Unit 6: Biofluidic mechanics

Biofluid mechanics applies the laws of fluid statics and dynamics to biological flows—principally the transport of blood through the cardiovascular network. Its foundation is the continuum hypothesis: fluid is treated as a continuous medium whose local properties (velocity, pressure, density) are defined at every point. The governing conservation statements are continuity (mass) and the Navier–Stokes equations (momentum).

• Continuity (incompressible flow): mass is conserved, so ∇·v = 0, where v is the velocity vector field (m/s).
• Navier–Stokes: momentum balance for a Newtonian fluid: ρ(∂v/∂t + v·∇v) = −∇p + μ∇²v + f; ρ = density (kg/m³), p = pressure (Pa), μ = dynamic viscosity (Pa·s), f = body force per volume.
• Reynolds number: Re = ρvD/μ predicts laminar (Re < 2000) vs turbulent flow; large arteries operate around Re ≈ 1000–4000.
• No-slip condition: fluid velocity equals the wall velocity at the boundary, giving parabolic velocity profiles in vessels.
• Newton's law of viscosity: shear stress relates to shear rate, τ = μ(du/dy), the reference behaviour against which all blood rheology is judged.

II. Nature of Fluids and Their Classification

Defining what a fluid is and how viscosity distinguishes fluid types.

A. Nature of fluids

A fluid is a substance that deforms continuously under any applied shear stress, however small.

• Inability to sustain shear: unlike a solid, a fluid cannot resist static shear—it flows indefinitely until the stress is removed.
• Shear stress definition: τ = F/A, tangential force per unit area (Pa); it is the driver of all fluid deformation.
• Shear rate: γ̇ = du/dy, the velocity gradient normal to flow (s⁻¹); rate of angular deformation.
• Compressibility: liquids (blood, plasma) are treated as incompressible (ρ constant); gases are compressible. Biofluid work assumes incompressibility.
• Continuum assumption: valid because vessel diameters (µm–cm) vastly exceed molecular scales, except in the smallest capillaries where cells approach vessel size.

B. Newtonian viscous fluid

A Newtonian fluid is one whose viscosity is constant, independent of shear rate.

• Constitutive law: τ = μγ̇; a straight line through the origin on a τ–γ̇ plot, slope = μ.
• Constant viscosity: μ depends only on temperature and pressure, not on how fast the fluid is sheared.
• Examples: water (μ ≈ 0.001 Pa·s at 20 °C), plasma (μ ≈ 0.0012–0.0015 Pa·s), air; plasma alone behaves near-Newtonian.
• Poiseuille flow: for steady laminar flow in a rigid tube, Q = πΔPr⁴/(8μL), where Q = volumetric flow (m³/s), ΔP = pressure drop, r = radius, L = length. The r⁴ term shows radius dominates resistance.
• Worked example: halving vessel radius reduces flow to (1/2)⁴ = 1/16 of the original at fixed ΔP—the basis of vasoconstriction as a flow-control mechanism.

C. Non viscous fluid

A non-viscous (inviscid) fluid is an idealisation with zero viscosity, μ = 0.

• No shear stress: τ = 0 regardless of shear rate; no internal friction, no energy dissipation.
• Governing equation: the Euler equation, ρ(∂v/∂t + v·∇v) = −∇p + f—Navier–Stokes with the μ∇²v term dropped.
• Bernoulli's principle: along a streamline, p + ½ρv² + ρgh = constant; valid for inviscid, incompressible, steady flow. Used to relate pressure and velocity across a stenosis.
• Limitation in biology: real blood always has viscosity, so inviscid results (e.g. flat velocity profile, no wall drag) approximate only the fast core of large-artery flow.

III. Rheological Properties of Blood

How blood, a concentrated cell suspension, departs from Newtonian behaviour.

A. Rheological properties of blood

Blood is a non-Newtonian, shear-thinning suspension of cells in plasma whose apparent viscosity varies with flow conditions.

• Composition basis: ~45 % cells by volume (hematocrit) in plasma; red cells (erythrocytes) dominate rheology.
• Shear-thinning: apparent viscosity falls as shear rate rises—at low γ̇, red cells aggregate into stacks (rouleaux) raising viscosity; at high γ̇, cells disperse and deform, lowering it.
• Yield stress: blood behaves like a Casson fluid, needing a threshold stress to start flowing: √τ = √τ_y + √(μ_c γ̇), where τ_y = yield stress (~0.005 Pa), μ_c = Casson viscosity.
• Fåhraeus–Lindqvist effect: in vessels below ~300 µm, apparent viscosity drops as diameter falls because cells migrate to the axis leaving a cell-free plasma layer at the wall.
• Hematocrit dependence: whole-blood viscosity rises sharply with hematocrit; at 45 % it is ~3–4× that of plasma (~0.003–0.004 Pa·s).
• Cell deformability: the biconcave red cell (~8 µm) deforms to pass ~5 µm capillaries; loss of deformability (e.g. sickling) raises effective viscosity.

