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.

1. Introduction

Incompressible fluid motion in “self-acting thrust bearings with spiral grooves” (SATBESPIG) is a complex motion in thin layers [1–5] bounded by two solid surfaces in relative rotation one to another. The incompressible fluid motion in thin layers has as mathematical consequence the simplification and particularization of the general equations of motion. We made this simplification of the general equations of motion by neglecting the terms smaller by more than one order of magnitude [1–7] than the other terms. Starting from this observation, the paper approaches the theoretical study of incompressible fluid motion in SATBESPIG with inner and exterior pumping, working in the laminar regime.

Some authors have developed comparative theoretical and experimental studies concerning different aspects of spiral groove bearings functioning in the laminar regime, with equations for their functioning using magnetic fluids, cavitation phenomenon, and the geometry of the bearings with spiral/axial multiples grooves.

In [8, 9], the author studied the static and dynamic performances of the bearings with spiral grooves and inner and exterior pumping, working with compressed air. A comparative study of the theoretical and experimental results was presented.

The diversification and optimization of the geometry of the bearings with spiral or axial multiples of grooves were approached in [10–12]. In [13], the author analyzed the cavitation phenomenon theoretically and experimentally in bearings with spiral grooves working in mineral oil. The unfavorable consequences of the cavitation were evaluated.

We consider the theoretical results as a first step in our intention to approach the functionality of the hydrodynamic bearings in general and of the thrust bearings with spiral grooves in particular in the stationary/nonstationary turbulent regime. The viscous compressible/incompressible fluid motion can be approached by taking into consideration the thermo-viscous-inertial phenomena ensemble that exists in bearings.

2. Constructive Geometric and Cinematic Elements of the SATBESPIG

The SATBESPIG (Figures 1 and 2) are special hydrodynamic bearings that have the specific effect of “autopumping” [1–3, 5, 7], which is why we are interested in the incompressible fluid motion in the laminar and permanent regime and, taking into consideration the influence of the inertial forces, in the length of the spiral groove (pumping direction 𝜓 from Figures 3, 4, and 5). In the literature [4, 7, 10, 12–15], there are only a few studies and a small amount of theoretical research concerning this subject. These studies approach the incompressible fluid motion only in the radial direction (𝑟) without taking into consideration the effects of the inertial forces.

Figure 1 

SATBESPIG and inner pumping.

Figure 2 

SATBESPIG and external pumping.

Figure 3 

Relations between the general and local coordinate systems (in an arbitrary point, 𝑃). At point 𝑃, the coordinate 𝜉 is normal to the 𝜓 direction of the spiral channel.

Figure 4 

Radial step of the inner pumping bearing.

Figure 5 

Radial step of the external pumping bearing.

In Figures 1 and 2, (i) 𝑎₁ is the width of the spiral channel, measured on the circle arc contour; (ii) 𝑎₂ is the width of the spiral threshold, measured on the circle arc contour; (iii) Δ𝜃 is the center angle corresponding to a channel—threshold pair; (iv) 𝛽₀ is the generator angle of the logarithmic spiral, which describes the form of the cannels; (v) 𝛽₁ and 𝛽₂ are the input and output angles, respectively, in and from the channel (according to the accepted notations from the general study of the hydraulic machineries); (vi) 𝜔₁ is the bearing angular rotation speed; (vii) 𝑟_𝑖 and 𝑟_𝑒 are the inner and external radius of the bearing, respectively; (viii) 𝑟_𝑐 is the radius marking the zone of the spiral channels; (ix) 𝑟 is the current radius of the bearing; (x) ℎ₁ is the lubrication film height over the bearing channels; (xi) ℎ₂ is the lubrication film height over the bearing thresholds; (xii) 𝑊 is the bearing load (the weight); and (xiii) by definition, 𝛼=𝑎₁/(𝑎₁+𝑎₂).

With the geometrical dimensions in Figure 1 (or Figure 2) we can write Δ𝜃=2𝜋/𝑛_𝑝, where 𝑛_𝑝 is the number of pairs channel—thresholds, 𝑎₁=𝛼𝑟Δ𝜃, 𝑎₂=(1−𝛼)𝑟Δ𝜃, 𝑎=𝑎₁+𝑎₂=𝑟Δ𝜃, and 𝜔₂=0 (the grooved surface is fixed).

3. Coordinate Systems, Control Volume, Speed Distributions, and Mass Flow Rate in Laminar Regime

To study the mathematical model, the coordinate systems and the control volume must be determined so as to define the speed distributions and the fluid mass flow rate [3, 5, 15]. The control volume (Vol) is the volume between the bearing surfaces, between the one channel surface and one consecutive threshold surface of the stator and the horizontal bottom surface of the rotor (Figures 1 and 2). In Figure 3, we show the general (𝑦,𝑟,𝜃) and local (𝜓,𝑦,𝜉) coordinate systems used for the motion study.

