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A falling ball viscometer was designed at ACBM at Northwestern University using a scale with accuracy of 0.001g, an elastic tensile spring, and steel balls of various diameters (Fig. 1). When a steel ball is suspended by the spring and is allowed to move in the fluid, then the forces acting on the ball can be resolved into four components. As shown in Fig. 2, these components are gravity (W), tensile force (T), buoyancy (B), and drag force (D). During the measurement, the spring is hooked to a sensor that is located at the bottom of the scale. By suspending the steel ball with the spring, the tensile force in the spring during the downwards movement of the ball can be continuously recorded by reading the numbers shown on the scale. Once the tensile force is known, the displacement of the ball can be computed and this allows for the calculation of the velocity and acceleration of the ball. Hence, the only force left to be determined is the drag force, and it can be solved using the equation of motion shown in Fig. 2. In steady state condition and when Reynolds number (R_n) is less than 0.5, the drag force can be linearly related to the velocity by applying Stoke’s Law for a spherical particle as shown in equation 1:

(2.1)

where, η is the viscosity of the measured liquid; r is the radius of the ball; v is the velocity of the ball; and ρ is the density of the measured liquid. From the equation, it can be noted that the drag force is linearly related to the velocity of the ball and the size of the moving ball.

Fig. 1 Configuration of the

falling ball viscometer

Fig. 2 Free body diagram of the ball

Equation 1 is valid for a spherical particle moving with a constant velocity in a Newtonian fluid. However, with the configuration that introduced, the velocity of a moving ball changes with time. This is attributed to the changing of the tensile force as the spring elongates according to the ball movement. Thus, to make sure the used theory is still valid for the used configuration, a calibration for the designed viscometer is necessary.

Various Newtonian fluids with known viscosities were used for the system calibration. It was found that the viscosity of the measured liquid can be determined through equation 2, where K_e is the slope of the drag force-ball velocity curve that determined experimentally:

(2.2)

This calibration is proved to be efficient for Newtonian fluids. It is suggested that the calibration should also be valid for non-Newtonian fluids, if the material behaves according to Bingham model. For a Bingham material, drag force D is linearly related to the velocity of the moving ball, however, the linear line has an intersection with the y axis as shown in Fig. 3, where D₀ can be correlated to the yield stress of the material. The solid line in the figure can be shifted down until it intersects with the origin. This implies that the calibration of the system should also be valid for Bingham fluids. Fresh concrete is normally regarded as a Bingham fluid, thus, the viscometer has a high potential to be applied to concrete. The yield stress acts tangentially to the surface of the steel ball. Thus, the relationship between the initial resistance to motion(subsequently called initial drag force (D₀)) and the yield stress (Ty) of the measured liquid can be expressed as follows: