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4 Fluid Mechanics Applied to Biosystems
4.3.3 Conservation of Mass
Conservation of volume in blood flow follows from the more fundamental conservation of mass-energy. In the realm of biochemistry, the conversion between mass and
energy is not noticeable. Mass is effectively fixed in any closed system. As water is
practically incompressible, mass conservation makes blood volume conserved.
The ‘equation of continuity’ expresses local conservation of mass within any
fluid volume. It states that any change in the mass of a fluid volume must come
from a flow of mass into or out of the volume. The rate of flow of mass through any
surface is the density of the fluid times the volume which passes that surface in a
given time. For a surface oriented in the xy plane, this will be ρv z dt dx dy/ dt. For
arbitrary orientation,
d
dt
V
ρdV = −
S
ρ v · n dA = −
S
ρ v · dS ,
(4.33)
which is the integral form of the mass conservation law. Here, dS is an infinitesimal
‘surface vector’ n dA, i.e. a vector with size equal to a small area dA on the
bounding surface and pointing outward perpendicular to that surface, that direction
denoted by the unit vector n.
Applying Gauss’ Theorem (See Appendix G.5) to the surface integral, and using
the fact that the volume is arbitrarily chosen within the fluid will give
∂ρ
∂t
+ ∇ · (ρ v) = 0 ,
(4.34)
which is the ‘local’ form of the mass conservation law.
If the mass density, ρ, is taken as constant, the fluid is assumed effectively
incompressible, and then, according to Eq. (4.34), the velocity field must satisfy
∇ · v = 0.
(4.35)
This divergence-free condition on the velocity is equivalent to the statement that
the volume of any mass of fluid does not change as the fluid moves. As we
know, incompressible fluids do not exist, because there are no absolutely rigid
bodies. However, many fluids, such as water, have a relatively large bulk modulus,
making the density of water increase by less than 0.3% over increased pressure
from atmospheric to 1000 times atmospheric going from the surface of the ocean
to its greatest depths. The density of the salt water next to a deep-sea fish is not
much different from fish in the ocean surf. For considerations of fluids consisting
mostly of water, such as blood, assuming incompressibility introduces little error.
An exception would occur if the fluid contained gas bubbles, which do get smaller
under increased pressure.
4 Fluid Mechanics Applied to Biosystems
4.3.3 Conservation of Mass
Conservation of volume in blood flow follows from the more fundamental conservation of mass-energy. In the realm of biochemistry, the conversion between mass and
energy is not noticeable. Mass is effectively fixed in any closed system. As water is
practically incompressible, mass conservation makes blood volume conserved.
The ‘equation of continuity’ expresses local conservation of mass within any
fluid volume. It states that any change in the mass of a fluid volume must come
from a flow of mass into or out of the volume. The rate of flow of mass through any
surface is the density of the fluid times the volume which passes that surface in a
given time. For a surface oriented in the xy plane, this will be ρv z dt dx dy/ dt. For
arbitrary orientation,
d
dt
V
ρdV = −
S
ρ v · n dA = −
S
ρ v · dS ,
(4.33)
which is the integral form of the mass conservation law. Here, dS is an infinitesimal
‘surface vector’ n dA, i.e. a vector with size equal to a small area dA on the
bounding surface and pointing outward perpendicular to that surface, that direction
denoted by the unit vector n.
Applying Gauss’ Theorem (See Appendix G.5) to the surface integral, and using
the fact that the volume is arbitrarily chosen within the fluid will give
∂ρ
∂t
+ ∇ · (ρ v) = 0 ,
(4.34)
which is the ‘local’ form of the mass conservation law.
If the mass density, ρ, is taken as constant, the fluid is assumed effectively
incompressible, and then, according to Eq. (4.34), the velocity field must satisfy
∇ · v = 0.
(4.35)
This divergence-free condition on the velocity is equivalent to the statement that
the volume of any mass of fluid does not change as the fluid moves. As we
know, incompressible fluids do not exist, because there are no absolutely rigid
bodies. However, many fluids, such as water, have a relatively large bulk modulus,
making the density of water increase by less than 0.3% over increased pressure
from atmospheric to 1000 times atmospheric going from the surface of the ocean
to its greatest depths. The density of the salt water next to a deep-sea fish is not
much different from fish in the ocean surf. For considerations of fluids consisting
mostly of water, such as blood, assuming incompressibility introduces little error.
An exception would occur if the fluid contained gas bubbles, which do get smaller
under increased pressure.