Next, we start from the fact that the incompressible fluid laminar motion, between the two quasiparallel surfaces of the bearing, is described by the following speed profiles:𝑢𝑦1≅−12𝜂𝑟𝜕𝑝𝑦+𝑦𝜕𝜃ℎ−𝑦ℎ𝜔₁𝑣𝑟,_𝑛𝑦𝑤𝑦1≅0,≅−2𝜂𝜕𝑝𝑦,𝑣𝜕𝑟ℎ−𝑦_𝜓𝑦≅12𝜂𝑟sin𝛽₀𝜕𝑝𝑦+𝑦𝜕𝜓𝑦−ℎℎ𝜔₁𝑟cos𝛽₀,𝑣_𝜉𝑦≅12𝜂𝑟sin𝛽₀𝜕𝑝𝑦−𝑦𝜕𝜉𝑦−ℎℎ𝜔₁𝑟sin𝛽₀. (1a)(1b)(1c)(1d)(1e)

In (1a)–(1e), (i) 𝑢(𝑦) is the fluid speed component in the 𝑥 (or 𝜃) direction, (ii) 𝑣_𝑛(𝑦) is the fluid speed component in the normal direction 𝑦, (iii) 𝑤(𝑦) is the fluid speed component in the 𝑧 (or 𝑟) direction, (iv) 𝑣_𝜓(𝑦) is the fluid speed component in the 𝜓 curb direction, (v) 𝑣_𝜉(𝑦) is the fluid speed component in the 𝜉 direction, (vi) 𝑝 is the pressure in the fluid, (vii) ℎ is the height of the lubrication film, and (viii) 𝜂 is the fluid dynamical viscosity.

The speed distributions, given by (1a)–(1e), are typical for the noninertial motion case initially considered, meaning that these are parabolic speed profiles [3, 5, 6]. Observing the geometry of the bearings in Figures 1 and 2, it is possible to establish some functional mathematical relations between the coordinates. Thus, the following operational expressions can be found [3, 5]:𝜕𝜕𝑟=cos𝛽₀𝜕𝜕𝜉+sin𝛽₀𝜕,𝜕𝜕𝜓≡1𝜕𝑥𝑟𝜕𝜕𝜃=−sin𝛽₀𝜕𝜕𝜉+cos𝛽₀𝜕,𝜕𝜕𝜓𝜕𝜉=cos𝛽₀𝜕𝜕𝑟−sin𝛽₀1𝑟𝜕,𝜕𝜕𝜃𝜕𝜓=sin𝛽₀𝜕𝜕𝑟+cos𝛽₀1𝑟𝜕.𝜕𝜃 (2a)(2b)(2c)(2d)

Using relation (1e), the fluid mass flow rate in the pumping direction 𝜓 (Figures 4 and 5), further denoted by ℳ̇_𝜓, can be expressed by the integral representation [3, 5]: ℳ̇_𝜓≅_𝜃^𝜃+Δ𝜃𝑟₀sin𝛽₀𝑑𝜃^ℎ₀𝜌𝑣_𝜓𝑦𝑑𝑦, (3) where 𝜌 is the fluid density and 𝑟₀ is the “reference” radius (𝑟₀∈[𝑟_𝑖,…,𝑟_𝑒], and further 𝑟₀ will be denoted by 𝑟).

4. Differential Equation for Pressure Distribution in the 𝜓 Direction

Observing the bearing geometry (Figures 1 and 2), we admit that the angle Δ𝜃 is infinitely small, meaning that there exist an infinite number of spiral channels. Given the relations (1e) and (3), the physical natural condition is that the mass flow rate ℳ̇_𝜓 is constant. With these conditions we obtain ℳ̇_𝜓≅Δ𝜃𝐶𝜕𝑝𝜕𝜓+𝐷𝑟², (4) where𝜌𝐶=−12𝜂𝛼ℎ³₁+(1−𝛼)ℎ³₂,𝜌𝐷=2𝜔₁sin𝛽₀cos𝛽₀𝛼ℎ₁+(1−𝛼)ℎ₂. (5a)(5b)

