in the differential form of the balance equations.
The purpose of the present chapter is twofoldr(l) to provide an alternate approach to the
formulation of field equations expressing the general principles, and (2) to apply the global
theorems to obtain approximate solutions of some engineering problems, using the concept
of control volumes, moving or fixed.
We shall begin by proving Green's theorem, from which the divergence theorem, which we
shall need later in the chapter, will be introduced through a generalization (without proof).

7.1 Green's Theorem

Let/*(*,)>), dP/dx and dP/dy be continuous functions of jcandy in a closed region.R bounded
by the closed curve C. Let n = n^ej+n^ be the unit outward normal of C. Then Green's
theorem* states that

and

t The theorem is valid under less restrictive conditions on the first partial derivative.

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