Homework #10 Systems Bioengineering I, Fall 2013Page 1 of 111Homework 8:Application of Poiseuille’s Law, Compliance, and Starling’s Filtration Hypothesis(43 points total – will be scaled to 100 points)Background on Poiseuille’s LawPoiseuille’s Law gives the relationship between flow and the radius of a tube, the length of a tube, andthe viscosity of a fluid.This equation is derived from the Navier-Stokes equations.The Navier-Stokesequations apply to every possible type of flow in every direction.Deriving Poiseuille’s Law involvesmaking a number of assumptions about the flow.Specifically, we assume:•Flow is only in one-direction (the x-direction)•The tube is horizontal so that gravitational effects will not affect flow•Flow is steady and does not vary with time (v= constant and dv/dt = 0)•Tube walls are rigid so the radius of the tube is not changed by the flow•Flow is laminar (no turbulence)•Vessels do not taper (radius stays constant)•No branching•No curvature of vessels•Relatively long tube lengths•Circular vessel geometry (vessels are not elliptical or partially collapsed)•There is no slip at the wall•Smooth internal surface of the vessel (no plaque or stenosis)Within the systemic circulation, many of the above assumptions do not hold.Our blood flow ispulsatile, our heart ejects into compliant vessels that change their size, and vessels curve and branch anddevelop plaque.Within the pulmonary circulation, virtually none of the above assumptions hold.Nevertheless, Poiseuille’s law is a useful tool for approximating the flow, and the change in flow, whichoccurs in most of our blood vessels.The general form of the Navier-Stokes equations can be found in any advanced fluids textbook.Theequation for flow in the x-direction is:(1)𝛿𝑢𝛿𝑡+𝑢𝛿𝑢𝛿𝑥+𝑣𝛿𝑢𝛿𝑦+𝑤𝛿𝑢𝛿𝑧=𝑋 −1𝜌𝛿𝜌𝛿𝑥+µ𝜌∇2µwhere∇2=𝜕2𝜕𝑥2+𝜕2𝜕𝑦2+𝜕2𝜕𝑧2= the Laplacian operatorX= body force per unit volume (such as gravity)µ=viscosity of the fluidρ=density of the fluidu, v, w = velocity of the fluid in the x, y, and z directions respectively.Using all of the assumptions stated above, equation (1) can be simplified to Poiseuille’s Law (you do notneed to know the Navier-Stokes equations for the exam, but you should understand we made a largenumber of assumptions to use them!):Poiseuille’s Law:QRP=∆