Sources and References

• White, F. M. Fluid Mechanics. McGraw-Hill.
• Fox, R. W., McDonald, A. T., and Pritchard, P. J. Introduction to Fluid Mechanics. Wiley.
• Munson, B. R., Young, D. F., and Okiishi, T. H. Fundamentals of Fluid Mechanics. Wiley.
• Truskey, G. A., Yuan, F., and Katz, D. F. Transport Phenomena in Biological Systems. Pearson.
• Fournier, R. L. Basic Transport Phenomena in Biomedical Engineering. CRC Press.
• Bird, R. B., Stewart, W. E., and Lightfoot, E. N. Transport Phenomena. Wiley.
• Fung, Y. C. Biomechanics: Circulation. Springer.
• Nichols, W. W., O’Rourke, M. F., and Vlachopoulos, C. McDonald’s Blood Flow in Arteries. CRC Press.
• Ku, D. N. “Blood Flow in Arteries.” Annual Review of Fluid Mechanics.
• MIT OpenCourseWare 2.006 Thermal-Fluids Engineering II and 2.25 Advanced Fluid Mechanics.
• Stanford ME 351A/B Fluid Mechanics lecture notes.

---

Chapter 1: Fluid Statics

Fluid statics is the branch of fluid mechanics concerned with fluids at rest, or more precisely, fluids in which there is no relative motion between adjacent fluid particles. Under these conditions shear stresses vanish, and the only stress that can exist within the fluid is an isotropic normal stress called the pressure. Although statics is the simplest chapter in the subject, it supplies essential tools for understanding the pressures that drive blood through vessels, the gauge pressures read by clinical transducers, and the hydrostatic corrections that must be applied when interpreting a patient’s arterial pressure measurement at different heights above the heart.

1.1 Pressure and the Continuum Hypothesis

Before writing any equation, it is worth reminding ourselves that fluid mechanics treats matter as a continuum. We do not track individual water molecules or red blood cells; instead, we assign to each point \( \mathbf{x} \) in space a set of smooth fields such as density \( \rho(\mathbf{x},t) \), velocity \( \mathbf{u}(\mathbf{x},t) \), and pressure \( p(\mathbf{x},t) \). This approximation holds when the smallest length scale of interest is much larger than the molecular mean free path, or in the case of blood, much larger than the diameter of a red cell, which is roughly \( 8\;\mu\mathrm{m} \). Once we approach capillary length scales near \( 5\;\mu\mathrm{m} \), the continuum description of blood starts to break down and we must resort to two-phase or particulate models.

In a fluid at rest, the pressure at a point is isotropic. This is Pascal’s principle, and it follows from a force balance on an infinitesimal wedge-shaped element. Because shear stresses are zero in static fluids, the stress tensor reduces to

\[ \boldsymbol{\sigma} = -p\,\mathbf{I}, \]

where \( \mathbf{I} \) is the identity tensor. The minus sign reflects the convention that compressive stresses are taken negative in continuum mechanics but that fluid pressures are reported as positive quantities.

1.2 The Hydrostatic Equation

Consider a differential fluid element of volume \( \mathrm{d}V \) in a gravitational field \( \mathbf{g} \). Summing pressure forces on its faces and balancing them against gravity gives

\[ \nabla p = \rho\,\mathbf{g}. \]

If gravity points in the negative \( z \)-direction and \( \rho \) is uniform, this reduces to the familiar scalar form

\[ \frac{\mathrm{d}p}{\mathrm{d}z} = -\rho g, \]

so that \( p(z) = p_0 - \rho g (z - z_0) \). Every clinician who has ever zeroed an arterial line at the level of the right atrium is using this equation. If a transducer is placed \( 20\;\mathrm{cm} \) above the heart, the measured pressure is depressed by roughly \( 15\;\mathrm{mmHg} \), a clinically significant offset.

1.3 Manometry and Gauge Pressure

A manometer uses a column of liquid to measure pressure differences by exploiting the hydrostatic equation in reverse. For a U-tube manometer connecting a vessel at pressure \( p_1 \) to the atmosphere at \( p_{\mathrm{atm}} \), following the fluid columns and applying \( \Delta p = \rho g h \) at each change of fluid yields the gauge pressure \( p_1 - p_{\mathrm{atm}} \). Clinical sphygmomanometers are mercury-column manometers in disguise, and the conventional unit \( \mathrm{mmHg} \) (or torr) is literally the height of a mercury column that the pressure will support. One \( \mathrm{mmHg} \) corresponds to about \( 133.3\;\mathrm{Pa} \).

1.4 Forces on Submerged Surfaces and Buoyancy

Integrating the pressure distribution over a submerged surface yields the hydrostatic force. For a flat plate at angle \( \theta \) to the horizontal with centroidal depth \( \bar{h} \), the magnitude of the resultant force is \( F_R = \rho g \bar{h} A \), and it acts below the centroid at a location shifted by \( I_{xc}/(\bar{h}_c A) \) where \( I_{xc} \) is the second moment of area about the centroidal axis. For curved surfaces, the horizontal and vertical components are computed separately, with the vertical component equal to the weight of fluid (real or virtual) displaced above the surface.