The fluid mass conservation in the 𝜓 direction can be expressed as follows: 𝜕(𝑚)+𝜕𝜕𝜏ℳ̇𝜕𝜓^𝜓Δ𝜓≅0, (6) where 𝜏 is the time, and the fluid mass 𝑚, contained in the control volume Vol, is given by the relation 𝑚≅Δ𝑟Δ𝜃𝜌𝑟𝛼ℎ₁+(1−𝛼)ℎ₂. (7) Or, using relations (4) and (7), (6) becomes 𝜕𝜕𝜏𝛼ℎ₁+(1−𝛼)ℎ₂𝜌+sin²𝛽₀1𝜓𝜕𝐶𝜕𝜓𝜕𝑝1𝜕𝜓+𝐷sin²𝛽₀𝜓²≅0. (8)

For the permanent motion regime, (8) becomes 1𝜓𝜕𝐶𝜕𝜓𝜕𝑝+1𝜕𝜓sin²𝛽₀𝐷𝜓²≅0. (9)

Using nondimensional variables [1, 3–5, 7, 10, 14, 15], (8) can be written as 𝜕𝜕𝜁𝑃𝐻³𝐾₁𝜁𝜕𝑃𝜕𝜁+𝐾₂ΩΛ𝜁²𝐻⁻²𝜕−𝜎𝜁𝜕𝑡𝑃𝐻𝐾₃≅0. (10)

For the stationary motion regime, (10) becomes [1, 2] 𝜕𝑃𝐾𝜕𝜁¹𝜁𝜕𝑃𝜕𝜁+𝐾₂ΩΛ𝜁²≅0. (11)

5. Integration of the Differential Equation of the Pressure Distribution in the 𝜓 Direction

In the 𝜓 direction, the fluid film ℎ varies rapidly from ℎ₁ to ℎ₂, at the frontier 𝑟≅𝑟_𝑐 (Figures 4 and 5). The existing radial step with length in the pumping direction 𝜓 produces a pressure jump from 𝑝_ℎ1 to 𝑝_ℎ2. This pressure jump has different values as a function of (i) the flow regime through bearing (laminar, transition, or turbulent regime), (ii) the value of the rapport ℎ₁/ℎ₂, (iii) the fact that we take (or not) into consideration the inertial forces, and (iv) the fluid type [2, 3, 5, 16, 17].

Integrating differential equation (9) and observing the limit conditions for pressures [3, 5] and the notations from Figures 4 and 5 (𝑝_supp. is the supply pressure of the lubrication fluid and 𝑝_atm. is the atmospheric pressure), we obtain the mathematical relations for the pressure distributions in the SATBESPIG: 𝑝𝜓≅𝑝_ℎ2+𝜓−𝐿₂+𝜓₀𝜓₀−𝐿₂+𝜓₀×𝑝_supp.−𝑝_ℎ2−2𝜂cos𝛽₀𝜔₁ℎ²₂sin𝛽₀𝜓³₀−𝐿₂+𝜓₀³+2𝜂cos𝛽₀𝜔₁ℎ²₂sin𝛽₀⋅𝜓³−𝐿₂+𝜓₀³. (12) Relation (12) presents the pressure distribution in the laminar and permanent flow regime in the smooth region of the inner pumping bearing surface, where ℎ=ℎ₂. Consider 𝑝𝜓≅𝑝_supp.+𝜓−𝐿+𝜓₀𝐿₂+𝜓₀−𝐿+𝜓₀×𝑝_ℎ1−𝑝_supp.−2𝜂cos𝛽₀𝜔₁𝛼ℎ₁+(1−𝛼)ℎ₂sin𝛽₀𝛼ℎ³₁+(1−𝛼)ℎ³₂×𝐿₂+𝜓₀³−𝐿+𝜓₀³+2𝜂cos𝛽₀𝜔₁𝛼ℎ₁+(1−𝛼)ℎ₂sin𝛽₀𝛼ℎ³₁+(1−𝛼)ℎ³₂𝜓³−𝐿+𝜓₀³. (13) Relation (13) presents the pressure distribution in the region with spiral channels of the inner pumping bearing surface, where ℎ=ℎ₁.

