Flow, deformation, and transport - Physical Processes in Earth and Environmental Sciences - page 111

frictional forces and (2) the free stream fluid outside the
boundary layer where viscous forces are negligible.
The audience of applied mathematicians who listened to the
young Prandtl give his 10 min paper in Heidelburg in 1904

witnessed the birth of a concept that was to revolutionize
the study of fluid mechanics. The first major challenge was
to apply the boundary layer concept to the study of natural
and experimental turbulent flows (Figs 4.12 and 4.13).

4.4 Laminar flow

Laminar flow is rare in the atmosphere and hydrosphere,
but the unseen laminar flow of mantle and lower crust
turns over a far greater annual discharge of material, not to
mention subsurface movement of molten magma and sur-
face flows of high viscosity substances, like mud, debris,
and lava. As we have seen, laminar flow is characterized by
individual particles of fluid following paths which are par-
allel and do not cross the paths of neighboring particles;
therefore, no mixing occurs. All forces set up within and at
the boundaries to flow are due to molecular viscosity and
three-dimensional (3D) flow patterns essentially conform
to the shape of the vessel through which the fluid happens
to be passing. In a wide channel, for example, the flow may
be imagined to comprise a multitude of parallel laminae
while in a pipe-like conduit the fluid layers comprise a
series of coaxial tubes (Fig. 4.14). In all cases the Reynolds
number is small and thus viscous forces predominate over
inertia forces and prevent 3D particle mixing. In a steady
laminar flow any instantaneous measurement of velocity at
a point will be exactly the same at that point every time.

concentrate on the behavior of small fluid volumes
(“particles”). Molecular diffusion would surely destroy the
color bands in the photo opposite line of bubbles in Fig.
4.11, given enough time, but we are dealing with bulk flow
velocities that are rapid compared to the diffusion time of the
fluid molecules. This approach to fluid flow is termed the
continuum approach . However, random molecular move-
ments still occur and are influenced by the fluid velocity: wit-
ness the interrelationships between flow velocity and fluid
pressure inherent in Bernoulli's equation (Section 3.12).

4.4.2 Viscous shear across a boundary layer

For each and every moving layer in a laminar flow there
must exist a shear stress due to the displacement of one
layer over its neighbors. Newton's relation for this in the
case of flow over a plane solid boundary orientated in the
zx plane (Sections 3.9 and 3.10) is

zx u / z . This says

that “the coefficient of molecular viscosity, , induces a

shear stress, zx (subsequently referred to as the shear

stress, ), in any fluid substance when the substance is

placed in a gradient of velocity.” We must emphasize how-
ever that this relation is only true when momentum is
transported by molecular transfer.
How does the shear stress vary across the laminar flow
boundary layer? Since viscosity is constant for the
experimental conditions, the viscous shear stress depends
only on the gradient of velocity which, as we have seen
(Section 3.10), decreases away from the solid flow bound-
ary. So, the stress must also die away across the boundary
layer in direct proportion to the velocity gradient. The
greatest value of stress,

4.4.1 The continuum approach to fluid flow

But, steady on, you might exclaim! What is all this talk of
nonmixing and “particles”? Surely, fluid molecules are the
only “particles” present in a laminar flow, or any other flow
come to that, and these are whizzing around randomly all
the while. These cogent points are why we continue to dis-
regard molecular scale processes until we return to the sub-
ject of heat conduction. When dealing with bulk flow we
must treat the fluid as if the molecules did not exist and

0 , will occur at the solid boundary
itself (Fig. 4.15). In the free stream, where the velocity
gradient has disappeared, or at least greatly diminished,
there is no or little viscous stress. In such areas of flow,
remote from boundary layers, the flow is said to be invis-
cid or ideal (Section 2.4) and the property of viscosity can
be neglected entirely.
In the area of the boundary layer between the solid
boundary and the free stream the simplest assumption
concerning the falloff of stress with distance (Fig. 4.15)
would be to assume that it changes at a constant rate

(a) (b)

Fig. 4.14 3D laminar flow: (a) Couette flow: shearing laminae in a
wide channel, (b) Poiseuille flow: shearing concentric cylinders in a
pipe or conduit.