Biomedical Engineering Reference
In-Depth Information
estimation is made for this that is updated in case the constraints are violated. In
this way an iterative solution can eventually be determined.
13.3
Solution strategies for viscous fluids
Consider a fixed volume
V
in three-dimensional space, through which (or within
which) a certain amount of material flows, while this material can be considered
as an incompressible viscous fluid (see Section
12.5
). Because, for such a fluid a
(possibly defined) reference state is not of interest at all, an Eulerian description
is used for relevant fields within the volume
V
.
The velocity field has to fulfil the incompressibility constraint at each point in
time. The current velocity field fully determines the deviatoric part of the stress
state (via the constitutive modelling). The hydrostatic part of the stress field cannot
be determined on the basis of the velocity field. The stress field (the combination
of the hydrostatic and deviatoric part) has to satisfy the momentum balance equa-
tion (see Section
11.3
). The problem definition is completed by means of initial
conditions and boundary conditions. The initial velocity field has to be described
in consistency with the incompressibility constraint and along the boundary of
V
for every point in time velocities and/or stresses have to be in agreement with real-
ity. It should be emphasized explicitly, that considering a fixed volume in space
implies a serious limitation for the prospects to apply the theory.
The goal of the present section is to outline a routine to formally determine the
velocity field and the (hydrostatic) stress field, both as a function of time, such
that all the above mentioned equations are satisfied. However, for (almost all)
realistic problems it is not possible to derive an exact analytical solution, not even
via assumptions that simplify the mathematical description drastically. A global
description will be given of strategies to derive approximate solutions.
In Section
13.3.1
the general (complete) formulation of the problem will be
given, including the relevant equations. Thereupon, in the following sections the
complexity of the formulation will be gradually reduced. For this, firstly in Section
13.3.2
the material will be modelled as a Newtonian fluid (see Section
12.6
). This
leads to the so-called Navier-Stokes equation (an equation with the pressure field
and the velocity field as unknowns) that has to be solved in combination with the
continuity equation (mass balance). In section
13.3.3
the limitation for a stationary
flow is dealt with (the time dependency, including the need for initial conditions
is no longer relevant). After a section on boundary conditions, a few elementary
analytical solutions of the equations for a stationary viscous flow are presented in
Section
13.3.5
.