In a similar way, for the SATBESPIG with exterior pumping (Figure 5), we obtain 𝑝𝜓≅𝑝_ℎ2+𝜓−𝐿₁+𝜓₀𝐿+𝜓₀−𝐿₁+𝜓₀×𝑝_supp.−𝑝_ℎ2−2𝜂cos𝛽₀𝜔₁ℎ²₂sin𝛽₀𝐿+𝜓₀³−𝐿₁+𝜓₀³+2𝜂cos𝛽₀𝜔₁ℎ²₂sin𝛽₀𝜓³−𝐿₁+𝜓₀³. (14) Relation (14) presents the pressure distribution in the laminar and permanent flow regime in the smooth region of the external pumping bearing surface, where ℎ=ℎ₂. Consider 𝑝𝜓≅𝑝_supp.+𝜓−𝜓₀𝐿₁+𝜓₀−𝜓₀×𝑝_ℎ1−𝑝_supp.−2𝜂cos𝛽₀𝜔₁𝛼ℎ₁+(1−𝛼)ℎ₂sin𝛽₀𝛼ℎ³₁+(1−𝛼)ℎ³₂×𝐿₁+𝜓₀³−𝜓³₀+2𝜂cos𝛽₀𝜔₁𝛼ℎ₁+(1−𝛼)ℎ₂sin𝛽₀𝛼ℎ³₁+(1−𝛼)ℎ³₂𝜓³−𝜓³₀. (15) Relation (15) presents the pressure distribution in the laminar and permanent motion regime in the spiral grooves region of the bearing surface with exterior pumping, where ℎ=ℎ₁.

In the relations (12)–(15), all the constants are known (for a designed and realized SATBESPIG), the exception being the extreme pressures 𝑝_ℎ1 and 𝑝_ℎ2. If we do not take into consideration the influence of the inertial forces, then the pressures 𝑝_ℎ1 and 𝑝_ℎ2 are equal. Thus, 𝑝_ℎ1≡𝑝_ℎ2, Δ𝑝=𝑝_ℎ2−𝑝_ℎ1=0.

6. Calculus Relations for the Extreme Pressures 𝑝_ℎ1 and 𝑝_ℎ2

To express the pressure distribution in the 𝜓 direction and the extreme pressures 𝑝_ℎ1 and 𝑝_ℎ2, we must analyze the liquid motion in the fluid film existing between the quasiparallel surfaces of the SATBESPIG [1–3, 5, 7]. On the other hand, the inertial effects (which considerably influence the extreme pressures 𝑝_ℎ1 and 𝑝_ℎ2) exist on all the surfaces of the SATBESPIG in the 𝜓 direction, but the maximum effect is concentrated in the zone of the radial step, 𝑟≅𝑟_𝑐 (Figures 4 and 5) [2, 3, 5, 16, 17].

Some theoretical results concerning the motion of liquids in similar bearings with the consideration of the influence of inertial forces have been presented in the literature [3, 5–7, 14]. It is possible to demonstrate [3, 5] that, for the case of the stationary motion regime and when only the smooth surface is in a rotation with 𝑛₁=constant [rot/min], the equation that describes the viscous fluid motion in 𝜓 direction is 𝜌𝜓𝑑𝛼𝑑𝜓⁰𝑄²_𝜓ℎ𝜓²Δ𝜃²−𝛾𝑄_𝜓𝜔₁Δ𝜃cos𝛽₀sin𝛽₀+𝛽𝜔²₁𝜓²cos²𝛽₀sin²𝛽₀ℎ+2𝜌𝛿𝑄_𝜓𝜔₁sin𝛽₀𝜓²Δ𝜃cos𝛽₀−3𝜌𝛽𝜔²₁ℎ+𝜌𝛼₀𝑄²_𝜓ℎ𝜓⁴Δ𝜃²−𝜌𝛾𝑄_𝜓𝜔₁cos𝛽₀𝜓²Δ𝜃sin𝛽₀+𝜌𝛽𝜔²₁cos²𝛽₀sin²𝛽₀1ℎ+𝜓𝑑𝑝ℎ+𝑑𝜓12𝜂ℎ𝑄_𝜓−𝜔ℎ𝜓Δ𝜃¹𝜓cos𝛽₀2sin𝛽₀≅0, (16) where𝑄_𝜓=𝑈_av.ℎℎ≅−³12𝜂𝑟sin𝛽₀𝜕𝑝+ℎ𝜕𝜓2𝜔₁𝑟cos𝛽₀(),𝑈=const.volumicflowrate_av.=1ℎ^ℎ₀𝑣_𝜓𝑦𝛼⋅𝑑𝑦=averagespeedinthefluidfilmbythecurbdirection𝜓,₀=652,𝛽=115,𝛾=51,𝛿=.10 (17a)(17b)(17c)The constructive angle of spiral groove is 𝛽₀=17^∘ [3, 5, 14].

