Laminar Motion of the Incompressible Fluids in Self-Acting Thrust Bearings with Spiral Grooves

Abstract

We analyze the laminar motion of incompressible fluids in self-acting thrust bearings with spiral grooves with inner or external pumping. The purpose of the study is to find some mathematical relations useful to approach the theoretical functionality of these bearings having magnetic controllable fluids as incompressible fluids, in the presence of a controllable magnetic field. This theoretical study approaches the permanent motion regime. To validate the theoretical results, we compare them to some experimental results presented in previous papers. The laminar motion of incompressible fluids in bearings is described by the fundamental equations of fluid dynamics. We developed and particularized these equations by taking into consideration the geometrical and functional characteristics of these hydrodynamic bearings. Through the integration of the differential equation, we determined the pressure and speed distributions in bearings with length in the “pumping” direction. These pressure and speed distributions offer important information, both quantitative (concerning the bearing performances) and qualitative (evidence of the viscous-inertial effects, the fluid compressibility, etc.), for the laminar and permanent motion regime.

Turbulent Motion of Liquids in Hydraulic Resistances with a Linear Cylindrical Slide‐Valve

“…The equations describing the motion of the viscous Newtonian incompressible fluid in the annular space from the controllable hydraulic resistances with the cylindrical slide-valve are the Navier-Stokes equation and the continuity equation. In a transient motion regime and considering the dynamical viscosity η ≅const., these equations have the known form [ 13 – 18 ] as follows: …”

“…We will use ( 6a ), ( 6b ), ( 6c ), and ( 6d ) which will be integrated along the length of the thickness of the fluid layer. We will perform an integration of the equations with partial derivatives ( 6a ), ( 6b ), ( 6c ), and ( 6d ) using the approximate method proposed by S. M. Targ and N. A. Slezkin and found in literature [ 13 , 14 , 17 , 18 ]. We consider that the fluid motion in the annular space between the slide-valve and the body ( Figure 1 ) is described by the velocity parabolic profiles u ( y ), v ( y ), and w ( y ), which qualitatively maintain the same appeal as the noninertial case: where the slide-valve is considered to have a small amplitude translational motion in the horizontal direction (closing/opening), along the z -axis, with a linear velocity v p .…”

“…The velocity distributions ( 7a ), ( 7b ), and ( 7c ) describe a laminar motion regime [ 11 , 12 , 15 , 16 , 18 ]. In the turbulent motion regime case, the velocity distributions can be described as follows [ 12 , 15 – 17 ]: …”

“…In fact, we consider the application of the method used in the hydrostatical or hydrodynamic lubrication field in a laminar or turbulent flow regime [ 12 – 18 ] to the case of the hydraulic resistance with a predominant Poiseuille motion.…”

“…First, this type of approach is motivated by the fact that the radial clearance δ from the hydraulic resistance is comparable to the height dimension h of the fluid film between the bearing surfaces [ 1 , 2 , 4 , 14 – 18 ] ( δ ⇔ h ≅(10 −6 ,…, 10 −5 ) [m]). Additionally, in actual functional conditions, the annular space δ has different values in either axial or radial directions [ 2 , 10 , 11 , 19 , 20 ].…”