Research Article Laminar Motion of the Incompressible Fluids in … · 2019. 7. 31. · Research...

Research ArticleLaminar Motion of the Incompressible Fluids in Self-ActingThrust Bearings with Spiral Grooves

Cornel Velescu1 and Nicolae Calin Popa2

1 Hydraulic Machinery Department, Mechanical Faculty, “Politehnica” University of Timisoara, Boulevard Mihai Viteazul No. 1,300222 Timisoara, Romania

2 Center for Advanced and Fundamental Technical Research (CAFTR), Romanian Academy-Timisoara Branch,Boulevard Mihai Viteazul No. 24, 300223 Timisoara, Romania

Correspondence should be addressed to Nicolae Calin Popa; [email protected]

Received 4 October 2013; Accepted 28 November 2013; Published 2 January 2014

Academic Editors: M. Kuciej, F. Liu, and L. Nobile

Copyright © 2014 C. Velescu and N. C. Popa. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We analyze the laminar motion of incompressible fluids in self-acting thrust bearings with spiral grooves with inner or externalpumping. The purpose of the study is to find some mathematical relations useful to approach the theoretical functionality ofthese 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 someexperimental results presented in previous papers. The laminar motion of incompressible fluids in bearings is described by thefundamental equations of fluid dynamics. We developed and particularized these equations by taking into consideration thegeometrical 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 speeddistributions offer important information, both quantitative (concerning the bearing performances) and qualitative (evidence ofthe viscous-inertial effects, the fluid compressibility, etc.), for the laminar and permanent motion regime.

1. Introduction

Incompressible fluid motion in “self-acting thrust bearingswith spiral grooves” (SATBESPIG) is a complex motion inthin layers [1–5] bounded by two solid surfaces in relativerotation one to another. The incompressible fluid motionin thin layers has as mathematical consequence the sim-plification and particularization of the general equations ofmotion. Wemade this simplification of the general equationsof motion by neglecting the terms smaller by more than oneorder of magnitude [1–7] than the other terms. Starting fromthis observation, the paper approaches the theoretical studyof incompressible fluidmotion in SATBESPIGwith inner andexterior pumping, working in the laminar regime.

Some authors have developed comparative theoreticaland experimental studies concerning different aspects ofspiral groove bearings functioning in the laminar regime,with equations for their functioning using magnetic fluids,

cavitation phenomenon, and the geometry of the bearingswith spiral/axial multiples grooves.

In [8, 9], the author studied the static and dynamic per-formances of the bearings with spiral grooves and inner andexterior pumping, working with compressed air. A compar-ative study of the theoretical and experimental results waspresented.

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

We consider the theoretical results as a first step in ourintention to approach the functionality of the hydrodynamicbearings in general and of the thrust bearings with spiralgrooves in particular in the stationary/nonstationary turbu-lent regime. The viscous compressible/incompressible fluid

Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 478401, 10 pageshttp://dx.doi.org/10.1155/2014/478401

2 The Scientific World Journal

W

Flow direction

AA

y

h1h2

rirc

re

a1

a2

r

r

Δ𝜃

𝛽0 ≡ 𝛽1

𝜔1A-A

Figure 1: SATBESPIG and inner pumping.

motion can be approached by taking into consideration thethermo-viscous-inertial phenomena ensemble that exists inbearings.

2. Constructive Geometric and CinematicElements of the SATBESPIG

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

In Figures 1 and 2, (i) 𝑎1is the width of the spiral channel,

measured on the circle arc contour; (ii) 𝑎2is the width of

the spiral threshold, measured on the circle arc contour; (iii)Δ𝜃 is the center angle corresponding to a channel—thresholdpair; (iv) 𝛽

0is the generator angle of the logarithmic spiral,

which describes the form of the cannels; (v) 𝛽1and 𝛽

2are the

input and output angles, respectively, in and from the channel(according to the accepted notations from the general studyof the hydraulic machineries); (vi) 𝜔

1is the bearing angular

rotation speed; (vii) 𝑟𝑖and 𝑟𝑒are the inner and external radius

of the bearing, respectively; (viii) 𝑟𝑐is the radius marking the

zone of the spiral channels; (ix) 𝑟 is the current radius of thebearing; (x) ℎ

1is the lubrication film height over the bearing

channels; (xi) ℎ2is the lubrication filmheight over the bearing

thresholds; (xii)W is the bearing load (the weight); and (xiii)by definition, 𝛼 = 𝑎

1/(𝑎1+ 𝑎2).

With the geometrical dimensions in Figure 1 (or Figure 2)we can write Δ𝜃 = 2𝜋/𝑛

𝑝, where 𝑛

𝑝is the number of pairs

W

Flow direction

AA

y

h1h2

rirc

re

a1

a2

r

r

Δ𝜃𝛽0 ≡ 𝛽2

𝜔1

A-A

Figure 2: SATBESPIG and external pumping.

dx

dr

≡ r · d𝜃−dx ≡ − r · d𝜃

𝛽0

𝛽0

𝛽0

𝜓

𝜉

y−d𝜓

d𝜓

r (or z)

d𝜉

𝜋

2− 𝛽0

P

x (or 𝜃)

Figure 3: Relations between the general and local coordinatesystems (in an arbitrary point, P). At point P, the coordinate 𝜉 isnormal to the 𝜓 direction of the spiral channel.

channel—thresholds, 𝑎1= 𝛼𝑟Δ𝜃, 𝑎

2= (1 − 𝛼)𝑟Δ𝜃, 𝑎 = 𝑎

1+

𝑎2= 𝑟Δ𝜃, and 𝜔

2= 0 (the grooved surface is fixed).

