When fluid flows through a tube or pipe, there is friction between the moving fluid and pipe wall. The amount of friction is primarily due to the viscosity of the fluid (B. Munson, T. H. Okiishi, W. Huebsch, and D. Young, “Fundamentals of Fluid Mechanics,” ed: New York: John Wiley & Sons Inc, 2013). In fluid dynamics, the viscosity μ is defined as the proportionality constant between wall shear stress τwall and wall shear rate {dot over (γ)}wall (B. Munson, T. H. Okiishi, W. Huebsch, and D. Young, “Fundamentals of Fluid Mechanics,” ed: New York: John Wiley & Sons Inc, 2013).τwall=μ·[{dot over (γ)}wall]  Eq. (1)where the wall shear stress [N/m2] is a tangential or frictional force per unit area, and the wall shear rate [s−1] is a velocity gradient at pipe wall, i.e., [du/dr]wall. Note that u or u(r) indicates the axial flow velocity profile in pipe flow, and r represents the radial coordinate in the pipe flow. The wall shear rate can be conceptually described as the ratio of the mean flow velocity V to pipe diameter d, i.e., ˜V/d. For Newtonian fluids, the wall shear rate is analytically determined as 8 V/d in a laminar flow in a circular pipe (B. Munson, T. H. Okiishi, W. Huebsch, and D. Young, “Fundamentals of Fluid Mechanics,” ed: New York: John Wiley & Sons Inc, 2013).
The viscosities of water and air do not vary with shear rate, whereas the viscosities of fluids such as paint, tomato ketchup, blood, synthetic engine oil, and polymer solutions vary with the shear rate. The liquids whose visocisties are independent of shear rate are referred to as Newtonian fluids, while the liquids whose viscosities are dependent on the shear rate are described as non-Newtonian fluids (B. Munson, T. H. Okiishi, W. Huebsch, and D. Young, “Fundamentals of Fluid Mechanics,” ed: New York: John Wiley & Sons Inc, 2013).
In the field of engineering, the viscosity of non-Newtonian fluids (e.g. synthetic oils and polymer solutions) is usually measured with a rotating-type viscometer such as the Brookfield cone and plate viscometer (S. Kaya and A. R. Tekin, “The effect of salep content on the rheological characteristics of a typical ice-cream mix,” Journal of Food Engineering, vol. 47, pp. 59-62, 2001). However, it is not practical to use such rotating viscometers for blood viscosity measurements in clinical settings because the test section in rotating viscometers must be manually cleaned after each viscosity test, a procedure that is unsafe to the operator due to the potential risk of making contact with contaminated blood.
The prior art describes the use of a U-shaped tube assembly, where a capillary tube is positioned horizontally between two vertical tubes (K. Kensey and Y. Cho, “Method for determining the viscosity of an adulterated blood sample over plural shear rates,” 2004; K. Kensey, W. N. Hogenauer, S. Kim, and Y. Cho, “Dual riser/single capillary viscometer,” 2002). This method presents capillary-based viscometry capable of measuring viscosity across a range of shear rates using a single sweeping scan. The U-shaped tube assembly in this prior art is disposable, enabling the technician or operator to avoid direct contact with blood. While this prior art provided improvements over earlier viscometer methods, it has a number of profound limitations that limit its functionality, accuracy, and practical use.
In a case where a liquid having an unknown viscosity fills the left-side vertical tube first to a predetermined height, the liquid begins to fall at the beginning of the test by gravity, moving through the capillary tube, and rising in the right-side vertical tube. In an idealized scenario, the liquid levels in the two vertical tubes become equal as time goes to infinity (i.e. the end of the test). However, due to the difference in the surface tension in the two vertical tubes, the left and right liquid levels do not perfectly equilibrate, even at t=infinity. The reason is as follows: as the liquid falls in the left-side vertical tube, the inner wall of the vertical tube is fully wet during the operation of the test. On the other hand, as the liquid rises in the right vertical tube, the inner wall of the right vertical tube remains fully dry prior to and at the interface of liquid movement during operation of the test. Accordingly, the surface tension in the liquid-falling tube is consistently larger than in the liquid-rising tube because the magnitude of the surface tension in a wet tube is significantly greater than that in a dry tube (B. Munson, T. H. Okiishi, W. Huebsch, and D. Young, “Fundamentals of Fluid Mechanics,” ed: New York: John Wiley & Sons Inc, 2013).
Accordingly, the liquid level in the liquid-falling tube is significantly higher than that in the liquid-rising tube at the end of the test. This liquid height difference is caused by the different surface tensions in the two vertical tubes and can affect the accuracy of viscosity measurement of the liquid near the end of the test, especially at low-shear ranges of less than 10 s−1. Since the difference in the surface tensions in the two vertical tubes could not be resolved in the prior art, the surface tension term was numerically isolated in the calculation of viscosity in S. Kim, Y. Cho, W. Hogenauer, and K. Kensey, “A method of isolating surface tension and yield stress effects in a U-shaped scanning capillary-tube viscometer using a Casson model” Journal of Non-Newtonian Fluid Mechanics, vol. 103, pp. 205-219, 2002. More specifically, the Casson model was used to relate the shear stress and shear rate using two unknown constants: Casson constant k and yield stress τy. The surface tension term was added as the third unknown constant, which had to be numerically determined in the minimization algorithm (i.e., minimizing the sum of the error in the curve fitting of experimental data of the liquid level variation over time). While this prior art describes a capillary-based method for measuring viscosity across multiple shear rates, there is still a need to measure more accurately the surface tension of the liquid.
Another limitation of the prior art is associated with the liquid-falling vertical tube. In the clinical case of hyperviscosity syndrome, common in polycythemias, multiple myeloma, leukemia, monoclonal gammopathies such as Waldenström macroglobulinemia, sickle cell anemia, and sepsis, whole blood often becomes very sticky. As blood falls in the liquid-falling vertical tube, a small droplet of blood can stick to the vertical wall, leaving a streak of blood, which becomes a significant source of error in the viscosity measurement.
Furthermore, the prior art utilized a gravity-driven flow in a U-shaped tube, where blood falls from one vertical tube by gravity, passes through the capillary tube, and then rises in another vertical tube. Blood moves through the capillary tube by the height difference between the two vertical tubes. In clinical cases of hyperviscosity syndrome (as well as with other very thick liquids such as yogurt, grease, and slurry of suspended particles), gravity alone cannot push the liquid through the capillary tube of the viscometer described in these prior arts because of increased friction caused by the thick liquid and small diameter of the capillary tube (i.e., less than 0.8 cm). Hence, the method in the prior art cannot measure the viscosity of thick liquids using the U-shaped tube beyond a certain threshold viscosity.
In summary, the prior art utilizes a gravity-driven flow in a U-shaped tube, where changes in the liquid height over time within the two vertical tubes are measured. The liquid viscosity is determined from the first derivative (i.e., slope) dh(t)/dt of the height change h(t). The procedure of taking the first derivative of the height change for multiple time points is calculation-intensive, requiring a microprocessor for the data reduction process. Furthermore, the need to calculate the first derivative significantly increases the potential for error in the viscosity measurement, because the liquid height changes in the two vertical tubes are naturally punctuated and are often not smooth. For example, a small pause of liquid motion or small sudden drop will be magnified in the first derivative of the height change.
There is a need in the art for an improved viscometers and methods for more accurately measuring and calculating liquid viscosity. The present invention meets this need.