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Pressure distribution in a porous squeeze film bearing lubricated by a Vočadlo fluid
Applied Mathematical Modelling 37 (2013) 9295–9307
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Pressure distribution in a porous squeeze film bearing lubricated by a Vocˇadlo fluid A. Walicka ⇑, P. Jurczak University of Zielona Góra, Faculty of Mechanical Engineering, Ul. Szafrana 2, P.O. Box 47, 65-516 Zielona Góra, Poland
a r t i c l e
i n f o
Article history: Received 24 May 2012 Received in revised form 12 March 2013 Accepted 19 April 2013 Available online 10 May 2013 Keywords: Vocˇadlo lubricant Porous bearing Squeeze film Thrust bearing
a b s t r a c t The influence of a wall porosity on the pressure distribution in a curvilinear squeeze film bearing lubricated by a lubricant being a viscoplastic fluid of a Vocˇadlo type is considered. After general considerations on the flow of viscoplastic fluid (lubricant) in a bearing clearance and in a porous layer a modified Reynolds equation for the curvilinear squeeze film bearing with a Vocˇadlo lubricant is given. The solution of this equation is obtained by a method of successive approximation. As a result one obtains a formula expressing the pressure distribution. The example of squeeze film in an axial bearing (modeled by two parallel disks) is discussed in detail. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Squeeze flow, a simplified model based on the Reynolds’ lubrication theory, has been widely used in many applications, such as petroleum and chemical technology, food engineering, pharmaceutical manufacture, etc., involving a variety of fluids, including non-Newtonian fluids. Many investigators [1–6] have researched on the subject of squeeze flow. These flows are found in fabrication operations such as stamping, injection molding, and sheet forming. Also, material properties of highly viscous fluids are measured with a device called the ‘‘plastometer’’ which incorporates a parallel-disk squeeze flow geometry [2,7,8]. In addition, such flows are encountered in lubrication systems, and there is a considerable interest as to the degree which viscoplastic additives enhance the load-bearing capacity of lubricant. The flows of Newtonian fluids in the clearance of a thrust bearing with impermeable surfaces have been examined theoretically. The bearing walls have been modeled as two disks, two conical or spherical surfaces. The more general case is established by the bearing formed by two surfaces of revolution [9]. Porous bearings have been widely used in industry for a long time [10–14]. Basing on the Darcy model Morgan and Cameron [12] first presented theoretical research on these bearings. Recently the problem of slide bearings with porous walls lubricated by Bingham fluid was taken up by Walicki et al. [15] but the problem of a porous squeeze film bearing lubricated by the same lubricant was presented by Walicka [16]. From various models of fluids with a yield shear stress the Vocˇadlo fluid (sometimes called Robertson and Stiff [17] fluid) flow frequently appears in many industrial branches: food processing, metal processing, petroleum industry. It also appears in tribology modeling semi-fluid lubricants [18,19]. The purpose of this study is to investigate the pressure distribution in a clearance of squeeze film bearing formed by two surfaces of revolution, having common axis of symmetry, as shown in Fig. 1; the lower one of these surfaces is connected
⇑ Corresponding author. E-mail addresses: [email protected] (A. Walicka), [email protected] (P. Jurczak). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.04.046
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Nomenclature C ðnÞ ðx; tÞ; C kðnÞ ðx; tÞ auxiliary functions given by formulae (3.10)1 and (3.12)1, respectively H thickness of a porous layer ~ non-dimensional thickness of a porous layer H h bearing clearance thickness K non-dimensional capillary radius of a porous matrix n exponents in a Vocˇadlo fluid p pressure Q flow rate R; RðxÞ local bearing radius rc capillary radius of a porous matrix SV Saint–Venant plasticity number V squeeze velocity tx ; ty components of the flow velocity e non-dimensional axial bearing clearance thickness Kyx component of shear stress s shear stress s0 yield shear stress sw shear stress on the bearing clearance wall UðnÞ auxiliary function in the plane flow of a Vocˇadlo fluid given by Eq. (2.13) WðnÞ auxiliary function in the capillary flow of a Vocˇadlo fluid given by Eq. (3.3)1
with a porous layer. The analysis is based on the assumption that the porous matrix consists of a system of capillaries of very small radii restricting the lubricant flow through the matrix in only one direction. The lubricant is assumed to be the Vocˇadlo type for which the constitutive equation has a following form [20,21]:
1
n
s ¼ s0n þ lc_ ;
ð1:1Þ
where s is the shear stress, s0 is the yield shear stress, l is the coefficient of plastic viscosity, c_ is the rate of deformation, m is the exponent in the Vocˇadlo fluid.
