TALLINN UNIVERSITY OF TECHNOLOGY, DIVISION OF PHYSICS 13. STOKES METHOD

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1 13. STOKES METHOD 1. Objective To determine the coefficient of viscosity of a known fluid using Stokes method.. Equipment needed A glass vessel with glycerine, micrometer calliper, stopwatch, ruler. 3. Theory Viscosity refers to the friction within a fluid from flowing freely and is essentially a frictional force between different layers of fluid as they move past one other. The tangential force F required to move a layer of area S and located a perpendicular distance x from an immobile surface is given by: where is the coefficient of viscosity. dv F S, (1) dx Provided the flow the velocity is not too large, the fluid flows smoothly and the flow is said to be laminar. The mutual influences of layers of fluid which move past one another are conditioned by the molecule intermediate attraction of fluid. This prevents also the motion of solid body in fluid because the whole surface of body is covered with thin molecular layers by molecules of fluid. Therefore the force, which prevents the motion of a body in a fluid, can be found by the force of viscosity. It is possible only then, when the velocity of the body is smaller than the velocity of laminar flow of fluid. Otherwise the whirls will arise and the formula (1) given by Newton is useless. In general it is complicated to find the formula of frictional force F. In case of regular bodies the problem will simplify. For a spherical bodies Stokes derived the following formula: t F t 6rv, () where is the coefficient of viscosity, r is the radius of the sphere and v is the velocity of the sphere. In this experiment the formula () is used to determine the coefficient of viscosity. Computing the frictional force 13.1). F t the sphere dropping through a column of liquid is observed. (see figure 1

2 There are three forces acting on the sphere ball dropped into the liquid (figure 13.1). 1) The force of gravity F1 mg Vg, (3) where V is the volume of the sphere ball, is the density of the sphere ball and g is acceleration of gravity. ) Buoyancy force F Vg, (4) where is the density of the liquid. 3) Viscous drag force F 3 6rv (5) Both forces F and F 3 act upwards buoyancy tending to float the sphere and the drag force resisting the acceleration of gravity. The only force acting downwards is the body force resulting from gravitational attraction ( mg ). By summing forces in the vertical direction the following equation can be written: F 1 F F3 (6) or The volume of a sphere is written as V Vg Vg 6rv 4 r 3 3. Rearranging and regrouping the terms from the above equation the following relationship will be arrived: r ( ) g (7) 9v The formula (7) is valid in case of the volume of liquid is infinity large. In reality there is to do with liquid located in a definitive size of vessel. For that reason the velocity gradient of layers of liquid is greater than dropping in infinity large volume of fluid. Therefore the

3 viscous drag force acting on the sphere will increase. That is why in the real trial the size and shape of the vessel must take into account. It can be shown that when the sphere is dropping in a cylindrical vessel with radius R the following formula is used: ( ) gr 9 r v(1,4 ) R (8) All the quantities in the formula (8) can be determined experimentally and therefore the coefficient of viscosity can be calculated by the formula (8). If the flow velocity is large the flow will become turbulent and Stokes Law will no longer hold: the flow rate will drop. The onset of turbulence is often abrupt and characterised by a dimensionless number, the Reynolds number (Re), where: vr Re (9) Experiment show that the flow is laminar if Re is less than about and turbulent if it exceeds this value. The viscosity depends to a large extent on temperature and pressure. The coefficient of viscosity for gas decreases proportionally to velocity of molecules, when temperature is diminishing i.e. proportionally to square root of temperature. But in case of liquids it grows exponentially according to the law: kt e, (1) where k is Boltzmann s constant, T is absolute temperature of the liquid and energy of molecules. is transition 4. Experimental procedure 1. Determine the diameter d and the mass m of the sphere balls.. Familiarise yourself with the provided stopwatch. 3. Drop the ball through the hole into the glass cylinder and measure the time t it takes the ball to sink the given distance s through the glycerine. 3

4 4. Repeat the procedure for another 3 5 different steel balls. Record the results into Data Table The densities of the glycerine and steel ball are given at the workplace. 5. Compute the average value of the coefficient of viscosity. Treat the values of as the direct measured quantities. Find it s A-type uncertainty. 6. Calculate the Reynolds number by the formula (9) and draw a conclusion about conditions of this experiment. Figure 13.1 The number of trial Determination of the coefficient of liquids viscosity. l d, mm m, mg t, s, kg/m 3, Pa*s Table 13.1 i, Pa*s Re... 4

5 5. Questions 1) What is viscosity, the coefficient of viscosity? ) How depends the coefficient of viscosity on temperature? 3) What kind of flow is said to be laminar? 4) Formulate the formula of Newton and Stokes. 5) Upon what depend the distance of accelerated falling ball? 6) What does the Reynolds number characterise? 7) What does the velocity gradient show? 8) Derive the units of the coefficient of viscosity in SI- and CGS system. 9) Is it a great mistake if we use the formula (8) instead of the formula (7) in our work? 1) How is it possible to measure the coefficient of viscosity on given experiment when the density of the ball is smaller than the liquids one? 6. Literature Halliday, D., Resnick, R, Walker, J. Fundamentals of Physics 8th ed. Hoboken (N. J.), John Wiley & Sons, Inc., 8,

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