Fluids in Motion Supplement I

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Fluids in Motion Supplement I Viscosity: – SI units of viscosity, η = kg m -1 s -1 or Pa·s (cgs unit: Poise, P = g cm -1 s -1 = 0.1 Pa s) – Viscosity varies with Temperature Material Viscosity (Pa·s) Air (20 C): 0.000018 Water (20°C): 0.001 Pa s (=1 cP) Water (40°C): 0.00065 Ethanol: 0.0012 Methanol: 0.00058 Glycerin (0°C): 12.0 Glycerin (30°C): 0.629 Oil, Olive (20°C): 0.084 30% aqueous Sucrose: 0.003 Molasses (45-50% Sucrose) 0.005-0.010 70% aqueous Sucrose: 0.481 Example S1 Reynolds Number for a baseball pitch Baseball pitchers can routinely throw baseballs at speeds of 90 mph (40.2 m/s). The diameter of a baseball is supposed to be 7.37 cm1.8 x 10 -5 Pa·s. The density of air is 1.29 kg/m 3 and the viscosity of air is 1.8 x 10 -5 Pa·s. a) What is the Reynolds Number for a fastball? b) Will the flow past the ball be laminar or turbulent? Reasoning: This is a straight plug-in problem using the definition of Reynolds Number. From the notes on the previous page, the definition of Reynolds number is: ρvD Re = η where ρ is the fluid density (not the baseball density), v is speed of the air around the baseball, or, equivalently for the baseball problem, the speed of the baseball through the air D is diameter of the baseball (given as 7.37 cm) η is the viscosity of the air (from the information given, η = 1.8 x 10 -5 Pa·s). Solution: a) The Reynolds number can be calculated using the values above: 3 ρvD (1.29 kg/m )(40.2 m/s)(0.0737 m) Re = = = 2.12× 10 -5 η 1.8× 10 kg/(m ⋅s) b) Since Re is 212,000, which is much greater than 1000 (the Reynolds Number above which flow becomes turbulent) the flow around the baseball will be turbulent. 5
Fluids in Motion Supplement I Some examples of Reynolds Numbers for biological organisms moving through fluids (air or water) are shown in the table below. These values were calculated in much the same way as for the baseball, but they use very rough estimates for the size of the organisms. A large whale swimming at 10 m/s 300,000,000 A duck flying at 20 m/s 300,000 A large dragonfly moving at 7 m/s 30,000 Flapping wings of the smallest flying insect 30 An invertebrate larva, 0.3 mm long, at 1 mm/s 0.3 A sea urchin sperm moving at 0.2 mm/s 0.03 A bacterium, swimming at 0.01 mm/s 0.00001 (From S. Vogel, Life in Moving Fluids, Princeton University Press, 1994, Table 5.1) One can find the size of the whale implied by the first item in the table above from the definition of the Reynolds Number. If Re = 3 x 10 8 , then using values for the density of water from Table 11.1 in Cutnell & Johnson (ρ water = 1000 kg/m 3 ) and the viscosity of water from the table on the previous page of this supplement (η water = 0.001 Pas): 8 Re η (3× 10 )(0.001 kg/(m ⋅s)) D = = = 30 m 3 ρv (1000 kg/m )(10 m/s) A size of 30 m is consistent with the length of a large whale. For a complex shape like a whale the size scale to use in the calculation of the Reynolds Number is a matter of debate, even among experts. It is not always clear whether one should one set D equal to the width of the whale, or the length of the whale, or the height of the whale (or some combination of the three). In most cases, however, the Reynolds Number will not change enough to affect what type of flow (laminar or turbulent) is predicted by the Re value if the width of the whale is used instead of the length. The true flow pattern can only be found by experiment or from the study of scale models in wind or water tunnels. The idea that if Reynolds Number is the same for two objects of different sizes, flow patterns will be the same leads to the idea of dynamic similarity. This implies that one can measure the flow pattern for a scale model of an object in a wind tunnel and expect that the flow pattern around the full-sized object will be similar if the density, viscosity or speed of the wind tunnel fluid are adjusted to give the same Reynolds Number as the full-sized object.
Page 1: Fluids in Motion Supplement I Cutne
Page 5 and 6: Drag on an Object Bernoulli’s Law
Page 7 and 8: Sedimentation of a Sphere at high R

Fluids in Motion Supplement I

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