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Hemodynamics: An Introduction

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PanVascular Medicine

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Abstract

The cardiovascular transport circuit is involved in both mass and heat transfer. It carries blood cells as well as oxygen and nutrients to cells of the body’s organs through the perfusing systemic arterial bed and wastes produced by working cells to their final destinations through draining veins. Blood flows throughout the body in the vasculature due to a pressure difference between the ventricular outlet and atrial inlet. Blood is propelled in the systemic and pulmonary circulation by the synchronized action of the left and right apposed cardiac pumps, respectively. Hemodynamics is related to the flow features in the heart and blood vessels, in normal and pathological conditions, in particular the pressure–flow relations and transport of substances by blood to given target organs. It can be required in therapy planning and optimization.

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Notes

  1. 1.

    From German Windkessel: air chamber and hence compliant reservoir analogous to an electrical capacitor

Abbreviations

Augmentation index (Alx):

Is the ratio of pressure difference between the shoulder (automatically determined from the time derivative of the pressure wave) of the ascending pressure phase and peak pressure to the pulse pressure used to assess the wave reflection effect.

Boundary layer:

Is the layer of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. This near-wall layer of a flowing fluid is characterized by a more or less strong spatial velocity gradient at a given time that decays from the wall to the flow core (vessel axis). The pressure gradient varies nonlinearly in the vessel entrance segment, in which the boundary layer grows, and linearly when the flow becomes fully developed, i.e., downstream from the station in which the cross-sectional distribution of the velocity becomes invariant, whatever the tube shape and local curvature.

The Blasius boundary layer refers to the similarity solution near a semi-infinite flat plate in a unidirectional flow. Within the boundary layer, viscous effects are balanced by convective inertia.

The Stokes boundary layer, or oscillatory boundary layer, refers to the thin shear layer that develops close to a solid wall in a laminar oscillatory flow of a viscous fluid or to a plate wetted by a viscous fluid at rest oscillating in the direction of its plane. A unidirectional motion of a wetted plate (boundary tangential velocity \( u(t)=\widehat{U} \cos \left(\omega\;t\right) \)) causes a fluid streaming. In heat and mass transfer, the temperature and mass transition from the wall surface to the free stream occur across a very thin region close to the wetted wall, in the so-called thermal and massic boundary layer.

Steady Two-Dimensional Boundary Layer Equations. The boundary layer theory states that ν x and ν y change much more rapidly with the normal direction y than with the streamwise direction x. Hence, |∂ y υ x | ≫ |∂ x υ x |, that is, using the length (L), pressure (P), and velocity (V x and V y in the x- and y-directions) scales, V x /δ ≫ V x /L, and δ ≪ L. The equation of mass conservation yields

$$ \frac{\partial {\upsilon}_x}{\partial x}+\frac{\partial {\upsilon}_y}{\partial y}=0. $$
(1)

Hence, V x /L ∼ V y /δ and V y  ≪ V x . Moreover, ∂ 2 x υ x  ≪ ∂ 2 y υ x .

The x-velocity component obeys the equation of momentum conservation:

$$ {\upsilon}_x\frac{\partial {\upsilon}_x}{\partial x}+{\upsilon}_y\frac{\partial {\upsilon}_x}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu {\partial}^2{\upsilon}_x\partial {x}^2+\nu {\partial}^2{\upsilon}_x\partial {y}^2, $$
(2)

the terms of which are proportional to V 2 x /L, P/ρL, νV x /L 2, and νV x /δ 2, respectively.

The y-velocity component obeys the corresponding equation of momentum conservation:

$$ {\upsilon}_x\frac{\partial {\upsilon}_y}{\partial x}+{\upsilon}_y\frac{\partial {\upsilon}_y}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu {\partial}^2{\upsilon}_y\partial {x}^2+\nu {\partial}^2{\upsilon}_y\partial {y}^2, $$
(3)

the terms of which are proportional to V 2 x δ/L 2, P n /ρδ, νV x δ/L 3 and νV x /(L δ), respectively. In addition, P/ρL ∼ P n /ρδ and P n  ≪ P. Therefore, the y-component of the equation of momentum conservation in a boundary layer can be neglected. Therefore, the equation of momentum conservation is

$$ {\upsilon}_x\frac{\partial {\upsilon}_x}{\partial x}+{\upsilon}_y\frac{\partial {\upsilon}_x}{\partial y}=-\frac{1}{\rho}\frac{\partial p}{\partial x}+\nu {\partial}^2{\upsilon}_x\partial {y}^2. $$
(4)
The boundary layer thickness (δ):

Is the distance normal to the wall from it to a point where the flow velocity reaches the free-stream velocity (u w⊥ = 0.99 u core ). Using the abovementioned phenomenological analysis, as the convection balances the viscous dissipation, that is, V 2 x /L ∼ νV x /δ 2, thus (δ/L)2, ∼ 1/Re.