Calculating the derivative of the relation (16) versus the variable 𝜓 and taking into consideration the constants from above, the following expression can be found: 𝑑𝑝≅𝑑𝜓𝜌𝛼₀𝑄²_𝜓ℎ³𝜓²Δ𝜃²−𝜌𝛽𝜔²₁𝜓²ℎcos²𝛽₀sin²𝛽₀𝑑ℎ+𝑑𝜓𝜌𝛼₀𝑄²_𝜓ℎ²𝜓³Δ𝜃²+3𝜌𝛽𝜔²₁𝜓1−cos²𝛽₀sin²𝛽₀−2𝜌𝛿𝑄_𝜓𝜔₁sin𝛽₀ℎ𝜓Δ𝜃cos𝛽₀+𝜌𝛾𝑄_𝜓𝜔₁cos𝛽₀ℎ𝜓Δ𝜃sin𝛽₀−12𝜂𝑄_𝜓ℎ³+Δ𝜃6𝜂𝜔₁𝜓²cos𝛽₀ℎ²sin𝛽₀. (18)

We obtain a similar but more precise relation to (18) by introducing a supplementary coercive term [1, 2, 4, 9]. In this case, relation (18) becomes 𝑑𝑝≅𝑑𝜓𝜌𝑄²_𝜓ℎ³𝜓²Δ𝜃²𝛼₀+𝜀𝜓𝑑ℎ−𝑑𝜓𝜌𝛽𝜔²₁𝜓²ℎcos²𝛽₀sin²𝛽₀𝑑ℎ+𝑑𝜓𝜌𝛼₀𝑄²_𝜓ℎ²𝜓³Δ𝜃²+3𝜌𝛽𝜔²₁𝜓1−cos²𝛽₀sin²𝛽₀−2𝜌𝛿𝑄_𝜓𝜔₁sin𝛽₀ℎ𝜓Δ𝜃cos𝛽₀+𝜌𝛾𝑄_𝜓𝜔₁cos𝛽₀ℎ𝜓Δ𝜃sin𝛽₀−12𝜂𝑄_𝜓ℎ³+Δ𝜃6𝜂𝜔₁𝜓²cos𝛽₀ℎ²sin𝛽₀, (19) where 𝜀=2/15 [1, 2, 4].

Integrating the differential equation (19), for both regions of the SATBESPIG, where ℎ=ℎ₁=const. and ℎ=ℎ₂=const. (Figures 4 and 5), we obtain the calculus relations for the pressures 𝑝_ℎ1 and 𝑝_ℎ2: 𝑝_ℎ1≅𝑝_supp.+𝜌𝛼₀𝑄²_𝜓2ℎ²₁Δ𝜃²1𝐿+𝜓₀²−1𝐿₂+𝜓₀²+32𝜌𝛽𝜔²₁1−cos²𝛽₀sin²𝛽₀𝐿₂+𝜓₀²−𝐿+𝜓₀²−2𝜌𝛿𝑄_𝜓𝜔₁sin𝛽₀ℎ₁Δ𝜃cos𝛽₀𝐿ln²+𝜓₀𝐿+𝜓₀+𝜌𝛾𝑄_𝜓𝜔₁cos𝛽₀ℎ₁Δ𝜃sin𝛽₀𝐿ln²+𝜓₀𝐿+𝜓₀+12𝜂𝑄_𝜓ℎ³₁Δ𝜃𝐿+𝜓₀−𝐿₂+𝜓₀−2𝜂𝜔₁cos𝛽₀ℎ²₁sin𝛽₀⋅𝐿+𝜓₀³−𝐿₂+𝜓₀³,𝑝_ℎ2≅𝑝_supp.−𝜌𝛼₀𝑄²_𝜓2ℎ²₂Δ𝜃²1𝐿₂+𝜓₀²−1𝜓²₀+32𝜌𝛽𝜔²₁1−cos²𝛽₀sin²𝛽₀𝐿₂+𝜓₀²−𝜓²₀+2𝜌𝛿𝑄_𝜓𝜔₁sin𝛽₀ℎ₂Δ𝜃cos𝛽₀𝜓ln⁰𝐿₂+𝜓₀−𝜌𝛾𝑄_𝜓𝜔₁cos𝛽₀ℎ₂Δ𝜃sin𝛽₀𝜓ln⁰𝐿₂+𝜓₀+12𝜂𝑄_𝜓ℎ³₂𝜓Δ𝜃⁰−𝐿₂+𝜓₀+2𝜂𝜔₁cos𝛽₀ℎ²₂sin𝛽₀𝐿₂+𝜓₀³−𝜓₀³. (20) These relations are valid for the SATBESPIG with inner pumping.

