1. The Stokes-Oseen formula for drag force Fon a sphere of diameter D in a fluid stream of low velocity V, density p, and viscosity u is: F = 3TuDV + 16PVD Is this formula dimensionally homogenous?
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Q: The Stokes-Oseen formula for drag force F on a sphere of diameter D in a fluid stream of low…
A: Given data F = 3πμDV +9π16ρV2D2
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A: F= cd 12ρ A V2 cd = coeff of drag ρ = density A = reference areaV = velocity
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- - Assume the input power to a pump P (M L T-¹) is depend on Q- Flowrate (L3 T-¹), H-pump head (L), p- fluid density (ML-³) Create a relation by dimensional analysis between the power and other variables by Buckingham Theorem. Assume the input power to a pump P (M L T-¹) is depend on Q- Flowrate (L3 T-¹), H-pump head (L), p- fluid density (ML-³) Create a relation by dimensional analysis between the power and other variables by Rayleigh Method.Q.5 A plate 1 mm distance from a fixed plate, is moving at 500 mm/s by a force induces a 2 shear stress of 0.3 kg(f)/m. The kinematic viscosity of the fluid (mass density 1000 kg/ 3. m) flowing between two plates (in Stokes) isWhen a person ice skates, the surface of the ice actuallymelts beneath the blades, so that he or she skates on a thinsheet of water between the blade and the ice.( a ) Find an expression for total friction force on the bottomof the blade as a function of skater velocity V , bladelength L , water thickness (between the blade and theice) h , water viscosity μ , and blade width W .( b ) Suppose an ice skater of total mass m is skatingalong at a constant speed of V 0 when she suddenlystands stiff with her skates pointed directly forward,allowing herself to coast to a stop. Neglecting frictiondue to air resistance, how far will she travelbefore she comes to a stop? (Remember, she iscoasting on two skate blades.) Give your answer forthe total distance traveled, x , as a function of V 0 , m ,L , h , μ , and W .( c ) Find x for the case where V 0 = 4.0 m/s, m = 100 kg,L = 30 cm, W = 5.0 mm, and h = 0.10 mm. Do youthink our assumption of negligible air resistance is agood one?
- Ex. The force required to tow a 1:30scale model of a motor boat in a lake at a speed of 2m/s is 0.5 N, Assuming that the viscosity resistance due to water and air is negligible in comparison with the wave resistance, calculate the corresponding speed of the prototype for dynamically similar conditions. What would be the force required to propel the prptype at that velocity in the same lake? Ans.: 10.95m/s, 13500NA milkshake has fairly similar density to that of water (m s = 1200 kg/m3) but is far more viscous ( = 1kg/ms) (a) Say you try to drink a milkshake through a straw that is 30 cm long and 5 mm in diameter. Your lungs are capable of creating a vacuum pressure of 3000 Pa. (Vacuum pressure just means a pressure below that of the atmosphere, so plung = patm 3000 Pa.) You Önd that if you place the straw just at the surface of the liquid, you are unable to suck the milkshake through the straw, but if you push the straw deeper into the shake, you can. To what depth, d, would you need to push the straw in order to just start to sip the milkshake?(b) Suppose you push the straw to a depth of 10 cm and suck with a suction pressure of 3000 N/m2. What volume áow rate of milkshake can you produce through the strawMott ." cometer, which we can analyze later in Chap. 7. A small ball of diameter D and density p, falls through a tube of test liquid (p. µ). The fall velocity V is calculated by the time to fall a measured distance. The formula for calculating the viscosity of the fluid is discusses a simple falling-ball vis- (Po – p)gD² 18 V This result is limited by the requirement that the Reynolds number (pVD/u) be less than 1.0. Suppose a steel ball (SG = 7.87) of diameter 2.2 mm falls in SAE 25W oil (SG = 0.88) at 20°C. The measured fall velocity is 8.4 cm/s. (a) What is the viscosity of the oil, in kg/m-s? (b) Is the Reynolds num- ber small enough for a valid estimate?
- Pipelines are cleaned by pushing through them a closefitting cylinder called a pig . The name comes from thesquealing noise it makes sliding along. Reference 50describes a new nontoxic pig, driven by compressed air, forcleaning cosmetic and beverage pipes. Suppose the pigdiameter is 5-15/16 in and its length 26 in. It cleans a6-in-diameter pipe at a speed of 1.2 m/s. If the clearance isfi lled with glycerin at 20 8 C, what pressure difference, inpascals, is needed to drive the pig? Assume a linear velocityprofi le in the oil and neglect air drag.P1.20 A baseball, with m = 145 g, is thrown directly upward from the initial position z = 0 and Vo = 45 m/s. The air drag on the ball is CV², as in Prob. 1.19, where C~ 0.0013 N: s*/m". Set up a differential equation for the ball motion, and solve for the instantaneous velocity V(t) and position z(1). Find the maximum height zmax reached by the ball, and compare your results with the classical case of zero air drag.A viscous fluid of density 1200 kg/m3 and viscosity 0.9 Pa.s is confined in a spacing of 0,01 m between two parallel and horizontal plates of length 0,6 m and width 0,6 m. The upper plate is pulled with a uniform and steady speed of 0,7 m/s in its own plane. The lower plate is stationary. Viscous fluid flow is laminar steady and incompressible. a) What is the magnitude of mass flow rate in spacing (in kg/s)? b) What is the magnitude of maximum shearing stress in the flow field (in Pa)? c) What is the frictional force over the lower plate surface (in N )? d) What is the frictional power lost in pulling the upper plate (in W)?
- Q1: Consider laminar flow over a flat plate. The boundary layer thickness o grows with distance x down the plate and is also a function of free-stream velocity U, fluid viscosity u, and fluid density p. Find the dimensionless parameters for this problem, being sure to rearrange if neessary to agree with the standard dimensionless groups in fluid mechanics. Answer: Q2: The power input P to a centrifugal pump is assumed to be a function of the volume flow Q, impeller diameter D, rotational rate 2, and the density p and viscosity u of the fluid. Rewrite these variables as a dimensionless relationship. Hint: Take 2, p, and D as repeating variables. P e paD? = f( Answer:An incompressible fluid flows in a linear porous medium with the following properties: Lenth = 3000 ft k = 100 md p1 = 2000 psig p2 = 1980 psig height = 25 ft porosity = 20% width = 300 ft viscosity = 2 cP Assume the dimension is slanted, i.e., a dip angle of 5 degrees (downward from p1 location to p2 location), what is the apparent fluid velocity under this new boundary condition?(b) A Formula 1 team tests their car in a wind tunnel at a scale of 50%, at a speed of Vwinarunnet = 50 m/s. The measured downforce at this speed is Foownforce.WT = 3000 N and the model reference area is Aref wT = 0.5 m?. 1. Find the downforce magnitude on the full-scale race car at the same Reynolds number. Assume constant air density p = 1.2 kg/m. II. It is said that F1 cars could travel upside down, as shown below. At what speed should the full-scale car travel to achieve this, if its mass is m = 740 kg? The acceleration of gravity is g = 9.81 m/s?. Downforce Weight