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# The Viscosity of Water Using the Capillary Method

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Viscosity of a liquid

PURPOSE: To measure the viscosity of water by capillary flow method.

REFERENCES:
1) Sears, Zemansky and Young, Univ. Physics, 1982 Edition, Section 13-6.
2) The Physics Teacher, May 1990, pp. 297-299.

THEORY: The rate of flow of a liquid (cm3/s through a capillary is directly proportional to the pressure gradient across the capillary:

dV/dt = (PI/8)*(R^4/eta)*[(P1-P2)/L], (1)

where V is the volume of water, R is the inner radius of capillary, eta is the viscosity of the liquid, P1 and P2 are the pressures at the ends of the capillary, and L is the length of the capillary. Note that the rate of the flow depends on the inner radius of the capillary. As a result for instance, if the human heart is to provide a steady supply of blood to the brain, a 10% reduction of the inner radius of the capillaries carrying the blood would require the blood pressure to increase by 50% to maintain this supply of blood. The above equation is known as Poiseuiville's Law, which is applied in LAMINAR flow. Since the rate of the flow dV/dt=dQ is linearly proportional to the pressure difference delta_P = P1-p2, we have dQ/d(delta_P) = S = PI*R^4/(8*eta*L) and the viscosity eta can be obtained from eta = PI*R^4/(8*S*L). It is common to measure R, and L in cm and Q in cm^3/s and delta_P in Dynes to obtain eta in Poises = g/(cm*s).

PROCEDURE:
In the experiment, a reservoir of water, A, with a constant level x cm above the bench feeds into a capillary length L, which must be set perfectly horizontal using a level. The capillary discharges into a cylinder C with graduations on it. One measures the rate of flow through the capillary by measuring, with a stop watch, the time it takes until the cylinder C is filled up to a certain mark. This measurement should be repeated 5 times. The pressure difference between both ends of the capillary is expressed in gauge pressure, so it is zero at the open end, and (x-y)*g*rho at the other end, where g = 981 cm/s^2 is the gravitational constant, and rho is the density of water (1 g/cc). The measurements are repeated at least at 5 different values for y, the height of the capillary above the bench, to obtain the rate of flow of water at different pressures. x will be constant. One then plots dV/dt = dQ versus pressure (x-y)*g*rho and calculates the slope S from a best-fit straight line.

Note: Viscosity depends on temperature. The temperature of the tap water may fluctuate. Start the water flow - through the waste drain - 15 minutes before doing the experiment and keep the thermometer in the reservoir. Record your temperature. The experiment should be repeated for as many capillaries as possible. The requirement of LAMINAR flow restricts the dimensions of capillaries to specific values. A detailed discussion of this can be found in Ref. 2.

The attached file provides the same description in addition to a figure that describes the experimental setup in detail.

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#### Solution Summary

This solution determines the viscosity of water using the capillary method. The experimental setup is described in detail, and from the experimental results the viscosity of water is determined.

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