1Physiology 472/572 - 2010 - Quantitative modeling of biological systems Lecture 15: Flow in tubes Flow in a cylindrical tube • many examples in the body and in medical devices • start with a simple prototype, many complexities - steady (constant in time) - laminar (not turbulent) - fully developed (no entrance effects) - uniform cylindrical tube (no taper, etc.) - rigid tube (neglect wall deformation) - Newtonian fluid (no non-Newtonian effects) Experiments • Poiseuille 1840, physician, used water and mercury • found Q ∝ d4 Δp/L where Δp = p2 - p1 • Q is volume flow rate (volume/unit time) Theory (Neumann and Hagenbach, 1858) • μΔπ=Ldp128Q4 Balance of forces on an inner cylindrical region • no acceleration, zero net force • force = stress × area • πr2 p2 - πr2 p1 - 2πrLτ = 0, i.e., τ = rΔp /2L2• this result is independent of fluid properties • when r = a, we get the wall shear stress τw = aΔp /2L Calculation of velocity profile • if the fluid is Newtonian, τ = - μ du/dr where u(r) is the velocity • minus sign because τ is defined in the minus direction • so du/dr = - τ/μ = - rΔp/2Lμ • integrate, apply non-slip condition • u = (Δp/4Lμ) (a2 - r2) Calculation of volume flow rate • for a uniform velocity, Flow rate = velocity × area • for a non-uniform velocity, integrate velocity with respect to area • μΔπ=π=∫Ldp128drr2)r(uQ4a0 • diameter d =2a • known as Poiseuille's law, although Poiseuille did not derive it • this kind of flow is known as Poiseuille flow Flow resistance • 4dL128QpRμπ=Δ= • proportional to L and μ, proportional to d-4, very sensitive function of d • importance for regulation of blood flow - small changes in d give large changes in R3Mean flow velocity • defined byμΔ===∫∫L8paA/QdA/dA)r(uu2AA • maximum velocity is u2L4pau2max=μΔ= • significant fact in discussing blood flow in microvessels Reynolds number • consider relative order of magnitude of viscous and inertial terms in steady flow • inertial term of Navier-Stokes equations, ρu ∂u/∂x ~ ρU2/L where U is a typical velocity, L is a typical length • viscous term μ ∂2u/∂x2 ~ μ U/L2 • ratio (inertial term)/(viscous term) ~ (ρU2/L)/(μU/L2) = ρUL/μ • i.e., Re = ρUL/μ is the dimensionless Reynolds number • indicates ability of viscosity to damp out disturbances and keep fluid flowing smoothly • low Re - large viscous effect, disturbances decay • high Re - little damping, disturbances grow, turbulence possible Values of Reynolds number • blood: ρ = 1 g/cm3, μ = 0.05 dyn.s/cm2 • for flow in tubes, convention is to use U = mean velocity, L = diameter = d • large artery, U = 100 cm/s, d = 1 cm, Re = Ud/ν = 100/0.05 = 2000 • capillary, U = 0.1 cm/s, d = 10 μm = 0.001 cm, Re = 0.002 • note orders of magnitude difference; inertial effects negligible in capillaries4 Turbulence • laminar flow is smooth, orderly, relatively predictable • turbulent flow has random, time-varying fluctuations of velocity • no complete theory for turbulence, use a mixture of theory and experimental information to predict turbulent flows • a relatively small viscous effect is enough to prevent turbulence, so it occurs only at high Reynolds numbers • for flow in a uniform pipe, turbulence occurs for Re > about 2000 • in irregular geometries, turbulence can occur at lower Re • in the human body, most flows are laminar, turbulence can occur in pathological