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

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

The Scientific World Journal 3

Next, we start from the fact that the incompressible fluidlaminarmotion, between the two quasiparallel surfaces of thebearing, is described by the following speed profiles:

𝑢 (𝑦) ≅ −1

2𝜂

1

𝑟

𝜕𝑝

𝜕𝜃𝑦 (ℎ − 𝑦) +

𝑦

ℎ𝜔1𝑟, (1a)

V𝑛(𝑦) ≅ 0, (1b)

𝑤 (𝑦) ≅ −1

2𝜂

𝜕𝑝

𝜕𝑟𝑦 (ℎ − 𝑦) , (1c)

V𝜓(𝑦) ≅

1

2𝜂𝑟 sin𝛽0

𝜕𝑝

𝜕𝜓𝑦 (𝑦 − ℎ) +

𝑦

ℎ𝜔1𝑟 cos𝛽

0, (1d)

V𝜉(𝑦) ≅

1

2𝜂𝑟 sin𝛽0

𝜕𝑝

𝜕𝜉𝑦 (𝑦 − ℎ) −

𝑦

ℎ𝜔1𝑟 sin𝛽

0. (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 inthe 𝑧 (or 𝑟) direction, (iv) V

𝜓(𝑦) is the fluid speed component

in the𝜓 curb direction, (v) V𝜉(𝑦) is the fluid speed component

in the 𝜉 direction, (vi) 𝑝 is the pressure in the fluid, (vii) ℎis the height of the lubrication film, and (viii) 𝜂 is the fluiddynamical viscosity.

The speed distributions, given by (1a)–(1e), are typicalfor the noninertial motion case initially considered, meaningthat these are parabolic speed profiles [3, 5, 6]. Observing thegeometry of the bearings in Figures 1 and 2, it is possible toestablish some functionalmathematical relations between thecoordinates. Thus, the following operational expressions canbe found [3, 5]:

𝜕

𝜕𝑟= cos𝛽

0

𝜕

𝜕𝜉+ sin𝛽

0

𝜕

𝜕𝜓, (2a)

𝜕

𝜕𝑥≡1

𝑟

𝜕

𝜕𝜃= − sin𝛽

0

𝜕

𝜕𝜉+ cos𝛽

0

𝜕

𝜕𝜓, (2b)

𝜕

𝜕𝜉= cos𝛽

0

𝜕

𝜕𝑟− sin𝛽

0

1

𝑟

𝜕

𝜕𝜃, (2c)

𝜕

𝜕𝜓= sin𝛽

0

𝜕

𝜕𝑟+ cos𝛽

0

1

𝑟

𝜕

𝜕𝜃. (2d)

Using relation (1e), the fluid mass flow rate in thepumping direction 𝜓 (Figures 4 and 5), further denoted byṀ𝜓, can be expressed by the integral representation [3, 5]:

Ṁ𝜓≅ ∫

𝜃+Δ𝜃

𝜃

𝑟0sin𝛽0𝑑𝜃∫

ℎ

0

𝜌V𝜓(𝑦) 𝑑𝑦, (3)

where 𝜌 is the fluid density and 𝑟0is the “reference” radius

(𝑟0∈ [𝑟𝑖, . . . , 𝑟

𝑒], and further 𝑟

0will be denoted by 𝑟).

4. Differential Equation for PressureDistribution in the 𝜓 Direction

Observing the bearing geometry (Figures 1 and 2), we admitthat 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 flowrate Ṁ

𝜓is constant. With these conditions we obtain

Ṁ𝜓≅ Δ𝜃 [𝐶

𝜕𝑝

𝜕𝜓+ 𝐷𝑟2] , (4)

where

𝐶 = −𝜌

12𝜂[𝛼ℎ3

1+ (1 − 𝛼) ℎ

3

2] , (5a)

𝐷 =𝜌

2𝜔1sin𝛽0cos𝛽0[𝛼ℎ1+ (1 − 𝛼) ℎ

2] . (5b)

The fluid mass conservation in the 𝜓 direction can beexpressed as follows:

𝜕

𝜕𝜏(𝑚) +

𝜕

𝜕𝜓(Ṁ𝜓) Δ𝜓 ≅ 0, (6)

where 𝜏 is the time, and the fluid mass 𝑚, contained in thecontrol volume Vol, is given by the relation

𝑚 ≅ Δ𝑟Δ𝜃𝜌𝑟 [𝛼ℎ1+ (1 − 𝛼) ℎ

2] . (7)

Or, using relations (4) and (7), (6) becomes

𝜕

𝜕𝜏{[𝛼ℎ1+ (1 − 𝛼) ℎ

2] 𝜌}

+ sin2𝛽0

1

𝜓

𝜕

𝜕𝜓{𝐶

𝜕𝑝

𝜕𝜓+ 𝐷

1

sin2𝛽0

𝜓2} ≅ 0.

(8)

For the permanent motion regime, (8) becomes

1

𝜓

𝜕

𝜕𝜓{𝐶

𝜕𝑝

𝜕𝜓+

1

sin2𝛽0

𝐷𝜓2} ≅ 0. (9)

Using nondimensional variables [1, 3–5, 7, 10, 14, 15], (8)can be written as

𝜕

𝜕𝜁[𝑃𝐻3(𝐾1𝜁𝜕𝑃

𝜕𝜁+ 𝐾2ΩΛ𝜁2𝐻−2)] − 𝜎𝜁

𝜕

𝜕𝑡[𝑃𝐻𝐾

3] ≅ 0.

(10)

For the stationary motion regime, (10) becomes [1, 2]

𝜕

𝜕𝜁[𝑃(𝐾

1𝜁𝜕𝑃

𝜕𝜁+ 𝐾2ΩΛ𝜁2)] ≅ 0. (11)

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

In the𝜓 direction, the fluid film ℎ varies rapidly from ℎ1to ℎ2,

at the frontier 𝑟 ≅ 𝑟𝑐(Figures 4 and 5).The existing radial step

with length in the pumping direction 𝜓 produces a pressurejump from 𝑝

ℎ1to 𝑝ℎ2. 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 rapportℎ1/ℎ2, (iii) the fact that we take (or not) into consideration

the inertial forces, and (iv) the fluid type [2, 3, 5, 16, 17].


