Conservation Laws - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 48

Y

dy

∂ p dx
∂ x 2

→

∂ p dx
∂ x 2

→

p +

dydz ( − i )

p − dydz ( i )

p

dz

dx

X

Z

FIGURE 3.1

X-direction pressure forces that act on a differential fluid element. The same pressure forces can
be derived for the other Cartesian directions.

Because the fluid is under static flow conditions, there are no shear forces applied to
the fluid element. Therefore, the only surface force acting on the element is the pressure
force. Pressure varies with position throughout the entire fluid. The total pressure that acts
on the differential element is equal to the summation of the pressure acting on each face
of the differential element. Let us define the pressure at the center of the differential ele-
ment to be p ( Figure 3.1 ). The pressure at each face would be equal to p plus or minus the
particular directional pressure gradient multiplied by the distance between the center of
the element and the face. For instance, in the x-direction, the pressure on the right face in
the current orientation, shown in Figure 3.1 , would be

1 @

p dx

2

p

@ x

whereas the pressure on the left face would be

dx
2
Remember that pressure has the same unit as stress. In order to determine the force
that the pressure exerts on each face, one must multiply the stress by the area over which
it works (the two forces that act in the x-direction are shown in Figure 3.1 ). In this figure,
the pressure force is also multiplied by a unit vector indicating the direction that the force
acts within. Remember, for pressure there is a sign convention; a positive pressure is a
compressive normal stress, and these are the forces that are shown on the differential ele-
ment in Figure 3.1 . To balance forces, the differential element would produce an equal and
opposite force on the adjacent fluid element.

2 @

p

p

@ x