P. Pallav, PhD

Introduction

Velocity Distribution Profiles (=VDP) are a graphical representation of the velocities in each layer of a fluid. 

During chewing, the pressure between the molars pushes the food out. In fluid mechanics this is known as pressure flow. Especially with this type of flow VDP's are a very powerful means to visualize the flow.

Flow Parameters

Velocity Distribution Profiles show three important flow parameters at the same time:

1. The amplitude (width) is proportional to the velocity (v) in each layer.
2. The area, in fact the integral of the velocity from one surface to the other, is proportional to the flow rate (Q), i.e. the volume, which passes per unit time.
3. The velocity gradient (v') is the derivative of the VDP. This is the tangent of the angle between the profile and a line square to the direction of the flow. (v' = tan(a))

Of special interest is the velocity gradient (v') at the wearing surfaces, because it is more or less proportional to the shear stress with which the food moves over the surfaces (Shearing Action and Erosive Activity).

Basic Shape

The relationships between the flow rate, maximum velocity (V_max), and the velocity gradient depend on the type of fluid. This is because the type of fluid is of influence on the shape of a Vdp. With power law fluids the basic shape varies with the value of the exponent (b).

Plastic	Pseudo Plastic	Viscosity	Dilatant
b ≈ 0 Vav≈ Vmax v' ≈ ∞	b = 0.5 Vav= 3/4·Vmax v' = 6·Vmax/h	b = 1 Vav= 2/3·Vmax v' = 4·Vmax/h	b = 2 Vav= 3/5·Vmax v' = 3·Vmax/h

The flow rate does not depend on the maximum velocity (V_max) at the peak of the Vdp, but on the average velocity (V_av).

The velocity gradient (v', tan(a)) at the lower surface.

The velocity gradient generates a shear stress at the surface (equation 3 in Shearing Action), which is the reaction force to the pressure (p), which is required to sustain the flow. In the figures on this page the flow is from left to right, so obviously the pressure to the left is greater than to the right. x is in the direction of the flow.

Examples

The examples below consider a fluid, which flows through a parallel gap. The width of the gap, square to the plane of the figures is much greater than the height (h). The fluids are assumed to behave as described by the power law, with a value of the exponent (b) of 1 (viscosity) and 0.5 (pseudo plasticity).

Example 1

The first example is a simple increase in flow rate (Q). In this case the flow rate was doubled. This means that the area becomes twice as great. Because the height is still the same, the width must be twice as great. With this being the only change, all velocity related parameters (Q, v, and v') are twice as great.

Example 1: The influence of the flow rate (Q) on the other parameters
Qa	Qb = 2 · Qa	Fluid type> Viscosity Pseudo Plasticity Pa : Pb 1 : 2 1 : 1.4 va : vb 1 : 2 1 : 2 v'a : v'b 1 : 2 1 : 2

Example 2

This illustrates the effect when the height (h) is decreased to half its original value. In this case the flow rate (Q) is assumed to be constant.

Example 2: The influence of the height (h) when the flow rate (Q = area of vdp) is constant
Qa	Qb = Qa, hb = 0.5 · ha	Fluid type> Viscosity Pseudo Plasticity Pa : Pb 1 : 8 1 : 2.8 va : vb 1 : 2 1 : 2 v'a : v'b 1 : 4 1 : 4

Example 3

Again the height (h) is decreased to one half, but now the pressure (P) is constant.

Example 3: The influence of the height (h) when the driving pressure (P = P1 − P2) is constant
Pa	Pb = Pa, hb = 0.5 · ha	Fluid type> Viscosity Pseudo Plasticity Qa : Qb 8 : 1 16 : 1 va : vb 4 : 1 8 : 1 v'a : v'b 2 : 1 4 : 1