Exam Cardiovascular Fluid Mechanics 8W090 sheet 1/4 on Thursday 5th of june 2005, 9-12 o clock

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1 EINDHOVEN UNIVERSITY OF TECHNOLOGY DEPARTMENT OF PHYSICAL TECHNOLOGY, Fluid Mechanics group DEPARTMENT OF BIOMEDICAL ENGINEERING, Cardiovascular Biomechanics group Exam Cardiovascular Fluid Mechanics 8W090 sheet 1/4 The exam has a maximum of 30 points. Distrubution of the points is indicated for each question. 1. Answer the following question with either yes or no and provide a short argumentation. a. Is the pressure in the aorta during the ejection phase of the heart at all times less than the pressure in the left ventricle? b. Assuming a blood vessel behaves like a linear elastic tube with a thin wall which has Young s Modulus E and Piosson ratio μ, is it true distensibility of the tube is solely determined by the thickness of the wall h? c. Does the presence of collagen fibers in blood vessels cause the compliance to increase in case of an increasing transmural pressure? d. Given an aortic flow in the windkessel model, is it true the average aortic pressure during the heart s cycle is determined solely by the peripheral resistance? e. Studying velocity profiles in a fully developed oscillating tube flow, can the influence of viscosity be determined by varying the oscillation frequency? f. Given a stationary flow in a curve, is the Dean number independent of the viscosity of the fluid? g. Will in case of a fully developed stationary flow in a curve, the axial velocity in the inner curve have a higher value than the axial velocity in the outer curve? h. Is it possible to estimate the velocity profiles in an oscillating flow within an elastic tube by those in a rigid tube when the wavelength of the pressure waves is large in comparison to the radius of the tube? i. Is the velocity of pressure waves in an elastic tube filled with fluid at a Womersley number of α = O(1) approximatly half the velocity at extremely high values of the Womersley number? j. Are flat velocity profiles measured in microcirculation because in general the flow of a microcirculation hasn t been fully developed? 1

2 Exam Cardiovascular Fluid Mechanics 8W090 sheet 2/4 2. Endothelial cells appear to orient themselves in the direction of a shear stress they undergo. In order to load a monolayer of endothelial cells with an oscillating shear stress τ w, a rigid system representing the situation in blood vessels is studied (see figure). r A z a 0 z L 0 substraat L The setup consists of a rigid tube with an internal radius of a 0 and length 2L. At z = L a piston with surface A z = π moves with velocity v z = v z0 cos(ωt) around its starting positionänd generates an oscillating flow q(t). The incompressible Newtonian fluid moved by the piston has density ρ and dynamic viscosity η. At z = 0 a harmonic pressure p m = p m0 cos(ωt + φ) independent of the r-position in thetubeismeasured. Forz>0 the flow is seen as fully developed. The channel has a free outflow at z = L where a constant pressure of (p z=l = 0) is present. For a fully developd tube flow the Navier-Stoke equation becomes: v t = 1 ρ z + ν v (r r r r ) 0 = 1 ρ r with ν = η/ρ representing the kinematic viscosity. a. Provide an equation for the pressure and flow, expressed in A z, v z0, L, p m0, ω and φ for z>0. b. Show for complex notation (e ix =cosx + i sin x) the following is true: z = ˆp z eiωt met ˆp z = p m0 L eiφ q =ˆqe iωt met ˆq = π v z0 c. Scale the Navier-Stoke equation given above with v = v/v z0, r = r/a 0, z = z/l, t = ωt en p = p/p 0 and show the Womersley number α = a 0 ω/ν is the only dimensionless expression for p 0 = ηlv z0 /. 2

3 Exam Cardiovascular Fluid Mechanics 8W090 sheet 3/4 d. Show the estimate velocity profile for a frictionless flow is flat and that it s given by: v(r, t) = p m0 ρωl cos(ωt) Also show that in this case the following is valid: φ = π/2 andp m0 = ρωlv z0. Give an expression for the wall shear stress based on this profile. Does this make sense in case of a frictionless flow? e. Show the velocity profile becomes parabolic for a friction-dominated flow. Also show it s given by: v(r, t) = 1 4 η ( ) 1 r2 z and φ =0andp m0 =8ηLv z0 / is valid in this case Given this profile, what is the wall shear stress? f. For arbitrary values of α (take α = 5) an instationary boundary layer with a thickness of δ a 0 α will form at z>0. Prove this and draw the velocity profiles for several points of time in the flow cycle. g. Use the equilibrium of forces on a disk with thickness dz to show the following is an α-independent valid expression for τ w : τ w = a ( ) 0 pm0 2 L cos(ωt + φ)+ρωv z0 sin ωt which can be determined from a pressure measurement at z =0givenaknown piston movement. 3

