from the fluid. This force is usually supported by special thrust bearings.
It is desirable to keep this axial force as small as possible. For this
reason the sides of the rotor are often fully or partially acted on by the
fluid. By properly choosing thes
torque T that the vane ring exerts on the fluid, as well as the tangential
velocity components cuo and cui are to be taken as positive in an
agreed sense. The surprisingly simple Eq. (2.95) will also be found for
the axial component of the torque on a rot
here by 2R. The force acting inwards is the centripetal force which is
exactly the same size as the centrifugal force experienced by the
rotating observer. In this example the reference frame of the observer
at rest, that is the earth, can be taken as an
continuum mechanical analogue to the first law of classical
thermodynamics. In the first law, de = w + q, (2.120) de is the change
in the internal energy in the time dt, w is the work done in this time,
and q is the gain of heat in this time (each per uni
whose single members have arisen from monomers and still show a
similar structure. Silicon oil (polydimethylsiloxane), for example,
consists of chain molecules of the form Si CH3 CH3 O Si CH3 CH3 O Si
CH3 CH3 O , which arise, through polymerization, from
particle can be calculated from the values of s and v, irrespective of
where the particle is or what its motion is. This assumption is
equivalent to the assumption that Gibbs relation is valid for irreversible
processes. For from the material derivative o
system, so that we finally reach the equation Dw Dt A = k + T a +
2 w + ( x) + d dt x ! . (2.68) (Note here that (2.68) is a
vector equation where k and T have meanings independent of frame,
i. e. they are the same arrows in all frames. Where written as a
viscosity, which is almost proportional to the molecular weight of the
fluid. The aligned molecules try to retangle them- 3 Constitutive
Relations for Fluids 83 selves and if this is hindered additional normal
stresses arise. In the extrusion process the
spinning top, the earth is a good enough inertial reference frame. But in
this system the tensor of the moments of inertia is time dependent, so
it is better to choose a reference frame attached to the top, where,
even though this is an accelerating refer
v ds + e v s dv (2.149) with Gibbs relation (2.132), we read off T = e
s v , (2.150) 2.8 Thermodynamic Equations of State 73 and p = e v
s . (2.151) The right-hand sides of (2.150) and (2.151) are functions of s
and v. If we consider both relationships to
(2.141) are sufficient for this. 2.8 Thermodynamic Equations of State
The principles we have discussed so far in Chap. 2 form the basis of
continuum mechanics. These principles represent a summary of our
experience of the behavior common to all bodies. Al
that a thermodynamic state is uniquely defined by a certain number of
independent variables 72 2 Fundamental Laws of Continuum Mechanics
of state. For the single component material to which we shall restrict
ourselves, two independent variables of state a
sea, and it is even larger in atmospheric flows. The earth rotates about
2 in one sidereal day (which with 86164 s is somewhat shorter than a
solar day of 86400 s), so it moves with an angular velocity of =
2/86164 7.29 105s1. Since the angular velocity i
stresses on the material particle at time t), d) hold locally (thus, for
example, the stress at a material particle 3 Constitutive Relations for
Fluids 77 depends only on the motion of material particles in its
immediate neighborhood), e) reflect the symm
q = 1 qi xi , (2.127) where each are per unit time and mass. The
work per unit time and mass can be split up into the reversible work as
in (2.124), and the irreversible work. The latter contribution is
irreversibly changed into heat by the action of the
Dt I = - Db Dt A + b . (2.63) If b = the changes in the inertial
reference frame and in the frame moving relative to it are equal: - D
Dt I = - D Dt A = ddt . (2.64) This obviously holds only for the
angular velocity or for vectors which are always parall
which idealize the behavior of the actual material but which in more
general circumstances do describe it as accurately as possible. Let us
adopt the viewpoint of an engineer who forecasts the flow of a given
fluid from the balance laws on the basis of th
Yet the opposite is actually the case. Many technically important, real
flows are described very well using this assumption. This has already
been stressed in connection with the flow through turbomachines.
Indeed the flow past a flying object can often b
particle, which indeed represent local rigid body motion. If the stress on
the material particle only depends on the instantaneous value of the
rate of deformation tensor, as is the case, for example, in CauchyPoissons law, then an observer-fixed referenc
flows. They show normal stress effects, of which the best known is the
Weissenberg effect. Contrary to what a Newtonian fluid does, some
non-Newtonian fluids climb up a rotating rod which is inserted
perpendicular to the free surface. This effect, which o
relative system be steady. Neglecting the moment of the volume forces,
the integral form of the balance of angular momentum then becomes
(S) (x c )(w n) dS + (V ) (x c) dV = (S) x t dS . (2.76) The middle
term contains a volume integral, but it is zero if
dS = 0 , (2.83) or, using the concept of mass flux , to m = (Ao) c n dS =
(Ai) c n dS . (2.84) The notation m used in the literature is not very
well chosen: it has nothing to do with the rate of change of the mass,
which is of course zero, but with the
given by c1i and by +c1o at the exit surface. At the blade itself, cn
vanishes. Since there is no flow through the blade, the normal
component of the velocity is in any case zero. By assumption, the flow
at the inlet and outlet surfaces is homogeneous, an
control volume is chosen as shown in Fig. 2.9: it starts at the outlet
surface Ao, goes along the side of a narrow slit to a vane, and around
the other side 2.5 Applications to Turbomachines 61 Fig. 2.9. Radial
machine with control volume in the guiding c
Turbomachines 63 T is the torque exerted on the fluid by the rotor; T is
the torque exerted on the rotor by the fluid. The inlet and outlet
surfaces are surfaces of rotation (Fig. 2.10), so that the vector x n is
perpendicular to e, and the pressure integ
e12 . (3.21) The tensorial generalization of this equation is the
constitutive relation of the linear viscoelastic fluid: Pij + 0 Pij t = 2
eij . (3.22) We can call the characteristic time 0 the memory span of
the fluid. As 0 0, we obtain from (3.22) the
a single cascade at rest Fig. 2.8. The equations which follow also hold
for a rotating cascade in an axial turbomachine, since Fig. 2.8. Control
volume for applying the momentum balance 58 2 Fundamental Laws of
Continuum Mechanics by the earlier arguments
same flow is found in all sections parallel to the plane of Fig. 2.6. In
reality the flow passages between blades become wider in the radial
direction, so that the assumption of plane cascade flow represents the
limit r with constant blade height. For the
frame at the position x, to give: c = w+ x + v . (2.58) Fig. 2.3. Moving
reference frame 2.4 Momentum and Angular Momentum in an
Accelerating Frame 49 From (2.55) to (2.58) we get the basic formula
for the rate of change of the vector x in the two referen
velocity is dependent on the reference frame, as is the change of any
vector (with an exception, as we shall see). 48 2 Fundamental Laws of
Continuum Mechanics First we shall turn towards the differential form
of the balances of momentum and angular momen