In laminar boundary layers over a flat plate at length ℓ, the Blasius solution gives

$$ {\delta}_{\mathrm{B}}\approx 4.91{\left(\frac{\nu z}{u_{core}}\right)}^{1/2}\approx 4.91\frac{\ell }{{\mathrm{Re}}_{\ell}^{1/2}}, $$
(5)

where Re ℓ  = u core ℓ/ν is the Reynolds number associated with the tube length.

The Stokes boundary layer thickness induced by a unidirectional oscillatory plate is given by

$$ {\delta}_{\mathrm{S}}\propto {\left(\frac{2\nu }{\omega}\right)}^{1/2} $$
(6)
Bulk modulus (B):

Parameter quantifies the reaction of a material to a volume change when it is subjected to a given load (B = p/(dV/V)). Its physical magnitude is homogeneous to a pressure.

Complex viscoelastic shear modulus (G ∗):

Parameter is related to a sinusoidal shear (strain e ∗ and stress c ∗) applied on a material:

$$ \begin{array}{c}{e}^{\ast }=\widehat{e} \exp \left\{\iota \left(\omega t\right)\right\}={C}^{*}\left(\omega \right){c}^{*}(t),\\ {}\hfill {c}^{*}=\widehat{c} \exp \left\{\iota \left(\omega t+\varphi \right)\right\}={G}^{*}\left(\omega \right){e}^{*}(t),\hfill \\ {}\hfill {G}^{*}\left(\omega \right)=\mathrm{\Re}\mathrm{e}\left[G\left(\omega \right)\right]+\iota \mathfrak{F}\mathrm{m}\left[G\left(\omega \right)\right]={G}^{\prime}\left(\omega \right)+\iota {G}^{{\prime\prime}}\left(\omega \right),\hfill \end{array} $$

where G′(ω) is the storage modulus and G″(ω) the loss modulus.

Compliance (C):

Property of a deformable vessel refers to variations of cross-sectional surface area (A) of the vascular lumen due to changes in transmural pressure (p) at a given vessel station:

$$ \mathrm{C}(p)=\frac{\partial A}{\partial p} $$
(7)
Darcy friction factor (f D):

Coefficient corresponds to the flow coefficient (Λ), or friction head loss coefficient, and depends on the conduit features and fluid flow velocity:

$$ {f}_{\mathrm{D}}={H}_f\frac{d}{L}\frac{2g}{V_q^2}, {f}_{\mathrm{D}}=\Delta p\frac{d}{L}\frac{2}{\rho {V}_q^2}, $$
(8)

where H f is the frictional head loss, Δp = ρgH f the pressure loss, d and L the pipe length and hydraulic diameter, g the gravitational acceleration, ρ the fluid density, and V q the cross-sectional average fluid velocity (volumetric flow rate per unit cross-sectional wetted area).

The Darcy friction factor is 4 times the dimensionless Fanning friction factor (f F ; aka friction coefficient and skin friction coefficient) that is related to the wall shear stress:

$$ {f}_{\mathrm{F}}={c}_w\frac{2}{\rho {V}_q^2}={H}_f\frac{d}{L}\frac{g}{2{V}_q^2} $$
(9)

hence,

$$ {f}_{\mathrm{D}}=\frac{8{c}_w}{\rho {V}_q^2}. $$
(10)

In laminar flow through ducts of circular cross section, the Darcy friction factor is given by

$$ {f}_{\mathrm{D}}=\frac{64}{\mathrm{Re}}, $$
(11)

where Re is the Reynolds number.

Darcy–Weisbach equation:

Is a phenomenological equation that relates the head loss, or pressure loss, due to friction along a given pipe length to the average fluid flow velocity. It yields the dimensionless Darcy friction factor:

$$ {H}_f={f}_{\mathrm{D}}\frac{L}{d}\frac{V_q^2}{2g}, \Delta p={f}_{\mathrm{D}}\frac{L}{d}\frac{\rho {V}_q^2}{2}. $$
(12)

Resistance to motion of a solid in a fluid increases with fluid density and viscosity and solid surface area and shape.