L y

u = 𝜔1r

y1

𝜔1

O1

O1

𝜓1

𝜓1

𝜓2

𝜓2

𝜓

𝜓

h1h2

h1h2

L1 L2

Pumping

Pumping

Detail

≅ psupp.patm.

ph1

ph2Δp = |ph2 − ph1|

1

1

2

2

𝜓0O

𝜓0O

Δ𝜓

d𝜓

𝜓0 ≤ 𝜓2 ≤ L2 = L − L1; L − L1 = L2 ≤ 𝜓1 ≤ L

Figure 4: Radial step of the inner pumping bearing.

Integrating differential equation (9) and observing thelimit conditions for pressures [3, 5] and the notations fromFigures 4 and 5 (𝑝supp. is the supply pressure of the lubricationfluid and 𝑝atm. is the atmospheric pressure), we obtain themathematical relations for the pressure distributions in theSATBESPIG:

𝑝 (𝜓)

≅ 𝑝ℎ2+𝜓 − (𝐿

2+ 𝜓0)

𝜓0− (𝐿2+ 𝜓0)

× {(𝑝supp. − 𝑝ℎ2) −2𝜂 cos𝛽

0𝜔1

ℎ22sin𝛽0

[𝜓3

0− (𝐿2+ 𝜓0)3

]}

+2𝜂 cos𝛽

0𝜔1

ℎ22sin𝛽0

⋅ [𝜓3− (𝐿2+ 𝜓0)3

] .

(12)

Relation (12) presents the pressure distribution in the laminarand permanent flow regime in the smooth region of the innerpumping bearing surface, where ℎ = ℎ

2. Consider

𝑝 (𝜓)

≅ 𝑝supp. +𝜓 − (𝐿 + 𝜓

0)

(𝐿2+ 𝜓0) − (𝐿 + 𝜓

0)

× {(𝑝ℎ1− 𝑝supp.) −

2𝜂 cos𝛽0𝜔1[𝛼ℎ1+ (1 − 𝛼) ℎ

2]

sin𝛽0[𝛼ℎ3

1+ (1 − 𝛼) ℎ3

2]

L y

u = 𝜔1r

y1

𝜔1

O1𝜓1𝜓2

𝜓1𝜓2

𝜓

h1h2

h1h2

L1L2

Pumping

Pumping

Details

≅ psupp.patm.

ph1

ph2Δp = |ph2 − ph1|

1

1

2

2

𝜓0O

O1𝜓 𝜓0O

Δ𝜓

d𝜓

𝜓0 ≤ 𝜓1 ≤ L1 = L − L2; L − L2 = L1 ≤ 𝜓2 ≤ L

Figure 5: Radial step of the external pumping bearing.

× [(𝐿2+ 𝜓0)3

− (𝐿 + 𝜓0)3

]}

+2𝜂 cos𝛽

0𝜔1[𝛼ℎ1+ (1 − 𝛼) ℎ

2]

sin𝛽0[𝛼ℎ3

1+ (1 − 𝛼) ℎ3

2]

[𝜓3− (𝐿 + 𝜓

0)3

] .

(13)

Relation (13) presents the pressure distribution in the regionwith spiral channels of the inner pumping bearing surface,where ℎ = ℎ

1.

In a similar way, for the SATBESPIG with exteriorpumping (Figure 5), we obtain

𝑝 (𝜓)

≅ 𝑝ℎ2+

𝜓 − (𝐿1+ 𝜓0)

(𝐿 + 𝜓0) − (𝐿

1+ 𝜓0)

×{(𝑝supp. − 𝑝ℎ2) −2𝜂 cos𝛽

0𝜔1

ℎ22sin𝛽0

[(𝐿 + 𝜓0)3

− (𝐿1+ 𝜓0)3

]}

+2𝜂 cos𝛽

0𝜔1

ℎ22sin𝛽0

[𝜓3− (𝐿1+ 𝜓0)3

] .

(14)


Relation (14) presents the pressure distribution in the laminarand permanent flow regime in the smooth region of theexternal pumping bearing surface, where ℎ = ℎ

2. Consider

𝑝 (𝜓)

≅ 𝑝supp. +𝜓 − 𝜓

0

(𝐿1+ 𝜓0) − 𝜓0

× {(𝑝ℎ1− 𝑝supp.) −

2𝜂 cos𝛽0𝜔1[𝛼ℎ1+ (1 − 𝛼) ℎ

2]

sin𝛽0[𝛼ℎ3

1+ (1 − 𝛼) ℎ3

2]

× [(𝐿1+ 𝜓0)3

− 𝜓3

0]}

+2𝜂 cos𝛽

0𝜔1[𝛼ℎ1+ (1 − 𝛼) ℎ

2]

sin𝛽0[𝛼ℎ3

1+ (1 − 𝛼) ℎ3

2]

[𝜓3− 𝜓3

0] .

(15)

Relation (15) presents the pressure distribution in the laminarand permanent motion regime in the spiral grooves region ofthe bearing surface with exterior pumping, where ℎ = ℎ

1.

In the relations (12)–(15), all the constants are known (fora designed and realized SATBESPIG), the exception beingthe extreme pressures 𝑝

ℎ1and 𝑝

ℎ2. If we do not take into

consideration the influence of the inertial forces, then thepressures 𝑝

ℎ1and 𝑝

ℎ2are equal. Thus, 𝑝

ℎ1≡ 𝑝ℎ2, Δ𝑝 =

𝑝ℎ2− 𝑝ℎ1= 0.