IV. Blood Vessel Architecture

The layered wall that carries and adapts to the flow described above.

A. Structure and composition of blood vessels

Vessel walls are three concentric tunicae whose relative thickness matches their mechanical role.

• Tunica intima: innermost; a single endothelial layer on a basement membrane, in direct contact with blood; senses shear stress and regulates tone.
• Tunica media: middle; smooth muscle plus elastin and collagen; thickest in arteries, controlling diameter and elastic recoil.
• Tunica externa (adventitia): outer collagenous coat anchoring the vessel and limiting overstretch.
• Structural proteins:
  • Elastin: low stiffness (E ≈ 0.3–0.6 MPa), gives reversible recoil at low strain.
  • Collagen: stiff (E ≈ 1 GPa), recruited at high strain to prevent rupture—source of the nonlinear stress–strain curve.
• Consequence: the elastin-then-collagen recruitment produces the characteristic J-shaped, stiffening stress–strain response of arterial wall.

B. Remodeling of blood vessels

Remodeling is the active, long-term restructuring of the wall in response to sustained mechanical stimuli.

• Mechanotransduction: endothelial cells convert wall shear stress and cyclic circumferential stress into biochemical signals altering matrix synthesis.
• Wall shear stress target: vessels adjust radius to hold τ_w ≈ 1.5–2.0 Pa; per Poiseuille, τ_w = 4μQ/(πr³), so a lasting flow rise widens the lumen (outward remodeling).
• Circumferential stress and Laplace's law: σ_θ = Pr/t, where P = transmural pressure, r = radius, t = wall thickness; chronic hypertension thickens t to normalise σ_θ (hypertrophic remodeling).
• Types: inward vs outward (lumen change) and eutrophic, hypertrophic or hypotrophic (mass change).
• Pathology: maladaptive remodeling underlies atherosclerosis, aneurysm (weakened wall, σ_θ rises with r) and restenosis.

V. Mechanical Properties of Arterioles, Capillary Vessels and Veins

Comparing wall behaviour across the microcirculation and low-pressure return.

A. Mechanical properties of arterioles

Arterioles are the primary resistance vessels, tuned for active diameter control.

• Thick muscular media: high smooth-muscle-to-lumen ratio allows large diameter changes.
• Resistance dominance: since R ∝ 1/r⁴, small radius changes give the largest pressure drop of the circuit—arterioles set peripheral resistance and mean arterial pressure.
• Myogenic tone: raised transmural pressure triggers reflex constriction, autoregulating downstream flow.
• Metabolic control: local vasodilators (CO₂, adenosine, low O₂) relax muscle to match perfusion to tissue demand.

B. Capillary vessels

Capillaries are the exchange vessels, mechanically simple but hydraulically decisive.

• Wall structure: endothelium plus basement membrane only, ~1 µm thick—no media—maximising diffusion.
• Laplace balance: despite tiny wall thickness they resist rupture because small r (~4 µm) keeps σ_θ = Pr/t low.
• Starling exchange: transcapillary flow J_v = K_f[(P_c − P_i) − σ(π_c − π_i)], where P = hydrostatic and π = oncotic pressures, σ = reflection coefficient, K_f = filtration coefficient—filtration at the arterial end, reabsorption at the venous end.
• Single-file flow: red cells traverse in single file, the regime where the Fåhraeus–Lindqvist effect is strongest.

C. Veins

Veins are high-compliance capacitance vessels returning blood at low pressure.

• Thin, distensible wall: little muscle, much collagen; large lumen holds ~65 % of blood volume.
• High compliance: C = ΔV/ΔP; a small pressure rise stores a large volume, buffering venous return.
• Collapsible tube behaviour: at low transmural pressure the lumen collapses to an elliptical or dumbbell cross-section, sharply raising resistance—unlike rigid-tube Poiseuille flow.
• Valves and pumps: one-way valves with the skeletal-muscle pump drive flow against gravity in a low-pressure system.

VI. Propulsion in Fluid Medium

How organisms and pumps generate net motion within a fluid.

A. Propulsion in fluid medium

Propulsion arises from momentum exchange with the surrounding fluid, governed by the Reynolds-number regime.

• Reynolds-regime split:
  1. High Re (fish, birds): inertia dominates; reciprocal strokes work because coasting between strokes carries the body forward.
  2. Low Re (sperm, bacteria): viscosity dominates; the scallop theorem forbids reciprocal motion, so propulsion needs non-reversible strokes such as helical flagellar rotation or travelling flagellar waves.
• Thrust by reaction: by Newton's third law, accelerating fluid backward (F = ṁΔv) pushes the swimmer forward; ṁ = mass flow rate of displaced fluid.
• Drag balance: steady swimming means thrust equals drag; at low Re, drag is linear in velocity (Stokes drag F = 6πμrv for a sphere).
• Cardiac analogue: the heart propels blood by cyclic pressure generation, imparting the momentum that sustains flow against vascular resistance and wall compliance.