Unit 6: Biofluidic mechanics - Subjective Questions

A Newtonian viscous fluid is a fluid in which the shear stress is linearly proportional to the rate of shear strain (velocity gradient), and the constant of proportionality (viscosity) is independent of the shear rate.

Constitutive relation:

where:

• = shear stress
• = dynamic (absolute) viscosity — a constant at a given temperature
• = velocity gradient (shear rate)

Key characteristics:

• The relationship between and shear rate is a straight line passing through the origin.
• Viscosity does not change with applied shear stress or time.

Examples:

• Water
• Air
• Plasma (approximately Newtonian at physiological shear rates)

A non-viscous fluid (also called an ideal or inviscid fluid) is a hypothetical fluid that has zero viscosity (), meaning it offers no resistance to shear and no internal friction between adjacent fluid layers.

Properties of a non-viscous fluid:

• No shear stress is developed regardless of the velocity gradient: .
• No energy is lost due to friction.
• Flow is frictionless and often assumed incompressible.

Why it is a useful idealization:

• Simplifies the governing equations (e.g., Euler's equation instead of Navier–Stokes).
• Allows application of Bernoulli's equation to estimate pressure–velocity relationships in large vessels.
• Provides a first-approximation model for flow in large arteries where viscous effects are relatively small compared to inertial effects (high Reynolds number regions).

Limitation: Real biological fluids like blood always have viscosity, so this model fails in small vessels (capillaries, arterioles) where viscous effects dominate.

Newtonian fluids have a constant viscosity independent of shear rate, whereas non-Newtonian fluids exhibit a viscosity that varies with the applied shear rate.

Constitutive relations:

• Newtonian:
• Non-Newtonian (power law): , where

Shear stress vs shear rate diagram (conceptual):

• Newtonian: straight line from origin.
• Shear-thinning (pseudoplastic): concave-down curve (viscosity decreases with shear).
• Bingham plastic: requires a yield stress before flow begins.

Where blood fits:

• Blood is a non-Newtonian, shear-thinning fluid. At low shear rates, red cells aggregate (rouleaux) raising viscosity; at high shear rates, cells align and deform, lowering viscosity.
• It also exhibits a small yield stress, so it can be approximated as a Casson fluid in small vessels.

Rheology is the study of deformation and flow of matter. Blood is a non-Newtonian, shear-thinning fluid and its rheological behaviour is complex due to its composition (plasma + cellular elements).

Key rheological properties of blood:

• Shear-thinning behaviour: Apparent viscosity decreases as shear rate increases.
• Yield stress: A minimum stress is required to initiate flow due to red cell aggregation.
• Thixotropy: Viscosity depends on the history of shearing (time-dependent).
• Viscoelasticity: Blood exhibits both viscous and elastic behaviour due to deformable RBCs.

Factors influencing blood viscosity:

• Hematocrit (Hct): The volume fraction of red blood cells is the single most important factor — viscosity rises sharply with increasing hematocrit.
• Shear rate: At low shear, rouleaux formation increases viscosity; at high shear, viscosity drops.
• Temperature: Viscosity decreases as temperature increases.
• Vessel diameter (Fåhraeus–Lindqvist effect): In vessels < 300 μm, apparent viscosity decreases with decreasing diameter.
• Plasma protein concentration: Fibrinogen and globulins promote RBC aggregation, raising viscosity.
• RBC deformability and aggregation: Rigid cells (e.g., in disease) increase viscosity.

Casson model is often used to describe blood:

where is the yield stress and is the Casson viscosity.

The Fåhraeus–Lindqvist effect describes the observation that the apparent viscosity of blood decreases as the diameter of the tube (blood vessel) decreases, for vessel diameters below approximately 300 μm (down to about 10 μm).

Explanation:

• In small vessels, red blood cells tend to migrate toward the centre of the vessel, leaving a cell-free (plasma) layer near the wall.
• This plasma-rich marginal layer has a lower viscosity and acts as a lubricating layer.
• As a result, the effective (apparent) viscosity of the flowing blood is reduced.

