TECHNISCHE UNIVERSITEIT EINDHOVEN Department of Biomedical Engineering, section Cardiovascular Biomechanics Exam Cardiovascular Fluid Mechanics (8W9) page 1/4 Monday March 1, 8, 14-17 hour Maximum score is 3 points. The partial scores are indicated in the margin of each question. Answers can be given in Dutch or English. 1. Answer the following questions with yes or no and give a short motivation. a. Is it correct that the relatively high pressure in the arterial system is mainly caused by the visco-elastic properties of the large arteries? b. Is it correct that the admittance at a certain location in the arterial system only depends on the material properties of blood and the local material properties of the artery? c. Does the non-linear stress-strain behavior of collagen in an artery result in a compliance that increases with increasing pressure? d. Suppose we model blood as a Newtonian fluid with a viscosity equal to the viscosity of plasma. Knowing that blood behaves like a shear-thinning fluid, will the Reynolds number in our model be overestimated? e. Is it justified to consider the flow in the micro-circulation as quasi-static? f. In rheometric measurements of generalied Newtonian fluids it is assumed that the viscosity depends on the local shear rate γ. Is this a result from the fact that in general the viscosity must depend on the second invariant II D of the rate of deformation tensor D? g. Is it correct that aggregation and deformation of red blood cells will have only a minor contribution to the shear-thinning property of blood? h. In a study of the steady flow in a slightly curved artery the flow in a curved tube of a fluid (water) with a viscosity that is 4 times smaller than the viscosity of blood is investigated. Is it possible to compensate for this mismatch in viscosity by investigating a flow with the same inflow velocity as in the artery but now in a curved tube with the same curvature but a radius that is 4 times smaller than the artery? i. Is it correct that in the entrance region of a curved tube flow with a flat inlet profile friction forces can be neglected, and that the velocity at the inner curve of the tube is larger than at the outer curve? j. Does the Moens-Korteweg wave speed correspond with the wave speed in an elastic tube for the case that the Womersley number is very small (α )? 1
Exam Cardiovascular Fluid Mechanics (8W9) page /4 Monday March 1, 8, 14-17 hour. Consider a tapering converging tube with circular cross-sectional area A() =πa () given by: A() =A (1 k/l) <<L, k<1 By means of a piston at the entrance of the tube a harmonic flow will be induced. The piston moves with a velocity V according to: V (t) =V cos ωt r A() A() = = L V L a. Show that for a rigid tube and an incompressible fluid, mass conservation can be reduced to: with q the flow defined by q = v(, t)a(). With v(, t) the mean velocity over a cross-section. So: v(, t) = π a() v (r,, t)rdr π a() rdr = q(, t) A(), a() = A/π tip: take a segment between and +Δ. b. The continuity equation in cylindrical coordinates is given by: v r r + v r r + 1 v φ r φ + v We assume frictionless flow with an approximately flat velocity profile v = (v r,,v )withv r = v r (r,, t), v φ = and v = v(, t). Give an expression for v(, t) andv r (r,, t) in the tapering artery as a function of the given A() and V (t) and make a drawing of the velocity vectors at two arbitrary values of ( <<L).
c. The momentum equation in -direction for cilindrical coordinates is given by: ( v t + v v r r + v v = 1 p 1 ρ + ν r r (r v ) v r )+ here we assumed axi-symmetry of the problem (v φ =, =). φ Show that for frictionless flow with a flat velocity profile v =(v r,, v) themomentum equation in -direction reduces to: ( ) q t A A + A p. d. For which value of V and given values for ω, A, k and L, is it allowed to neglect the non-linear (second) term with respect to the unsteady inertia term in the momentum equation given above. e. Write p =ˆpe iωt and q =ˆqe iωt and show that the solution of the linearied momentum equation: t + A() p is given by: p(, t) = ωρlv ( ) 1 k/l ln sin ωt k 1 k f. Make a graph (sketch) of the amplitude of p and of p coordinate. as a function of the axial 3