6. Calculus Relations for the ExtremePressures 𝑝

ℎ1and 𝑝

ℎ2

To express the pressure distribution in the 𝜓 direction andthe extreme pressures 𝑝

ℎ1and 𝑝

ℎ2, wemust analyze the liquid

motion in the fluid film existing between the quasiparallelsurfaces of the SATBESPIG [1–3, 5, 7]. On the other hand,the inertial effects (which considerably influence the extremepressures 𝑝

ℎ1and 𝑝

ℎ2) exist on all the surfaces of the

SATBESPIG in the 𝜓 direction, but the maximum effect isconcentrated in the zone of the radial step, 𝑟 ≅ 𝑟

𝑐(Figures 4

and 5) [2, 3, 5, 16, 17].Some theoretical results concerning themotion of liquids

in similar bearings with the consideration of the influence ofinertial forces have been presented in the literature [3, 5–7,14]. It is possible to demonstrate [3, 5] that, for the case of thestationary motion regime and when only the smooth surfaceis in a rotation with 𝑛

1= constant [rot/min], the equation

that describes the viscous fluid motion in 𝜓 direction is

𝜌

𝜓

𝑑

𝑑𝜓(𝛼0𝑄2

𝜓

ℎ𝜓2Δ𝜃2−𝛾𝑄𝜓𝜔1

Δ𝜃

cos𝛽0

sin𝛽0

+ 𝛽𝜔2

1𝜓2 cos2𝛽0sin2𝛽0

ℎ)

+2𝜌𝛿𝑄

𝜓𝜔1sin𝛽0

𝜓2Δ𝜃 cos𝛽0

− 3𝜌𝛽𝜔2

1ℎ +

𝜌𝛼0𝑄2

𝜓

ℎ𝜓4Δ𝜃2

−𝜌𝛾𝑄𝜓𝜔1cos𝛽0

𝜓2Δ𝜃 sin𝛽0

+ 𝜌𝛽𝜔2

1

cos2𝛽0

sin2𝛽0

ℎ +1

𝜓

𝑑𝑝

𝑑𝜓ℎ

+12𝜂

ℎ(𝑄𝜓

ℎ𝜓Δ𝜃−𝜔1𝜓 cos𝛽

0

2 sin𝛽0

) ≅ 0,

(16)

where

𝑄𝜓= 𝑈av.ℎ ≅ −

ℎ3

12𝜂𝑟 sin𝛽0

𝜕𝑝

𝜕𝜓

+ℎ

2𝜔1𝑟 cos𝛽

0= const. (volumic flow rate) ,

(17a)

𝑈av.

=1

ℎ∫

ℎ

0

V𝜓(𝑦) ⋅ 𝑑𝑦

=average speed in the fluid film by the curb direction 𝜓,(17b)

𝛼0=6

5, 𝛽 =

2

15, 𝛾 =

1

5, 𝛿 =

1

10. (17c)

The constructive angle of spiral groove is 𝛽0= 17∘ [3, 5, 14].

Calculating the derivative of the relation (16) versus thevariable 𝜓 and taking into consideration the constants fromabove, the following expression can be found:

𝑑𝑝

𝑑𝜓≅ (

𝜌𝛼0𝑄2

𝜓

ℎ3𝜓2Δ𝜃2−𝜌𝛽𝜔2

1𝜓2

ℎ

cos2𝛽0

sin2𝛽0

)𝑑ℎ

𝑑𝜓+𝜌𝛼0𝑄2

𝜓

ℎ2𝜓3Δ𝜃2

+ 3𝜌𝛽𝜔2

1𝜓(1 −

cos2𝛽0

sin2𝛽0

) −2𝜌𝛿𝑄

𝜓𝜔1sin𝛽0

ℎ𝜓Δ𝜃 cos𝛽0

+𝜌𝛾𝑄𝜓𝜔1cos𝛽0

ℎ𝜓Δ𝜃 sin𝛽0

−12𝜂𝑄𝜓

ℎ3Δ𝜃+6𝜂𝜔1𝜓2 cos𝛽

0

ℎ2 sin𝛽0

.

(18)

We obtain a similar but more precise relation to (18) byintroducing a supplementary coercive term [1, 2, 4, 9]. In thiscase, relation (18) becomes

𝑑𝑝

𝑑𝜓≅ [

𝜌𝑄2

𝜓

ℎ3𝜓2Δ𝜃2(𝛼0+ 𝜀𝜓)]

𝑑ℎ

𝑑𝜓−𝜌𝛽𝜔2

1𝜓2

ℎ

cos2𝛽0

sin2𝛽0

𝑑ℎ

𝑑𝜓

+𝜌𝛼0𝑄2

𝜓

ℎ2𝜓3Δ𝜃2+ 3𝜌𝛽𝜔

2

1𝜓(1 −

cos2𝛽0

sin2𝛽0

)

−2𝜌𝛿𝑄

𝜓𝜔1sin𝛽0

ℎ𝜓Δ𝜃 cos𝛽0


ℎ𝜓Δ𝜃 sin𝛽0

−12𝜂𝑄𝜓

ℎ3Δ𝜃

+6𝜂𝜔1𝜓2 cos𝛽

0

ℎ2 sin𝛽0

,

(19)

where 𝜀 = 2/15 [1, 2, 4].


Integrating the differential equation (19), for both regionsof the SATBESPIG, where ℎ = ℎ

1= const. and ℎ = ℎ

2=

const. (Figures 4 and 5), we obtain the calculus relations forthe pressures 𝑝

ℎ1and 𝑝

ℎ2:

𝑝ℎ1≅ 𝑝supp. +

𝜌𝛼0𝑄2

𝜓

2ℎ21Δ𝜃2[

1

(𝐿 + 𝜓0)2−

1

(𝐿2+ 𝜓0)2]

+3

2𝜌𝛽𝜔2

1(1 −

cos2𝛽0

sin2𝛽0

) [(𝐿2+ 𝜓0)2

− (𝐿 + 𝜓0)2

]

−2𝜌𝛿𝑄

𝜓𝜔1sin𝛽0

ℎ1Δ𝜃 cos𝛽

0

ln𝐿2+ 𝜓0

𝐿 + 𝜓0


ℎ1Δ𝜃 sin𝛽

0

ln𝐿2+ 𝜓0

𝐿 + 𝜓0

+12𝜂𝑄𝜓

ℎ31Δ𝜃

[(𝐿 + 𝜓0) − (𝐿

2+ 𝜓0)]

