fluid flow and Circulatory system

Transcription

1 Importance of the physics of fluids fluid flow and Circulatory system Miklós Kellermayer I. Hemodynamics E.g.: What are the characteristics of blood flow in the circulatory system? II. Motion in viscous fluids E.g.: What is the force exerted by a single spermatocyte during its motion? Biophysics of fluid flow and blood circulation Basic principles Types of fluids Types of fluid flow Laws of fluid flow Blood as a fluid; determinants of blood viscosity The circulatory system; blood vessels as elastic tubes uxiliary forces of circulation streamlines Basic principles I. Volumetric flow rate (Q): ΔV Q = ΔV Δt = Δs Δt = v Δs verage velocity: v = Q flow velocity

2 y v y / Δv/Δy Basic principles II. Viscosity (internal friction) = shear force = area of fluid layer = viscosity = flow velocity = distance between fluid layers = shear stress (τ) = velocity gradient (D) Units of viscosity: v y vmin = Δv Δy = τ D Δv vmax (Newton s friction law) 1Pas = 1 Ns = 10P( poise) 2 m Viscosity of distilled water (25 C): 1 mpas (1 centipoise) 1. Ideal frictionless, incompressible ρ = constant, = 0 2. Non-ideal (real) a. Newtonian (viscous) independent of shear stress b. Non-newtonian (anomalous) changes with shear stress Types of fluids D Relationship between velocity gradient and shear stress in real fluids St. Venant τf Newtonian Bingham Casson τ f = flow threshold Viscoelastic materials: combination of viscous and elastic properties (e.g., solution of polymers, macromolecules) Stress-relaxation: decay of stress in rapidly deformed viscoelastic body. Blood is a non-newtonian fluid; it displays viscoelastic properties. τ Model of viscoelastic body - Kelvin-body: spring and dashpot coupled in parallel Types of fluid flow 1. Stationary Volumetric flow rate stays constant (parameters characterizing flow remain unchaned). 2. Laminar luid layers do not mix. 3. Turbulent luid layers mix. laminar flow turbulent flow Laws of flow in ideal fluids I. Continuity equation Osborne Reynolds ( ) laminar Reynolds number (R): R = vrρ R c 2000 v = flow rate (m/s) r = tube radius (m) ρ = density of fluid (kg/m 3 ) = viscosity (Ns/m 2 ) turbulent R 1 v 1 2 v 2 1v1 = 2v2 = constant = cross-sectional area v = flow velocity "When I meet God, I am going to ask him two questions: Why relativity? nd why turbulence? I really believe he will have an answer for the first." (Werner Heisenberg)

3 Laws of flow in ideal fluids II. Bernoulli s law - law of conservation of energy Laws of flow in viscous fluids I. Stokes law Daniel Bernoulli ( ) p ρv2 + ρgh = konst p = static pressure 1 2 ρv2 ρgh = dynamic pressure = hydrostatic pressure = γv = 6rπv Georg Gabriel Stokes ( ) = force γ = drag coefficient (shape factor) v = flow rate r = radius of sphere = viscosity Venturi effect Stokes force Hydrodynamic drag force (Stokes force): = γv = 6rπv Laws of flow in viscous fluids II. Hagen-Poiseuille s law Thermodynamic current Relevant intensive variable (its difference maintains current) Current density Physical law G.H.L. Hagen ( ) J.-L.-M. Poiseuille ( ) Volumetric flow Pressure (p) J V = R2 Δp Hagen-Poiseuille 8 Δx ffects stationary objects in moving fluid (microbead captured in optical trap) ffects objects moving in stationary fluid (moving spermatocytes) Rigid-wall tube p 1 Paraboloid flow profile R p 2 J V = V t = R2 Δp 8 Δx r = 1.6 μm = 1.6 x 10-6 m v = 50 μm/s = 5 x 10-5 m/s = 10-3 Pas r How much force is exerted by a single spermatotyce during its movement? γ = 6rπ = π 10 3 = Ns/m = γv = Ns/ m m /s = N =1.5 pn V t R p x = volume = time = tube radius = viscosity = pressure = tube length x V/t = Q = volumetric vlow rate Δp/Δx = pressure gradient, maintained by p 1 -p 2 = cross-sectional area of tube JV = flow intensity = R 2 π V t = R4 π Δp 8 Δx

