Analytical calculation of flow profiles and forces between teatcup and teat

Abstract. It can be observed that the teat cups migrate up the teat during milking process. To explain this observation, it is supposed that there are short time intervals during the milking process where the positive locking between the teat and teatcup does not exist. In this case, a flow is formed between the teat and teat rubber which generates a force.

The force is considered on the basis of the flow between teat and inner wall of the teat cup depending on various parameters. It is necessary to calculate and study the flow profile in this area. The flow profile is calculated by the Navier-Stokes equations in cylindrical coordinates. With the assumptions that the flow is stationary, the velocity components in r- and phi- direction are zero and the flow is cylindrically. The solution of this equation gives us the velocity in z-direction depending on the pressure, the radius and two more variables. These variables can be calculated by using the no-slip condition on the boundary surface, which says that the flow rate of a viscous fluid at a solid boundary surface is equal to zero or is zero relative to the velocity of the boundary surface. In the size range of common teatcup rubber the result is nearly a square flow profile.