important cases: The first case is the application of Bernoullis equation
to stream filament theory (see discussion in connection with Fig. 1.7). In
this theory, the representative streamline is fixed by the shape of the
streamtube which does not change i

form (3.22) only holds in systems which rotate and translate with the
particle, where the translation is taken 86 3 Constitutive Relations for
Fluids into account if the partial derivative in (3.22) is the material
derivative. It would appear obvious at f

u) ( u) , (4.10) which is easily verified in index notation, and
which reduces the application of the Laplace operator to operations
with even in curvilinear coordinates. Because u = 0, we then have
u = 2 . (4.11) This equation makes it clear that in
inc

expression Pij t = DPij Dt + Pmj um xi + Pim um xj . (3.30) This is
also found when the objective symmetric tensor Pmj emi + Pimemj is
added to the right-hand side of Eq. (3.29). Then, besides the spin tensor,
the rate of deformation tensor also appears.

(4.42) and the component form of Eulers equation in natural
coordinates, with u = |u|, becomes u t + u u = k 1 p ,
(4.43) u2 R = kn 1 p n (4.44) 0 = kb 1 p b . (4.45) As already
noted, ignoring the viscosity is physically akin to ignoring the heat
conduct

situation it is therefore necessary to check carefully whether a flow
calculated under the assumption of zero viscosity is actually realized. On
the other hand, the discussion here has shown that the assumption of
inviscid flow often allows a realistic de

ij if 1 2 ij ij 2, (3.61) where = 1 + /(2e ije ij ) 1/2 . (3.62) The
incompressible Bingham material is determined by the three material
constants G, and 1. Wherever it flows it behaves as a fluid with
variable viscosity , which depends on the second inva

irrotational fluids. As we know, equations which express physical
relationships and which are dimensionally homogeneous (only these
are of interest in engineering) must be reducible to relations between
dimensionless quantities. Using the typical velocity

u = D/Dt we can shorten this to D Dt = u + . (4.16) This
equation takes the place of the Navier-Stokes equation, and is often
used as a starting point for, in particular, numerical calculations.
Because 2 = curl u, (4.16) represents a differential equatio

in viscous flow this can occur only through diffusion of the angular
velocity from the wall. The order of magnitude of the typical time
for the diffusion of the angular velocity from the surface of the plate to
a point at distance (x1) can be estimated f

steady flow this even gives rise to a pure algebraic relationship between
the velocity, the potential of the mass body force and the pressure
function (in incompressible flow, the pressure). In order to apply
Bernoullis equation in potential theory, the s

also in two phase flows if one phase is gaseous, and the other liquid or
solid, but also if small solid particles are suspended in liquid. Flows
through porous media, for example ground water flows, also fall into
this category.) From (4.34), Re 0 charact

p + u . (4.9b) Often the density distribution is homogeneous when
the incompressible fluid is set in motion. Because D/Dt = 0, this
homogeneity remains for all time, so that the condition incompressible
flow can be replaced by the condition constant densi

is nonsingular since the Jacobian J = det(xi/j ) is not equal to zero, a
fact which was used in Sect. 1.2 and in the discussion of the Bingham
material in Chap. 3. The material derivative of (4.19) leads to the
relation Di Dt = Dcj Dt xi j + cj D Dt xi j

similar manner, and the kinematic viscosity or the velocity U are
simultaneously changed so 102 4 Equations of Motion for Particular
Fluids that the Reynolds number stays the same. As long as the
Reynolds number remains constant, nothing changes in the
ma

lm; this is also known as Almansis strain tensor . The symmetric tensor
(k/xl)(k/xm) is Cauchys deformation tensor, and it is the Eulerian
counterpart to Greens deformation tensor. We also express the
deformation tensors using the displacement vector y =

thermal equation of state p = p(, T ) and the caloric equation of state e =
e(, T ) appear also. This set of equations forms the starting point for the
calculation of frictional compressible flow. By (4.1) the Navier-Stokes
equations are given in Cartesia

together with the initial condition, but (4.27) shows us clearly that the
deformation gradient xi/j also must remain finite. A flow which
develops discontinuities is in general no longer irrotational. 4.1.3 Effect
of Reynolds Number In viscous flow, the t

solutions, and will be treated further in Chap. 6. If we denote the flow
direction with the unit vector e1, the direction of velocity change with
e2 and the direction orthogonal to these by e3, the first and second
Rivlin-Ericksen tensors take on the form

, (3.32) can be written in the form T = p I + A(1) , (3.39a) or ij = p ij
+ A(1)ij . (3.39b) Since A(1)ij = 2eij, we recognize the Cauchy-Poisson
law (3.1) for incompressible Newtonian fluids, which we have reached
here for the limiting case of very slow

the components in the rotating system, and to represent this derivative
in quantities and components of the frame fixed in space, since the
other tensors are already in the fixed frame. We reach the required
formula for the derivative if, starting with th

experienced at time t . Consider the fluid motion x = x(, t) and the
position of the material point at time t < t, i.e. x = x(, t ). If we replace
here by = (x, t) to obtain x = x(x, t, t ) we are actually using the
current configuration as the reference

ikjmdx kdx m = A(n)ijdx idx j , hence the above equivalence.) If we
truncate the series at the nth term (either because the higher
RivlinEricksen tensors become very small, as according to (1.68) is the
case if the change of the material line element occu

motions of small amplitude very well. Both of the models discussed are
examples from the many non-Newtonian fluid models, which are, as a
rule, all of empirical nature. On the basis of a simple fluid, a number of
these constitutive relations can be system

, (3.32b) furnishes the definition of a second order fluid: T = p I +
A(1) + A2 (1) + A(2) . (3.40) The coefficients , and here are
material dependent constants (where, from measurements, turns out
to be negative and should not be confused with the shear

lm Dt = 1 2 ul xm + um xl = elm . (3.55) In rheology it is usual to
denote the negative mean normal stress as the pressure, and we shall
follow this usage here, noting however that the mean normal stress in
general includes isotropic terms which are depen

aikajl DP kl Dt = DPij Dt + Pmjmi + Pimmj . (3.28) 3 Constitutive
Relations for Fluids 87 The right-hand side of (3.28) already is the
required rate of change of the tensor P kl in the system rotating with
the material particle, given in components of the

increasingly interesting. Even the behavior of grease used as a means of
lubricating ball bearings, can be described with the Bingham model. We
can gain considerable insight into the behavior of Bingham materials
behavior looking at the simple shearing fl

deformation gradient. Thus we write for the square of the element of
length |dx| dxidxi = xi j xi k djdk (3.44) as well as for the
difference |dx| 2 |d| 2 = xi j xi k jkdjdk (3.45) and we
shall denote the half of the expression in parantheses as Lagrangia

(4.9b). For reasons of clarity, we shall use symbolic notation here. We
assume further that k has a potential (k = ), and use the identity
(4.11) in Eq. (4.9b). In addition, we make use of (1.78) to obtain the
Navier-Stokes equations in the form 1 2 u t u

with actual material behavior is in any case to be checked
experimentally (as is also done with the Cauchy-Poisson law). The
materials mentioned until now have been pure fluids, that is materials
where the shearing forces vanish when the rate of deformati