Mechanics of Fluids

Integrating this between θ = 0 and θ = π, that is, from h = c(1 + ε)
to h = c(1 − ε), gives a total volume flow rate �Rεcz. Consequently, at
one end of the bearing, where z = L/2, the lubricant escapes at the rate
�RεcL/2; a similar amount escapes at the other end, and thus the total
volumetric supply rate is �RεcL.
In conclusion we may remark that the hydrodynamics of lubrication is
not, in any event, the whole story. Two oils may have the same viscosity yet
not be equally effective as lubricants in a particular application. There are
other characteristics – mainly in the field of physical chemistry – that are of
great importance. Also, conditions of low speed or high load may reduce the
thickness of the lubricant film to only a few molecules and it may then take
on properties different from those of the normal fluid. And, the dissipation
of energy as heat may not only result in non-uniform viscosity of the fluid,
but may also cause distortion of the journal and the bush. For these further
aspects of the subject, more specialist works should be consulted.
6.8 LAMINAR FLOW THROUGH POROUS MEDIA
There are many important instances of the flow of fluids through porous
materials. For example, the movement of water, oil and natural gas through
the ground, seepage underneath dams and large buildings, flow through
packed towers in some chemical processes, filtration – all these depend on this
type of flow. The velocity is usually so small and the flow passages so narrow
that the flow may be assumed to be laminar without hesitation. Rigorous
analysis of the flow is not possible because the shape of the individual flow
passages is so varied and so complex. Several approximate theories have,
however, been formulated, and we shall briefly examine the principal ones.
In 1856 the French engineer Henri Darcy (1803–58) published experimental
results from which he deduced what is now known as Darcy’s
Law:
u =−C ∂p∗
(6.67)
∂x
In this expression x refers to the average direction of flow, u and ∂p∗ /∂x
represents respectively the steady mean velocity and the rate of increase of
piezometric pressure in this direction, and C is a constant at a given temperature
for a particular fluid (free from suspended solid particles) and for
the piece of porous medium concerned. To calculate the mean velocity we
consider a cross-sectional area �A (perpendicular to the x direction) through
which a volumetric flow rate of fluid is �Q. Then, if �A is large enough
to contain several flow passages, u = �Q/�A. The minus sign appears in
eqn 6.67 because p∗ decreases in the mean flow direction, that is, ∂p∗ /∂x is
negative. The direct proportionality between the mean velocity and ∂p∗ /∂x is
characteristic of steady laminar flow. Further experiments on porous media
have shown that, for a Newtonian fluid, the mean velocity is inversely proportional
to the dynamic viscosity µ, and this result too is characteristic of
steady laminar flow.
Laminar flow through porous media 239