PROBLEM SET 5. SOLUTIONS March 16, 2004

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Lorraine Briggs
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1 Havad-MIT ivision of Health Sciences and Technology HST.54J: Quantitative Physiology: Ogan Tanspot Systems Instuctos: Roge Mak and Jose Venegas MASSACHUSETTS INSTITUTE OF TECHNOLOGY epatments of Electical Engineeing, Mechanical Engineeing, and the Havad-MIT ivision of Health Sciences and Technology 6.0J/.79J/BEH.37J/HST54J: Quantitative Physiology: Ogan Tanspot Systems PROBLEM SET 5 SOLUTIONS Mach 6, 004

2 Poblem A. The hypodemic needle in the figue below contains a saline solution. If a plunge of aea A is pushed in at a steady ate (V ), what is the mean exit velocity (V e ) of solution leaving the needle of aea A e? Assume no leakage past the plunge. Using consevation of mass: AV V e = A e V e = V A A e B. If thee is leakage back past the plunge equal to one-thid the volume flow ate fom the needle, find an expession fo V e. If one-thid of the needle flow ate leaks back past the plunge we would have: AV = A e V e + 3 A ev e = 4 3 A ev e V e = 3 4 V A A e C. Neglecting leakage past the plunge, find an expession fo the pessue at the face of the plunge if the fluid exits the needle at atmospheic pessue and the fluid can be teated as though it wee inviscid. The flow can be teated as steady. A e V V e Aea = A Using Benoulli s equation between a point on the plunge and the end of the needle: P + ρv = P atm + ρv e P P atm = ( ) ρ Ve V but V, V e wee given above in A. P = ρv [ ( A A e ) ] 6.0j 004: Solutions to Poblem Set 5

3 assuming P atm 0 004/ j 004: Solutions to Poblem Set 5

4 Poblem A common type of viscomete consists of a cone otating against a fixed plate, as shown in Figue. Show fom physical aguments (o othewise) that the shea ate is independent of. (Hint: v φ = A( )z.) Explain how this viscomete can be used to constuct the stess-stain elationship of a non-newtonian fluid like blood, when the toque T on the cone and the angula speed ω ae known. Figue : ω T R z In ode to constuct a flow cuve fo the unknown fluid we must measue both the shea ate and the shea stess of the fluid. We ae given the cone viscomete with angula velocity ω and toque T. The cone angle, θ, is vey small, so we make the following appoximations: tan θ sin θ θ; cos θ ; R cos θ R The angula velocity ω should tell us about shea ate. Let us conside a band of fluid at a distance fom the cente that is d wide and h high. (See Figue.) h is given by the geomety of the device and is h = tan θ = θ The velocity of the cone would be ω at the selected adius. The shea ate, γ, would then be v φ ω γ = = = ωθ z θ Note that γ is independent of the adius,. Next we need to elate the applied toque, T, to the shea stess, τ. The shea foce acting on the diffeential suface ing of width d and adius would be df = τ( )π d In ou case, τ( ) is actually not a function of. v φ µω τ = µ = z θ 6.0j 004: Solutions to Poblem Set 5 4

5 Figue : Side view z ω d h Top view ω d The contibution of df to the toque would then be The total toque would then be dt = df = πτ d So R R πτ R 3 T = πτ d = πτ d = T τ = π R 3 Finally, we can use the measued toque and angula velocity to measue viscosity, µ. 004/43 µω 3T τ = = θ π R 3 3T θ µ = πµω 5 6.0j 004: Solutions to Poblem Set 5

6 Poblem 3 Conside lamina viscous flow in a cylindical vessel. Show that the magnitude of the shea ate at the wall is given by: v 8v γ = = wall whee v is the aveage flow velocity though the vessel and is the diamete. v γ = =a The velocity pofile fo lamina viscous flow (Poiseuille flow) has the paabolic pofile whee u(0) is the centeline velocity ( ) u( ) = a u(0) du = u(0) d a dv = u(0) a d =a Noting that mean velocity, v, is half the centeline velocity: dv d =a ū = u(0) = a 8v = The magnitude is simply the absolute value, which is γ = v =a = 8 v 004/4 6.0j 004: Solutions to Poblem Set 5 6

7 Poblem 4 One simple and instuctive model of the flow of eythocytes though the capillaies is shown in the sketch below. The eythocyte fills the tube so that a bolus of plasma is tapped between each pai of cells and tavels with the cells. If the distance between cells, l, is lage compaed to the capillay diamete, the velocity pofile in the plasma between the cells is nealy that of a Poiseuille flow. Show that the plasma centeline velocity, V, is twice the eythocyte velocity V 0. l u() V 0 V 0 V Only when the cells ae fa apat will the flow be fully developed as given in the poblem. Since a bolus of fluid is tapped, the flow ates of fluid nea the cells and in the middle must be equal. ( ) ] u() = v [ Poiseuille Flow ( ) u()d dθ = π v 0 / ( ) ] ( ) π v [ d = π v 0 πv [ 0 ( ) ( ) 4 ( ) ] 4 = π v = v 0 ) v 0 ( P.S. The flow behind the ed cell is complicated. Nevetheless, the aveage flow ate must equal v / 7 6.0j 004: Solutions to Poblem Set 5

8 Poblem 5 A. A patient has a diseased aotic valve. The valve does not leak, but it has stenosis leading to maximum velocity of 5 mete/sec exiting the valve. If the peak flow ate in systole though the valve is 350 ml/sec and the left venticula outflow tact aea is 3. squae centimetes, what is the maximal systolic gadient acoss the valve? What ae the assumptions that you made, and why ae they easonable? Benoulli: tempoal tem out valid at peak since d v dt = 0, so okay inetia (ove shea) dominates check Re, know fom lectue the Re, aota huge! So okay A = 3. cm V max = 5 m/s Q peak = 350ml/sec Q = A V = A V 350ml/sec = A (5m/sec)(00cm/m) A =.7cm ( ) P = ρ v v ( ) = () (500cm/s) (09cm/s) (dyne/cm ) = dyne/cm = 89.5mmHg 6.0j 004: Solutions to Poblem Set 5 8

9 B. Occasional patients have stenosis of the aotic valve, but also have a naowed left venticula outflow tact just poximal to the valve. If the maximal systolic velocity exiting the aotic valve stenosis is 5 metes/sec and the flow ate is 350 ml/sec but the outflow tact aea is now.5 squae centimetes, what is the maximal gadient acoss the valve? 350ml/sec = A V = (.5cm )V V = 33.3cm ( ) P = ρ v v ( ) = () (500cm/s) (33.3cm/s) (dyne/cm ) = 97778dyne/cm = 73.5mmHg C. Fo both of the above cases, calculate the aea of the vena contacta (the aea of the smallest egion of the jet). Is the TRUE valve aea lage o smalle than the vena contacta aea? case 350ml/sec = A (500cm/sec) A =.7cm case 350ml/sec = A (500cm/sec) A =.7cm 004/558 valve vena contacta The tue valve is bigge than the vena contacta that s why the Golin constant is diffeent fo diffeent valve geometies j 004: Solutions to Poblem Set 5

Stress, Cauchy s equation and the Navier-Stokes equations

Stress, Cauchy s equation and the Navier-Stokes equations Chapte 3 Stess, Cauchy s equation and the Navie-Stokes equations 3. The concept of taction/stess Conside the volume of fluid shown in the left half of Fig. 3.. The volume of fluid is subjected to distibuted