Distensibility property:

Also called specific compliance refers to deformation of the cross-sectional surface area of the vascular lumen (A i) in the range of positive transmural pressures:

$$ D=\frac{1}{A_{ref}}\frac{\partial A}{\partial p}=C/A $$
(13)
Drag coefficient (C d):

In a steady motion of a Newtonian incompressible fluid is given by C d = F d/(πR 2p )(ρν 2/2) (F d: Stokes drag force on a rigid spherical particle).

Dynamic viscosity (μ):

Is the ratio of shearing stress to shear rate.

Elastic modulus (E):

Coefficient of the stress–strain relation is obtained using different types of loading (traction, bending, and torsion). A material is linearly elastic in a given loading range if the elastic modulus remains constant, the stress being proportional to the strain (“ut tensio, sic vis”; the Hooke law): E = C ii /E ii . The elastic modulus is then a measure of recoverable deformation when a force is applied; the deformation reverses when the force is removed.

Friction:

Is a resistive force that opposes the motion of an object in contact with another one.

Impedance (Z):

Is the ratio between the pressure drop and flow rate. It measures opposition to motion of a fluid subjected to a pressure, that is, the strength with which the fluid resists motion. It depends on the frequency (ω) of the applied pressure.

Incremental elastic modulus (E inc):

Is a modulus that is considered constant in small loading range, as the nonlinear stress–strain relation is decomposed into small intervals (piecewise constant elastic modulus). For a point of the outer surface of the vessel wall, which is easy to observed experimentally, Bergel (1961) proposed the following formula:

$$ {E}_{\mathrm{i}\mathrm{nc}}\left({R}_{\mathrm{i}}\right)=\frac{2\left(1-{\nu}_{\mathrm{P}}\right){R}_{\mathrm{i}}^2{R}_{\mathrm{e}}}{R_{\mathrm{e}}^2-{R}_{\mathrm{i}}^2}\frac{\Delta p}{\Delta {R}_{\mathrm{e}}}, $$
(14)

and for a point of the wetted surface, which is easy to target by medical imaging,

$$ {E}_{\mathrm{i}\mathrm{nc}}\left({R}_{\mathrm{e}}\right)=\frac{\left(1+{\nu}_{\mathrm{P}}\right){R}_{\mathrm{i}}}{R_{\mathrm{e}}^2-{R}_{\mathrm{i}}^2}\left(\left(1-2{\nu}_{\mathrm{P}}\right){R}_{\mathrm{i}}^2+{R}_{\mathrm{e}}^2\right)\frac{\Delta p}{\Delta {R}_{\mathrm{e}}}. $$
(15)
Inertia:

Is the property of an object to resist any change in its state of motion.

Kinematic viscosity (ν):

Is the ratio of dynamic viscosity to fluid density that is equivalent to momentum diffusivity.

Mach number (Ma):

Is the ratio of the fluid flow velocity to the propagation speed (Ma = V q /c). The wave speed depends on the involved deformable parts, fluid compressibility (K), and vessel distensibility (D; with K ≪ D even for inhaled air). The usual speed is the speed of sound. In blood circulation, Ma is related to the propagation speed of pressure waves.

Mechanical stress:

Is force per unit surface area producing a deformation usually decomposed into tangential (shear) and normal (pressure and stretch) components. In any point of the flowing blood and within the wall of any part of the cardiovascular circuit, it is represented by the stress tensor (C).

Mechanotransduction:

Is a process used by cells to convert a mechanical stimulus into an electrochemical and/or chemical signal by gating mechanosensitive ion channels and/or launching chemical reaction cascade from proper receptors at the cell surface.

Momentum:

Is a conserved vector quantity (total momentum of any closed system that is not affected by external forces cannot change). Linear momentum is the product of the mass and velocity of an object. Angular momentum is the product of rotation inertia and angular velocity.

Navier number:

\( \left(\mathrm{N}\mathrm{a}=\frac{P^{\ast }}{\mu {V}^{\ast }/{L}^{\ast }}\right) \) is the ratio between the pressure gradient (∇ x p; normal stress component) and friction (μ∇ 2 x v; tangential stress component).

