P. Pallav, PhD

Introduction

On this page we start with a theoretical set up where an upper and a lower molar move closer.
At the end of this page you will find several algebraic equations, which deal with this set up.

Vertical cross-section through the hypothetical setup with an upper and a lower molar, and a vertical aspect of one of the molars. The lower molar moves up, which causes the fluid (food) to flow away from the center (thin arrows).

The space between the molars is filled with a pseudo plastic material (the food). The molars are cylindrical and on this page the occlusal surfaces are still absolutely flat. In the next pages we will gradually add more complexity to the occlusal surfaces and wearing restorations.

Minimum Height

Because the fluid resists the compression, pressure is needed. If the height would decrease all the way down to zero, the pressure in our set up would even rise asymptotically (Equation 3 below). However, when you observe a real life wax bite, you will see that there is still a small volume of wax left between the molars at centric occlusion. Therefore we will take it that the vertical closing motion of our set up stops when the volume is equal to the volume of the wax of the wax bite between the molars. Of course the diameters of the molars must be similar too.
This means that the average height (vertical clearance) between the real molars is the same as in the set up. So, if the thickness of the wax bite ranges from 0 millimeter at contacts to say 1 millimeter at the thickest parts, we might say that the upward motion of the lower molar of the set up should stop when the surfaces are still say half a millimeter apart.

Pressure

The figure shows the development of the pressure during the last millimeters of the closing motion, based on Equation 3 below. Two important things can be seen.

1. The pressure increases strongly as the molars come closer. 
2. How much (vertical) pressure is required depends on how 'rigid' (µ) the food is. The two pressure curves are for two values of µ in Equation 3.

Erosive Activity

What is not drawn is that the velocity gradient (v') at which the fluid moves over the surfaces increases in a similar way as the pressure when the molars come closer.

As a consequence the Erosive Activity,  p x v', to which the wear rate is proportional, increases even stronger towards the end of the closing motion.

The straight line representing the height indicates that the molars move closer at a uniform velocity. It is of course highly questionable how realistic this is, but if the closing motion slows down at the end, the pressure and velocity gradient will be present during a longer time, rather than at a higher level.

Therefore it is clear that most of the CFA wear occurs during the last few tenths of a millimeter of the closing motions during chewing.

Equations

Key parameters of the theoretical set up at or shortly before the end of a chewing motion. R is the radius of the molars, Vv is the vertical closing velocity at which the lower molar moves closer to the upper molar. Because of this, the distance between the surfaces represented by h (height) decreases and the fluid (food) flows out sideways at an average velocity, vav, which varies with the distance from the center r where it is zero.

The flow rate (Q), the volume per unit time escaping at the circumference, is generated by the vertical speed (V_v) of the occlusal area (p·R²):

At the circumference, the average velocity (v_av,R) of the food is calculated as the flow rate (Q) divided by the cross-sectional area (width x height) of the parallel gap at the circumference. Of course the width of the gap is equal to the perimeter (2pR) of the set up.

The same is true at any distance (r) from the center

With the Equations in the pages Erosion, Shearing Action, and Velocity in Fluids, several things can be derived.

The average pressure on the surfaces and in the fluid (P_av), required to maintain the closing velocity (V_v)

See Shearing Action for a description of µ and b. Because we assume pseudo plastic flow of the food (b = ½), Equation 3 simplifies to

Note that Equations 3 and 3a are limited to plane and parallel occlusal surfaces.