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thus the positions of the yield surfaces depend parametrically on x, the
velocities at the yield surfaces are functions of x. The contradiction
becomes clear if we use the Bingham constitutive relation (3.63) and
(3.64). Since the rigid solid body here on

on its walls. Such deposits naturally increase the thickness of the wall,
so that an increased pressure difference is required to maintain the
same volume flux through the wall. The filter must be renewed when
this pressure loss impairs engine performance

algorithms are required for purely numerical integration.) Figure 8.9
shows a typical example of such a solution. Obviously there is a length
of filter for which the pressure loss is a minimum for some given filter
geometry and mass flux: a longer filter

technically important flows which on a superficial glance have nothing
in common with lubrication theory. A typical feature of these flows is
the gradual thinning of the film flow, which creates a locally-valid film of
constant thickness. As an example of

gradient in (6.19) to the pressure distribution (8.74), and the volume
flux in (6.21) to the load-bearing capacity (8.76). In the fluid film of a
cylindrical slider with circular cross-section A = R2 we find from the
analogy with (6.53) the pressure distr

condition (6.28) and the continuity of the stress vector on the free
surface (6.29) we obtain: u(0) = 0 , (8.102) nj ji(1) = nj ji(2) . (8.103)
Fig. 8.8. Film flow over a horizontal plate. 250 8 Hydrodynamic
Lubrication Using formula (4.164) we find the n

can be locally considered as unidirectional flows. Therefore the
equations of motion are valid in the form (6.190) and (6.191), which
reduce to (8.7) and (8.9) wherever the material flows, since the material
then behaves like a generalized Newtonian fluid

clearly the first integral of (8.113). The expression V x3/g has
dimension (length)3, so that a characteristic length L for this problem
may be defined as: L = V x g !1 3 . Integrating (8.113) once again, we
get: h4 L4 = 12 * x L + c + , (8.115) which, fo

however even for a given geometry a calculation of the velocity field or
the pressure field is not justified, owing to the complex geometry of the
numerous pores. We must therefore restrict ourselves to treating mean
values over numerous pores. Thus, volu

sense) is: U = V tot F = N R4 8 p x F , (8.129) where N is the
number of holes crossing the surface F and in which (6.63) gives the
volume flux through a single hole. The ratio N/F is at the same time the
ratio of the voids in the material NR2 L to the en

inertia forces do not come into play in some other flows; thus: for
laminar rectilinear flows the inertia terms vanish on kinematic grounds
irrespective of the Reynolds number; for locally rectilinear flows inertia
terms may be neglected because the produ

y h(x) . (8.108) To calculate the film height, we use the kinematic freesurface boundary condition (4.170), which becomes in steady flow: u
F =0= v(x,y) + h (x)u(x,y) , (8.109) and on y = h(x) this reduces to: h (x)
= v(x,y) u(x,y) . (8.110) We can obtai

takes the form: kij = kij , (8.126) then it follows that: Ui = k p xi
(8.127) and so: U = k p . (8.128) It is therefore the same
relationship between mean pressure gradient and mean velocity, which
also arises in lubrication theory (8.12) and in rectiline

(8.91) and then with this to h p x = 2 1 + A 3 1/2 x L 1 2 1/2 ! for
A 0 , (8.92) where we choose the sign in (8.91) so that the load, or
the magnitude of the pressure gradient, increases as A becomes larger.
For very large A we directly 248 8 Hydrodynami

somewhat simpler problem if we lay down the so-called Reynolds
boundary condition. To do this we assume that the cavitation region
always ends at the widest part of the lubricating film ( = ), so that the
pressure build up begins there and the appropriate

the entry tube is, over a distance dx, 4V c dx; this equals the change
in volume flux in the entry tube, which is dUinc2, and it follows that
dUin dx = 4V c . (8.117) Similarly the change in mean velocity at the
corresponding position in the exit tube is

a linear function of x. The system is in fact a fourth-order boundary
value problem, with prescribed boundary conditions at x = 0 and x = L.
As the volume flux v on entry to an entry tube may be found by
dividing the total entry flux by the number of entr

= h x L 2 , (8.87) or h p x3 + 3 121hL h2 x L 1 2 ! h
p x2 4=0 , (8.88) where, because of the symmetry already
mentioned, we have restricted ourselves to the region L/2 x L. To
calculate the pressure distribution from this differential equation we
would

by experiment. If greater accuracy is required, the constant c may be
found from measured values. When the local Reynolds number is
defined by: Re = u(x, h) h , we find that Re = h (x)Re = 9 2 V 2 x h3g ,
which is independent of viscosity. 252 8 Hydrodyna

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where u and v are in ft/s and x and y are in feet. Plot the sham»
line that passes through ‘2: = O and y = 0. Compare this stream-
line with the streakline through the origin.
a = X , v ‘-” X {X '

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17‘} A 6-in.—-diameier piston is lucated within a cylinder
which is connected to a i-inmdiameter inclined-tube manometer I
a_s shown in Fig. 132.7%}. The ﬂuid in the cylinder and the ma- I
nometer is oil (speciﬁc weight '=’;JS§ lb/ft’). When i Reig'ht W '