130 Experiment-366 F MEASUREMENT OF VISCOSITY OF LIQUIDS BY THE STOKE S METHOD Jeethendra Kumar P K, Ajeya PadmaJeeth and Santhosh K KamalJeeth Instrumentation & Service Unit, No-610, Tata Nagar, Bengaluru-560092. INDIA. Email: labexperiments@rediffmail.com Abstract Using a photo-gated digital time interval measuring clock, acrylic spheres dropped in a vertical tube filled with castor oil such that they move through the oil under gravity, the viscosity of the oil is determined by applying the Stoke s law. The value of viscosity obtained is compared with the standard value for castor oil. The experimental set-up is now available in two models. Of the two models, the one explained in this paper, which can be interfaced with a computer is accurate up to the third decimal resolution for time measurement, provides reasonably accurate value of viscosity. Introduction Viscosity is a measure of the resistance of a fluid to a force applied to it or experienced by a body moving through it. In other words, it indicates the thickness (or stickyness) of a liquid. Thus, water having low viscosity is "thin", whereas honey with high viscosity is "thick". Cooking oils with intermittent thickness have viscosity value in between water and honey. By heating the liquid, its thickness gets reduced or in other words its viscosity is reduced. This indicates that viscosity is a function of temperature. Viscosity also varies with pressure. George Gabriel Stokes, an Irish-born mathematician, spent most of his research effort on describing fluid properties. His most significant contribution is perhaps the work on the motion of a sphere through a viscous fluid. This work led to the development of Stokes Law, a mathematical description of the force required to move a sphere through a quiescent, viscous fluid at a specific velocity. This law forms the basis of the laboratory investigations carried out in course of this work [1]. Stokes law is usually written as, F d = 6πηrv 1 where F d is the drag force of the fluid on a sphere moving through it, η is the fluid viscosity, v is the velocity of the sphere relative to the fluid, and r is the radius of the sphere.
131 Using this equation, along with other well-known principles of physics, one can write an expression that describes the rate at which the sphere falls through a quiescent, viscous fluid. The drag force (F d ), acts upward just like friction. Along with it the buoyancy force (F b ), the thrust exerted by the liquid on the sphere also acts upward. The two forces acting upward are balanced by the weight of the sphere. Figure-1 shows a pictorial representation of the various forces acting on the sphere falling through the liquid. The buoyancy force (F b ) tends to keep the sphere afloat and the drag force (F d ) resists the acceleration due to the gravity. By summing up the forces in the vertical direction, one can write the following equation F b F d Spherical ball Figure-1: The forces acting on a sphere in a liquid F b + F d = mg 2 The buoyancy force is simply the weight of the displaced fluid. The volume of a sphere (V sphere ) is given by V sphere = 3 where r is the radius of the sphere. Combining the volume with the mass density of the fluid, ρ fluid, one can write the buoyancy force as the product F b = m df g = 4 where g is the acceleration due to Earth s gravity, and ρ is the density of the liquid Combining the various forces acting on the sphere moving through a fluid, one can write the following expression πr ρ g +6ηπvr = mg 5 mg
132 where mg is the weight of the ball = volume of the ball x density of the ball. mg = ρ Sphere g 6 Substituting for mg in Equation- 5 and rearranging πr ρ g + 6ηπvr = πr ρ Sphere g 7 6ηπvr = πr ρ Sphere g - πr ρ g = (ρ Sphere ρ fluid ) πr g η = 2gr 8 where v is the velocity of the sphere moving through the liquid Hence if the velocity of the sphere in the liquid is known, one can find the viscosity of the liquid. As the sphere enters a liquid, its velocity is not uniform initially. As it travels through the liquid for a certain distance, the velocity becomes uniform which can be determined accurately. By determining the travel time of the sphere for a fixed distance, one can calculate its velocity. In lab experiments this is done by using a mechanical stop clock or a digital clock. We have