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## Fluid Dynamics

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**Fluid Dynamics**Fluid dynamics is the study of how fluids (gases or liquids) flow. Because water is such a common fluid, fluid dynamics is often called hydrodynamics.**Discharge**When fluid flows through a pipe, the flow or discharge (J) is the mass {J} (or volume {Q}) of fluid that passes a given point per unit of time.**Discharge**Mathematically: Q = Av**Discharge**Describing discharge as mass per unit time is actually more correct, but if the pipe is full of an incompressible (constant r) fluid then either description is fine.**Discharge**i.e. J = Dm/t = rDV/t = rADl/t = rAv Since r is usually constant, discharge in terms of volume is… Q = Av**Equation of continuity**If an incompressible fluid fills a pipe and flows through it, the discharge stays constant even if the diameter of the pipe changes.**Equation of continuity**Mathematically: Q = A1v1 = A2v2 = constant**Wind Tunnels**We can study fluid flow patterns with wind tunnels:**Wind Tunnels**There are many types.**Wind Tunnels**Some are wicked cool!**Wind Tunnels**Some contain wicked cool things!**Wind Tunnels**After designing models based on computed calculations of flow characteristics, the predictions can be checked with a flow test. http://www.esa.int/esaCP/ESA9DBG18ZC_index_0.html**Types of Fluid Flow**There are two main types of fluid flow: Laminar and Turbulent**Laminar Flow**Laminar flow (AKA streamline flow) occurs when the particles of the fluid follow smooth, noncrossing paths.**Laminar Flow**Note that during laminar flow, neighboring layers of the fluid slide by each other smoothly.**Laminar Flow**Note that this is a shearing process. F**Laminar Flow**To study this process, two plates are separated by a thin layer of liquid.**Laminar Flow**A force is applied to the top plate to make it move. F**The rapidity of the shearing motion is characterized by the**shear rate of the two plates and the fluid between them. F**Shear rate =speed of top plate**distance between plates Shear rate = v/L = Des / t F**The viscosity of a fluid is the shear stress required to**produce a unit shear rate. F**h = viscosity = shear stress/shear rate.**h = (F/A) / (v/L) F**h = viscosity = shear stress/shear rate.**h = (F/A) / (v/L) = (FL) / (vA) FYI: h = the lower Greek letter eta F**F a vA / L for any given fluid. So the larger the h value,**the greater the force resisting the attempted shear under a given set of conditions. (i.e. The fluid is stickier.) F**For liquids, the viscosity results from attractive forces**between the molecules. For gases, the viscosity results from collisions between the molecules. F**The SI unit for h is N.s/m2, or Pa.s. This is called the**poiseuille (Pl). Other units are the cgs unit the poise (P), for which 1 P = 0.1 Pl, and prefix versions of each. F**The greater the viscosity in a fluid, the greater the heat**generated as it is sheared under a given set of conditions. F**Because of viscosity, it takes a pressure difference at the**ends of a horizontal pipe to have laminar fluid flow through it at a steady rate. A French scientist named Poiseuille studied this in the 1800s and developed the formula that bears his name: Q = (pr4(P1-P2)) / (8hL)**Q = (pr4(P1-P2)) / (8hL)**All My Loving Pi r fourth delta P Over 8 eta L Shows how fast V’s flowing… That’s Q**Turbulent Flow**When fluid flows beyond a certain speed, the laminar flow breaks down into turbulent flow.**Turbulent flow is characterized by small whirlpools called**eddies, which consume an enormous amount of energy. This increases the drag on the object in the fluid flow far above the drag created by viscosity during streamline (laminar) flow. For liquids in a pipe, this translates to a need for a much higher (and less predictable) ____________ to maintain the flow. pressure**Reynold's Number**The Reynold’s Number (NR) is a dimensionless experimental number that gives an indication of the velocity at which turbulence will occur in a fluid.**Reynold's Number**Mathematically: NR = rvD/h Fluid flow will usually be laminar if NR does not exceed about 2000 for fluid flowing through a pipe, or about 10 for obstacles.**Bernoulli's Principle**In the 1700s, a mathematician named Daniel Bernoulli studied the pressure associated with moving fluids and came to a startling conclusion: http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Bernoulli_Daniel.html**Bernoulli's Principle**Bernoulli’s Principle basically states that… As a fluid’s velocity increases, its internal pressure decreases!**Bernoulli's Principle**Bernoulli’s Principle applies to a variety of phenomena.**Bernoulli's Equation**Mathematically: P1 + ½ rv12 + rgh1 = P2 + ½ rv22 + rgh2 OK. But why???**Bernoulli's Equation**Bernoulli’s Equation is really a restatement of the Law of Conservation of Energy: The total energy of a closed system remains constant. (This is true unless there is a ____________ change.) P1 + ½ rv12 + rgh1 = P2 + ½ rv22 + rgh2 nuclear**Since work is done whenever a force is applied through a**distance, work is done whenever pressure forces a volume of fluid to move as well. P1 + ½ rv12 + rgh1 = P2 + ½ rv22 + rgh2 W = F x d W = P x V Note:W = (F/A) x A x Dl AA**Also, since work must be done to accelerate an object,**faster moving objects have more kinetic energy. By replacing the m in the equation with rADl, we can see that P1 + ½ rv12 + rgh1 = P2 + ½ rv22 + rgh2 W = DKE = ½ mv2 W = DKE = ½ rVv2**Lastly, since work must be done to raise an object,**potential energy may be exchanged for kinetic energy. By replacing the m in the equation with rADl, we can see that P1 + ½ rv12 + rgh1 = P2 + ½ rv22 + rgh2 W = DPE = DKE = mgDh W = DPE = DKE = rVgDh**So all the terms in Bernoulli’s Equation are really energy**terms associated with a given volume movement. P1V + ½ rVv12 + rVgh1 = Constant This becomes: P1 + ½ rv12 + rgh1 = Constant / V**Bernoulli's Equation**Note that Bernoulli’s Equation ignores viscosity and compressibility. Reality is more closely modeled with the Navier-Stokes equation, but that is beyond the scope of this course.**"That we have written an equation does not remove from the**flow of fluids its charm or mystery or its surprise." --Richard Feynman [1964] http://jef.raskincenter.org/published/coanda_effect.html http://en.wikipedia.org/wiki/Richard_Feynman**Torricelli's Theorem**Long before Bernoulli entered the world, Torricelli realized that if a fluid were to flow from a w-i-d-e barrel, the fluid velocity would depend on the height of the fluid above the spigot. He determined that the formula was… v =**Torricelli's Theorem**Long before Bernoulli entered the world, Torricelli realized that if a fluid were to flow from a w-i-d-e barrel, the fluid velocity would depend on the height of the fluid above the spigot. He determined that the formula was… v = √2gh**Torricelli's Theorem**Why would that be? Well, if we modify Bernoulli we can derive this! (Note: essentially we are giving up _______, and gaining _______.) PE KE P1 + ½ rv12 + rgh1 = P2 + ½ rv22 + rgh2 Since the air pressure doesn’t change much,P1+ ½ rv12 + rgh1 = P2+ ½ rv22 + rgh2**Torricelli's Theorem**Since the top is still and the bottom is… the bottom…P1+½ rv12+rgh1 = P2+ ½ rv22+ rgh2 Since the fluid is considered to be incompressible… P1 + ½ rv12 +rgh1 = P2 + ½ rv22+ rgh2**Torricelli's Theorem**By rearrangement v = √2gh So Torricelli is an example of Bernoulli. How about others?