4 Medical significance of fluid flow Bernoulli s law: ormation of aneurysm (pathological expansion of blood vessel): Expansion of vessel: diameter increases low rate decreases, accoring to continuity equation Static pressure increases due to Bernulli s law neurysm pregrediates - positive fedback mechanism leading to catastrophy Hagen-Poiseuille s law: V Bifurcating vessel neurysm t = R4 π Δp 8 Δx low intensity, hence the delivered oxygen quantity, may be drastically reduced in certain pathological conditions: constriction of blood vessels (e.g., diabetes, Bürger s disease) change in blood viscosity (e.g., fever, anaemia) Reduction of vessel diameter by half leads to a reduction of volumetric flow by 1/16! Blood: Blood as a fluid 55-60% of body mass is water 42 kg (70 kg body mass) 2/3 intracellular 28 kg 1/3 plasma 4-5 kg 1/3 extracellular 14 kg 2/3 intersticium 9-10 kg verage volume: 5 l verage viscosity: 5 mpas verage density: 1.05 g/cm 3 Composition: % corpuscular, % plasma 1. Hematocrit (htc, φ): Normal range: htc = V cells V total 2. Plasma viscosity Depends on plasma proteins. In paraproteinaemias (e.g. myeloma multiplex or plasmocytoma) the concentration of immunoglobulins is high, leading to increased viscosity. Viscosity of blood as suspension (in the physiologically relevant htc range): s =suspension viscosity, B=empirical constants lg s = + Bφ Plasma protein Normal concentration lbumin g/l 55% % ratio unction maintenance of colloind osmotic pressure, transport Globulins g/l 38% Part of the immune system ibrinogen g/l 7% Blood coagulation

5 ibrinogen, fibrin 47,5 nm ibrinogen: MW = Da In plasma: 2-4 g/l 10 μm verage nearest-neighbor distance 55 nm 3. Plasticity of red blood cells 65% suspension of blood-cell-size particles is rock hard. By contrast, a 95% blood suspension if fluid, with viscosity of ~20 mpas! Deformation of red blood cells: droplet, parachute, arrowhead shapes. 3 μm Red blood cells in fibrin meshwork ibrin polymerized in vitro (M) Extensibility of fluorescently labeled fibrin fibrils Disc-shaped cell with 7-11 μm diameter Deformation of a RBC with optical tweezers ixed RBC maintaining impression (M) 4. ggregation of red blood cells Stack or roleaux formation. More pronounced at low flow rates 5. low rate, velocity gradient 6. Blood vessel diameter (mpas) 5 (mpas) 5 2 Roleaux (stack) V (m/s) Diameter (μm) N.B.: With a decrease of vessel diameter, the anomalous (non-newtonian) behavior of blood becomes more pronounced. xial migration: the red blood cells line up in the axis of the vessel (Bernoulli s law). In the axis the velocity gradient decreases, and near the vessel wall it increases. Increase in velocity gradient decreases apparent viscosity (åhraeus-lindquist effect). 5

6 The vascular system is a closed vessel system returning into itself. unction: Maintenance of environmental parameters of cells ( steady state ) Transport: Gases Metabolites Hormones, signal transmitters Immunoglobulins Heat B. Hemodynamic requirements: Slow (matches diffusion-driven processes) Steady (no fluctuations) Unidirectional luid flow in bifurcating vessel system 1 v 1 2 v 2 1 v 1 Continuity equation 2 v 2 3 v 3 1 v 1 = 2 v 2 = const = cross-sectional area v = flow rate 1 v 1 = Σ () v average = const Σ = total cross-sectional area Structure and physical properties of the vascular system Total crosssectional Diameter area Pressure 100 Hgmm 2500 cm 2 Wall tension and bood pressure Circumferential stress (σ θ) depends on blood pressure: (Young-Laplace - equation) Vessel wall displays non-linear elastic properties low rate 0.33 m/s 0.22 m/s Crosssectional area 2.5 cm m/s orta rteries rterioles Capillaries Veins Pressure on blood vessel wall: "blood pressure". Pressure drop along vessel maintains blood flow. Reason of pressure drop: flow resistance - most of energy is converted to heat. low rate and total cross-sectional area change inversely (based on equation of continuity, v = constant). low rate typically does not exceed the critical (see Reynolds number), and flow remains laminar. (Exceptions: behind aortic valve, constricted vessels, low-viscosity conditions, Korotkoff sound). rterioles (vessels containing smooth muscle, under vegetative innervation) are pressure-regulators: "resistance vessels." Most of blood volume in veins: "capacitance vessels." 8 cm 2 P = blood pressure r = radius of tube t = wall thickness = force l = tube length Wall tension or circumferential stress is the average force exerted circumferentially (perpendicular to both the axis and the radius) in the cylinder wall. Wall tension Volume Determinants of vascular elasticity: Elastin Collagen Smooth muscle Implications of vascular easticity: Storage of potential (elastic) energy Dampening of pressure pulses Constant flow rate

7 Dynamic pressure-changes in the arterial system uxiliary factors of circulation Harvey s experiment (1628) 1. rterial elasticity elastic fibers storage of potential energy 2. Venous valves (Harvey's experiment) On the Circulation of the Blood (1628). Systole Diastole Because of vessel wall elasticity, pressure fluctuations are dampened. 3. Muscle action 4. Negative intrathoracic pressure 5. Up-and-down" movement of atrioventricular plane