−2𝜂𝜔1cos𝛽0

ℎ21sin𝛽0

⋅ [(𝐿 + 𝜓0)3

− (𝐿2+ 𝜓0)3

] ,

𝑝ℎ2≅ 𝑝supp. −

𝜌𝛼0𝑄2

𝜓

2ℎ22Δ𝜃2[

1

(𝐿2+ 𝜓0)2−1

𝜓20

]

+3

2𝜌𝛽𝜔2

1(1 −

cos2𝛽0

sin2𝛽0

) [(𝐿2+ 𝜓0)2

− 𝜓2

0]

+2𝜌𝛿𝑄

𝜓𝜔1sin𝛽0

ℎ2Δ𝜃 cos𝛽

0

ln𝜓0

𝐿2+ 𝜓0


ℎ2Δ𝜃 sin𝛽

0

ln𝜓0

𝐿2+ 𝜓0

+12𝜂𝑄𝜓

ℎ32Δ𝜃

[𝜓0− (𝐿2+ 𝜓0)]

+2𝜂𝜔1cos𝛽0

ℎ22sin𝛽0

[(𝐿2+ 𝜓0)3

− 𝜓0

3] .

(20)

These relations are valid for the SATBESPIG with innerpumping.

In a similar way, for the SATBESPIG with exteriorpumping, we obtain


𝜌𝛼0𝑄2

𝜓

2ℎ21Δ𝜃2[1

𝜓20

−1

(𝐿1+ 𝜓0)2]

+3

2𝜌𝛽𝜔2

1(1 −

cos2𝛽0

sin2𝛽0

) [(𝐿1+ 𝜓0)2

− 𝜓2

0]

+2𝜌𝛿𝑄

𝜓𝜔1sin𝛽0

ℎ1Δ𝜃 cos𝛽

0

ln𝜓0

𝐿1+ 𝜓0


ℎ1Δ𝜃 sin𝛽

0

ln𝜓0

𝐿1+ 𝜓0

+12𝜂𝑄𝜓

ℎ31Δ𝜃

[𝜓0− (𝐿1+ 𝜓0)]


ℎ21sin𝛽0

[𝜓3

0− (𝐿1+ 𝜓0)3

] ,

(21)


𝜌𝛼0𝑄2

𝜓

2ℎ22Δ𝜃2[

1

(𝐿 + 𝜓0)2−

1

(𝐿1+ 𝜓0)2]

+3

2𝜌𝛽𝜔2

1(1 −

cos2𝛽0

sin2𝛽0

) [(𝐿1+ 𝜓0)2

− (𝐿 + 𝜓0)2

]

−2𝜌𝛿𝑄

𝜓𝜔1sin𝛽0

ℎ2Δ𝜃 cos𝛽

0

ln𝐿1+ 𝜓0

𝐿 + 𝜓0


ℎ2Δ𝜃 sin𝛽

0

ln𝐿1+ 𝜓0

𝐿 + 𝜓0

+12𝜂𝑄𝜓

ℎ32Δ𝜃

[(𝐿 + 𝜓0) − (𝐿

1+ 𝜓0)]

+2𝜂𝜔1cos𝛽0

ℎ22sin𝛽0

[(𝐿1+ 𝜓0)3

− (𝐿 + 𝜓0)3

] .

(22)

In relations (20)–(22), all the variables are known exceptthe volumetric flow rate 𝑄

𝜓.

7. Calculus Relation for the Fluid VolumetricFlow Rate 𝑄

𝜓

Thecalculus relation for the fluid volumetric flow rate𝑄𝜓will

be established using differential equation (19) again. To findthe calculus relation, at the position of the radial step of thebearing we suppose that 𝑑ℎ/𝑑𝜓 ̸= 0. So, in this zone of thebearing the inertial effects are dominant in comparison to theviscous effects. In other words, in the radial step zone of thebearing, the liquid moves approximately like an ideal but notviscous fluid [3, 5].

Integrating (19) in the vicinity of the radial step of thebearing grooved surface (Figures 4 and 5), we obtain arelation between 𝑝

ℎ1, 𝑝ℎ2, and 𝑄

𝜓:

Δ𝑝 = 𝑝ℎ1− 𝑝ℎ2≅

𝜌𝑄2

𝜓

𝜓2𝑐Δ𝜃2(𝛼0+ 𝜀𝜓𝑐) (

1

2ℎ22

−1

2ℎ21

)

− 𝜌𝛽𝜔2

1𝜓2

𝑐

cos2𝛽0

sin2𝛽0

ln ℎ1ℎ2

,

(23)

where 𝜓𝑐is the length measured by the 𝜓 coordinate cor-

responding to the grooves radius of the profiled surface, 𝑟𝑐

(Figures 4 and 5).Relation (23) has some limits, especially at high values of

ℎ1/ℎ2, when ℎ

2→ 0, or, in other words, at the heavy regimes


for the bearing functionality. If we know the other variables,including the extreme pressures 𝑝

ℎ1and 𝑝

ℎ2, relation (23)

offers the flow rate 𝑄𝜓.

