364 8.4  Reaction, Diffusion, and Flow

8.4.4  FLUID TRANSPORT IN BIOLOGY

Previously in this book, we discussed several examples of fluid dynamics. The core math­

ematical model for all fluid dynamics processes is embodied in the Navier–​Stokes equation.

The Navier–​Stokes equation results from Newton’s second law of motion and conservation

of mass. In a viscous fluid environment under pressure, the total force at a given position in

the fluid is the sum of all external forces, F, and the divergence of the stress tensor σ. For a

velocity of magnitude v in a fluid of density, ρ this leads to

(8.99)

ρ

σ

d

d

v

t

v

v

F

+ ⋅∇



^



^{= ∇⋅}

+

The left-​hand side of this equation is equivalent to the mass multiplied by total acceleration

in Newton’s second law. The stress tensor is given by the grad of the velocity multiplied by

the viscous drag coefficient minus the fluid pressure, which after rearrangement leads to the

traditional form of the Navier–​Stokes equation of

(8.100)

d

d

v

t

v

v

p

v

F

= −

⋅∇

(

)⋅−

∇+ ∇

+

1

2

ρ

γ

For many real biological processes involving fluid flow, the result is a complicated system

of nonlinear PDEs. Some processes can be legitimately simplified, however, in terms of the

mathematical model, for example, treating the fluid motion as an intrinsically 1D problem, in

which case the Navier–​Stokes equation can be reduced in dimensionality, but often some of

the terms in the equation can be neglected, and the problem reduced to a few linear differ­

ential equations, and sometimes even just one, which facilitates a purely analytical solution.

Many real systems, however, need to be modeled as coupled nonlinear PDEs, which makes

an analytical solution difficult or impossible to achieve. These situations are often associated

with the generation of fluid turbulence with additional random stochastic features, which are

not incorporated into the basic Navier–​Stokes equation becoming important in the emer­

gence of large-​scale pattern formation in the fluid over longer time scales. These situations

are better modeled using Monte Carlo–​based discrete computational simulations, in essence

similar to the molecular simulation approaches described earlier in this chapter but using the

Navier–​Stokes equation as the relevant equation of motion instead of simply F =​ ma.

In terms of simplifying the mathematical complexity of the problem, it is useful to

understand the physical origin and meaning of the three terms on the right-​hand side of

Equation 8.100.

The first of these is −(v·∇)·v. This represents the divergence on the velocity. For example,

if fluid flow is directed to converge through a constriction, then the overall flow velocity will

increase. Similarly, the overall flow velocity will decrease if fluid flow is divergent in the case

of a widening of a flow channel.

The second term is −∇p/​ρ. This represents the movement of diffusing molecules with

changes to fluid pressure. For example, molecules will be forced away from areas of high-​

pressure changes, specifically, the tendency to move away from areas of higher pressure. If

the local density of molecules is high, then a smaller proportion of molecules will be affected

by changes in the local pressure gradient. Similarly, at low density, there is a greater propor­

tion of the molecules that will be affected.

The third term is γ∇²v. This represents the effects of viscosity between neighboring

diffusing molecules. For example, in a high viscosity fluid (e.g., the cell membrane), there

will be a greater correlation between the motions of neighboring biomolecules than in a low

viscosity fluid (e.g., the cell cytoplasm). The fourth term F as discussed is the net sum of all

other external forces.

In addition to reducing mathematical descriptions to a 1D problem where appropriate,

further simplifications can often involve neglecting the velocity divergence and external