Related — Fåhraeus effect: The average hematocrit within a small tube is less than that of the feed reservoir, because faster-moving central RBCs spend less time in the tube.

Physiological significance:

• Reduces the workload on the heart, since blood flows more easily through the microcirculation than a constant-viscosity model would predict.
• Facilitates perfusion of tissues through narrow capillaries and arterioles.
• Below ~10 μm the effect reverses (inversion) because cells must squeeze single-file, increasing resistance.

Blood vessel walls (arteries and veins) are generally composed of three concentric layers (tunics), each with a distinct structure and function.

1. Tunica Intima (innermost layer):

• A single layer of endothelial cells resting on a basement membrane.
• Provides a smooth, low-friction surface for blood flow.
• Regulates vascular tone, permeability, and prevents clotting.

2. Tunica Media (middle layer):

• Composed mainly of smooth muscle cells, elastin, and collagen fibres.
• Thickest layer in arteries; controls vasoconstriction and vasodilation.
• Elastin provides recoil (important in large arteries); collagen provides tensile strength.

3. Tunica Adventitia (Externa) (outermost layer):

• Composed of collagen and elastic fibres, connective tissue.
• Contains the vasa vasorum (small vessels supplying the wall) and nerves.
• Anchors the vessel to surrounding tissue and prevents overstretching.

Compositional building blocks:

• Elastin: low stiffness, highly extensible, provides recoil.
• Collagen: high stiffness, provides strength at high pressures.
• Smooth muscle: active tension control.
• Endothelium: biochemical regulation.

The relative proportion of these components varies with vessel type — elastic arteries are elastin-rich, muscular arteries and arterioles are smooth-muscle-rich, and veins have thinner walls with more collagen.

Vascular remodeling refers to the active structural adaptation of a blood vessel — changes in its geometry (diameter, wall thickness) and composition — in response to sustained changes in the mechanical or biochemical environment. It is a dynamic process involving cell growth, death, migration, and changes in extracellular matrix.

Types of remodeling:

• Hypertrophic: wall thickens (increase in mass).
• Eutrophic: rearrangement without change in mass.
• Hypotrophic: wall thins (decrease in mass).
• Inward vs outward: lumen narrows or widens.

Mechanical stimuli driving remodeling:

• Wall shear stress (): exerted by flowing blood on the endothelium.
  
  Vessels adjust radius to maintain a constant baseline shear stress. Increased flow → outward remodeling (larger radius).
• Circumferential (hoop) stress: due to blood pressure acting on the wall.
  
  Increased pressure → wall thickening to normalize hoop stress.
• Axial (longitudinal) stress: governs vessel length adaptation.

Significance:

• Maintains mechanical homeostasis (constant shear and hoop stress).
• Underlies adaptation in hypertension, atherosclerosis, and after surgery/exercise.
• Explains growth-related and disease-related changes in vessel architecture.

A fluid is a substance that continuously deforms (flows) under the action of a shear stress, however small. Fluids include both liquids and gases. Their nature can be classified along several dimensions.

A. Classification based on viscosity:

• Ideal (non-viscous) fluid: ; no shear resistance; a hypothetical fluid.
• Real (viscous) fluid: possesses viscosity and offers resistance to shear.
• Newtonian fluid: ; viscosity constant (e.g., water, plasma).
• Non-Newtonian fluid: viscosity varies with shear rate (e.g., blood).
  • Shear-thinning (pseudoplastic)
  • Shear-thickening (dilatant)
  • Bingham plastic (needs yield stress)

B. Classification based on compressibility:

• Incompressible fluid: density essentially constant (most liquids, blood).
• Compressible fluid: density varies with pressure (gases).

C. Other properties defining fluid nature:

• Density () and specific weight
• Surface tension
• Vapour pressure
• Elasticity / bulk modulus

In biofluid mechanics, blood is typically treated as an incompressible, non-Newtonian fluid, while air (in respiration) is treated as compressible and Newtonian.

Propulsion in a fluid medium refers to the mechanisms by which organisms (or devices) generate thrust to move through a fluid such as water or air. The propulsion strategy strongly depends on the Reynolds number, which characterizes the ratio of inertial to viscous forces.

