In Silico Biofluid Mechanics - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 335

As we know, the Reynolds number can be defined as
Re 5 ρvL
μ

ð 13 : 1 Þ

is the density of the fluid, v is the mean fluid velocity, L is the diameter of the ves-
sel (or some characteristic length to describe the vessel), and

ρ

where

is the viscosity of the fluid.
The equation for the Reynolds number can be rearranged to the following form:

μ

Re 5 ρ v 2
μU

inertial forces
viscous forces

L 5

ð 13 : 2 Þ

v

Indeed, the Reynolds number reveals an important relationship between the inertial
forces and the viscous forces applied to the fluid elements. For slow flows or flows with
small Reynolds numbers, the viscous terms dominate; therefore, we may ignore the iner-
tial terms and state that the viscous forces are more important than fluid momentum
(flows within the microcirculation may use this assumption). For faster flows or flows
with higher Reynolds numbers, we may potentially ignore the viscous resistance,
because momentum dominates the flow conditions. By using the Bernoulli equations
under these conditions, we may obtain an approximate solution to these types of flows
(see Section 3.8). However, for flows where the viscous terms have a similar weight to
the inertial terms, we cannot ignore either of the terms in the governing fluid dynamics
equations, and it typically becomes very difficult to solve the Navier-Stokes equations by
hand. For example, when the fluid is relatively compressible (e.g., gases), the inertial
terms and the viscous terms have a similar weight; therefore, not only do the Navier-
Stokes equations need to be solved, but the energy and thermodynamic relationships are
needed as well. However, most of the fluids we discussed in this textbook (e.g., blood
and interstitial fluid, among others) are usually considered incompressible within the
range of applied pressures.
When it becomes very difficult to solve partial differential equations analytically, the
Finite Element Method (FEM) or Finite Difference Method (FDM) is often used. FEM is
most often used to solve solid mechanics problems, due to its advantage in handling com-
plex geometries. FDM and its related Finite Volume Method (FVM) are often used to solve
fluid mechanics problems. By using FDM, differential equations can be solved numerically
by using a finite value for

Δx approaches 0. The mathematical equa-
tions behind this can be developed from the Taylor series expansion for multiple indepen-
dent variables. This method is derived by gridding (or meshing) the volume of interest with
equal spacing in each direction, although in general the grid does not need to be equally
spaced. These variable meshes are used to put more calculation points in the region of inter-
est and less calculation points in other, not as critical, areas. In two dimensions, the rectan-
gular grid in the x/y directions corresponds to i and j steps, respectively ( Figure 13.1 ). The
spacing between the mesh are denoted as

Δx rather than having

Δy , in the i and j directions, respectively.
Therefore, the location of subsequent points within the mesh can be identified by

Δx and

x n 5x 0 1nΔx

ð 13 : 3 Þ

y n 5y 0 1nΔy