VISCOSITY OF A LIQUID. To determine the viscosity of a lubricating oil. Time permitting, the temperature variation of viscosity can also be studied.

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1 VISCOSITY OF A LIQUID August 19, 004 OBJECTIVE: EQUIPMENT: To determine the viscosity of a lubricating oil. Time permitting, the temperature variation of viscosity can also be studied. Viscosity apparatus Tensor lamp Stop watch Vernier depth gauge Various masses Oil, beakers DISCUSSION: The coefficient of viscosity (often called simply the viscosity) is defined as shearing stress rate of shear (1) where shearing stress is force per unit area, and the rate of shear is velocity per unit length, thus f/a fr = v/r Av () The SI unit of viscosity is N 1 m = 1 Ns/ m 1 m 1 m/ sec 1 (3) A viscosity of 1 poise is one that requires a tangential force of 1 dyne for each square centimetre of surface to maintain a relative velocity of 1 cm/sec between two planes separated by a layer of fluid 1 cm thick. Therefore 1 poise = 0.1 Nsm -. A stoke is related to the poise by the density of the fluid. You can think of the stoke as the kinematic viscosity. The stoke is defined as 10-4 m s -1. Thus, the relationship between poise and stoke is poise 3 stoke= 10 density (4) In the coaxial-cylinder method of determining viscosity, a shearing takes place in which concentric cylindrical layers of liquid slip over each other. The angular velocity ω increases progressively from zero at the stationary cylinder A to ω D at the rotating one, B Viscosity-1

rate of shear (1) where shearing stress is force per unit area, and the rate of shear is velocity per unit length, thus f/a fr = v/r Av () The SI unit of viscosity is N 1 m = 1 Ns/ m 1 m 1 m/ sec 1

2 August 19, 004 in Fig. 1. The linear velocity at an intermediate surface SS is v = ωr. The velocity gradient at SS is then d dω (5) ( ω r)= ω + r dr dr Fig. 1: Shearing stress in a liquid confined between concentric cylinders. The first term ω on the right in Eq.(3) represents the rate of increase in v with r when all portions of the substance move with the same angular velocity. When shearing occurs, however, each cylindrical layer has an angular velocity greater than that of the one just inside it; the second term r dω/dr represents the variation in v due to the variation in ω. In a rigid body this term would be zero. In a fluid it represents the velocity gradient due to relative movement of adjacent layers. Substituting this term for the velocity gradient v/r in Eq.(1) f /A r (dω/dr) (6) or f dr η dω = A r (7) If the torque applied to the rotating cylinder is L, the tangential force at any layer SS is L/r, and the tangential force per unit area is L/πr l where l is the length of cylinder in contact with the fluid. Substitution in Eq.(5) gives L η ω = πl dr d 3 r (8) Integrating between limits r = a and r = b gives Viscosity-

(3) represents the rate of increase in v with r when all portions of the substance move with the same angular velocity.

3 or August 19, 004 L 1 1 η ω B = - 4πl a b (9) b - a 4π a b L l ω (10) B The viscosity η is thus determined from the constants of the apparatus and from the experimentally determined ratio L/ω B. In the apparatus used for this experiment, the inner cylinder A rotates inside a stationary cylinder B. However, the analysis above holds regardless of which cylinder is rotated, since it is the relative velocity that is important. There is a torque not yet considered due to the viscous drag between the ends of the cylinders. For a given apparatus, this end effect is a constant which can be treated as a correction to be made to the length of the cylinder. Its effective length is then the immersed length l plus a factor e (e is negative) to be determined experimentally. Eq.(3) then becomes b - a L 4 π a b ( l - e ) ω (11) B The torque is applied to the inner cylinder A (Fig. ) by a fine cord that passes over a series of pulleys and carries a mass m. The shearing torque in absolute units is L = mgk, where k is the radius of the drum. The angular velocity is ω = s/kt, where s is the distance the mass m descends in time t. Substituting these relations in Eq.(9) η ( b - a ) k g 4 π a b s ( l - e = m t ) (1) Eq.(10) shows that the viscosity can be found from a series of measurements of the time required for a fixed mass to fall a distance s with a known oil depth. The end correction can also be found from these measurements. A second method to measure viscosity uses a constant oil depth; then once again the time is recorded for a series of masses. Viscosity-3

However, the analysis above holds regardless of which cylinder is rotated, since it is the relative velocity that is important.

4 PROCEDURE: NOTE: August 19, 004 Throughout this experiment, keep the apparatus in the tray and make every effort to minimize the mess with the oil. Wipe up any spills immediately. 1) Make all necessary measurements on the apparatus. Note that you need to know a, b, k, s and be able to calculate l for each run.. ) Pour in oil to the depth of about 1.5 cm. It is easy, though slow, to determine the distance to the oil surface from the lip with the depth gauge. Shine a bright light down the oil space and slowly extend the gauge. When the gauge touches the oil surface, the distortion of the surface can be seen as a meniscus is formed. In this way depths can be measured to ± 0.5 mm. 3) Put 0 g of weight on the end of the string. Wind the string on the drum until the weight is above the upper marker. Release the weight and time the fall between the markers or from the upper marker to the floor. Alternatively, time the fall between the upper marker and the sound of the mass hitting the floor. Add some oil from the beaker and repeat for between 5 and 10 runs until the oil level is near to, but not covering, the upper cylinder surface. Calculate l for each run. 4) Plot a graph of t vs. l and determine the viscosity and the length correction e from the plot. 5) With the oil near, but not covering the upper surface of the cylinder, make a series of timings of the descent of the weight for 10 g up to 60 g in steps of 10 g. Plot t vs. 1/m and determine η from the graph using the value of e obtained in step 4. Viscosity-4

It is easy, though slow, to determine the distance to the oil surface from the lip with the depth gauge. Shine a bright light down the oil space and slowly extend the gauge.

5 August 19, 004 Fig. : Vertical cross-section of concentric cylinder viscosity apparatus. Viscosity-5

6 August 19, 004 Figure 3: APPROXIMATE VISCOSITY EQUIVALENTS (Ref.: Esso Petroleum) Viscosity-6

7 August 19, 004 WRITE-UP GUIDELINES FOR RESULTS SECTION OF VISCOSITY EXPERIMENT Your Results section on this report should contain the following: 1. A table showing your measurements of the apparatus, oil depths and times for each fall.. A plot of t vs. l (or vice versa - you may have good reasons to put one or the other on the y-axis) including linear regression line. 3. A plot of t vs. 1/m (or vice versa - you may have good reasons to put one or the other on the y-axis) including linear regression line. 4. Calculations of η and e from the t vs. l graph and calculation of η from the t vs. 1/m graph and comparison of the two values of η. Note that the units can cause difficulty in these calculations so be extra careful to carry all units all the way through these calculations. The error calculations on η and e are quite involved. Maple is recommended. Include a printout of your Maple program (with comments!) as a sample calculation. These should be the only sample calculations necessary in this lab. 5. Discussion of calculation techniques, error estimates, etc. Page limit (entire lab - Results and Conclusions): 10 Viscosity-7

l (or vice versa - you may have good reasons to put one or the other on the y-axis) including linear regression line. 3. A plot of t vs.

The Viscosity of Fluids

Experiment #11 The Viscosity of Fluids References: 1. Your first year physics textbook. 2. D. Tabor, Gases, Liquids and Solids: and Other States of Matter (Cambridge Press, 1991). 3. J.R. Van Wazer et