MOTION OF FLUIDS (part II) M Theory Constants n = 4- 10-3 Parsec is viscosity of blood; p= 1,05 - 10° kg/m blood density; g = 9,8 m/sec is free fall acceleration. hz Formulas 1. Law of energy conservation for fluid: E = Ex + Ep + E, = const, Here Eg fluid intemal energy. is fluid kinetic energy (m = pV); E, = mgh is fluid potential energy; E, = PV is 2. Bernoulli's Equation The relationship between pressure and velocity in fluids is described quantitatively by Bernoulli's equation, named after its discoverer, the Swiss scientist Daniel Bemoulli (1700–1782). Bemoulli's equation states that for an incompressible, frictionless fluid, the following sum is constant: + pgh + P = const, where P is the absolute pressure, p is the fluid density, v is the velocity of the fluid, h is the height above some reference point, and g is the acceleration due to gravity. If we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Let the subscripts 1 and 2 refer to any two points along the path that the bit of fluid follows; Bernoulli's equation becomes: Pri + pgh, + P, = i+ pghz + P.. 1.1 Bemoulli's Equation for static fluid, when v, = v2: pgh, + P, = pghz + P2 Po AP PA R2 P R Pieg - Pneart = PgAh Pheart - Phead = pgAh and Pa 1.2 Bemoulli's Equation at constant height h, = h2: pvž. -+ P =- + P2 3. Laminar and turbulent flow, Re Laminar flow is characterized by the smooth flow of the fluid in layers that do not mix. Turbulent flow, or turbulence, is characterized by eddies and swirls that mix layers of fluid together. An indicator called the Reynolds number Re can reveal whether flow is laminar or turbulent. For flow in a tube of uniform diameter, the Reynolds number is defined as: Laminar Turbulent Transitional- oscillates between laminar and turbulent pvd Re = where p is the fluid density, v its speed, n its viscosity, and d the tube diameter. The Reynolds number is a unitless quantity. Experiments have revealed that Re is related to the onset of turbulence. For Re below 2300, flow is laminar. For Re above 2300, flow is turbulent.. 4. Work of heart per one beat is: W = 1,2V,(P + p is the density of blood, v is the velocity of the blood flow, V, is P is the arterial blood pressure, stroke volume of blood. 5. Power of heart: =, here W is work of heart, t = 0.3 sec is duration of one stroke. 6. Law of continuity: v,A, = V½A2.