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(a)
δ y
p
x
x
p
δ x
p
p
2
2
x
p
δ z
x
δ x
(b)
∂τ x
δ
z
x
2
z
δ y
δ z
x
x
z
x
x
x
2
z
Figure 3.3 (a) Pressure and (b) shear forces acting on a cuboid in the x direction.
the net force on a particle of fluid, we need to know the gradients of pressure within
the fluid. Figure 3.3a shows that the forces acting on a small cuboid of fluid in the
x direction are p LH dydz =(p
dx(
@
p/
@
x)/2) dydz on the left-hand face and p RH
dydz
x)/2) dydz on the right-hand face. These forces act in opposite
directions, and if there is no gradient of pressure (p LH ¼
¼
(p
þ
dx(
@
p/
@
p RH ), there will be no pressure
force in the x direction on the cuboid. In the presence of a gradient of pressure
(p LH
p RH ), the forces on the two faces will differ by:
¼ @
p
@
p
ð
p LH
p RH Þ
y
z
x
x
y
z
x per unit volume
:
ð
3
:
11
Þ
@
@
Similarly, the forces in y and z directions are determined by the y and z pressure
gradients.
The effect of the frictional stresses on a fluid particle can be determined in a similar
way. Consider again a small cuboid of fluid as shown in Fig. 3.3b . Think of this
cuboid as situated in a horizontal flow in the x direction in which the velocity u(z)
varies with height (i.e. the flow is sheared in the z direction). A frictional stress
between layers of fluid arises because they are moving relative to each other; a faster
layer of water will tend to drag an adjacent slower layer along with it, transferring
some of its momentum to the lower layer. In a laminar flow, the stress will be due
simply to the molecular viscosity of the fluid. However, in the ocean, where the flow is
often turbulent, much larger turbulent stresses are involved. For the moment we shall
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