designed a photo-gated time interval measuring stop clock from which the travel time of a moving object can be determined easily and the human reflex error controlling the digital or mechanical stop clock is eliminated. This makes the operation simple and measurements more accurate. Clock for measurement of time interval Figure-2: Time interval measuring clock This is an IR (infrared) sensor based digital stop clock. Two sensor pairs are used; one of these is fixed on to the upper clamp holding the tube and second pair is fitted on to the lower clamp holding the tube. When the sphere passes across the upper sensor, the clock is reset (to 00.00) and it passes through the lower sensor it halts the clock (i.e. the clock stops counting time). In this manner the time taken by the sphere to travel from the first sensor to the second
Lab Experiments 133 sensor is noted. The tube is graduated in centimeters; hence velocity v of the sphere can be determined. Figure-2 shows the time interval measuring clock. v= 9 ୲ A 75mm diameter, 600mm long acrylic tube is used as a container for the oil for the sphere to move through it. Larger the tube diameter more accurate will be the measurements. A funnel is provided to guide the ball along the axis of the tube. Apparatus used Digital Stokes viscosity apparatus, time interval measuring clock, spheres of aluminum, acrylic and marble, digital vernier, and digital scale (to measure up to 200gm). The complete experimental set-up is shown in Figure-3. Figure-3: Stoke s apparatus The diameter and mass of the three spheres are determined first using the digital vernier and digital scale, from which the density of the spheres is calculated and present in Table-1. Material Aluminum Acrylic Marble Diameter (mm) Table-1 Radius (r) Volume X10-3(m) x10-6 (m3) Mass (m) x10-3 (kg) Density (ρ) (Kg/m3) 21.31 11.865 6.997 23.73 3045 20.01 10.005 4.19 9.09 2169 16.47 8.235 2.34 5.79 2475 Density of different spherical balls used in the experiment Experimental procedure Velocity (m/s) 0.769 0.2685 0.2667
134 1. The given liquid (about 2 liters of castor oil) is poured into the acrylic tube which is placed on the stand on which the sensor is fixed with two clamps, as shown in Figure- 3. The density of the liquid at room temperature, as given in the Clark s table, is ρ castor oil = 961 kg/m 3 2. The top clamp of the set-up is fixed at the 0 mark on the graduation tube and the bottom sensor is clamped at the 40 cm mark. Hence the distance travelled by the ball S = 40cm = 0.4m 3. The funnel is now placed above the acrylic tube and the aluminium ball is dropped through the funnel and time taken by it to travel a distance of 40cm is noted and its velocity is determined. The spheres of marble and acrylic are also dropped into the tube one by one and time of their travel is also noted. 4. The above trial is repeated 5-6 times and the average time for each sphere is calculated and the velocity is determined and tabulated in Table-1. V aluminum = =. = 0.769 m/s. V marble = =. = 0.2685 m/s. V acrylic = =. = 0.2667 m/s. 5. Viscosity is calculated using Equation-8. η = 2gr η = 2gr η = 2gr =... =... =... =.. = 0.831 =.., = 0.98 =.. = 0.838 The average value of η = 0.883 Ns/m 2 or η = 0.883 Poiseuille (denoted as symbol Pl). 10 Poise = 1 Poiseulle, or η = 8.83 Poise Result and Discussion The value of viscosity of castor oil (η) obtained = 0.883Pl = 8.83 Poise, which is compares reasonably well with the standard value of η (=9.86 Poise) for castor oil. The acrylic sphere, which is of more perfectly spherical shape compared to the other two spheres (aluminum and
135 marble), yielded more accurate value of viscosity (9.8 Poise) compared to the value obtained from the other two spheres. Therefore, perfectness of the spherical shape is very important for this experiment. To verify this, the experiment was repeated with a steel ball bearing which gave the value of viscosity as 10.0 Poise. In the case of the steel ball, the time taken for travelling the distance of 0.4m is 0.377s. The value of viscosity obtained (6.8 Poise) employing smaller diameter (38mm) acrylic tube is also not accurate. Hence spherical shape and large diameter tubes are necessary for accurate viscosity measurements. References [1] C L Arora, BSc Practical Physics, S Chand & Company, Page-85, 2000.