Archimedes’ principle follows directly from integrating \( -p\,\mathbf{n} \) over a closed surface: the net upward buoyant force on a submerged body equals the weight of the displaced fluid. In blood, erythrocytes sediment slowly because their density, about \( 1100\;\mathrm{kg/m^3} \), is only slightly greater than the surrounding plasma at roughly \( 1025\;\mathrm{kg/m^3} \), and this small buoyancy-corrected weight is balanced against Stokes drag in the erythrocyte sedimentation rate (ESR) test.

---

Chapter 2: Fluid Kinematics and the Control Volume Approach

Kinematics is the description of motion without regard to the forces that cause it. In fluid mechanics kinematics is unusually rich because a fluid continuum has infinitely many degrees of freedom, and describing its motion requires careful choices of reference frame and field quantities.

2.1 Lagrangian and Eulerian Descriptions

The Lagrangian description follows individual fluid particles, assigning to each particle a position \( \mathbf{X}(\mathbf{X}_0,t) \) as a function of its initial location and time. The Eulerian description, which dominates practical fluid mechanics, instead reports field quantities at fixed spatial points. The connection between the two is the material derivative,

\[ \frac{\mathrm{D}}{\mathrm{D}t} = \frac{\partial}{\partial t} + \mathbf{u}\cdot\nabla, \]

which gives the rate of change following a fluid particle. The first term captures local unsteadiness and the second the convective change experienced as the particle moves through spatial gradients. In pulsatile arterial flow both terms are substantial; in steady laminar flow in a rigid capillary the local term vanishes.

2.2 Streamlines, Pathlines, and Streaklines

Three curves characterize fluid motion. A streamline is everywhere tangent to the instantaneous velocity field, a pathline is the actual trajectory of a particular fluid particle, and a streakline is the locus of all particles that have previously passed through a fixed point. In steady flow these three curves coincide. Doppler ultrasound essentially visualizes an instantaneous velocity field, so the flow patterns it reveals are streamline-like, whereas dye injection during cardiac catheterization produces streaklines.

2.3 The Reynolds Transport Theorem

The bridge between a system, which is a fixed collection of matter, and a control volume, which is a fixed region of space, is the Reynolds transport theorem. If \( B \) is any extensive property of the fluid with intensive counterpart \( b = B/m \), then

\[ \frac{\mathrm{d}B_{\mathrm{sys}}}{\mathrm{d}t} = \frac{\partial}{\partial t}\int_{CV}\rho\,b\;\mathrm{d}V + \int_{CS}\rho\,b\,(\mathbf{u}\cdot\mathbf{n})\;\mathrm{d}A. \]

The first term accumulates \( B \) within the control volume, and the second accounts for the net convective flux of \( B \) across its bounding surface. This identity is the starting point for the integral forms of the conservation laws we derive in the next chapters.

---

Chapter 3: Conservation of Mass

3.1 Integral Form

Applying the Reynolds transport theorem with \( B = m \) and \( b = 1 \), and noting that the mass of a closed system is constant, we obtain the integral continuity equation

\[ \frac{\partial}{\partial t}\int_{CV}\rho\;\mathrm{d}V + \int_{CS}\rho\,(\mathbf{u}\cdot\mathbf{n})\;\mathrm{d}A = 0. \]

For steady flow through a control volume with a single inlet and single outlet of cross-sectional areas \( A_1 \) and \( A_2 \), this reduces to \( \rho_1 \bar{u}_1 A_1 = \rho_2 \bar{u}_2 A_2 \). For incompressible flow the densities cancel, leaving \( \bar{u}_1 A_1 = \bar{u}_2 A_2 = Q \), where \( Q \) is the volumetric flow rate.

The cardiovascular system provides a striking illustration. The aorta, with cross-sectional area of roughly \( 3\;\mathrm{cm}^2 \), carries the full cardiac output of about \( 5\;\mathrm{L/min} \), implying a mean velocity near \( 0.3\;\mathrm{m/s} \). The capillaries, in contrast, have individual areas of order \( 3\times 10^{-7}\;\mathrm{cm}^2 \), but there are about \( 10^{10} \) of them in parallel, and the total cross-sectional area is therefore of order \( 3000\;\mathrm{cm}^2 \). By continuity the mean velocity must drop by a factor of \( 1000 \), to about \( 0.3\;\mathrm{mm/s} \) — slow enough to permit diffusive exchange of oxygen and carbon dioxide.