Therefore, using (20) and then (21), (22), and (23), weobtain a typical second degree algebraic equation, (24), fromwhich we obtain the fluid flow rate 𝑄

𝜓:

𝜌𝑄2

𝜓

Δ𝜃2{𝛼0[1

2ℎ21

(1

(𝐿 + 𝜓0)2−

1

(𝐿2+ 𝜓0)2)

+1

2ℎ22

(1

(𝐿2+ 𝜓0)2−1

𝜓20

)]

−1

𝜓2𝑐

(𝛼0+ 𝜀𝜓𝑐) (

1

2ℎ22

−1

2ℎ21

)}

+𝑄𝜓

Δ𝜃⋅[𝜌𝛾𝜔1cos𝛽0

sin𝛽0

(1

ℎ1

ln𝐿2+ 𝜓0

𝐿 + 𝜓0

+1

ℎ2

ln𝜓0

𝐿2+ 𝜓0

)

−2𝜌𝛿𝜔1sin𝛽0

cos𝛽0

(1

ℎ1

ln𝐿2+ 𝜓0

𝐿 + 𝜓0

+1

ℎ2

ln𝜓0

𝐿2+ 𝜓0

)

+12𝜂(𝐿 − 𝐿2

ℎ31

+𝐿2

ℎ32

)] − 𝑇1 + 𝑇2 ≅ 0,

(24)

where

𝑇1 =3

2𝜌𝛽𝜔2

1(1 −

cos2𝛽0

sin2𝛽0

) [(𝐿 + 𝜓0)2

− 𝜓2

0] ,

𝑇2 = 𝜌𝛽𝜔2

1𝜓2

𝑐

cos2𝛽0

sin2𝛽0

ln ℎ1ℎ2


sin𝛽0

× {1

ℎ21

[(𝐿 + 𝜓0)3

− (𝐿2+ 𝜓0)3

]

+1

ℎ22

[(𝐿2+ 𝜓0)3

− 𝜓0

3]} .

(25)

Relation (24) is valid for the SATBESPIGwith inner pumping.For the SATBESPIG with exterior pumping, we obtain a

similar relation:

𝜌𝑄2

𝜓

Δ𝜃2{𝛼0[1

2ℎ22

(1

(𝐿 + 𝜓0)2−

1

(𝐿1+ 𝜓0)2) (26)

−1

2ℎ21

(1

𝜓20

−1

(𝐿1+ 𝜓0)2)] (27)

+1

2𝜓2𝑐

(𝛼0+ 𝜀𝜓𝑐) (

1

ℎ22

−1

ℎ21

)}

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

(𝜓 − 𝜓0)/L

Up

av./(U

av.)

max

;(𝜓

)/p

supp

.

p: h1 = 0.7mm; h2 = 0.3mmU: h1 = 0.7mm; h2 = 0.3mmp: h1 = 0.9mm; h2 = 0.5mmU: h1 = 0.9mm; h2 = 0.5mmp: h1 = 1.1mm; h2 = 0.7mmU: h1 = 1.1mm; h2 = 0.7mmp: h1 = 1.3mm; h2 = 0.9mmU: h1 = 1.3mm; h2 = 0.9mm

Figure 6: Pressure and speed distributions in the SATBESPIG withinner pumping. (See details in Figures 7 and 8.)

+𝑄𝜓

Δ𝜃⋅[𝜌𝛾𝜔1cos𝛽0

sin𝛽0

(1

ℎ1

ln𝜓0

𝐿1+ 𝜓0

+1

ℎ2

ln𝐿1+ 𝜓0

𝐿 + 𝜓0

)

−2𝜌𝛿𝜔1sin𝛽0

cos𝛽0

(1

ℎ1

ln𝜓0

𝐿1+ 𝜓0

+1

ℎ2

ln𝐿1+ 𝜓0

𝐿 + 𝜓0

)

+12𝜂(𝐿 − 𝐿1

ℎ32

+𝐿1

ℎ31

)] − 𝑇1 − 𝑇3 ≅ 0,

(28)

where

𝑇3 = 𝜌𝛽𝜔2

1𝜓2

𝑐

cos2𝛽0

sin2𝛽0

ln ℎ1ℎ2


sin𝛽0

{1

ℎ21

[𝜓0

3− (𝐿1+ 𝜓0)3

]

+1

ℎ22

[(𝐿1+ 𝜓0)3

− (𝐿 + 𝜓0)3

]} .

(29)

Both (24) and (28) are classical algebraic equations ofsecond degree in 𝑄

𝜓. If we denote by 𝑄𝐼

𝜓and 𝑄𝐼𝐼

𝜓the two

solutions of the every algebraic equation (24) or (28), using𝑄𝐼

𝜓and 𝑄𝐼𝐼

𝜓, and if we take into consideration that the


0.25 0.26 0.27 0.28 0.29 0.304.504.524.544.564.584.604.624.644.664.684.704.724.744.764.784.80

(𝜓 − 𝜓0)/L

p: h1 = 0.7mm; h2 = 0.3mmp: h1 = 0.9mm; h2 = 0.5mmp: h1 = 1.1mm; h2 = 0.7mmp: h1 = 1.3mm; h2 = 0.9mm

p(𝜓

)/p

supp

.

Figure 7: Detail of Figure 6 concerning the pressure jump betweenℎ1and ℎ

2. (The marked points are those where we made the

calculation.)

SATBESPIG realizes inner pumping and exterior pumping,it is not possible to have a negative fluid flow rate from aphysical point of view.Thus, the algebraic solution, which hasphysical meaning, is the positive solution [3, 5].

The numerical evaluation of 𝑄𝐼𝜓and 𝑄𝐼𝐼

𝜓algebraic solu-

tions, for 𝑄𝐼𝜓and 𝑄𝐼𝐼

𝜓using medium (normal) values for the

physical and geometrical dimensions [1, 2] of (24) and (28),leads to 𝑄𝐼

𝜓< 0 and 𝑄𝐼𝐼

𝜓> 0. Thus, the mathematical relation

for the calculus of the fluid volumetric flow rate 𝑄𝜓is

𝑄𝜓≡ 𝑄𝐼𝐼

𝜓, (30)

where, in conformity to the devoted notations from theclassical algebra, the solution 𝑄𝐼𝐼

𝜓is

𝑄𝐼𝐼

𝜓=−�̈� − √�̈�2 − 4 ̈𝑎 ̈𝑐

2 ̈𝑎, (31)

where ̈𝑎, �̈�, and ̈𝑐 are the coefficients of the algebraic equations(24) or (28).

