Lecture #19: Fluid Dynamics I

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Presentation transcript:

Lecture #19: Fluid Dynamics I 7 fundamental concepts of fluid mechanics 1. dynamic viscosity Area = S L velocity, U force ~ S x U x L force force = m S x U x L, where m = viscosity (dynamic viscosity) Fundamental units: M / LT SI unit: Pascal-second cgs unit: Poise

2. density 3. kinematic viscosity n = density, r = mass / volume Fundamental units: M / L3 3. kinematic viscosity dynamic viscosity density n = Ratio of the fundamental properties of a fluid fundamental units: L2 / T cgs unit: Stokes Air 18 x 10-6 1.2 15 x 10-6 Water 1 x 10-3 1 x 103 1 x 10-6 dynamic viscosity (Pa-s) kinematic viscosity (kg m-3) density

4. no slip condition (boundary layer) solid surface Velocity, u = u (mean stream flow) o u = 0 (no slip condition) boundary layer Laminar flow over solid surface laminar flow through pipe diffusion convection

uinf d (x) = 0.95 x uinf y flat plate with upstream edge x Ground effect

5. Law of continuity (conservation of mass) mass flux = r2 s2u2 mass flux = r1 s1u1 S1U1 s2u2 gravity accelerates water downward For incompressible fluid r1 = r2 Thus, s1u1=s2u2 = r2 s2u2 r1 s1u1 by conservation of mass:

lung right ventricle body right ventricle

constricting streamlines = higher velocity flow tank pump top view dye injector stream line fish small fluid volume constricting streamlines = higher velocity

6. Bernoulli Equation (conservation of energy) consider flow in one stream tube: Apply conservation of energy within tube: kinetic energy pressure energy gravitational energy divide by volume: consider two points along tube:

u1 _ + u2

u 7. Reynolds number two sources of force: viscosity force = m u S / L 2) inertial force = r u2 S Force sensor u length, L S Ratio = Re (Reynolds number) = r u L / m = u L / v Re is a dimensionless number that captures relative importance of viscous and inertial forces: Low Re Hi Re Force ~ u2 Force ~ u Flows are reversible Flows are irreversible

Forces proportional to u Low Reynolds numbers Forces proportional to u

Copepod Re = 300 Sperm whale Re = 300,000,000 Thrips Re = 30 Tuna Re = 30,000,000 Whale sperm Re = 0.00003 Duck Re = 300,000 Dragonfly Re = 30,000 Bacterium Re = 0.00003