3.2 Differential Form

Shrinking the control volume to a point and applying the divergence theorem converts the integral statement into the partial differential continuity equation,

\[ \frac{\partial \rho}{\partial t} + \nabla\cdot(\rho\,\mathbf{u}) = 0. \]

For an incompressible fluid, where \( \mathrm{D}\rho/\mathrm{D}t = 0 \), this simplifies to the solenoidal velocity field condition \( \nabla\cdot\mathbf{u} = 0 \). Blood, like most liquids under physiological stresses, is incompressible to within better than one part in a thousand, so the divergence-free condition is an excellent approximation throughout circulatory flow.

---

Chapter 4: Conservation of Momentum

4.1 Integral Momentum Equation

Applying the Reynolds transport theorem with \( B = m\mathbf{u} \) and using Newton’s second law yields

\[ \sum \mathbf{F}_{\mathrm{ext}} = \frac{\partial}{\partial t}\int_{CV}\rho\,\mathbf{u}\;\mathrm{d}V + \int_{CS}\rho\,\mathbf{u}\,(\mathbf{u}\cdot\mathbf{n})\;\mathrm{d}A. \]

The right-hand side is the sum of the unsteady momentum accumulation inside the control volume and the net rate of momentum flux out of it. The left side includes body forces (usually gravity) and surface forces (pressure and viscous stresses). This integral form is powerful for computing, say, the reaction force exerted on a vascular graft anastomosis, or the thrust produced by a left-ventricular-assist pump outlet.

4.2 The Navier–Stokes Equations

For a Newtonian fluid of constant density and viscosity, Cauchy’s equation of motion together with the Newtonian constitutive law gives the Navier–Stokes equations

\[ \rho\left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot\nabla\mathbf{u}\right) = -\nabla p + \mu\,\nabla^{2}\mathbf{u} + \rho\,\mathbf{g}. \]

Four terms describe four distinct physical effects. The left side, the inertia, combines local acceleration and convective acceleration. On the right side, the pressure gradient drives flow, the viscous term diffuses momentum, and gravity is a body force. Together with continuity, \( \nabla\cdot\mathbf{u} = 0 \), these form a closed system for the velocity and pressure fields, subject to no-slip boundary conditions on solid walls and appropriate inlet or outlet conditions.

4.3 Dimensionless Form and the Reynolds Number

Nondimensionalizing the Navier–Stokes equations by a characteristic length \( L \), velocity \( U \), and pressure \( \rho U^2 \) yields

\[ \frac{\partial \mathbf{u}^{*}}{\partial t^{*}} + \mathbf{u}^{*}\cdot\nabla^{*}\mathbf{u}^{*} = -\nabla^{*} p^{*} + \frac{1}{\mathrm{Re}}\,\nabla^{*2}\mathbf{u}^{*}, \]

where the dimensionless Reynolds number is

\[ \mathrm{Re} = \frac{\rho U L}{\mu} = \frac{UL}{\nu}. \]

The Reynolds number measures the ratio of inertial to viscous forces. In the aorta \( \mathrm{Re}\approx 4000 \), in medium arteries several hundred, in arterioles of order unity, and in capillaries well below one. Low Reynolds number flow in the microcirculation is a Stokes flow regime where inertia is negligible and the Navier–Stokes equations reduce to the linear Stokes equations \( \nabla p = \mu\,\nabla^{2}\mathbf{u} \).

---

Chapter 5: Conservation of Energy

5.1 First Law for a Control Volume

Applying Reynolds transport with \( b = e + u^2/2 + gz \) and equating the material derivative to the net rate of heat addition minus work done yields the control-volume form of the first law. For steady flow with one inlet and one outlet and no external heat or shaft work, this reduces to the Bernoulli equation along a streamline,

\[ \frac{p}{\rho} + \frac{u^{2}}{2} + g z = \mathrm{const}. \]

Each term has units of energy per mass. The pressure term is flow work, the kinetic term is directed kinetic energy, and the gravitational term is potential energy.

Bernoulli is the foundation of Doppler echocardiography’s pressure gradient estimation. Across a stenotic valve where the upstream velocity is negligible, the simplified equation \( \Delta p \approx \tfrac{1}{2}\rho u_{\max}^{2} \) becomes, with \( \rho \) for blood and convenient units,

\[ \Delta p\;[\mathrm{mmHg}] \approx 4\,u_{\max}^{2}\;[\mathrm{m/s}]^{2}. \]

This modified Bernoulli relation is how a cardiologist infers that a peak jet velocity of \( 4\;\mathrm{m/s} \) corresponds to a transvalvular gradient of \( 64\;\mathrm{mmHg} \) — severe aortic stenosis.

5.2 Head Losses and the Extended Bernoulli Equation

When viscosity dissipates mechanical energy, or when pumps add energy, Bernoulli must be extended. Writing each term in units of head (length) and adding a head-loss term \( h_L \) and a pump head \( h_p \),

\[ \frac{p_1}{\rho g} + \frac{u_{1}^{2}}{2g} + z_1 + h_p = \frac{p_2}{\rho g} + \frac{u_{2}^{2}}{2g} + z_2 + h_L. \]

The head loss itself decomposes into major losses due to wall friction over the length of the conduit, and minor losses concentrated at fittings, bends, and expansions.

