Mechanics of Fluids

200 Laminar flow between solid boundaries
Fig. 6.6
magnitude τ on the lower face and τ + δτ, say, on the upper face in the
directions shown in Fig. 6.6. (The indicated directions of the stresses follow
the convention mentioned in Section 1.6.1 by which a stress in the x direction
acts on the surface facing the direction of increase of y.) Let the piezometric
pressure be p ∗ at the left-hand end face, and p ∗ + δp ∗ at the right-hand
end face.
Then, if the width of the element in the direction perpendicular to the page
is δz, the total force acting on the element towards the right is
{p ∗ − (p ∗ + δp ∗ )}δyδz +{(τ + δτ) − τ}δxδz
But for steady, fully developed flow there is no acceleration and so this total
force must be zero.
∴ −δp ∗ δy + δτδx = 0
Dividing by δxδy and proceeding to the limit δy → 0, we get
δp ∗
δx
= ∂τ
∂y
(6.16)
For laminar flow of a Newtonian fluid the stress τ = µ∂u/∂y. Hence eqn 6.16
becomes
δp∗ �
∂
= µ
δx ∂y
∂u
�
∂y
(6.17)
As p∗ nowhere varies in the y direction, δp∗ /δx is independent of y and
eqn 6.17 may be integrated with respect to y:
δp∗ y = µ∂u + A
δx ∂y
If µ is constant, a further integration with respect to y gives
�
δp∗ �
y2 = µu + Ay + B (6.18)
δx 2
Since the portion of fluid studied is very far from the edges of the planes,
A and B are constants, independent of both x and z.
To determine these constants the boundary conditions must be considered.
If both the planes are stationary the velocity of the fluid in contact with each