Nodal cells:

Are specialized cardiac cells involved in the generation and transmission of action potentials, hence responsible for the cardiac automatism and intrinsic conduction under the control of the nervous system. Nodal cells of the sinoatrial node, the cardiac natural pacemaker, trigger action potentials that then spread through both atria and reach the atrioventricular node and then His bundle and its branches and Purkinje fibers to produce successively atrial and ventricular contraction.

Oscillatory shear index (OSI) is given by the formula \( {\displaystyle {\int}_{t_{\mathrm{ri}}}^{t_{\mathrm{rf}}}\left|{c}_w\right|dt}/{\displaystyle {\int}_0^T\left|{c}_w\right|}\;dt \), where t ri and t rf are the initial and final instants of the reversed flow period.

Poisson ratio (νP):

Is the ratio of the relative contraction in the transverse direction j to the relative deformation in the direction i of the applied load νP = E jj /E ii (i ≠ j). One-dimensional extension is characterized by (1) a longitudinal lengthening e ℓ  = ΔL/L and (2) a transverse shortening e t  = Δw/w = − (ν P/E)c (transverse strain-to-axial strain ratio).

Pressure (p):

Is the ratio of force to the area over which it is distributed, i.e., force over an area (scalar quantity; [N/m2 or Pa]) applied to an object in a direction perpendicular to the surface. The stress tensor that relates the force vector f to the area vector A incorporates in its diagonal components the pressure.

Hydrostatic pressure is the pressure due to the weight of a fluid: p = ρgh (ρ, fluid density; g, acceleration due to gravity – approximately 9.81 m/s2 on earth’s surface – h, height of the fluid column).

Stagnation pressure is the pressure a fluid exerts when it is forced to stop moving (immersed body, branching apex, etc.).

Dynamic pressure is the pressure that results from fluid motion 1/(2ρν2). In incompressible fluid flow, it is the difference between total pressure and static pressure.

Total pressure is the pressure determined by all moving fluid particles crossing a virtual surface that equals stagnation or impact pressure:

Pulse pressure maximum (peak systolic) minus minimum (diastolic) pressure.

Reactive hyperemia index (RHI):

Is the ratio of the post- to preocclusion pulse volume amplitude of the tested upper limb during reactive hyperemia elicited by the removal of an upper arm blood pressure cuff inflation to suprasystolic pressure during 5 mn.

Reynolds number (Re = V*L*/v):

Is the ratio between convective inertia and viscous effects applied on a unit of fluid volume (ν = μ/ρ; L*: length scale, often the vessel radius [R]; V*: velocity scale, often the cross-sectional average velocity [V q ]). It is also the ratio between the momentum diffusion time scale and the convection characteristic time Re = (R2/ν)/(R/U). In pulsatile flows, both mean \( \overline{\mathrm{Re}}=\mathrm{R}\mathrm{e}\left(\overline{V_q}\right) \) and peak Reynolds numbers \( \widehat{\mathrm{Re}}=\mathrm{R}\mathrm{e}\left(\widehat{V_q}\right) \), proportional to the mean and the peak cross-sectional average velocity, respectively, can be calculated. This parameter controls flow pattern transition. Re δ  = Re/Sto is used for flow stability study (δ: boundary layer thickness). Branching pulsatile flows are currently based on stem peak Reynolds number \( \widehat{\mathrm{Re}} \) calculated from \( \widehat{V_{ q}} \), trunk radius R, and blood kinematic viscosity (ν = 4 × 10−6m2 s−1), assuming a Newtonian blood. In bend flows, a secondary motion-associated Reynolds number can be introduced, using the velocity scale V ∗2 , when centrifugal forces ρV 2 / R c are balanced by local inertia effects ρωV *2  : Re2 = V 2 R/(ωνR c), i.e., V 2/(ων) × κ c  = Re/St. The wall shear Reynolds number is defined by u ∗ δ/ν.

Shear rate (\( \dot{\gamma} \)):

Is the rate at which a progressive shear is applied to a material. In the vascular lumen, the flowing blood experiences a shear stress associated with a shear rate given by the velocity distribution at any cross section and phase of the cardiac cycle (locally given by the spatial velocity gradient).

Shear-thinning material:

Is a non-Newtonian medium that has an apparent viscosity that decreases with an increasing shear rate.

Stokes drag force (F d):

On a rigid spherical particle of radius (R p) and the drag coefficient are given by F d = 6πR p μν.

