FLUID MECHANICS  THEORY



In this section, attention is focused on
the discussion of a fluid at rest or moving in a way such
that there is no presence of shearing stresses. The variations of pressure
with direction and depth in a fluid will be examined.






Pressure at a Point 
Force Balance on a
Triangular Element


Consider a small triangular wedge of
fluid taken from an arbitrary location within a fluid mass to determine the variation of pressure with direction. With the absence of shear stresses, the only forces acting on the fluid element are
the normal and gravitational forces. Note that the forces in the x direction
are not shown for clarity. According to Newton’s
second law, the forces in the y and z directions can be summed as
ydirection:
p_{y} ΔxΔz
 p_{s} ΔxΔs
sinθ = ρ (ΔxΔyΔz/2)
a_{y}
zdirection:
p_{z} ΔxΔy
 p_{s} ΔxΔs
cosθ  ρ (ΔxΔyΔz/2)
g
= ρ (ΔxΔyΔz/2)
a_{z}
Since Δz = Δs sinθ
and Δy = Δs cosθ,
the above equations reduce to
p_{y}
 p_{s} = ρ(Δy/2)
a_{y}
p_{z}
 p_{s} = ρ (Δz/2)
(g + a_{z})
By shrinking the fluid element to a point, i.e., Δx, Δy, and Δz
approach zero, it can be seen that
p_{y} = p_{z} = p_{s}
These results are known as Pascal’s law, which states that the
pressure at a point in a static fluid is independent of direction. In
other words, pressure is a scalar for fluids.






Pressure Field 
Force Balance on a
Rectangular Element 

Next, consider a small rectangular fluid
element taken from an arbitrary location within a fluid mass. Let the pressure
at the center of the element be p. Then, use the first order Taylor series
expansion to determine the forces at the faces of the fluid element (forces
in the xdirection are not shown for clarity). For the force at the x+ face:
Similarly, the forces at other faces can be written as:
By assuming the only body force acting on the fluid element is due to
gravity (i.e., the weight of the fluid element) and the
absence of shearing forces (i.e., inviscid assumption), then applying
Newton’s
second law in the xdirection and yields
xdirection:






Similarly, according to Newton's second law, the force balances in the
y and z directions are given by
ydirection:
zdirection:
The above equations can be simplified to obtain
xdirection:
ydirection:
zdirection:
These equations can be recast into vector notation:
where is the gradient operator and is the acceleration vector of the fluid element. 





Pressure Variation in a Static Fluid

Derivation of Hydrostatic
Pressure Distribution


For a fluid at rest (i.e.,
= 0), we have
or in scalar form
Hence, it can be concluded that p varies only in the zdirection
and is not a function of x and y, giving
dp/dz = ρg
The above equation determines how pressure changes with elevation
for a fluid at rest. Integration can be performed to obtain the changes
in pressure once the variation of gravity and fluid density
with elevation is known. For most practical engineering applications, the variation of g is negligible over the change in height. This gives,

p_{1}  p_{2} = ρg (z_{2}  z_{1})






Hydrostatic Pressure Distribution
in a Static Fluid 

Thus, pressure increases with depth in a linear fashion. This type of pressure distribution is called hydrostatic. We can also rearrange the above equation to yield an expression for the pressure head as follows:
h = (p_{1}  p_{2}) / ρg
Physically, the pressure head represents the height of a homogeneous
fluid column required to produce a pressure difference of (p_{1} 
p_{2}). The term ρg
is often written as γ (specific
weight) in civil engineering applications. 