---

Chapter 6: Dimensional Analysis and Similarity

Dimensional analysis is a way of extracting physical insight from the mere fact that physical laws must be dimensionally consistent. Its workhorse is the Buckingham \( \pi \) theorem, which states that a physical relation among \( n \) variables involving \( k \) fundamental dimensions can be rewritten as a relation among \( n - k \) dimensionless groups.

6.1 Key Dimensionless Groups in Biofluid Mechanics

Several dimensionless groups recur throughout biomedical transport.

6.2 Dynamic Similarity

Two flows are dynamically similar if they share all relevant dimensionless groups. Similarity is the foundation of all in vitro modeling of the cardiovascular system. A scaled-up glycerol-water model of the aorta can reproduce in vivo hemodynamics provided both \( \mathrm{Re} \) and \( \alpha \) match. Because glycerol-water has higher viscosity than blood, the match is achievable at enlarged scale and lower frequency. Particle-image-velocimetry data from such models are routinely used to validate computational fluid dynamics simulations of aneurysms and heart valves.

---

Chapter 7: Viscous Flow in Pipes

7.1 Laminar Flow: The Hagen–Poiseuille Solution

For steady, fully developed, laminar flow of a Newtonian fluid in a long cylindrical tube of radius \( R \), the Navier–Stokes equations reduce to a single ODE in the radial coordinate. Solving with no-slip at the wall and symmetry at the centerline gives the parabolic velocity profile

\[ u(r) = \frac{1}{4\mu}\left(-\frac{\mathrm{d}p}{\mathrm{d}z}\right)\left(R^{2}-r^{2}\right). \]

Integrating over the cross-section yields the volumetric flow rate

\[ Q = \frac{\pi R^{4}}{8\mu}\left(-\frac{\mathrm{d}p}{\mathrm{d}z}\right) = \frac{\pi R^{4}\,\Delta p}{8\mu L}. \]

This is the Hagen–Poiseuille equation, the single most used relation in biomedical fluid mechanics. Two features deserve emphasis. First, the dependence on \( R^{4} \) means that halving a vessel’s radius reduces its flow by a factor of sixteen at constant driving pressure, a fact that makes arteriolar smooth-muscle tone the principal regulator of peripheral resistance. Second, the equation expresses hydraulic resistance \( R_h = 8\mu L/(\pi R^4) \), analogous to electrical resistance, permitting series and parallel vascular network analysis.

7.2 Entry Length and Fully Developed Flow

Flow entering a tube from a reservoir requires an entry length \( L_e \) over which the velocity profile develops from roughly uniform to fully parabolic. Empirically, \( L_e/D \approx 0.06\,\mathrm{Re} \) for laminar flow. In the aorta with \( \mathrm{Re}\approx 4000 \), this implies \( L_e \) of order \( 240 \) diameters, which is far longer than the aorta itself. Arterial flow is therefore never truly fully developed, and in practice the entrance region dynamics dominate.

7.3 Transition and Turbulent Flow

Above \( \mathrm{Re}\approx 2300 \) pipe flow becomes transitional, and by \( \mathrm{Re}\sim 4000 \) it is fully turbulent. Turbulent pipe flow is characterized by a flattened mean velocity profile and a thin viscous sublayer near the wall. The mean wall shear stress is customarily expressed through the Darcy friction factor \( f \) defined by

\[ \Delta p = f\,\frac{L}{D}\,\frac{\rho \bar{u}^{2}}{2}. \]

For laminar flow \( f = 64/\mathrm{Re} \). For turbulent flow the Moody chart or the Colebrook equation provides \( f \) as a function of \( \mathrm{Re} \) and roughness ratio \( \varepsilon/D \). Turbulence is occasionally encountered in the ascending aorta at peak systole and routinely downstream of severely stenosed heart valves, where it contributes to murmurs and to mechanical erythrocyte damage.

---

Chapter 8: Boundary Layers

When a viscous fluid flows past a solid surface, the no-slip condition forces the velocity to match the surface locally, while far from the surface the flow remains approximately inviscid. In between lies a thin region called the boundary layer in which viscous effects are concentrated.

8.1 Prandtl’s Boundary-Layer Approximation

For large Reynolds number flow past a body, scaling arguments show that the boundary-layer thickness \( \delta \) grows as \( \delta/L \sim \mathrm{Re}^{-1/2} \). Within this layer, streamwise diffusion of momentum is negligible compared with wall-normal diffusion, and pressure is approximately uniform across the layer. These simplifications reduce the Navier–Stokes equations to the parabolic boundary-layer equations, which can be integrated from leading edge to trailing edge.