1. High Reynolds number regime (inertia dominated):

• Applies to large swimmers (fish, whales) and flyers (birds).
• Thrust is generated by imparting momentum to the fluid (Newton's third law) — pushing fluid backward propels the body forward.
• Uses undulatory motion, oscillating fins/tails, and lift-based propulsion.

2. Low Reynolds number regime (viscosity dominated):

• Applies to microorganisms (bacteria, spermatozoa).
• Viscous forces dominate; inertia negligible.
• Reciprocal (back-and-forth) motion produces no net movement (Purcell's scallop theorem).
• Requires non-reciprocal motion — e.g., rotating flagella (helical corkscrew) or beating cilia in coordinated metachronal waves.

Key principles:

• Conservation of momentum: thrust = rate of momentum imparted to fluid.
• Drag balance: at steady speed, thrust = drag.
• Efficiency depends on matching the propulsion mechanism to the flow regime.

Biological examples: fish tail propulsion (high Re), bacterial flagellar rotation (low Re), ciliary transport in respiratory tract.

Arterioles are small-diameter (~10–100 μm) branches of arteries that lead into capillaries. They possess distinctive mechanical properties that make them the principal regulators of blood flow and pressure.

Mechanical / structural properties:

• Thick smooth muscle layer relative to lumen size — the tunica media is proportionally the thickest, dominated by circular smooth muscle.
• High vasoactivity: capable of significant vasoconstriction and vasodilation via smooth muscle tone.
• Low elastin content compared with large elastic arteries.
• Wall behaves viscoelastically — exhibits both elastic recoil and time-dependent (viscous) response.
• Maintain a basal (myogenic) tone, contracting in response to stretch (Bayliss effect).

Why they are called resistance vessels:

• According to Poiseuille's law, resistance is inversely proportional to the fourth power of radius:
  
• Because arterioles can dramatically change their radius, even small changes cause large changes in resistance ().
• They contribute the greatest drop in blood pressure along the circulation.
• By regulating resistance, arterioles control local tissue perfusion and help maintain systemic arterial blood pressure.

Capillaries are the smallest blood vessels (diameter ~5–10 μm), forming the site of exchange between blood and tissues.

Structure:

• Wall consists of only a single layer of endothelial cells on a basement membrane — no tunica media or adventitia.
• Extremely thin wall (~0.5 μm) to minimize diffusion distance.
• Lumen barely wide enough for red blood cells to pass (often single-file).
• Types: continuous, fenestrated, and sinusoidal (discontinuous) — varying permeability.

Mechanical properties:

• Low distensibility / high relative stiffness — despite thin walls, they resist over-distension because they are supported by surrounding tissue and are of small radius.
• The Law of Laplace explains why thin capillary walls tolerate pressure:
  
  Because the radius is very small, wall tension remains low even at moderate pressure, so a thin wall suffices.
• Nearly inextensible compared with arteries; pressure changes cause little diameter change.

How structure suits function:

• Thin, single-cell wall → efficient diffusion of , , nutrients, and waste.
• Small radius → keeps wall tension low (Laplace) and slows flow, maximizing exchange time.
• Fenestrations/pores → selective filtration and reabsorption (governed by Starling forces).
• Huge total cross-sectional area → low velocity, ideal for exchange.

Veins are the vessels that return blood from tissues back to the heart. Their mechanical properties differ significantly from arteries.

Structural / mechanical properties:

• Thin walls with relatively little smooth muscle and elastin, but more collagen.
• Highly distensible (compliant) — large volume change for a small change in pressure.
• Operate at low internal pressure (~5–15 mmHg).
• Contain valves (in limb veins) that prevent backflow.
• At low pressures the cross-section is collapsed/elliptical, becoming circular as they fill — this gives a highly non-linear pressure–volume relationship.

Compliance:

Veins have a much higher compliance than arteries (roughly 20–24 times), meaning they accommodate large volumes with little pressure rise.

