Basics of Fluid Mechanics, 2014a

70 CHAPTER 4. FLUIDS STATICS
direction of the pressure derivative and dl is the infinitesimal length. This mathematical
statement simply requires that the pressure can deviate in such a way that the average
on infinitesimal area can be found and expressed as only one direction. The net pressure
force on the faces in the x direction results in
dF = −
( ∂P
∂x
)
dydx î (4.2)
In the same fashion, the calculations of the three directions result in the total net
pressure force as
∑
( ∂P
F = −
∂x î + ∂P
∂y ĵ + ∂P )
∂y ˆk
(4.3)
surface
The term in the parentheses in equation (4.3) referred to in the literature as
the pressure gradient (see for more explanation in the Mathematics Appendix). This
mathematical operation has a geometrical interpretation. If the pressure, P , was a
two–dimensional height (that is only a function of x and y) then the gradient is the
steepest ascent of the height (to the valley). The second point is that the gradient is a
vector (that is, it has a direction). Even though, the pressure is treated, now, as a scalar
function (there no reference to the shear stress in part of the pressure) the gradient is
a vector. For example, the dot product of the following is
î · gradP = î ·∇P = ∂P
(4.4)
∂x
In general, if the coordinates were to “rotate/transform” to a new system which
has a different orientation, the dot product results in
i n · gradP = i n ·∇P = ∂P
(4.5)
∂n
where i n is the unit vector in the n direction and ∂/∂n is a derivative in that direction.
As before, the effective gravity force is utilized in case where the gravity is the only
body force and in an accelerated system. The body (element) is in rest and therefore
the net force is zero
∑
F = ∑
F (4.6)
total
surface
F + ∑ body
Hence, the utilizing the above derivations one can obtain
−gradPdxdydz + ρg eff dx dy dz =0 (4.7)
or
Pressure Gradient
gradP = ∇P = ρg eff
(4.8)