Mathematical Models for Blood Coagulation - Fluid-Structure Interaction and Biomedical Applications - page 549

The parameters 1 resp. 2 denote the relaxation, resp. retardation times. Upper-
convected (time) derivative 98 of tensors is marked by the symbol O . The viscosity
(being constant in the original Oldroyd model) is defined as a product of a shear-
dependent apparent viscosity . P / and an activation dependent factor

Q . A /, i.e.,

D . P /Q . A /. For the shear-thinning part of viscosity, the model of Yeleswarapu

is suggested:

. P / D 1 C . 0 1 / 1 C ln.1 C P /
1 C P

(7.124)

where P is the shear-rate and ƒ is an adjustable parameter. The platelet activation-
dependent multiplicative factor / is defined as 99

A

Q .

n

1 for resting platelets

Q . A / D (7.125)

1 for activated platelets

A much more complex macroscopic coagulation biochemistry model, described
by a set of more than 20 coupled ADR equations, was introduced in [ 8 ]. To describe
the shear-thinning viscoelastic character of blood a nonlinear model was used based
on earlier works considering a novel thermodynamically consistent framework for
rate type fluid models [ 199 ]. Within this model the viscoelastic stress tensor T is
split as:

T D B p .t/ C 1 D (7.126)

where D denotes the symmetric part of the velocity gradient tensor and B p .t/ stands
for the elastic stretch tensor. 100

The upper-convected time derivative of the tensor

B p .t/ is then given by:

B p .t/ D2K tr.B p .t/ / 3 n B p .t/ 1

(7.127)

where the coefficient depends on the trace of the inverse of the tensor B p .t/
according to

3
tr B 1 p.t/

D : (7.128)

98 The upper-convected derivative M of a tensor M is defined using the classical material time-
derivative M and the symmetric resp. skew-symmetric parts of the velocity gradient D resp. W as
M D M WM C MW .DM C MD/.
99 See [ 6 ] for the complete algorithm of platelet activation depending on the values of A .
100 The subscript p .t/ is used to emphasize that the stretch is expressed with respect to natural
(time-dependent) configuration p .t/. This notation follows exactly the original papers [ 7 , 12 , 199 ]
and [ 34 ] where the model has been introduced and used.