8. Numerical and Experimental Results

The established mathematical relations allow the numericalcalculation of the pressure and speed distributions for severalSATBESPIG with inner or external pumping in permanentand laminar regimes. For these numerical calculations we

0.25 0.26 0.27 0.28 0.29 0.300.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

(𝜓 − 𝜓0)/L

U: h1 = 0.7mm; h2 = 0.3mmU: h1 = 0.9mm; h2 = 0.5mmU: h1 = 1.1mm; h2 = 0.7mmU: h1 = 1.3mm; h2 = 0.9mm

Uav

./(U

av.)

max

Figure 8: Detail of Figure 6 concerning the speed jump betweenℎ1and ℎ

2. (The marked points are those where we made the

calculation.)

used two different computer programs: one for the SATBE-SPIG with inner pumping and the other for the SATBESPIGwith external pumping.

In Figures 6 to 8, we present the calculated pressure andspeed distributions for a SATBESPIG with inner pumpingand 𝑟𝑒= 90mm, 𝑟

𝑐= 57.15mm, 𝑟

𝑖= 45mm, 𝑛

𝑝= 10, 𝛼 = 0.615,

𝛽0= 17∘, 𝜌 = 905 kg/m3, 𝜂 = 10−4 Pa⋅s, 𝑝supp. = 101325N/m

2,𝑛 = 900 rot/min, 𝛼

0= 6/5, 𝜀 = 2/15, 𝛾 = 1/5, 𝛿 = 1/10, 𝛽 = 2/15,

and the parameters h1and h2as written in the figures.

Figure 9 presents the calculated pressure and speed dis-tributions for a SATBESPIG with external pumping with thesame characteristics as the SATBESPIG with inner pumpingabove, with the exception that 𝑟

𝑐= 77.85mm.

In Figure 10, we compared the theoretical results and theexperimental measurements for the SATBESPIG with innerpumping and ℎ

1, ℎ2, and 𝑛 written in the figure.

In all our studies, the main constructive geometrical andfunctional parameters for the calculation of these bearingswere ℎ

1, ℎ2, n, 𝑟𝑖, and 𝑟

𝑒[3, 5].

9. Discussion and Conclusions

Theanalysis of the theoretical results allows some conclusionsto be drawn concerning the laminar motion of incompress-ible fluids in some models of a SATBESPIG. We analyzedtwo variants of the SATBESPIG with inner pumping, whichare called the First Variant and the Second Variant below.The difference between these bearings is the 𝑟

𝑐dimension.


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

(𝜓 − 𝜓0)/L

p: h1 = 0.7mm; h2 = 0.3mmU: h1 = 0.7mm; h2 = 0.3mmp: h1 = 0.9mm; h2 = 0.5mmU: h1 = 0.9mm; h2 = 0.5mmp: h1 = 1.1mm; h2 = 0.7mmU: h1 = 1.1mm; h2 = 0.7mmp: h1 = 1.3mm; h2 = 0.9mmU: h1 = 1.3mm; h2 = 0.9mm

Up

av./(U

av.)

max

;(𝜓

)/p

supp

.

Figure 9: Pressure and speed distributions in the SATBESPIG withexternal pumping.

The First Variant is the SATBESPIG from Figure 6 (𝑟𝑐=

57.15mm), and the Second Variant has 𝑟𝑐= 70.29mm.

(a) For all the rotation per minute (r.p.m.) ranges n andfor all the analyzed rapports ℎ

1/ℎ2, the First Vari-

ant SATBESPIG with inner pumping had superiorhydrodynamics performance compared to the SecondVariant SATBESPIG.Thus, high pressures in bearingscan be realized with an appropriate dimension ofthe bearing surface, especially the dimension of thegrooved surface, meaning the parameters 𝑟

𝑖, 𝑟𝑐, and

𝑟𝑒.

(b) Generally speaking, increasing other hydraulics per-formance of the SATBESPIG for the same dimensions𝑟𝑖, 𝑟𝑐, and 𝑟

𝑒can be realized by increasing the number

of the rotation per minute 𝑛. The modification of therapport ℎ

1/ℎ2has a small influence.

(c) The nondimensional speeds increase (and decrease)from the bearing input to the bearing output, with ajump at the radial step 𝑟 ≅ 𝑟

𝑐. The nondimensional

pressures change, but not linearly with the channellength, from the input, where𝑝imput ≅ 𝑝supp., up to thepressure 𝑝

ℎ1and from the pressure 𝑝

ℎ2to the output

pressure, where 𝑝output ≅ 𝑝supp..(d) The pressure jump Δ𝑝 = 𝑝

ℎ1− 𝑝ℎ2theoretically tends

to zero when the rapport ℎ1/ℎ2→ 1 and thus when

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

2.1

2.2

(𝜓 − 𝜓0)/L

Th.: h1 = 0.4mm; h2 = 0.1mm; 180 rot/minTh.: h1 = 0.38mm; h2 = 0.08mm; 280 rot/minTh.: h1 = 0.38mm; h2 = 0.08mm; 450 rot/minEx.: h1 = 0.4mm; h2 = 0.1mm; 180 rot/minEx.: h1 = 0.38mm; h2 = 0.08mm; 280 rot/minEx.: h1 = 0.38mm; h2 = 0.08mm; 450 rot/min

1.00.91.0

0.0

p(𝜓

)/p

supp

.

Figure 10: Pressure distributions in the SATBESPIG with innerpumping (comparison between the theoretical and experimentalcurves). The marked points are those where we performed theexperimental measurements.

the radial step vanishes or when 𝑝ℎ1≅ 𝑝ℎ2. The case

𝑝ℎ1

≅ 𝑝ℎ2

appears only when we do not take intoconsideration the inertial effects.

(e) The pressure jump Δ𝑝 depends not only on therapport ℎ

1/ℎ2but also on the value of the 𝑛 and on

the bearing geometry (𝑟𝑖, 𝑟𝑐, and 𝑟

𝑒).

