Question

In the human body, the aorta is the major artery that conducts blood from the heart to the rest of the body. The abdominal aorta is a relatively long straight section of the aorta that runs down from the chest to the pelvis. This section is prone to aneurisms, where a section of the aorta fails and stretches. In order to study how wall shear from blood flow in this portion of the aorta may contribute to aneurisms, a laboratory model must be created. (a) Use dimensional analysis to reveal that the nondimensional parameters that govern dynamic similarity. For this problem, the relevant parameters include the artery diameter D = 2 cm, the angular frequency of pulses w = 1.5 Hz (in radians per second), the velocity of the peak pulse U, and the kinematic viscosity v of blood, averaging 3 x 10 m /s. (b) You are asked to design a laboratory experiment for measurement of shear stress in the aorta that is as close to the conditions in the human body as possible, while using water instead of blood (v = 1 x 10 6 m/s). The laboratory-scale model will be life sized. You may assume that blood is a Newtonian fluid for this analysis, with a maximum velocity during the peak of the pulse of 0.5 m/s. If you match the diameter of the aorta, what velocity and pulsation frequency in Hz should be used?

Answer 1

(a, Total parameters =n=4 = f (D,w, u, v) Selecting ne netess = D and u Therefore total repeating parameters =m=2 i. Total non dimensional (a-groups) will be = nom=4+2=2 Now from Buckingham t-theorem we have T, -y [Da ub] → [Mºlº T'] = {T'IL L Toby » [M° LºTºJ = [[2 tarb.q-1-63 to a poco 7 6 =1 1 = (2+atb = 0 = 2ta-1 =) a = -1 3. " " Y[-o-'] e 12 Similanty T = w[oa uby [M°C T° Je T+ [La Louby | 2 [ML] - [+ --*] Jo=-1-6 b = -1

- L = Lath > 0 =9-1 → a=l och = w[st u-I] - Tz - We do ". Non dimensional parameters are t=2 and t = woo which govern the dynamic similarity ... Do 90 vorn inamic 1 Human body condition ... laboratory enperimit conditit Antory diameter =D = 2cm wo los Hz conditiny amparimet De =D=2cm U= Bosnian blood , V = 3x106m? U = ? water, vz 1x10m?

a for dynamic similarity equake x, and groups for human body conditions and laboratory ouperiments :. Гэто т. 1 "Р Бо . P. о. = 0 = Yux B x 0 > U = 1.****0.5 - [ U = 0.67 m/s similarity for the group → We swx B x W = 1.5x x 0.167 TWL = 0.5 H2 2 0.5