Following Stokes and others it will be presumed that the viscous stress tensor can be expressed as a linear combination of the deformation tensor and the divergence as follows
This combination of terms satisfies the requirement thta
be symmetrical, i.e. 
,
since both the deformation and the unit tensor
are symmetrical. It will be recalled
that the Kronecker delta is defined by
For the case of simple shear where v = w = 0 and u = u(z), the stresses reduce to
all all other stresses are zero. This is the law commonly applied in the case of flow in a thin lamina of fluid contained between two flat plates. For radially symmetric flow in a circular tube where u = u(r), the analogous special case for the stresses is
with all other stresses being zero (a x directed along the axis of the tube).
Another special case of (5.03-1) is for plane acoustical waves of the simplest type where v = w = 0, u = u(x,t). In this case
and all other stresses are zero. This shows that a different coefficient governs for the stress related to pure bulk deformation as compared with the stress related to pure shear. There is empirical evidence to support this result, and hence justifies the inclusion of the special term for divergence in the assumed general expression (5.03-1).
The general stress from (5.03-1) when written out in full are
The coefficents 
and 
, which will be referred to
as the kinematic shear viscosity and the
kinematic bulk viscosity, respectively, have the
dimensions of length times velocity like 
. The
value of is not completely independent of
but is restricted to certain values depending on the
value of . One possible relation between and
was given by Navier and Stokes, but Eckart has
proven that the Navier-Stokes relation is only a limiting
case of a more general relation (as Lamb already implied
in one of his cautious footnotes, p. 575). The proof given
below follows that of Eckart and is based upon the second
law of thermodynamics.
For convenience, the second law is restated as
In the discussion of the previous sections of this
chapter we have considered a 
which depends upon
and a 
which depends essentially
upon 
, and consequently the term involving these
quantities in (VI) must be independently positive or at
least zero since it is independent of the other
quantities. This is also true of the term involving
the viscous stress, since by (5.03-1) no dependence on
the gradients of 
or T has been assumed.
The situation in this case, however, is complicated by the
fact that the term in itself involves six different variables.
That is, the statement
when written out in full, employing the relation (5.03-1), becomes
where the subscripts are used as an abbreviation of the
appropriate partial derivatives (i.e. 
, etc.). This relation involves
a sum of squares and will certainly be positive if both
and are positive. However, the latter
restriction is too severe, for it is possible for
to be negative and still not violate the inequality
expressed in (5.03-8). The reason for this lies in the
fact that in the first line of (5.03-8) there are four
squares but only three different variables. It is known
from the theory of quadratic functions that the above sum
of four squares can be reduced to a sum of three squares
of functions which are simple linear combinations of the
three variables. In the second line of (5.03-8) we already
have three squares involving three variables, and this can
be reduced no further. Furthermore, since the three variables
in the second line (the three components of shear deformation)
are independent of the three variables in the first line, it
follows that the second line must satisfy the inequality
independently. Therefore, we obtain as the first result
, which agrees with empirical findings.
To find the restriction on we must first reduce the
expression
to the sum of three squares. Expanding the last term and rearranging gives
This suggests the possible form
Expanding this yields
which, when compared with (5.03-10), gives five equations for the five unknown coefficients in (5.03-11). The resulting values are
In order to satisfy the required inequality 
it is
therefore necessary that the coefficients 
, 
and
each be positive or at least zero. This
requirement is met if
for it is readily shown that the terms 
and
are positive under this condition. Thus, in
summary, it has been shown that for the assumed form
for as given by (5.03-2) the coefficients
are restricted by the conditions
in order that the second law of thermodynamics be satisfied.
The limiting condition on , that
, represents the celebrated Navier-Stookes
relation, which was arrived at by a different line of
reasoning (Lamb (1945), p. 574). As for empirical support
for the more general result, the following quotation is
taken from (), p. 87: ``In the last ten
years, experiments on the absorption of sound have
suggested that may be positive, and even be greater
than , by as much as a factor of ten in some liquids.
In the case of water, the factor appears to be about +1.''
In the equation of motion (III-1) or (III-3) the viscous stress enters in the form
This represents the contribution to the time rate of change in momentum of a unit volume of fluid due to viscous stresses or, in other words, the viscous force per unit volume. Employing Eq. (5.03-1) we find that
which can also be written in the more familiar vector notation as
where 
is the Laplacian operator. Another form
can be shown to be
The first of these equations is useful in dealing with
divergenceless motion, in which case the viscous force per
unit volume reduces to 
; while for
irrotational motion the second form shows that the viscous
force is 
. Note that
from the conditions (5.03-15)
In the case of water is nearly the same as as
noted earlier, hence
which differ vastly from the limiting Navier-Stokes relations.
Eckart has pointed out a somewhat paradoxical situation which
arises in connection with Laplacian flow for which both
and 
are zero,
resulting in the condition that 
satisfies the
Laplacian equation
In this case the viscous force 
vanishes identically. However, the stresses
are not zero and lead to dissipation of
kinetic energy into internal energy since the term
as long as differs from zero. The theory of gravity
waves and ripples is based almost exclusively on the
assumption of Laplacian flow and one may well ask how
a free wave train of permanent form can exist, as is possible
for Laplacian motion, when the kinetic energy of the system
is being reduced through dissipation. Reference to
Eq. (III-4) suggests that this is possible if the work
is done on the fluid by the surface stresses. In fact, the
vanishing of implies that
under conditions of Laplacian flow. However, this contradicts the statement that the waves are free (i.e., not forced). Thus there are still some paradoxical elements that require further study. The problem is perhaps most significant in connectino with the small capillary waves or ripples which evidently are subject to considerable damping and therefore may possess significant vorticity, contrary to the usual theories. Here is an inviting problem awaiting some student to explore! It derives its importance from the fact that the upper end of the frequency spectrum of surface wave motion, e.g. ripples, has been shown by recent studies of () to be a vital connecting link in the mechanism of energy transfer from air to water. The problem of generation, maintenance and decay of such waves is therefore a significant one from the practical standpoint.