$$ \newcommand{\partd}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\partdd}[2]{\frac{\partial^{2} #1}{\partial {#2}^{2}}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\Int}{\int\limits} \newcommand{\D}{\displaystyle} \newcommand{\ie}{\textit{i.e. }} \newcommand{\dA}{\; \mbox{dA}} \newcommand{\dz}{\; \mbox{dz}} \newcommand{\tr}{\mathrm{tr}} \renewcommand{\eqref}[1]{Eq.~(\ref{#1})} \newcommand{\reqs}[2]{\req{#1} and \reqand{#2}} \newcommand{\rthreeeqs}[3]{Eqs.~(\ref{#1}), (\ref{#2}), and (\ref{#3})} $$

 

 

 

1 Nomenclature

symbol definition
\( m \) Mass [kg]
\( \rho \) Mass density \( [kg/m^3] \)
\( \mathbf{P} \) Linear momentum \( [kg\, m/s] \)
\( \mathbf{v} \) Velocity or specific linear momentum [m/s]
\( K \) Kinetic energy \( [kg\, m^2/s^2] \)
\( \mathbf{v\cdot v}/2 \) Specific kinetic energy \( [m^2/s^2] \)
\( B \) Generic extensive property
\( \beta \) Generic specific intensive property per mass unit
\( \mathbf{n} \) Outward unit normal for the surface \( A(t) \)
\( \mathbf{e}_i \) Orthogonal unit vectors
\( T_{ik} \) Coordinate stresses [Pa]
\( T \) Stress matrix
\( T_{iI} \) Normal stresses
\( T_{ij} \) Shear stresses, when \( i\neq k \)
\( \mathbf{T} \) Stress tensor [Pa]
\( \gamma \) Shear strain []
\( \epsilon_v \) Volumetric strain []
\( \eta \) Modulus of elasticity [Pa]
\( \nu \) Poisson's ratio []
\( \varepsilon_v=E_{ii} \) Voumetric strain
\( w \) Stress work per unit volume
\( \boldsymbol{T}_0 \) The first Piola-Kirchhoff stress tensor
PKS Piola-Kirchhoff stress tensor
\( \boldsymbol{S} \) The second Piola-Kirchhoff stress tensor
\( p \) Thermodynamical pressure
\( R \) Gas constant for a particular gas
$ \alpha$ Womersley parameter
\( pp \) Pulse pressure [mmHg]
\( CO \) Cardiac output [ml/min]
\( \tau=RC \) Relaxation time \( s \)
\( \dot{V} \) Rate of change in volume
\( Q_{i} \) Flow rate into a volume
\( Q_o \) Flow rate out of a volume
\( Q \) Volumetric flow rate.
\( \delta \) Nonlinear velocity profile correction factor
\( \delta \) Correction factor
\( C_1 \) Arbitrary constant
\( \delta'=\delta+1 \) The modified correction factor
\( Z_c^f \) Characteristic impedance for a forward traveling wave
\( Z_c^b \) Characteristic impedance for a backward traveling wave
\( \rho_{w} \) Density of vessel wall
\( \tau_w = \tau_{rz}\vert_{r=r_i} \) Wall shear stress
\( \delta_{ij} \) Kronecker delta
\( e_{ijk} \) Permutation symbol []