Using Buckingham-pi theorem, find an expression for the drag force (D) of a sphere of diameter (do) falling slowly with a velocity (V) in a large vessel containing a liquid of density (ρ) and viscosity (μ).
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Using Buckingham-pi theorem, find an expression for the drag force (D) of a sphere of diameter (do) falling slowly with a velocity (V) in a large vessel containing a liquid of density (ρ) and viscosity (μ).
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- Assume an inviscid, incompressible flow. Also, standard sea level density and pressure are 1.23 kg/m3 (0.002377 slug/ft3) and 1.01 × 105 N/m2 (2116 lb/ft2), respectively. Consider the lifting flow over a circular cylinder of a given radius and witha given circulation. If V∞ is doubled, keeping the circulation the same,does the shape of the streamlines change? Explain.A capillary viscometer measures the time required for a specified volume υ of liquid to flow through a small-bore glasstube, as in Fig. . This transit time is then correlated withfluid viscosity. For the system shown, (a) derive an approximateformula for the time required, assuming laminar flowwith no entrance and exit losses. (b) If L = 12 cm, l = 2 cm,υ = 8 cm3, and the fluid is water at 20 °C, what capillary diameterD will result in a transit time t of 6 seconds?Suppose that it is desired to estimate volume fl ow Q in apipe by measuring the axial velocity u ( r ) at specifi c points.For cost reasons only three measuring points are to be used.What are the best radii selections for these three points?
- At low velocities (laminar flow), the volume flow Q through a small-bore tube is a function only of the tube radius R, the fluid viscosity ?܇, and the pressure drop per unit tube length dp/dx. Using the Buckingham pi theorem, find an appropriate dimensionless relationship. Make use of Table 1By an inkjet printerthe diameter (d) of the points created, the dynamic of the inkviscosity (µ), density (ρ), surface tension (σ), nozzlediameter (D), the distance of the nozzle from the paper surface (L) andink jet velocity is thought to depend on V.Get an expression to characterize the behavior of the ink jetplease.Liquid is pushed through the narrow gap formed between two glass plates. Thedistance between the plates is h and the width of the plate is b . Assuming that theflow is laminar, two-dimensional and fully developed, derive a formula for thelongitudinal pressure gradient in terms of the volumetric flow rate, Q, the fluid viscosityand the distances b and h .In an example the above conditions apply with b = 0.10 m and h = 0.00076 m. Theliquid is glycerine that has an absolute viscosity of 0.96 N s/m². A pressure differenceof 192 kN/m² is applied over a distance of 0.24 m. Calculate the maximum velocity thatoccurs in the centre plane of the gap and the mean velocity.Answer 0.06 m/s