Role as capacitance vessels:

• At rest, veins hold about 60–70% of the total blood volume, serving as a blood reservoir.
• Through venoconstriction, they can shift stored blood back to the central circulation, adjusting venous return and cardiac output.
• This capacitance function is vital for maintaining circulatory stability during hemorrhage, posture change, and exercise.

Aids to venous return: skeletal muscle pump, respiratory (thoracic) pump, and venous valves.

Consider a thin-walled cylindrical blood vessel of internal radius , wall thickness (with ), and length , subjected to an internal (transmural) pressure .

Setup:
Imagine cutting the cylinder longitudinally into two halves. For the upper half to be in equilibrium, the force due to internal pressure pushing the halves apart must be balanced by the circumferential (hoop) tension in the two cut wall edges.

Force due to pressure (acting on projected area):

(The projected area of the curved surface onto the diametral plane is .)

Force due to wall tension:
The hoop stress acts over the two wall cross-sections, each of area :

Equilibrium ():

Solving for the circumferential stress:

Wall tension form (Law of Laplace): If we define tension per unit length , then:

Interpretation:

• Hoop stress increases with pressure and radius, and decreases with wall thickness.
• Explains why aneurysms (large ) are prone to rupture, and why thick-walled or small-radius vessels (capillaries) sustain low wall stress.

The Reynolds number () is a dimensionless quantity that expresses the ratio of inertial forces to viscous forces in a flowing fluid.

where:

• = fluid density
• = mean flow velocity
• = characteristic diameter of the vessel
• = dynamic viscosity, = kinematic viscosity

Significance:

• Predicts the flow regime (laminar vs turbulent).
• Governs similarity/scaling in fluid experiments and models.
• Determines the dominant propulsion mechanism for organisms in fluids.

Critical values (flow in a pipe/vessel):

• → Laminar flow (smooth, orderly, layered flow).
• → Transitional flow.
• → Turbulent flow (chaotic, mixing).

In the circulation:

• Blood flow is normally laminar throughout most vessels.
• Turbulence may occur in the aorta during peak systole or at sites of stenosis/branching, producing audible murmurs/bruits.
• Low dominates in capillaries where viscous forces prevail.

Assumptions: Steady, laminar, fully developed flow of an incompressible Newtonian fluid through a rigid horizontal cylindrical tube of radius and length ; no slip at the wall.

Force balance on a coaxial fluid cylinder of radius :
The pressure force driving the flow is balanced by the viscous shear force on the surface:

Let . Rearranging:

Integrate with boundary condition at (no slip):

This is the parabolic velocity profile.

Volumetric flow rate by integrating velocity over the cross-section:

Relevance to blood flow:

• Shows flow — small radius changes (arterioles) drastically alter flow, explaining vascular resistance control.
• Hydraulic resistance: .
• Limitations: blood is non-Newtonian, vessels are elastic (not rigid), flow is pulsatile, and entrance effects exist — so Poiseuille's law is only an approximation for larger vessels at steady flow.

Blood vessel walls are viscoelastic — they exhibit both elastic (energy-storing, spring-like) and viscous (energy-dissipating, dashpot-like) responses to loading. This arises from the combination of elastin, collagen, smooth muscle, and interstitial fluid in the wall.

Key viscoelastic phenomena:

1. Hysteresis:

• The loading and unloading stress–strain curves do not coincide; they form a loop.
• The area within the loop represents energy dissipated as heat during each cycle.
• Physiologically important in pulsatile arterial loading.

2. Creep:

• Under a constant applied stress, strain (deformation) continues to increase gradually with time.
• Reflects the time-dependent rearrangement of wall constituents.

3. Stress Relaxation:

• Under a constant applied strain (fixed stretch), the stress within the wall decreases gradually over time.
• Important in maintaining stable wall tension.

Modeling:
Viscoelastic behaviour is commonly modeled using combinations of springs and dashpots:

• Maxwell model (spring + dashpot in series) — captures stress relaxation.
• Kelvin–Voigt model (spring + dashpot in parallel) — captures creep.
• Standard linear solid (Kelvin model) — captures both creep and relaxation more realistically.