In a similar way, for the SATBESPIG with exterior pumping, we obtain 𝑝_ℎ1≅𝑝_supp.+𝜌𝛼₀𝑄²_𝜓2ℎ²₁Δ𝜃²1𝜓²₀−1𝐿₁+𝜓₀²+32𝜌𝛽𝜔²₁1−cos²𝛽₀sin²𝛽₀𝐿₁+𝜓₀²−𝜓²₀+2𝜌𝛿𝑄_𝜓𝜔₁sin𝛽₀ℎ₁Δ𝜃cos𝛽₀𝜓ln⁰𝐿₁+𝜓₀−𝜌𝛾𝑄_𝜓𝜔₁cos𝛽₀ℎ₁Δ𝜃sin𝛽₀𝜓ln⁰𝐿₁+𝜓₀+12𝜂𝑄_𝜓ℎ³₁𝜓Δ𝜃⁰−𝐿₁+𝜓₀−2𝜂𝜔₁cos𝛽₀ℎ²₁sin𝛽₀𝜓³₀−𝐿₁+𝜓₀³,𝑝_ℎ2≅𝑝_supp.+𝜌𝛼₀𝑄²_𝜓2ℎ²₂Δ𝜃²1𝐿+𝜓₀²−1𝐿₁+𝜓₀²+32𝜌𝛽𝜔²₁1−cos²𝛽₀sin²𝛽₀𝐿₁+𝜓₀²−𝐿+𝜓₀²−2𝜌𝛿𝑄_𝜓𝜔₁sin𝛽₀ℎ₂Δ𝜃cos𝛽₀𝐿ln¹+𝜓₀𝐿+𝜓₀+𝜌𝛾𝑄_𝜓𝜔₁cos𝛽₀ℎ₂Δ𝜃sin𝛽₀𝐿ln¹+𝜓₀𝐿+𝜓₀+12𝜂𝑄_𝜓ℎ³₂Δ𝜃𝐿+𝜓₀−𝐿₁+𝜓₀+2𝜂𝜔₁cos𝛽₀ℎ²₂sin𝛽₀𝐿₁+𝜓₀³−𝐿+𝜓₀³. (21)(22)

In relations (20)–(22), all the variables are known except the volumetric flow rate 𝑄_𝜓.

7. Calculus Relation for the Fluid Volumetric Flow Rate 𝑄_𝜓

The calculus relation for the fluid volumetric flow rate 𝑄_𝜓 will be established using differential equation (19) again. To find the calculus relation, at the position of the radial step of the bearing we suppose that 𝑑ℎ/𝑑𝜓≠0. So, in this zone of the bearing the inertial effects are dominant in comparison to the viscous effects. In other words, in the radial step zone of the bearing, the liquid moves approximately like an ideal but not viscous fluid [3, 5].

Integrating (19) in the vicinity of the radial step of the bearing grooved surface (Figures 4 and 5), we obtain a relation between 𝑝_ℎ1, 𝑝_ℎ2, and 𝑄_𝜓: Δ𝑝=𝑝_ℎ1−𝑝_ℎ2≅𝜌𝑄²_𝜓𝜓²_𝑐Δ𝜃²𝛼₀+𝜀𝜓_𝑐12ℎ²₂−12ℎ²₁−𝜌𝛽𝜔²₁𝜓²_𝑐cos²𝛽₀sin²𝛽₀ℎln¹ℎ₂, (23) where 𝜓_𝑐 is the length measured by the 𝜓 coordinate corresponding to the grooves radius of the profiled surface, 𝑟_𝑐 (Figures 4 and 5).

Relation (23) has some limits, especially at high values of ℎ₁/ℎ₂, when ℎ₂→0, or, in other words, at the heavy regimes for the bearing functionality. If we know the other variables, including the extreme pressures 𝑝_ℎ1 and 𝑝_ℎ2, relation (23) offers the flow rate 𝑄_𝜓.

Therefore, using (20) and then (21), (22), and (23), we obtain a typical second degree algebraic equation, (24), from which we obtain the fluid flow rate 𝑄_𝜓: 𝜌𝑄²_𝜓Δ𝜃²𝛼₀12ℎ²₁1𝐿+𝜓₀²−1𝐿₂+𝜓₀²+12ℎ²₂1𝐿₂+𝜓₀²−1𝜓²₀−1𝜓²_𝑐𝛼₀+𝜀𝜓_𝑐12ℎ²₂−12ℎ²₁+𝑄_𝜓⋅Δ𝜃𝜌𝛾𝜔₁cos𝛽₀sin𝛽₀1ℎ₁𝐿ln²+𝜓₀𝐿+𝜓₀+1ℎ₂𝜓ln⁰𝐿₂+𝜓₀−2𝜌𝛿𝜔₁sin𝛽₀cos𝛽₀1ℎ₁𝐿ln²+𝜓₀𝐿+𝜓₀+1ℎ₂𝜓ln⁰𝐿₂+𝜓₀+12𝜂𝐿−𝐿₂ℎ³₁+𝐿₂ℎ³₂−𝑇1+𝑇2≅0, (24) where 3𝑇1=2𝜌𝛽𝜔²₁1−cos²𝛽₀sin²𝛽₀𝐿+𝜓₀²−𝜓²₀,𝑇2=𝜌𝛽𝜔²₁𝜓²_𝑐cos²𝛽₀sin²𝛽₀ℎln¹ℎ₂−2𝜂𝜔₁cos𝛽₀sin𝛽₀×1ℎ²₁𝐿+𝜓₀³−𝐿₂+𝜓₀³+1ℎ²₂𝐿₂+𝜓₀³−𝜓₀³. (25) Relation (24) is valid for the SATBESPIG with inner pumping.

