Answered: c. The volume rate of flow (Q) can be…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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C. The volume rate of flow (Q) can be determined according to:
=-jv2πrdr
Using the final velocity equation from part a) and equation 2, derive Poiseuille's law for flow in this tube:
π(AP) yª
8n(AL)
Show all work.
Q
[2]
[3]

Transcribed Image Text:4. Fluid Dynamics: Blood in a new pediatric blood cannula can be described by Equation 1 below. We assume that flow conditions are steady (time independent), fully developed (fluid layers moving consistently), with laminar flow properties (no mixing or rotational fluid dynamic properties). This fluid flow is often referred to as Poiseuille flow. The figure below illustrates the concept of Poiseuille flow and the characteristic parabolic velocity distribution. X 154 AL The governing fluid momentum equation in the x-flow direction is as follows: nd dv ΔΡ 74(*)= [1] r dr dr AL is the fluid viscosity, r refers to the radial position, and AP/AL signifies the where v corresponds to the velocity, imposed pressure gradient driving flow. a. Integrated the differential equation to find v(r). Using the following boundary conditions, determine the expressions for the constants of integration in the solution to part a) and the final velocity distribution equation (parabolic shape): v must be bounded at r = 0 (ie. note that A In r⇒0 due to the bounding constraint) v = 0 at r = y, no slip condition or velocity = 0 at the wall The constant of integration will be in terms of y, AP/AL, and n. b. Graph the velocity distribution as a function of radius for an artery with a diameter of 0.2 cm, fluid viscosity of 0.055 dyne-sec/cm² (a dyne is a unit of force = g-cm/sec²), a pressure drop (AP) of 0.05 mmHg (1 mmHg = 1330 dyne/cm²), and a vessel length of 3 cm. Plot this velocity distribution across the entire width of the artery. Comment on how your results compare to the velocity profile (parabolic) shown in the pipe figure above.

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