Stokes number (Sto = R(ω/ν)1/2):

Is a frequency parameter of pulsatile flows, also called Witzig–Womersley number (ω: pulsation of flow oscillation). It is the square root of the ratio between time inertia and viscous effects. The Stokes number is a Reynolds-like number for periodic flow (local acceleration replaces convective acceleration). The Stokes number is proportional to the ratio between the vessel hydraulic radius and the Stokes boundary layer thickness (Sto ∝ R h/δ S ) and the ratio between momentum diffusion time and the cycle period (Sto ∝ R 2/ν)/(1/ω) ≡ ((T diff/T)1/2).

Strain:

Is a variable that relates the deformed to the unstressed reference configuration. It results from change in length (normal strain) and angle (shear strain).

Strouhal number (St = ωL* / V*):

Is the ratio between time and convective inertia (St = Sto2/Re). In quasiperiodic flow in a branching vessel region, the peak Strouhal number is based on the trunk peak cross-sectional average velocity: \( \mathrm{St}=\omega R/\widehat{V_q} \). The Strouhal number is proportional to the ratio between the steady and the Stokes boundary layer thicknesses (St = δ / δ S ). The dimensionless stroke length \( \left(\widehat{V_q}/\left({R}_{\mathrm{h}}\omega \right)\right) \), which is also the ratio of the length scale of the axial displacement of a fluid particle to the vessel radius or the ratio between the flow cycle and the convection time scale, is the inverse of the Strouhal number. In the aorta at rest, \( \widehat{V_q}/{R}_{\mathrm{h}}\omega =11.1 \) with the following value set: f = 1 Hz, R h = 10 mm, and \( \widehat{V_q}=0.7\;\mathrm{m}/s \). Bend flows depend, for small values of the frequency parameter, on the Strouhal number for the secondary motion (when centrifugal forces ρV 2/R c are balanced by local inertia effects ρωV *2 , St2 = (ω 2 RR c)/V 2). In turbulent periodic flows, St = Rω/u′ (u′: turbulent intensity) is the ratio of the time scale of turbulent fluctuations R/u′ to the flow period. The turbulent Strouhal number \( \mathrm{St}=\omega R/{\overline{u}}_{*} \) can also be defined by the time mean friction velocity (\( {\overline{u}}_{*} \)).

Tube law:

Is the relation between the transmural pressure (p) and the luminal cross-sectional surface area (A i). It depends on various factors: vessel geometry, vascular wall rheology, prestresses, vicinity loadings, end effects, and developed stresses (tension, bending).

Velocity (v), linear velocity :

Is the time rate at which distance is covered by a moving object. Angular velocity is the number of rotations per unit time. Terminal velocity is the speed and direction of a falling object when friction balances weight.

Velocity amplitude ratio or modulation rate:

Is the ratio of peak to mean velocity of a pulsatile flow \( \left({\upgamma}_{\upupsilon}=\widehat{V_q}/\overline{V_q}\right) \). The pulsatile flow is characterized by less backflow when the modulation rate is lower.

Vorticity (ω):

Is the curl of the velocity (∇ × v). This property of a fluid particle is related to its angular velocity that describes its tendency to rotate. In a laminar flow, any infinitesimal viscous fluid particle that experiences a shearing torque applied by apposed inner and outer fluid particles does not rotate because of the uniform pressure exerted on its faces.

Viscous head loss:

Is the pressure reduction per unit length given by the Darcy–Weisbach formula.

Wall shear stress (c w ):

Is the product of the wall shear rate by the fluid kinematic viscosity. The wall shear stress exerted by blood is sensed by components of the luminal surface of endotheliocytes.

Wall stiffness:

Relates stress to strain. Due to a nonlinear rheological behavior, the stiffness of the vascular wall increases progressively with the strain.

Arterial wall stiffening increases the speed of pressure wave propagation and creates reflection points closer to the heart, causing premature wave reflection. Early reflected waves amplify the systolic pressure, whereas diastolic blood pressure falls.

Windkessel effect:

Is the effect related to the interaction between the stroke volume and the compliance of the aorta and large deformable elastic arteries that distend when the blood pressure rises during systole and recoil when the blood pressure falls during diastole. The systolic distension is associated with blood storage that is restituted during the following diastole. Therefore, the Windkessel effect dampens the fluctuation in blood pressure and enables continuous blood flow during the entire cardiac cycle, transforming the starting–stopping flow at the exit of the ventricle into a pulsatile flow that maintains organ perfusion during diastole.