(f) For the same constructive geometrical and functionalparameters (𝑟

𝑖, 𝑟𝑒, ℎ1/ℎ2, n,. . .), the SATBESPIG with

exterior pumping gives lower pressures than thebearing with inner pumping.

(g) The comparative analysis between the theoreticaland experimental results shows good correlation,especially at low 𝑛 and at middle values for therapport ℎ

1/ℎ2(ℎ1/ℎ2≅ 4). Some of the mathematical

relations established above can be used to approachthe theoretical functionality of the SATBESPIG hav-ing magnetic controllable fluids (magnetic fluids ormagnetorheological 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 interestsregarding the publication of this paper.


Acknowledgment

Thisworkwas partly supported by the Project PN II (NationalEducation Ministry—Romania) no. 157/2012 “MagNanoMi-croSeal” (2012–2015).

References

[1] V. N. Constantinescu, Al. Nica, M. D. Pascovici, G. Ceptureanu,and S. Nedelcu, Sliding Bearings , Tehnica Publishing House,Bucharest, Romania, 1980.

[2] V. N. Constantinescu, Dynamics of Viscous Fluids in LaminarRegime, Romanian Academy Publishing House, Bucharest,Romania, 1987.

[3] C. Velescu, The study of the viscous fluids motion in self-acting thrust bearings [Ph.D. thesis], “Politehnica” University ofTimisoara, 1998.

[4] A. V. Yemelyanov and I. A. Yemelyanov, “Physical models,theory and fundamental improvement to self-acting spiral-grooved gas bearings and visco-seals,” Proceedings of the Institu-tion of Mechanical Engineers J, vol. 213, no. 4, pp. 263–271, 1999.

[5] C. Velescu, Hydrodynamics of the Thrust Bearings With SpiralGrooves, Mirton Publishing House, Timisoara, Romania, 2000.

[6] C. Velescu, “Theoretically remarks on the thermal and visco—inertial effects in the self acting sliding conically radially—thrust bearings,” in Proceedings of the International Symposiumon Fluid Machinery and Fluid Engineering (ISFMFE ’96), Bei-jing, China, September 1996.

[7] N. Kawabata, I. Ashino, M. Sekizawa, and S. Yamazaki, “Spiralgrooved bearing utilizing the pumping effect of a herringbonejournal bearing. (Method of numerical calculation and influ-ences of bearing parameters),” JSME International Journal 3, vol.34, no. 3, pp. 411–418, 1991.

[8] S. M. Yao, “Aerostatic and aerodynamic performance of an out-pump spirally grooved thrust bearing: analysis and compar-isons to static load experiments,”Tribology Transactions, vol. 52,no. 3, pp. 376–388, 2009.

[9] S. M. Yao, “Aerostatic and aerodynamic performance of an in-pump spirally grooved thrust bearing: Analysis and compar-isons to static load experiments,” Transactions, vol. 51, no. 5, pp.679–689, 2008.

[10] L. Wu, P. Liu, Z. Liu, Z. Liu, Q. Yang, and Y. Wei, “Optimizationdesign of hydrodynamic bearing with multiple spiral grooves,”in Proceedings of the 7th International Conference on ProgressMachining Technology (ICPMT ’04), pp. 502–507, Suzhou,China, December 2004.

[11] B. C. Majumdar, R. Pai, and D. J. Hargreaves, “Analysis ofwater-lubricated journal bearings with multiple axial grooves,”Proceedings of the Institution of Mechanical Engineers J, vol. 218,no. 2, pp. 135–146, 2004.

[12] J. F. Zhou, B. Q. Gu, and C. Ye, “An improved design ofspiral groove mechanical seal,” Chinese Journal of ChemicalEngineering, vol. 15, no. 4, pp. 499–506, 2007.

[13] G. I. Broman, “Implications of cavitation in individual groovesof spiral groove bearings,” Proceedings of the Institution ofMechanical Engineers J, vol. 215, no. 5, pp. 417–424, 2001.

[14] S. B. Malanoski and C. H. T. Pan, “The static and dynamiccharacteristics of the spiral—grooved thrust bearing,” Journal ofBasic Engineering D—Transactions of the ASMED, vol. 87, no. 3,pp. 547–558, 1965.

[15] W. C. Wachmann, M. S. Malanoski, and V. J. Vohr, “Thermaldistortion of spiral-grooved gas-lubricated thrust bearings,”Journal of Lubrication Technology—ASME Transactions, vol. 93,no. 1, pp. 102–112, 1971.

[16] C. M. Rodkiewicz and M. I. Anwar, “Inertia and convectiveeffects in hydrodynamic lubrication of a slider bearing,” Journalof Lubrication Technology F—ASME Transactions, vol. 93, 1971.

[17] C. M. Rodkiewicz, J. C. Hinds, and C. Dayson, “Inertia,convection, and dissipation effects in the thermally boostedoil lubricated sliding thrust bearing,” Journal of LubricationTechnology F—ASME Transactions, vol. 97, no. 1, pp. 121–129,1975.

[18] I. De Sabata, N. C. Popa, I. Potencz, and L. Vékás, “Inductivetransducers with magnetic fluids,” Sensors and Actuators A, vol.32, no. 1–3, pp. 678–681, 1992.

[19] N. C. Popa, I. De Sabata, I. Anton, I. Potencz, and L. Vékás,“Magnetic fluids in aerodynamic measuring devices,” Journal ofMagnetism and Magnetic Materials, vol. 201, no. 1–3, pp. 385–390, 1999.

[20] M. Liţǎ, N. C. Popa, C. Velescu, and L. N. Vékás, “Investigationsof a magnetorheological fluid damper,” IEEE Transactions onMagnetics, vol. 40, no. 2, pp. 469–472, 2004.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Research Article Laminar Motion of the Incompressible Fluids in … · 2019. 7. 31. · Research...

Documents

Transcript of Research Article Laminar Motion of the Incompressible Fluids in … · 2019. 7. 31. · Research...