4 Exam Cardiovascular Fluid Mechanics 8W090 sheet 4/4 3. A long, thin, elastic tube with radius a =10 2 m and wall thickness h =10 3 m of an incompressible material (Poisson ratio μ = 0.5) will be used as a model representing a blood vessel. In a static experiment the tube will be connected to a fluid column on one side. the other end will be axially fixed and obstructed. By varying the hight of the column a compliance of C 0 = m 2 Pa 1 was measured from the pressure-diameter ratio. Answer the following questions and provide suitable argumentation. a. Give the values of the distensibility D 0 of the tube and the Young s Modulus E of the tube s material? The characteristics of the tube are to be researched at both high and low values of the Womersley number, by inducing a harmonic wave (p =ˆpe i(ωt kz) ) with angular frequency ω =1rad s 1. To facilitate this, the tube is filled first with water (kinematic viscosity ν =10 6 m 2 s 1,densityρ =10 3 kg m 3 ) and then with sugarwater (ν = m 2 s 1, ρ =10 3 kg m 3 ). b. Give the Womersley number in both cases. In which of these cases does friction play a role? c. The following system can be derived from the constitutive laws and the conservation of mass and momentum: C 0 t + q z =0 ρ q t + A 0 z = f 0q Give a good estimate of the friction function f 0 for both experiments. d. Give a good estimate of the wave velocity in both systems. Which one is called the Moens-Korteweg velocity? e. Give a good estimate of the wave admittances of the tube for these fluids. f. Determine the attenuation constant γ of the wave in both cases. 4

5 EINDHOVEN UNIVERSITY OF TECHNOLOGY DEPARTMENT OF PHYSICAL TECHNOLOGY, Fluid Mechanics group DEPARTMENT OF BIOMEDICAL ENGINEERING, Cardiovascular Biomechanics group 1. Solutios: Solutions for the exam Cardiovascular Fluid Mechanics (8W090) on Thursday 5th of June 2005, 9-12 o clock a. No, not during the last phase when a high aortic pressure is needed to decrease the velocity of the aortic flow. b. No, as determined by ratio a/h with a the radius of the tube. c. No, stretching of the collagen fibers causes the compliance to decrease. d. Yes, ˆp(ω =0)=R pˆq(ω = 0). e. Yes, increasing/decreasing α = a ω/ν ν 2x is identical to increasing/decreasing ω 2x.(1pnt) f. No. Dn = δ 1/2 Re = a 2aV R ν g. No, it is the other way around. h. Yes. Also for V/c 1. i. Yes. c = 1αc 2 0 j. No. The entrance length is small compared to the diameter, but the flattening is caused by the cell-free layer near the wall. 2. Answers: a. Rigid tube so q = q 0 cos ωt is valid everywhere. With q 0 = A z v z0. For z>0the flow is fully developed so =constant cos(ωt+ φ) = (p z m0/l)cos(ωt+ φ). Dus L z p = p m0 cos(ωt + φ). L b. = p m0 [cos(ωt + φ)+isin(ωt + φ)] = p m0 z L L ei(ωt+φ) = p m0 L eiφ e iωt = ˆp z eiωt q = q 0 [cos ωt + i sin ωt] =ˆqe iωt = π v z0. (1pnt) v c. ωv z0 = p t 0 ρl 1 r r (r v + νv z z0 1 r ). So p r 0 = ηlv z0 d. α is large so v = 1 t ρ i ( p m0 ρω L eiφ )sov = p m0 (r v ). Divide by νv r r z0 / gives a2 0 ω v = p 0a 2 ν t 0 ηlv z0.(1pnt) z +. Substitution of v z =ˆveiωt and p =ˆpe iωt gives ˆv = i ˆp = ρω z sin(ωt + φ) =v ρωl z0 cos ωt. Soφ = π/2 andp m0 = ρωlv z0. z + ν r r e. α issmallso0= 1 v (r ). Integrate twice + v(a) =0givesv(r, t) = ρ r 1 4 η (1 r 2 /a 2 ). Integrate over the cross section gives q = πa4 0 p m0 cos ωt. Also z 8η q = v z0 π cos ωt so p m0 = 8ηLv z0.(2pnt) f. The boundary layer is determined by the transition where viscous and inert forces have the same order. So O( νv z0 δ 2 )iso(ωv z0 ). Which gives δ = O(a/α). 5

6 g. For the disc ΣF = ma is valid, so: [p(z) p(z + dz)]a 2πa 0 dzτ w = ρadz v t other words 2 z a 0 τ w = ρ v t in 3. Answers: 1 μ 2 D a. D 0 = C 0 /A 0 = Pa 1. E = 2a =6 h 105 Pa b. α w = a ω = 10. ν α s = a ω =0.1. ν c. water: f 0 = 0. sugar: f 0 =8η s /a 2 = kgm 3 s 1 d. c w = c A 0 ρc 1 =6.32m/s. c s = 1α 2 sc 0 =0.32m/s. (2pnt) e. Y w = A ρc 0. Y s = 1+iαY 4 0 f. γ w =0. γ s =2π 6