Biofluid Flow - Principles of Biomedical Engineering - page 114

4.2.3 Conservation of Momentum
According to the conservation of momentum, the total momentum of a system
or an object is a constant (i.e., momentum is neither created nor destroyed). For
example, if a moving object collides with a stationary object, the total momentum
before the collision, plus any additional impulse from outside the system, will equal
the total momentum after the collision. Momentum is changed through the action
of forces based on Newton's second law of acceleration. In general, the change
in momentum must equal the sum of the forces acting on the mass. Consider an
incompressible steady laminar flow in the z -direction in a long rigid tube (Figure
4.2) of radius r . In this case, using cylindrical polar coordinates ( r ,

, z ) is simpler
than using Cartesian coordinates. Assume that the flow is a completely developed
laminar flow (i.e., velocity profile shows a parabolic shape and the entry effects are
stabilized). The forces on the fluid consist of the pressure force, the gravitational
force, and the viscous force. If the tube is horizontally placed, the gravitational
forces can be neglected. However, if the tube is oriented vertically or at an angle,
then gravitation forces (=

φ

gh ) need to be considered and the tube must be balanced
by the hydrostatic pressure. For a horizontal tube, the difference in the pressure
forces must be balanced by the difference in the viscous forces. Within the flowing
fluid, consider a differential element of length

ρ

Δ z in the direction of the flow and

radial thickness r at r radius in the flow system. The pressure force, given as the

product of the area and the pressure acting on it, is

Δ

Pressure force =Δ+Δ−Δ 2 π rrP ( P ) 2 π rrP

r exerts a shear stress on the fluid in the shell in the positive z -
direction, which is designated as

The fluid at r + Δ

τ rz + Δτ rz . This notation means “shear stress evalu-

ated at the location r r .” Multiplying the surface area of the differential element

at that location yields a force 2

+ Δ

r ) in the z -direction. In the same
way, the fluid at the location r exerts a stress t rz ( r ) on the fluid below (i.e., at a
smaller r ). As a result, the fluid below exerts a reaction on the shell fluid that has
the opposite sign. The resulting force is

π ( r + Δ r ) Δ z τ rz ( r + Δ

τ rz . Therefore, the total force arising
from the shear stresses on the two cylindrical surfaces of the shell fluid is written as

− 2 π r Δ z

Viscous forces = 2 ( π r + Δ+ Δ+ ΔΔ− r )( τ τ )( r r ) z r 2 π τ Δ z

rz rz rz

Using the conservation of momentum principles gives

Figure 4.2 Forces in fl uid fl owing in a tube.