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2.3. VISCOSITIES 49
2.3 Viscosities
Mechanics of continuous media teaches that the state of stresses at each point of a
medium is determined by a symmetric tensor of second order
⎛
⎞
p 11 p 12 p 13
⎜
⎟
τ e ≡ ⎝ p 21 p 22 p 23 ⎠ , (2.33)
p 31 p 32 p 33
of components p ij = p ji in a rectangular cartesian system of reference. 16 The physical
meaning of this tensor is the following: p ij is the component parallel to axis x j of the
stress acting upon a surface element normal to axis x i , see Fig. 2.4.
x
p
12
p
13
p
11
3
p
23
p
p 22
21
p 1
33
p31
x 2 p 32
Figure 2.4: Schematic diagram showing the components of the stress tensor τ e.
x
Stress ¯f acting upon a surface element dσ, see Fig. 2.5, whose orientation is
defined by the unit vector ¯n, is given by the expression 17
¯f = ¯n · τ e , (2.34)
whose components are
f i = ∑ j
n j p ji , (i = 1, 2, 3), (2.35)
where n j are the components of ¯n.
Kinetic Theory of gases shows 18 that in a dilute gas the stress originates from
transfer of momentum of the gas through a surface element dσ moving at velocity ¯v of
16 See, i.e., Ref. [15].
17 See appendix to chapter 3 for notation.
18 See Ref. [1], p. 31.