For the SATBESPIG with exterior pumping, we obtain a similar relation: 𝜌𝑄²_𝜓Δ𝜃²𝛼₀12ℎ²₂1𝐿+𝜓₀²−1𝐿₁+𝜓₀²−12ℎ²₁1𝜓²₀−1𝐿₁+𝜓₀²+12𝜓²_𝑐𝛼₀+𝜀𝜓_𝑐1ℎ²₂−1ℎ²₁+𝑄_𝜓⋅Δ𝜃𝜌𝛾𝜔₁cos𝛽₀sin𝛽₀1ℎ₁𝜓ln⁰𝐿₁+𝜓₀+1ℎ₂𝐿ln¹+𝜓₀𝐿+𝜓₀−2𝜌𝛿𝜔₁sin𝛽₀cos𝛽₀1ℎ₁𝜓ln⁰𝐿₁+𝜓₀+1ℎ₂𝐿ln¹+𝜓₀𝐿+𝜓₀+12𝜂𝐿−𝐿₁ℎ³₂+𝐿₁ℎ³₁−𝑇1−𝑇3≅0, (26) where 𝑇3=𝜌𝛽𝜔²₁𝜓²_𝑐cos²𝛽₀sin²𝛽₀ℎln¹ℎ₂−2𝜂𝜔₁cos𝛽₀sin𝛽₀1ℎ²₁𝜓₀³−𝐿₁+𝜓₀³+1ℎ²₂𝐿₁+𝜓₀³−𝐿+𝜓₀³. (27)

Both (24) and (26) are classical algebraic equations of second degree in 𝑄_𝜓. If we denote by 𝑄^𝐼_𝜓 and 𝑄_𝜓^𝐼𝐼 the two solutions of the every algebraic equation (24) or (26), using 𝑄^𝐼_𝜓 and 𝑄_𝜓^𝐼𝐼, and if we take into consideration that the SATBESPIG realizes inner pumping and exterior pumping, it is not possible to have a negative fluid flow rate from a physical point of view. Thus, the algebraic solution, which has physical meaning, is the positive solution [3, 5].

The numerical evaluation of 𝑄^𝐼_𝜓 and 𝑄_𝜓^𝐼𝐼 algebraic solutions, for 𝑄^𝐼_𝜓 and 𝑄_𝜓^𝐼𝐼 using medium (normal) values for the physical and geometrical dimensions [1, 2] of (24) and (26), leads to 𝑄^𝐼_𝜓<0 and 𝑄_𝜓^𝐼𝐼>0. Thus, the mathematical relation for the calculus of the fluid volumetric flow rate 𝑄_𝜓 is 𝑄_𝜓≡𝑄_𝜓^𝐼𝐼, (28) where, in conformity to the devoted notations from the classical algebra, the solution 𝑄_𝜓^𝐼𝐼 is 𝑄_𝜓^𝐼𝐼=−𝑏̈−𝑏̈²−4𝑎𝑐̈̈2𝑎̈, (29) where 𝑎̈, 𝑏̈, and 𝑐̈ are the coefficients of the algebraic equations (24) or (26).

8. Numerical and Experimental Results

The established mathematical relations allow the numerical calculation of the pressure and speed distributions for several SATBESPIG with inner or external pumping in permanent and laminar regimes. For these numerical calculations we used two different computer programs: one for the SATBESPIG with inner pumping and the other for the SATBESPIG with external pumping.

In Figures 6 to 8, we present the calculated pressure and speed distributions for a SATBESPIG with inner pumping and 𝑟_𝑒 = 90?mm, 𝑟_𝑐 = 57.15?mm, 𝑟_𝑖 = 45?mm, 𝑛_𝑝 = 10, 𝛼 = 0.615, 𝛽₀ = 17°, ? = 905?kg/m³, 𝜂 = 10^-4?Pa·s, 𝑝_supp. = 101325?N/m², 𝑛=900 rot/min, 𝛼₀ = 6/5, 𝜀 = 2/15, 𝛾 = 1/5, 𝛿 = 1/10, 𝛽 = 2/15, and the parameters ℎ₁ and ℎ₂ as written in the figures.