For flow over a flat plate at zero incidence, the Blasius solution yields \( \delta(x)/x = 5.0/\sqrt{\mathrm{Re}_x} \), skin-friction coefficient \( C_f = 0.664/\sqrt{\mathrm{Re}_x} \), and a wall shear stress

\[ \tau_w(x) = 0.332\,\rho U^{2}\,\mathrm{Re}_{x}^{-1/2}. \]

8.2 Separation

When the free-stream pressure rises in the streamwise direction (an adverse pressure gradient), the boundary layer can separate from the surface, forming a recirculating wake. Separation is central to understanding why bluff bodies have high drag and why certain arterial geometries are prone to flow recirculation. Downstream of a stenosis, or at the carotid bifurcation, separation zones create low and oscillatory wall shear stress, a condition that correlates strongly with atherosclerotic plaque localization.

8.3 Wall Shear Stress and the Endothelium

Wall shear stress is not just a quantity of engineering interest; it is a biological signal. Endothelial cells sense shear through mechanotransduction pathways involving ion channels, glycocalyx deformation, and focal-adhesion remodeling. Physiological shear stress is roughly \( 1\)–\(4\;\mathrm{Pa} \) in large arteries. Chronically low or oscillatory shear promotes proinflammatory endothelial phenotype, upregulation of adhesion molecules, and a permissive environment for atherogenesis.

---

Chapter 9: Lift and Drag

Forces on bodies immersed in a flowing fluid are customarily decomposed into a drag component parallel to the oncoming stream and a lift component perpendicular to it. Both are usually reported through dimensionless coefficients,

\[ C_D = \frac{F_D}{\tfrac{1}{2}\rho U^{2} A}, \qquad C_L = \frac{F_L}{\tfrac{1}{2}\rho U^{2} A}. \]

9.1 Stokes Drag and the Low-Reynolds Regime

For a sphere of radius \( a \) moving slowly through a fluid at \( \mathrm{Re}\ll 1 \), the Stokes solution gives

\[ F_D = 6\pi\mu a U. \]

Stokes drag governs the motion of red blood cells separating in a centrifuge, the sedimentation of platelets in a stopped vessel, and the transport of magnetic microspheres used as drug-delivery vehicles. The corresponding drag coefficient is \( C_D = 24/\mathrm{Re} \), a relation that breaks down as inertia becomes important above \( \mathrm{Re}\sim 1 \).

9.2 High-Reynolds Drag

At high Reynolds number, drag is dominated by pressure (form) drag from a separated wake, and the drag coefficient becomes nearly constant. For a smooth sphere \( C_D\approx 0.47 \) over a broad range above \( \mathrm{Re}\sim 10^{3} \), dropping sharply at the drag crisis near \( \mathrm{Re}\sim 3\times 10^{5} \) when the boundary layer transitions to turbulence and delays separation.

9.3 Lift on Red Cells and Leukocyte Margination

In shear flow near a wall, deformable red cells experience an inertial or deformability-induced lift that tends to push them toward the vessel center. Leukocytes, which are stiffer and rounder, are pushed laterally and can therefore migrate toward the vessel wall, a behavior known as margination. This hydrodynamic phenomenon is essential to the leukocyte rolling that precedes adhesion and extravasation at inflamed endothelium.

---

Chapter 10: Compressible Flow Basics

Although liquids are effectively incompressible, compressibility becomes important for gases and for rapid-transient pressure waves. In biomedical contexts, compressible flow enters through pulmonary airway dynamics, through acoustic propagation in tissue, and — more subtly — through the arterial pulse wave, which propagates through a distensible rather than compressible medium but is modeled by analogous mathematics.

10.1 Speed of Sound and Mach Number

In an ideal gas, the isentropic speed of sound is \( c = \sqrt{\gamma R_s T} \), where \( \gamma \) is the specific-heat ratio and \( R_s \) the specific gas constant. The ratio \( \mathrm{Ma} = U/c \) is the Mach number. Flows with \( \mathrm{Ma}<0.3 \) can be treated as incompressible to about one percent accuracy.

For airway flow during normal respiration, velocities in the trachea peak around \( 5\;\mathrm{m/s} \) and \( \mathrm{Ma} \) is small. During a forced cough, however, velocities can reach the local sound speed within the collapsed airway, leading to the well-known flow-limitation plateau in a maximum-expiratory-flow curve.

10.2 Pulse Wave Propagation

The arterial pulse propagates through the elastic arterial wall at the Moens–Korteweg wave speed

\[ c = \sqrt{\frac{E\,h}{2\rho\,R}}, \]

where \( E \) is Young’s modulus of the wall, \( h \) the wall thickness, \( \rho \) the blood density, and \( R \) the vessel radius. Typical values in the aorta give \( c \) between \( 5 \) and \( 10\;\mathrm{m/s} \), much larger than mean blood velocities but very slow compared with sound in water. Pulse-wave velocity is now a clinical biomarker of arterial stiffness, with accelerations associated with hypertension and aging.