Significance:

• Enables the vessel to smooth out pulsatile pressure (Windkessel effect).
• Protects the wall from sudden loading by dissipating energy.

Arteries and veins differ substantially in structure and mechanical behaviour because they operate under different pressure and flow conditions.

Summary:

• Arteries are stiffer, thicker, and elastic to withstand and smooth high pulsatile pressure (Windkessel effect).
• Veins are thin, compliant capacitance vessels that store blood and adjust venous return, relying on valves and muscle pumps to move low-pressure blood back to the heart.

The Windkessel effect (German for air chamber) describes how the elastic large arteries (e.g., the aorta) store energy during systole and release it during diastole, thereby converting the intermittent, pulsatile output of the heart into a more continuous, smoother flow in the peripheral circulation.

Mechanism:

• During systole: the heart ejects blood rapidly. The elastic aortic wall stretches and stores a portion of the stroke volume as elastic potential energy (the wall bulges outward).
• During diastole: the heart stops ejecting. The stretched arterial wall recoils, pushing the stored blood forward, maintaining pressure and flow to the tissues.

Role of elastic properties:

• Depends on the high elastin content and compliance of large arteries.
• Compliance — a higher compliance means more effective buffering.

Importance:

• Smooths pulsatile flow, reducing large pressure swings.
• Reduces cardiac workload by dampening peak pressures.
• Maintains diastolic pressure, ensuring continuous coronary and peripheral perfusion.

Clinical relevance:

• With ageing/arteriosclerosis, arteries stiffen (compliance ↓), the Windkessel effect weakens, systolic pressure rises, diastolic pressure falls, and pulse pressure widens — increasing cardiac load.

Blood is a suspension of cellular elements in plasma, making it a two-phase, non-Newtonian fluid. Its composition directly determines its rheological behaviour.

A. Plasma (~55% of blood volume):

• Composition: ~92% water, plus proteins (albumin, globulins, fibrinogen), electrolytes, nutrients, and wastes.
• Mechanical role: Plasma alone behaves as a nearly Newtonian fluid with a viscosity ~1.2 mPa·s. Proteins such as fibrinogen promote red cell aggregation, increasing whole-blood viscosity at low shear.

B. Formed elements (~45%):

1. Red Blood Cells (Erythrocytes) — most abundant (~99% of cells):

   • Determine the hematocrit, the dominant factor in blood viscosity.
   • Deformable biconcave discs — deformability lowers viscosity at high shear; aggregation (rouleaux) raises it at low shear.

2. White Blood Cells (Leukocytes):

   • Few in number; minor contribution to bulk viscosity but affect flow in capillaries (stiffer, larger).

3. Platelets (Thrombocytes):

   • Involved in clotting; negligible effect on normal viscosity.

Overall rheological consequence:

• The shear-thinning behaviour of blood arises from RBC aggregation/disaggregation and deformation.
• Yield stress arises from the network of aggregated RBCs at rest.
• Hematocrit is the single most influential compositional determinant of viscosity: viscosity rises steeply as hematocrit increases.

Thus, blood's mechanical behaviour is a direct result of the interaction between the Newtonian plasma and the deformable, aggregating cellular suspension.

Wall shear stress () is the tangential frictional force per unit area exerted by flowing blood on the endothelial surface of the vessel wall. It is a critical mechanical signal sensed by endothelial cells (mechanotransduction).

Derivation for Poiseuille flow:
For laminar flow of a Newtonian fluid in a tube of radius , the velocity profile is:

Wall shear stress is the viscosity times the velocity gradient at the wall ():

Expressing in terms of flow rate using :

Significance and role:

• Endothelial cells sense and release vasoactive substances (e.g., nitric oxide for vasodilation).
• Vessels remodel to restore a normal baseline shear stress (~1–2 Pa): increased flow → outward remodeling (radius ↑); decreased flow → inward remodeling.

Role in disease:

• Low or oscillatory shear stress regions (arterial bends, bifurcations) are prone to atherosclerotic plaque formation.
• Abnormally high shear can damage endothelium and platelets.
• Understanding distribution guides prediction of disease sites and design of stents/grafts.