Practice Problems on Viscosity. free surface. water. y x. Answer(s): base: free surface: 0

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1 viscosity_01 Determine the magnitude and direction of the shear stress that the water applies: a. to the base b. to the free surface free surface U y x h u water u U y y 2 h h 2 2U base: yx y 0 h free surface: 0 yx y h C. Wassgren, Purdue University Page 1 of 11 Last Updated: 2010 Aug 27

2 viscosity_02 When a vehicle such as an automobile slams on its brakes (locking the wheels) on a very wet road it can hydroplane. In these circumstances a film of water is created between the tires and the road. Theoretically, a vehicle could slide a very long way under these conditions though in practice the film is destroyed before such distances are achieved (indeed, tire treads are designed to prevent the persistence of such films). To analyze this situation, consider a vehicle of mass, M, sliding over a horizontal plane covered with a film of liquid of viscosity,. Let the area of the film under all four tires be A and the film thickness (assumed uniform) be h. a. If the velocity of the vehicle at some instant is V, find the force slowing the vehicle down in terms of A, V, h, and. b. Find the distance, L, that the vehicle would slide before coming to rest assuming that A and h remain constant (this is not, of course, very realistic). c. What is this distance, L, for a 1000 kg vehicle if A = 0.1 m 2, h = 0.1 mm, V = 10 m/s, and the water viscosity is = kg/(ms)? V F A h MhV L 0 A L = 10,000 m ( 6.2 miles) C. Wassgren, Purdue University Page 2 of 11 Last Updated: 2010 Aug 27

3 viscosity_03 Magnet wire is to be coated with varnish for insulation by drawing it through a circular die of 0.9 mm diameter. The wire diameter is 0.8 mm and it is centered in the die. The varnish, with a dynamic viscosity of 20 centipoise, completely fills the space between the wire and the die for a length of 20 mm. The wire is drawn through the die at a speed of 50 m/s. Determine the force required to pull the wire. 2 L i V F o i F = 1.0 N C. Wassgren, Purdue University Page 3 of 11 Last Updated: 2010 Aug 27

4 viscosity_04 A rotating disk viscometer has a radius, = 50 mm, and a clearance, h = 1 mm, as shown in the figure. fluid h h a. If the torque required to rotate the disk at = 900 rpm is N m, determine the dynamic viscosity of the fluid. You may neglect the viscous forces acting on the rim of the disk and on the vertical shaft. b. If the uncertainty in each parameter is ±1%, determine the uncertainty in the viscosity. Th 4 = 0.29 kg/(ms) or 0.29 Pas or 290 cp u 4.4% = Pas C. Wassgren, Purdue University Page 4 of 11 Last Updated: 2010 Aug 27

5 viscosity_05 The cone and plate viscometer shown in the figure is an instrument used frequently to characterize non-newtonian fluids. It consists of a flat plate and a rotating cone with a very obtuse angle (typically is less than 0.5 degrees). The apex of the cone just touches the plate surface and the liquid to be tested fills the narrow gap formed by the cone and plate. a. Derive an expression for the shear rate in the liquid that fills the gap in terms of the geometry of the system and the operating conditions. b. Evaluate the torque on the driven cone in terms of the shear stress and geometry of the system. c. Evaluate the torque on the driven cone in terms of the geometry, operating conditions, and fluid properties if the fluid is Newtonian. fluid du shear rate dy tan 2 T 3cos 3 y yh 3 2 T 3sin C. Wassgren, Purdue University Page 5 of 11 Last Updated: 2010 Aug 27

6 viscosity_06 A rotating cylinder viscometer is shown in the figure below. The inner cylinder has radius,, and height, H. An incompressible, viscous, Newtonian fluid of density,, and viscosity,, is contained between the cylinders. The narrow gap between the cylinders has width, r (<< and H). A torque, T, is required to rotate the inner cylinder at constant speed. Determine the fluid viscosity,, in terms of the other system parameters., T y H r r Newtonian fluid of density,, and viscosity, r T 2 r 3 1 H 4 C. Wassgren, Purdue University Page 6 of 11 Last Updated: 2010 Aug 27

7 viscosity_07 The viscosity of blood is to be determined from measurements of shear stress and shear rate obtained from a small blood sample tested in a suitable viscometer. Based on the data given in the table below, determine if the blood is a Newtonian or a non-newtonian fluid. Explain how you arrived at your answer. data set shear rate [s -1 ] shear stress [N/m 2 ] shear stress [N/m 2 ] shear rate [s -1 ] The slope of the blood data is not equal to one indicating that blood is non-newtonian. In fact, since the slope is less than one over most of the range of shear rate, blood is shear thinning. C. Wassgren, Purdue University Page 7 of 11 Last Updated: 2010 Aug 27

8 viscosity_08 The no-slip condition states that fluid sticks to solid surfaces. Two immiscible layers of Newtonian fluid are dragged along by the motion of an upper plate as shown in the figure. The bottom plate is stationary and the velocity profiles for each fluid are linear. The top fluid (fluid 1), with a specific gravity of 0.8 and kinematic viscosity of 1.0 cst, puts a shear stress on the upper plate, and the lower fluid (fluid 2), with a specific gravity of 1.1 and kinematic viscosity of 1.3 cst, puts a shear stress on the bottom plate. Determine the ratio of the shear stress on the top plate to the shear stress on the bottom plate m m 3 m/s fluid 1 fluid 2 2 m/s yx on top plate yx on bottom plate 1.0 C. Wassgren, Purdue University Page 8 of 11 Last Updated: 2010 Aug 27

9 viscosity_09 A viscous, Newtonian liquid film falls under the action of gravity down the surface of a rod as shown in the figure below. film film r atmosphere gravity a b The velocity of the fluid is given by: 2 g 2 2 gb r uz r a ln 4 2 a where is the liquid density, g is the acceleration due to gravity, is the liquid s dynamic viscosity, a is the radius of the rod, b is the radius to the free surface of the film, and r is the radius measured from the centerline of the rod. Determine: a. What is the shear stress, rz, at the free surface of the liquid? b. What force acts on the rod per unit length of the rod due to the viscous liquid? c. Set up, but do not solve, the integral for determining the volumetric flow rate of liquid flowing down the rod. 0 rz r b 2 F 2 b ga 1 L a rb Q uz 2 rdr ra da z C. Wassgren, Purdue University Page 9 of 11 Last Updated: 2010 Aug 27

10 viscosity_10 During a coating process, a thin, flat tape of width w is pulled through a channel of length L containing a Newtonian fluid of density and dynamic viscosity. The fluid is in contact with both sides of the tape. Estimate the force required to pull the tape through the channel if the tape has velocity V and the channel has height H. You may assume that the tape is much thinner than H. H L V, F channel of width w filled with a Newtonian fluid 4VLw Fshear H C. Wassgren, Purdue University Page 10 of 11 Last Updated: 2010 Aug 27

11 viscosity_11 Estimate the torque required to rotate a windshield wiper blade over the surface of a wet windshield in terms of the parameters given in the figure below, L w Here, is the radius of the inner most point swept by the wiper blade, L is the length of the wiper blade, is the angle swept by the blade in a time T, w is the width of the part of the blade in contact with the windshield, t is the thickness of the water layer, and is the viscosity of water. For a 1996 Toyota av 4, = 6, L = 11, = 110, T = 2 sec, w = ¼, and t = 1 / 16. Evaluate your expression to estimate the torque L 3 w M T t M 7 ft-lb f C. Wassgren, Purdue University Page 11 of 11 Last Updated: 2010 Aug 27

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