---

Chapter 11: Mass Transfer Fundamentals

The second major theme of the course is mass transfer: the movement of chemical species driven by concentration gradients, flow, or electrical potentials. Biomedical mass transfer links physiology (oxygen delivery, drug distribution, waste clearance) and engineering (dialyzers, oxygenators, controlled-release implants).

11.1 Fick’s First Law

The simplest constitutive law for mass transport is Fick’s first law, which states that the diffusive molar flux of species \( A \) in a binary mixture is proportional to the concentration gradient,

\[ \mathbf{J}_A = -D_{AB}\,\nabla c_A. \]

Here \( D_{AB} \) is the binary diffusivity, with SI units of \( \mathrm{m^{2}/s} \). For small solutes in water \( D \) is of order \( 10^{-9}\;\mathrm{m^{2}/s} \); for proteins, because of their larger hydrodynamic radii, \( D \) drops by one or two orders of magnitude. The Stokes–Einstein relation \( D = k_B T/(6\pi\mu a) \) estimates \( D \) from molecular size.

11.2 The Convection–Diffusion Equation

Combining Fick’s law with a species mass balance on a differential control volume yields

\[ \frac{\partial c_A}{\partial t} + \mathbf{u}\cdot\nabla c_A = D_{AB}\,\nabla^{2} c_A + r_A. \]

The first term is unsteady accumulation, the second convective transport, the third diffusion, and the last any homogeneous reaction. The balance between convection and diffusion is measured by the Péclet number \( \mathrm{Pe} = UL/D \). In capillaries \( \mathrm{Pe} \) is often of order unity to ten for small metabolites, so both mechanisms matter.

11.3 Fick’s Second Law and Simple Diffusion Problems

For purely diffusive transport with no reaction in a stationary medium,

\[ \frac{\partial c_A}{\partial t} = D_{AB}\,\nabla^{2} c_A, \]

a parabolic equation with the same mathematical structure as heat conduction. Classic solutions include one-dimensional diffusion into a semi-infinite medium, giving \( c_A(x,t) = c_{A,\infty} + (c_{A,0}-c_{A,\infty})\,\mathrm{erfc}\!\left(x/2\sqrt{Dt}\right) \), and diffusion in a slab, treated by separation of variables as an eigenfunction expansion. The characteristic diffusion time scale is \( \tau_D = L^{2}/D \). For oxygen through \( 100\;\mu\mathrm{m} \) of tissue with \( D\approx 2\times 10^{-9}\;\mathrm{m^{2}/s} \), \( \tau_D \) is around \( 5\;\mathrm{s} \); this is why the microcirculatory capillary spacing cannot much exceed this scale without metabolic penalty.

11.4 Mass-Transfer Coefficients and the Sherwood Number

For practical engineering correlations, one often writes the wall flux as \( J_A = k_c(c_{A,s}-c_{A,\infty}) \), where \( k_c \) is the mass-transfer coefficient. The dimensionless counterpart is the Sherwood number \( \mathrm{Sh}=k_c L/D \). For laminar flow in a circular tube with constant wall concentration and fully developed profiles the asymptotic \( \mathrm{Sh} \) is \( 3.66 \); for constant wall flux it is \( 4.36 \). Analogously to heat transfer, developing-flow correlations such as the Graetz solution give \( \mathrm{Sh} \) as a function of \( \mathrm{Re}\,\mathrm{Sc}\,D/L \).

---

Chapter 12: Cardiovascular System — Macrocirculation

12.1 Arterial Hemodynamics and the Windkessel

The aorta and large arteries are elastic reservoirs that store blood during systole and release it during diastole, smoothing the intermittent output of the ventricle into a more continuous peripheral flow. The simplest lumped-parameter description is the two-element Windkessel, an electrical analogy in which arterial compliance \( C \) is a capacitor and peripheral resistance \( R \) is a resistor. A step change in cardiac output then yields an exponential pressure decay during diastole with time constant \( \tau = RC \). Three- and four-element Windkessels improve the fit by adding a characteristic aortic impedance and an inertance term, and are the workhorses of reduced-order cardiovascular modeling.

12.2 Pulsatile Flow and the Womersley Solution

Steady Poiseuille flow is inadequate to describe the aorta, where the flow is periodic at roughly one Hertz. Writing the pressure gradient as a Fourier series and solving the Navier–Stokes equations in cylindrical coordinates, Womersley obtained for each harmonic a velocity profile involving Bessel functions,

\[ u_n(r,t) = \frac{\mathrm{Re}\!\left[\tilde{G}_n\left(1-\dfrac{J_0(i^{3/2}\alpha\, r/R)}{J_0(i^{3/2}\alpha)}\right)e^{i\omega_n t}\right]}{i\omega_n \rho}, \]

where \( \tilde{G}_n \) is the complex pressure-gradient amplitude and \( \alpha = R\sqrt{\omega\rho/\mu} \) is the Womersley number. For \( \alpha\ll 1 \) the profile is quasi-parabolic and tracks the pressure gradient in phase. For \( \alpha\gg 1 \) it is nearly flat except in a thin wall layer of thickness \( \sqrt{\nu/\omega} \), and velocity lags the pressure gradient by roughly \( 90^{\circ} \). In the human aorta at one Hertz \( \alpha\approx 20 \), firmly in the inertia-dominated regime, while in small arterioles \( \alpha<1 \).