Exam Cardiovascular Fluid Mechanics (8W9) page 3/4 Monday March 1, 8, 14-17 hour 3. A long thin-walled elastic tube with radius a =1 m and wall thickness h =1 3 m made of an incompressible material (Poisson ration μ =.5) will be used as a model for a blood vessel. In a static experiment the tube is at one side connected to a fluid reservoir at height H. The other end of the tube is fixed and closed. By varying the height H of the reservoir the pressure in the tube can be varied and from the pressure-diameter relation a compliance C =7.85 1 9 m Pa 1 is found. a. Give the values of the distensibility D and the Young s modulus E of the tube. We want to investigate the properties of the tube at both high and low values of the Womersley number. Hereto an harmonic pressure wave (p =ˆpe i(ωt k) )withan angular frequency ω =1rad s 1 will be generated. First the tube will be filled with water (kinematic viscosity ν w =1 6 m s 1,densityρ w =1 3 kg m 3 andinasecond case with syrup (ν s =1 1 m s 1, ρ s =1 3 kg m 3 ). b. Give for both cases the values of the Womersley number. In which of the two cases is friction of importance? c. For long waves conservation of mass and momentum reduce to: p C t + ρ t + A p = f q Give for both cases a good approximation for the friction function f. d. What is, in good approximation, the value of the wave speed in both cases and how do they compare to the Moens-Korteweg wave speed? e. Give expressions for the admittances Y w and Y s. f. Determine for both cases the attenuation constant γ of the wave. 4
TECHNISCHE UNIVERSITEIT EINDHOVEN Department of Biomedical Engineering, section Cardiovascular Biomechanics Answers belonging to the exam Cardiovascular Fluid Mechanics (8W9) Monday March 1, 8, 14-17 hour 1. Answers: a. No. By the high resistance of the peripheral micro-circulation. (1 pnt) b. Yes. Y = q/p = ωc/k only depends on local properties. (1 pnt) c. No. C = A p will decrease with increasing pressure. (1 pnt) d. Yes. η>η plasma so Re < Re plasma since Re = ρv a/η. (1 pnt) e. Yes. Since α = a ω/ν 1. (1 pnt) f. Yes. II D = 1 (tr(d )). (1 pnt) g. No. Hardened cells only give an increase in viscosity. Aggregation will increase the viscosity at low shear rate and deformation will lower the viscosity at high shear rates. (1 pnt) h. No. The Dean number Dn = δ 1/ Re is crucial: (a 1 /R 1 ) 1/ V 1 a 1 /ν 1 =(a /R ) 1/ V a /ν so a =(1/4) /3 a 1.() i. Yes. Then there is balance between pressure forces and centrifugal forces and Bernoulli will result in higher velocity at the inner curve. (1 pnt) j. No. For α.(). Answers: a. For a segment between and +Δ we have: ρ v(, t)a() ρ v( +Δ, t)a( +Δ) = ρ v(, t)a() ρ v( +Δ, t)a( +Δ) lim Δ Δ v(, t)a() ( pnt) = b. q is constant so v A() =V A cos ωt hence v () = v φ =and 1 rv r r r V A cos ωt A (1 k/l) = V cos ωt 1 k/l + v 5 =
v = k V cos ωt L (1 k/l) rv r r = r k V cos ωt L (1 k/l) rv r r = r k V cos ωt L (1 k/l) + C v r = r k V cos ωt L (1 k/l) + C/r v r () = C = v r = r k V cos ωt L (1 k/l) r ( pnt) c. v φ =, v = r v t + v v + 1 p multiply by A: A v t + A v A v A + A p t + q ( q A )+A p t + q ( 1 A )+ q A + A p t ( q A ) A ++A p ( pnt) d. Then O( q A ) O( ) and thus: t A ωq q A A = q k A L q 1 k ωa L (1 pnt) 6
e. iωˆq + A() ˆp ρ f. = ˆp = iωρ 1 ˆp = iωρˆq A ln(1 k/l)( L k )+ˆp boundary conditions: and A() ˆq = iωρ A (1 k/l) ˆq ˆp(L) = iωρlˆq ka ln(1 k) = ˆp = iωρlˆq ka ln(1 k) ˆp = iωρlˆq ka ln 1 k/l 1 k p(, t) = ωρlv k ( pnt) 6 dp/d ln 1 k/l 1 k sin ωt p 1/(1 k/l) 5 4 3 1.5 1 /L (1 pnt) ln(1 k/l) 1.5 1.5..4.6.8 1 /L 3. Answers: 1 μ D a. D = C /A =.5 1 5 Pa 1. E = a =6 h 15 Pa (1 pnt) b. α w = a ω = 1. ν α s = a ω =.1. (1 pnt) ν c. water: f =. syrup: f =8η s /a =8 1 5 kgm 3 s 1 ( pnt) d. c w = c A ρc 1 =6.3m/s. c s = 1 α sc =.3m/s. () e. Y w = A ρc. Y s = 1+iαY 4 ( pnt) f. γ w =. γ s =π ( pnt) 7