Mechanics of Fluids

Laminar flow between
solid boundaries 6
6.1 INTRODUCTION
In laminar flow, individual particles of fluid follow paths that do not cross
those of neighbouring particles. There is nevertheless a velocity gradient
across the flow, and so laminar flow is not normally found except in the
neighbourhood of a solid boundary, the retarding effect of which causes the
transverse velocity gradient. Laminar flow occurs at velocities low enough
for forces due to viscosity to predominate over inertia forces, and thus, if any
individual particle attempts to stray from its prescribed path, viscosity firmly
restrains it, and the orderly procession of fluid particles continues.
We recall from Section 1.6.1 that viscous stresses are set up whenever
there is relative movement between adjacent particles of fluid and that these
stresses tend to eliminate the relative movement. The basic law of viscous
resistance was described by Newton in 1687:
τ = µ ∂u
(6.1)
∂y
Here ∂u/∂y is the rate at which the velocity u (in straight and parallel flow)
increases with coordinate y perpendicular to the velocity, µ represents the
dynamic viscosity and τ the resulting shear stress on a surface perpendicular
to, and facing the direction of increase of y. The partial derivative ∂u/∂y is
used because u may vary not only with y but also in other directions.
We now consider a number of cases of laminar flow which are of particular
interest.
6.2 STEADY LAMINAR FLOW IN CIRCULAR PIPES:
THE HAGEN–POISEUILLE LAW
The law governing laminar flow in circular pipes was one of the first examples
to be studied. About 1840, experimental investigations of flow in straight
pipes of circular cross-section were carried out independently by two men.
The first results, published in 1839, were the work of the German engineer
G. H. L. Hagen (1797–1884). He had experimented with the flow of water
through small brass tubes, and his figures showed that the loss of head
experienced by the water as it flowed through a given length of the tube