Theory of Laminar Flows

1. Notations and units

Notation	Quantity	Unit
$\rho_f$	fluid density	$kg \cdot m^{-3}$
$\boldsymbol{u}_f$	fluid velocity	$m \cdot s^{-1}$
$\boldsymbol{\sigma}_f$	fluid stress tensor	$N \cdot m^{-2}$
$\boldsymbol{f}^t_f$	source term	$kg \cdot m^{-3} \cdot s^{-1}$
$p_f$	pressure fields	$kg \cdot m^{-1} \cdot s^{-2}$
$\mu_f$	dynamic viscosity	$kg \cdot m^{-1} \cdot s^{-1}$
$\bar{U}$	characteristic inflow velocity	$m \cdot s^{-1}$
$\nu$	kinematic viscosity	$m^2 \cdot s^{-1}$
$L$	characteristic length	$m$

Notation

Quantity

Unit

$\rho_f$

fluid density

$kg \cdot m^{-3}$

$\boldsymbol{u}_f$

fluid velocity

$m \cdot s^{-1}$

$\boldsymbol{\sigma}_f$

fluid stress tensor

$N \cdot m^{-2}$

$\boldsymbol{f}^t_f$

source term

$kg \cdot m^{-3} \cdot s^{-1}$

$p_f$

pressure fields

$kg \cdot m^{-1} \cdot s^{-2}$

$\mu_f$

dynamic viscosity

$kg \cdot m^{-1} \cdot s^{-1}$

$\bar{U}$

characteristic inflow velocity

$m \cdot s^{-1}$

$\nu$

kinematic viscosity

$m^2 \cdot s^{-1}$

$L$

characteristic length

$m$

2. Equations

Navier-Stokes model is used to model incompressible Newtonian fluid. It can be described by these conservative laws :

Momentum conservation equation

$\rho_{f} \left. \frac{\partial\mathbf{u}_f}{\partial t} \right|_\mathrm{x} + \rho_{f} \left( \boldsymbol{u}_{f} \cdot \nabla_{\mathrm{x}} \right) \boldsymbol{u}_{f} - \nabla_{\mathrm{x}} \cdot \boldsymbol{\sigma}_{f} = \boldsymbol{f}^t_f , \quad \text{ in } \Omega^t_f \times \left[t_i,t_f \right]$

Mass conservation equation

$\nabla_{\mathrm{x}} \cdot \boldsymbol{u}_{f} = 0, \quad \text{ in } \Omega^t_f \times \left[t_i,t_f \right]$

we complete this set of equations with the fluid constitutive law

Material constitutive equation

$\boldsymbol{\sigma}_{f} = -p_f \boldsymbol{I} + 2\mu_f D(\boldsymbol{u}_{f})$

with strain tensor $D(\boldsymbol{u}_{f})$ defined by :

Strain tensor

$D(\boldsymbol{u}_{f}) = \frac{1}{2} (\nabla_\mathrm{x} \boldsymbol{u}_f + (\nabla_\mathrm{x} \boldsymbol{u}_f)^T)$

An alternative model is the Stokes model. It is valid in the case of small Reynolds number. It corresponds to the same formulation than Navier-Stokes equations but without the convective term $\left( \boldsymbol{u}_{f} \cdot \nabla_{\mathrm{x}} \right) \boldsymbol{u}_{f}$ .

2.1. Generalized Newtonian fluid

A non newtonian fluid is characterized by a non constant viscosity, which is a function of strain rate $\boldsymbol{D}\left(\boldsymbol{u}_{f}\right)$ .

We start by introducing a metric of the rate of deformation, denoted by $\dot{\gamma}$ :

Rate of deformation

$\dot{\gamma} \equiv \sqrt{2 \ tr \left( \boldsymbol{D}\left(\boldsymbol{u}_{f}\right)^{2} \right) }$

We represent the viscosity $\mu_f$ as a function of $\dot{\gamma}$ . The deviatoric stress tensor $\boldsymbol{\tau}$ is obtained thanks to generalised Newtonian model, which takes the following form:

Deviatoric stress tensor

$\boldsymbol{\tau} = 2 \mu_f \left(\dot{\gamma} \right) \boldsymbol{D}\left(\boldsymbol{u}_{f}\right)$

The simplest example of a generalised Newtonian model is the power-law fluid, which has a viscosity given by:

Power law

$\mu_f \left(\dot{\gamma} \right) = k \dot{\gamma}^{n-1}$

where $k$ and $n < 1$ are two parameters related to fluid properties.

Blood flow viscosity

In the context of blood flow modeling, an extension of the power model was proposed by Walburn and Schneck.

The parameters $k$ and $n$ are related to the hematocrit $Ht$ and Total Proteins Minus Albumin (TPMA) as follows

$k = C_1 e^{C_2 Ht} e^{C_4 \text{TPMA} / Ht }, \quad\quad n = 1- C_3 H t$

and $C_i, i=1,..,4$ are parameters to fit with experimental data.

Another family of generalised Newtonian model can be defined from a function $\Phi$ express by:

$\Phi\left( \dot{\gamma}, \mu_{\infty},\mu_{0} \right) = \frac{\mu\left(\dot{\gamma}\right) - \mu_{\infty}}{\mu_{0}-\mu_{\infty}}$

where $\mu_0$ and $\mu_{\infty}$ are the asymptotic viscosities at zero and infinite shear rate.

Viscosity law	$\Phi\left( \dot{\gamma}, \mu_{\infty},\mu_{0} \right)$
Carreau	$\left(1+\left(\lambda\dot{\gamma}\right)^{2}\right)^{(n-1)/2}$
Carreau-Yasuda	$\left(1+\left(\lambda\dot{\gamma}\right)^{a}\right)^{(n-1)/a}$

Viscosity law

$\Phi\left( \dot{\gamma}, \mu_{\infty},\mu_{0} \right)$

Carreau

$\left(1+\left(\lambda\dot{\gamma}\right)^{2}\right)^{(n-1)/2}$

Carreau-Yasuda

$\left(1+\left(\lambda\dot{\gamma}\right)^{a}\right)^{(n-1)/a}$

3. Forces

The force $\mathbf{F}$ applied by the fluid flowing in $\Omega$ on a suface $\Gamma$ writes:

$\mathbf{F} = \int_\Gamma \boldsymbol{\sigma}_f(\mathbf{u},p) \mathbf{n}$

where

$\boldsymbol{\sigma}_f(\mathbf{u},p)$ is the stress tensor defined above
$\mathbf{n}$ is the unit outward normal to $\Gamma$

There two ways to compute $F$ :

direct: we compute the force from the formula above.
weak: we derive the force computation from the weak formulation.