Demonstration of Liquid/Solid Surface Slippage by Extrapolation of the Flow Pattern

Introduction

My method visualises the flow pattern of liquid being stirred by a rotating cylinder and finds a relationship between the distance of the liquid from the driving cylinder and the rotation speed of the liquid at that point. By extrapolating the mathematical expression for this relationship forwards and backwards, I have shown that there is movement between the liquid at the surface of the driving cylinder and also movement of the liquid against the wall of the container, and how this changes according to the cylinder’s surface type and the type of liquid.

The currently accepted model for viscosity

Figure 1: Is the Currently Accepted Model, to Which there are Many Published References

In this model, the top surface moves to the right, carrying the viscous liquid with it. The velocity of the liquid was assumed to decrease to zero at the bottom surface via a series of conceptual steps in an arithmetic progression, and all current calculations of viscosity originate from this model.

A position of zero velocity is exactly what is described in many diagrams of liquid flows: see “No Slip Condition” in Wikipedia. They assert, without much experimental proof, that there is no movement of the liquid next to the solid surface. The only experimental demonstration is provided by videos on YouTube, where the experimenter has placed ink on the surface of the solid and then stirred the liquid above it.

• https://www.youtube.com/watch?v=cUTkqZeiMow

• https://www.youtube.com/watch?v=brE-62QvuX4

• https://www.youtube.com/watch?v=xPahJBojSIs

The ink appears not to move at the solid surface, but visibly does so when the ink depositing tube is raised into the liquid flow. As will be seen from my later experiment (see below), the flow at the container surface is very slow, but it does move, and the velocities can be calculated for various surfaces. Also, ink contains dyes which are attracted towards surfaces: a property that enables them to stain cloth. Therefore, the YouTube demonstration only shows that dye does not move much against a surface while in a moving liquid.

Looking Again at the Currently Accepted Model

This model looks suspiciously like the model for an elastic solid, stretching to the right and remaining anchored to the lower surface. In an elastic solid, there are continuous lines of atoms joining the top and bottom surfaces, which stay present even after movement. This is not true for an element of liquid, where there are no permanent lines of atoms linking the layers. Instead, there must be transmission of movement from conceptual layer to conceptual layer. This will be found to reduce as the point of measurement moves away from the driving surface.

Looking at this model another way, one can understand that it requires a rectangular box. The top and bottom surfaces represent the moving and stationary surfaces respectively and the sides represent adjacent molecules of the liquid.

Figure 2

The box contains a viscous liquid, designated by the thin lines. If the top surface of the box is pushed to the right, the following

shape is obtained:

Figure 3

With this container, all the liquid is pushed to the right, and flows that way. But for the liquid to move, movement of the sides of the box is required in order to push the liquid. In this current model, there are no sides to the box, and therefore the liquid cannot move the way that is claimed, in this linear fashion, even as a very small element of a larger system. The sides of the box do not exist. The liquid can only move when the driving surface transmits some motion to the layer of liquid underneath, then that layer transmits movement to the layer beneath it, and so on.

A Proposed New Model

What must be the case around a rotating cylinder in a “Newtonian” liquid is that a series of conceptual concentric, infinitesimally thin rotating cylinders of liquid are present, with the rotational speed of each being in a geometric series as we get further away from the driving cylinder. (See example below.)

Figure 4

In this thought experiment, each notional cylinder must impart the same relative motion at a ratio of 0.5 to its neighbour, or whatever fraction pertains to the substance in question. Clearly, a thin liquid will experience a fraction approaching unity, and the velocity of the cylinders will gradually diminish the further away from the driving cylinder one measures.

A liquid with high viscosity will experience a low fraction, so that the velocities drop quickly. A material with non-Newtonian flow behaviour will set up conceptual concentric cylinders where the velocity of each cylinder will not decrease according to a geometric series but by some other function that can go to zero.

1.4. Published Work on Rotating Cylinders G.I. Taylor F.R.S studied liquid flow between cylinders and published in Philosophical Transactions of the Royal Society of London, Series A, Vol.223 Isue 605-615 VIII in 1922 in an article entitled “Stability of a Viscous Liquid contained between Two Rotating Cylinders”, but this study was concerned with the onset of instability of flow. The same subject was later worked on by Katherine J. Asztalos and Jorge Pulpeiro Gonzalez in a piece entitled “Stability Analysis of Taylor-Couette flow, May 5th 2017, but there was no experimental work described, only mathematical analysis. My geometric series model required testing in the following experiment: 1.5. Experiment to Find the Velocities of Concentric Circles in a Viscous Liquid Driven by a Rotating Cylinder The following apparatus was constructed: The dimensions of the apparatus are as follows: • Cylindrical tank height = 10.9 cm internal. PMMA (clear acrylic). • Internal diameter of tank = 10.4 cm. • Float, for sugar syrup = 2.8 cm, made from plastic tubing, sealed at bottom and weighted with bronze powder, to float vertically. • Float, for castor oil = 2.8 cm, uses a ballpoint pen ink tube, sealed, and weighted with bronze powder • Cylindrical spindle diameter = 1.86cm. • Cylindrical spindle height = 6.48 cm • Viscometer is a Chinese-made NDJ-1, capable of 6, 12, 30 and 60 rpm, and was checked for accuracy of speed. Composition of viscous, non-crystallising, liquid Glycerol 360g Sucrose 210g Dextrose 290g Water 181.75g Total in container 1041.75g. The composition is heated to clarity with stirring, cooled, and water loss replaced. Viscosity = 439cP at 15.7ºC. It is still somewhat hygroscopic.