Flow, deformation, and transport - Physical Processes in Earth and Environmental Sciences - page 112

(Fig. 4.16; see derivation in Cookie 10). A laminar,
non-Newtonian fluid flow has a characteristic plug-like
profile in the middle of the flow (Fig. 4.16) where there
is virtually no velocity gradient and hence no internal
shear.
What does all this mean in practice? Any laminar flow
will exert greatest stress, and therefore greatest strain,
across the boundary layer. At some point in the flow

throughout the boundary layer thickness, . A simple

expression for this is given by 0 (1 z / ); as z goes to

at the outer edge of the boundary layer, vanishes. As z

goes to zero at the solid boundary, 0 .

In fact, the linear assumption is untrue in detail for
laminar flows, though curiously enough it is true for the
very thin innermost layer of turbulent flow (Section 4.5).
Making use of the Navier-Stokes expressions for viscous
force balance, Reynolds originally deduced that the pro-
file of velocity across the laminar boundary layer for a
Newtonian fluid actually has the shape of a parabola

goes to

(a) Definitions for derivation of velocity profiles

y = + a

a = Half width

y

p 1

p 2

x u y = 0

m = Viscosity

u max

y = - a

t = 0

d , thickness of
boundary layer

(b) Computed velocity profiles

Non-Newtonian
fluid

t = t max (1- z / d )

u max

t max at t 0

Newtonian
fluid

u , Flow velocity t , Viscous shear stress

Fig. 4.16 Laminar flow boundary layers between parallel walls
(Coutte flow) for Newtonian and non-Newtonian fluids.

Fig. 4.15 Velocity and shear stress distribution (to first order only) in
laminar flow.

Fig. 4.17 A natural boundary layer “frozen in time.” This Namibian dyke intrusion ( Coward's dyke ) was once molten silicate magma flowing as
a viscous fluid along a crack-like conduit opening up in cool brittle “country-rock.” Gas exsolved from solution to form bubbles. These were
deformed (strained) into ellipses (see Section 3.14 on strain) by shear in the boundary layer until the whole flow solidified as heat was lost
outward by conduction. Note the greatest strains (more elongate ellipses) occurred at the dyke margin boundary layers where velocity
gradients were highest. The generalized shape of the whole flow boundary layer is shown by the dashed white line; it approximates to the
non-Newtonian case of Fig. 4.16.