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Further Reading

  • Details and associated references on specific topics at diverse length scales (micro-, meso-, and macroscopic scales) can be found in

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  • Thiriet M (2013) Tissue functioning and remodeling in the circulatory and ventilatory systems, vol 5, Biomathematical and biomechanical modeling of the circulatory and ventilatory systems. Springer, New York

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  • Thiriet M (2014) Anatomy and physiology of the circulatory and ventilatory systems, vol 6, Biomathematical and biomechanical modeling of the circulatory and ventilatory systems. Springer, New York

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  • Thiriet M (2008) Biology and mechanics of blood flows, part II: mechanics and medical aspects of blood flows, CRM series in mathematical physics. Springer, New York

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  • Thiriet M, Sheu TWH, Garon A (2014) Biofluid flow and heat transfer. In: Johnson RW (ed) Handbook of fluid dynamics, 2nd edn. CRC Press, Boca Raton. ISBN 9781439849552

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  • The anatomy of the cardiovascular system is covered in any human anatomy textbook. Gray’s Anatomy: Descriptive and Surgical (1858) is an revised open-access text on the web (education.yahoo.com/reference/gray)

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  • The physiology of the cardiovascular apparatus is described in the textbook of Medical Physiology by A.C. Guyton, updated in Guyton AC and Hall JE (2006; 7th edition) Elsevier – Saunders Other cardiovascular physiology textbooks include in particular An Introduction to Cardiovascular Physiology by J.R. Levick (2003), Cardiovascular Physiology by M.N. Levy and R.M. Berne (2007), and Cardiac Electrophysiology: from Cell to Bedside by D.P. Zipes (2004)

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  • Principles of Biochemistry by A.L. Lehninger (1993) that is updated enables a basic understanding of cell functioning. Details on cell signaling are given in

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  • Thiriet M (2012) Signaling at the cell surface in the circulatory and ventilatory systems, vol 3, Biomathematical and biomechanical modeling of the circulatory and ventilatory systems. Springer, New York

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  • Thiriet M (2012) Intracellular signaling mediators in the circulatory and ventilatory systems, vol 4, Biomathematical and biomechanical modeling of the circulatory and ventilatory systems. Springer, New York

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  • Textbooks on blood flow comprise McDonald’s Blood Flow in Arteries by W.W. Nichols and M.F. O’Rourke (1998) rewritten and upgraded from Blood Flow in Arteries by D.A. McDonald and Hemodynamics by W.R. Milnor (1989). Pulmonary Circulation: Basic Mechanisms to Clinical Practice by J.M.B. Hughes and N.W. Morrell (2001) emphasizes the pulmonary microcirculation. An introduction to the cardiovascular system and basic biomechanical principles is given in The Mechanics of the Circulation by C.G. Caro, T.J. Pedley, R.C. Schroter, and W.A. Seed (1978).

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  1. team INRIA-UPMC-CNRS REO, Laboratoire Jaques-Louis Lions, CNRS UMR 7598, Université Pierre et Marie Curie (Sorbonne Universités), F-75252, Paris, France

    Marc Thiriet

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  1. Division of Cardiovascular Disease Department of Internal Medicine, Health Care Center Bitterfeld, Bitterfeld-Wolfen, Germany

    Peter Lanzer

Notation Rules

Biochemical species usually possess many aliases. Conversely, a given alias can refer to various types of molecules. Aliases that designate different types of molecules as well as those that do not have an obvious meaning are not used in the present text.

Symbols for physical quantities are most often represented by a single letter of the Latin or Greek alphabet (J, flux; L, length; m, mass; p, pressure; T, temperature; t, time; u, displacement; v, velocity; x, space; μ, dynamic viscosity; ρ, mass density; etc.). These symbols are specified using sub- and superscripts (\( \mathcal{D} \): mass diffusivity and \( {\mathcal{D}}_T \): thermal diffusivity).

A physical quantity associated with a given point in space at a given time can be (1) a scalar uniquely defined by its magnitude; (2) a vector characterized by a magnitude, a support, and a direction represented by an oriented line segment defined by a unit vector; and (3) a tensor specified by a magnitude and a few directions. To ensure a straightforward meaning of symbols used for scalar, vectorial, and tensorial quantities, boldface upper- (T) and lowercase (v) letters are used to denote a tensor and a vector, respectively, whereas both roman-style upper- and lowercase letters designate a scalar.

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Thiriet, M. (2015). Hemodynamics: An Introduction. In: Lanzer, P. (eds) PanVascular Medicine. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37078-6_22

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