Figure 6 

Pressure and speed distributions in the SATBESPIG with inner pumping. (See details in Figures 7 and 8.)

Figure 7 

Detail of Figure 6 concerning the pressure jump between ℎ1 and ℎ2. (The marked points are those where we made the calculation.)

Figure 8 

Detail of Figure 6 concerning the speed jump between ℎ1 and ℎ2. (The marked points are those where we made the calculation.)

Figure 9 presents the calculated pressure and speed distributions for a SATBESPIG with external pumping with the same characteristics as the SATBESPIG with inner pumping above, with the exception that 𝑟_𝑐 = 77.85?mm.

Figure 9 

Pressure and speed distributions in the SATBESPIG with external pumping.

In Figure 10, we compared the theoretical results and the experimental measurements for the SATBESPIG with inner pumping and ℎ₁, ℎ₂, and 𝑛 written in the figure.

Figure 10 

Pressure distributions in the SATBESPIG with inner pumping (comparison between the theoretical and experimental curves). The marked points are those where we performed the experimental measurements.

In all our studies, the main constructive geometrical and functional parameters for the calculation of these bearings were ℎ₁, ℎ₂, 𝑛, 𝑟_𝑖, and 𝑟_𝑒 [3, 5].

9. Discussion and Conclusions

The analysis of the theoretical results allows some conclusions to be drawn concerning the laminar motion of incompressible fluids in some models of a SATBESPIG. We analyzed two variants of the SATBESPIG with inner pumping, which are called the First Variant and the Second Variant below. The difference between these bearings is the 𝑟_𝑐 dimension. The First Variant is the SATBESPIG from Figure 6 (𝑟_𝑐 = 57.15?mm), and the Second Variant has 𝑟_𝑐 = 70.29?mm.(a)For all the rotation per minute (r.p.m.) ranges 𝑛 and for all the analyzed rapports ℎ₁/ℎ₂, the First Variant SATBESPIG with inner pumping had superior hydrodynamics performance compared to the Second Variant SATBESPIG. Thus, high pressures in bearings can be realized with an appropriate dimension of the bearing surface, especially the dimension of the grooved surface, meaning the parameters 𝑟_𝑖, 𝑟_𝑐, and 𝑟_𝑒.(b)Generally speaking, increasing other hydraulics performance of the SATBESPIG for the same dimensions 𝑟_𝑖, 𝑟_𝑐, and 𝑟_𝑒 can be realized by increasing the number of the rotation per minute 𝑛. The modification of the rapport ℎ₁/ℎ₂ has a small influence.(c)The nondimensional speeds increase (and decrease) from the bearing input to the bearing output, with a jump at the radial step 𝑟≅𝑟_𝑐. The nondimensional pressures change, but not linearly with the channel length, from the input, where 𝑝_imput≅𝑝_supp., up to the pressure 𝑝_ℎ1 and from the pressure 𝑝_ℎ2 to the output pressure, where 𝑝_output≅𝑝_supp..(d)The pressure jump Δ𝑝=𝑝_ℎ1−𝑝_ℎ2 theoretically tends to zero when the rapport ℎ₁/ℎ₂→1 and thus when the radial step vanishes or when 𝑝_ℎ1≅𝑝_ℎ2. The case 𝑝_ℎ1≅𝑝_ℎ2 appears only when we do not take into consideration the inertial effects.(e)The pressure jump Δ𝑝 depends not only on the rapport ℎ₁/ℎ₂ but also on the value of the 𝑛 and on the bearing geometry (𝑟_𝑖, 𝑟_𝑐, and 𝑟_𝑒).(f)For the same constructive geometrical and functional parameters (𝑟_𝑖, 𝑟_𝑒, ℎ₁/ℎ₂, 𝑛,…), the SATBESPIG with exterior pumping gives lower pressures than the bearing with inner pumping.(g)The comparative analysis between the theoretical and experimental results shows good correlation, especially at low 𝑛 and at middle values for the rapport ℎ₁/ℎ₂ (ℎ₁/ℎ₂≅4). Some of the mathematical relations established above can be used to approach the theoretical functionality of the SATBESPIG having magnetic controllable fluids (magnetic fluids or magnetorheological fluids) as incompressible fluids, in the presence of a controllable magnetic field [18–20].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was partly supported by the Project PN II (National Education Ministry—Romania) no. 157/2012 “MagNanoMicroSeal" (2012–2015).