12.3 Wave Propagation, Reflections, and Pulse Amplification

In reality the arterial tree is a distensible, branched, and tapered network. The pulse wave propagates, reflects from bifurcations and terminations, and generates the complex patterns observed in pressure and flow waveforms. A surprising consequence is pulse-pressure amplification: systolic pressure in the brachial artery typically exceeds that in the aorta by \( 10\)–\(20\;\mathrm{mmHg} \) in a healthy young adult, as forward and reflected waves constructively interfere at the periphery. Arterial aging flattens this amplification, partly explaining why aortic and brachial pressures are less reliably distinguished in older subjects.

---

Chapter 13: Blood Rheology

Blood is a suspension of roughly forty-percent by volume cellular elements, primarily erythrocytes, in plasma. Its mechanical behavior cannot be captured by a single viscosity.

13.1 Non-Newtonian Character

At low shear rates below about \( 10\;\mathrm{s^{-1}} \), erythrocytes aggregate into rouleaux, raising apparent viscosity sharply. At moderate shear rates rouleaux break up and cells deform and align, lowering viscosity. At shear rates above \( 100\;\mathrm{s^{-1}} \), whole-blood viscosity approaches an asymptotic value near \( 3.5\;\mathrm{mPa\cdot s} \), roughly three to four times the plasma viscosity. Phenomenological constitutive models include the Casson model,

\[ \sqrt{\tau} = \sqrt{\tau_y} + \sqrt{\mu_C \dot{\gamma}}, \]

which introduces a yield stress \( \tau_y \), and the Carreau–Yasuda model, which fits the full shear-thinning curve with four parameters. For large arteries shear rates are high enough that a Newtonian approximation with \( \mu=3.5\;\mathrm{mPa\cdot s} \) is usually acceptable; in small arterioles and in stasis zones a non-Newtonian model is essential.

13.2 The Fåhræus and Fåhræus–Lindqvist Effects

In tubes smaller than about \( 300\;\mu\mathrm{m} \), two remarkable phenomena occur. The Fåhræus effect is the observation that the tube hematocrit is lower than the feed hematocrit: because red cells travel in the faster central core, they are overrepresented in the flux relative to their spatial average. The Fåhræus–Lindqvist effect is the related decrease of apparent viscosity with decreasing tube diameter, reaching a minimum near \( 6\)–\(10\;\mu\mathrm{m} \) (the cell diameter scale) before rising again in capillaries where red cells must squeeze single-file. Both effects arise from the cell-free plasma layer that forms near the wall due to lift forces on deformable erythrocytes.

---

Chapter 14: Microcirculation and Transport in Tissue

14.1 Capillary Exchange and the Starling Hypothesis

Solute and water exchange between capillary lumen and interstitium is governed by a balance of hydrostatic and osmotic (oncotic) pressures across the capillary wall. Starling’s principle states that the net filtration rate per unit area is

\[ J_v/A = L_p\left[(P_c - P_i) - \sigma\,(\pi_c - \pi_i)\right], \]

where \( L_p \) is the hydraulic conductivity, \( \sigma \) the reflection coefficient, \( P_c \) and \( P_i \) are capillary and interstitial hydrostatic pressures, and \( \pi_c \) and \( \pi_i \) are the corresponding oncotic pressures. The reflection coefficient encodes how selectively the wall retains plasma proteins: \( \sigma=1 \) for perfectly impermeable, \( \sigma=0 \) for freely permeable.

Classically one pictures net filtration at the arteriolar end and net reabsorption at the venular end. Modern revised Starling theory emphasizes the role of the endothelial glycocalyx as the true semipermeable barrier, with implications for why plasma-volume expansion by albumin infusion is less effective than originally predicted.

