4. [55 points] Consider steady, fully developed, laminar fluid flow through a rigid tube with radius, y. 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. AL The governing fluid momentum equation in the x-flow direction is as follows: n d dv AP r dr dr) AL [1] where v corresponds to the velocity, n is the fluid viscosity, r refers to the radial position, and AP/AL signifies the imposed pressure gradient driving flow. a. Integrate Equation 1 twice and obtain the general velocity distribution equation for fluid flow in this rigid tube. (Keep constants of integration) b. 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 2 The constant of integration will be in terms of y, AP/AL, and n. Graph the velocity distribution (solution found in part b) as a function of radius for an artery with a diameter of 0.4 cm, fluid viscosity of 0.024 dyne-sec/cm- (a dyne is a unit of force = g-cm/sec-), a pressure drop (AP) of 0.075 mmHg (1 mmHg = 1330 dyne/cm ), 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. d. The volume rate of flow (Q) can be determined according to: 0= -[v2zardr [2] Using the final velocity equation from part b) and equation 2, derive Poiseuille's law for flow in this tube: OR(AP) y [3] 8n(AL) Show all work. e. For a Newtonian fluid and Poiseuille flow, the viscosity (n) is defined as the ratio of the shear stress (t) to the shear rate (dv/dr). Thus, 1= 7 dv/ dr [4] Determine an equation for the shear stress by substituting for the differentiated velocity solution in part b. Calculate the maximum shear stress and state the location of this maximum level in the vessel for an imposed pressure gradient of 0.075 mmHg / cm and a vessel radius of 0.4 cm. f. The Reynold's number (Re), a non-dimensional term used widely throughout engineering and fluid mechanics, compares the inertial forces of the fluid flow to the viscous forces. Higher Re numbers signify turbulent flow conditions while lower values denote laminar flows: Re = - PDV V= [5] where D is the vessel diameter, v corresponds to the average velocity, p represents the fluid density, n refers to the fluid viscosity, Q is the volume flow rate, and A denotes the multiple parallel pathway cross sectional area. Calculate the Reynold's number, Re, for the following vessels: Vessel Diameter Area (cm') (cm) Additional parameters that you will need are: Aorta 1.6 2.0 Q = 2000 cm3/min Artery 0.4 40 Arteriole 0.003 60 n = 0.024 dyne-sec/cm2 Capillary 0.0006 1400 Venule 0.004 800 p = 1.04 g/cm3 Be very careful with units, a dyne is a unit of force = gm cm/sec. Would you expect laminar, turbulent, or transitional flow conditions in each vessel