Page 183 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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CONVECTION HEAT TRANSFER
7.2 Navier–Stokes Equations
The mathematical model of any fundamental fluid dynamics problem is governed by the
Navier–Stokes equations. These equations are important and represent the fluid as a contin-
uum. The equations conserve mass, momentum and energy, and can be derived following
either an integral or a differential approach. The integral form of the equations is derived
using Reynolds Transport Theorem (RTT) and is discussed in many standard fluid mechan-
ics texts (Shames 1982). The approach we follow in this book is the differential approach
in which a differential control volume is considered in the fluid domain and a Taylor
expansion is used to represent the variation of mass, momentum and energy.

7.2.1 Conservation of mass or continuity equation

The conservation of mass equation ensures that the total mass is conserved, or, in other
words, the total mass of a fluid system is completely accounted for. In order to derive
a general conservation of the mass equation, consider the differential control volume as
shown in Figure 7.2. The reader can assume the control volume to be infinitesimal for
a typical flow problem, such as flow in a channel (Figure 7.1), flow over a flat plate or
the temperature (or density) difference driven circulation of air inside a room as shown in
Figure 7.3.
Let us assume that the mass flux rate entering the control volume (Figure 7.2) is ρu 1
in the x 1 direction and ρu 2 in the x 2 direction. It is also assumed that there is no reaction
or mass production within the fluid domain. The Taylor series expansion may be used to
express the mass flux rate exiting the control volume as (refer to Figure 7.2)
2 2
x 1 ∂(ρu 1 ) x 1 ∂ (ρu 1 )
+ + + ··· . (7.2)
(ρu 1 ) x 1 + x 1 = (ρu 1 ) x 1 2
1! ∂x 1 2! ∂x
1
∂ru 2
ru +∆x 2 + ....
2
∂x 2

∂ru
ru 1 ∆x 2 ru +∆x 1 1 + ....
1
∂x 1
∆x 1

ru 2

Figure 7.2 Infinitesimal control volume. Derivation of conservation of mass in a flow field