14.2 Solute Transport Across Capillary Walls

For small lipophilic solutes, the capillary wall behaves as a thin homogeneous membrane, and the flux is \( J_A = P_m(c_{A,c}-c_{A,i}) \) where \( P_m \) is the permeability. For polar solutes and macromolecules, transport proceeds through water-filled pores or by transcytosis, and the Kedem–Katchalsky formalism combines diffusive and convective (solvent-drag) contributions,

\[ J_A = P_m\,\Delta c_A + (1-\sigma)\,J_v\,\bar{c}_A. \]

14.3 The Krogh Tissue Cylinder

In 1919 August Krogh proposed a simple but durable model of oxygen delivery. Tissue surrounding a capillary is idealized as a cylinder of radius \( R_t \), with the capillary at the axis. Oxygen diffuses radially outward into tissue, where it is consumed at a rate \( M \) (per unit volume). Assuming steady state and axial invariance, the diffusion equation becomes

\[ \frac{D}{r}\frac{\mathrm{d}}{\mathrm{d}r}\left(r\,\frac{\mathrm{d}c}{\mathrm{d}r}\right) = M. \]

Integrating with the boundary conditions \( c(R_c) = c_c \) and \( \mathrm{d}c/\mathrm{d}r|_{R_t}=0 \) yields a parabolic radial oxygen profile. The Krogh model predicts a critical tissue radius beyond which the cells at the periphery become hypoxic — the conceptual basis for why tumors larger than a few hundred micrometers must develop their own vasculature, the phenomenon of angiogenesis targeted by modern oncology drugs.

14.4 Convective Transport in Pores

The hydraulic conductivity of the capillary wall is itself computed from a pore model. For cylindrical pores of radius \( r_p \), the Hagen–Poiseuille relation gives the flow per pore, and the sum over \( N \) pores per unit area leads to \( L_p = N\pi r_p^{4}/(8\mu h) \), with \( h \) the wall thickness. Solute transport through such pores is restricted by steric and hydrodynamic interactions, producing a diameter-dependent reflection coefficient \( \sigma(r_s/r_p) \). This is the bridge from the macroscopic Starling equation to the molecular architecture of the endothelium.

---

Chapter 15: Artificial Organs and Clinical Applications

The fluid-mechanics and mass-transfer fundamentals developed in previous chapters underlie the design of many therapeutic devices.

15.1 Hemodialysis

A dialyzer is a hollow-fiber mass-exchanger in which the patient’s blood flows through the lumens of thousands of small semipermeable fibers and dialysate flows countercurrently on the shell side. The mass-transfer performance is characterized by the clearance

\[ K = Q_B\,\frac{c_{B,\mathrm{in}}-c_{B,\mathrm{out}}}{c_{B,\mathrm{in}}}, \]

and by the dimensionless number \( K_0 A/Q_B \), where \( K_0 A \) is the mass-transfer-area coefficient. For urea, clearances of \( 200\;\mathrm{mL/min} \) over a three-to-four-hour session are typical, and the prescribed dose is usually quantified by \( Kt/V \), the fraction of the body water pool cleared.

15.2 Blood Oxygenators

Membrane oxygenators for cardiopulmonary bypass use either microporous hollow fibers or nonporous silicone sheets. Oxygen flux is limited by a series combination of resistances: gas-side, membrane, plasma film, and red-cell internal diffusion bound by the reaction with hemoglobin. The dominant resistance is usually the blood-side boundary layer, and oxygenator designs prioritize blood-path geometries that promote secondary flows and reduce boundary-layer thickness without generating damaging shear.

15.3 Mechanical Circulatory Support

Ventricular assist devices are turbomachines, usually continuous-flow centrifugal or axial pumps, that augment or replace ventricular output. Their design must simultaneously deliver the required flow and pressure rise, avoid hemolysis (requiring shear stresses below about \( 150\;\mathrm{Pa} \) for sustained exposure), and minimize stagnation zones that could promote thrombosis. The power number and flow coefficient familiar from classical turbomachinery carry over directly to this biomedical domain, and similarity allows scaled-up in vitro testing of prototypes with blood analogs.

15.4 Controlled Drug Release

A final example synthesizes many of the course’s ideas. A drug-releasing implant — a stent coating, a subcutaneous depot, or a polymer matrix — releases drug by some combination of diffusion through the polymer, convection through surrounding tissue, and reaction or binding. The local concentration field is governed by the convection–diffusion–reaction equation developed in Chapter 11, and the therapeutic window is set by the interplay of release kinetics, tissue transport, and clearance. Designing a safe and effective implant therefore requires the full machinery of biomedical transport that this course has laid out.

---

Closing Perspective

The biomedical transport enterprise sketched here begins with the static pressure exerted by a column of mercury and ends with the shear-sensitive biochemistry of an endothelial cell. It is unified by a small set of conservation laws — mass, momentum, energy, and species — combined with constitutive relations and, crucially, the willingness to nondimensionalize. The Reynolds, Womersley, and Péclet numbers recur again and again as signposts that tell us which physical mechanisms matter and which can be safely ignored. A biomedical engineer who has internalized these dimensionless groups, together with the archetypal solutions — Poiseuille’s parabola, Blasius’s boundary layer, Stokes’s drag, Krogh’s cylinder, Womersley’s Bessel functions, and Starling’s balance — can confront almost any new circulatory or mass-transfer problem with an informed first estimate. The remainder is careful thinking and, when necessary, numerical simulation.