2. Analysis of the Vocˇadlo fluid flow in a bearing clearance The flow configuration is shown in Fig. 1. The upper boundary of a porous layer is described by function R(x), which denotes the radius of this boundary. The fluid film thickness is given by function h(x,t). An intrinsic curvilinear orthogonal coordinate system (x, #, y) is also depicted in Fig. 1.
Fig. 1. Configuration of a thrust bearing with one porous wall.
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By using the assumptions typical for the flows in a thin layer the equations of motion for the Vocˇadlo fluid for axial symmetry one can present in the form [22–24]:
1 @ðRtx Þ @ ty þ ¼ 0; R @x @y
ð2:1Þ
@p @ Kyx ¼ ; @x @y
ð2:2Þ
@p ¼ 0; @y
ð2:3Þ
where the non-zero component of stress tensor is [23,24,18]
8 1 n > < s0n þ l @@ytx Kyx ¼ 1 n > : sn l @ tx 0 @y
y 6 h0 ;
for
ð2:4Þ
y P h h0 :
In the flow of a fluid with yield shear stress there exists a quasi-solid core bounded by surfaces laying at
y ¼ h0 or y ¼ h h0 for which the shear stress is : jKyx j ¼ s0 :
ð2:5Þ
The problem statement is complete after specification of boundary conditions which are
tx ðx; 0; tÞ ¼ 0; tx ðx; h; tÞ ¼ 0; @h ; @t
ty ðx; 0; tÞ ¼ V; ty ðx; h; tÞ ¼ @p ¼ 0; @x x¼0
ð2:6Þ ð2:7Þ
pðxo Þ ¼ po ;
ð2:8Þ
here V is the lubricant velocity on the upper boundary of a porous matrix, po is the outlet pressure. Putting Eq. (2.4) into Eq. (2.2) and integrating the resulted equations with respect to y in the interval 0 6 y 6 h and determining arbitrary constants from the boundary conditions (2.5) and (2.6) we obtain [23–25]: - for shear flow (y 6 h0 or y P h h0 )
txs ¼
1 1 @p n 1þ1n 2ðn þ 1Þ 1þ1n n yðh 2h h ðh 2yÞ Þ 0 nþ1 @x n 2 n lðn þ 1Þ n
ð2:9aÞ
for y 6 h0 and
txs ¼
1 1 @p n 1þ1n 2ðn þ 1Þ 1þ1n n h ð2y hÞ ðh yÞðh 2h Þ 0 nþ1 @x n 2 n lðn þ 1Þ n
ð2:9bÞ
for y P h h0 ; - for core flow (h0 6 y 6 h h0 )
txc ¼
1 1 1 @p n 1þ1n 2ðn þ 1Þ h0 ðh 2h0 Þn : h ðh 2h0 Þ1þn n lðn þ 1Þ @x n
nþ1 n
2
ð2:10Þ
The flow rate across the clearance is given by
^x ¼ 2 Q c ¼ ht
Z
h0
txs dy þ 2
0
Z
h=2
txc dy ¼ 2
Z
h0
txs dy þ txc ðh 2h0 Þ;
ð2:11Þ
0
h0
^x is the mean velocity in the clearance; after calculations there is where t
^x ¼ ht
2nþ1 1 h n @p n UðnÞ ðvÞ: @x 12l
ð2:12Þ
Here
UðnÞ ðvÞ ¼
3 2
1 n
2n 1 1 1 vn þ v2þn 2n þ 1 2n þ 1
ð2:13Þ
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and
s0 sw s0 h h @p ; ; h0 ¼ ¼ ð1 vÞ; sw ¼ 2 2 @x sw ð @p Þ @x h 2h0 @p 2s0 ; finally v ¼ @p ; s0 ¼
v¼
2
@x
ð2:14Þ
h @x
s0 – is the yield shear stress, sw – is the shear stress on the clearance wall. 3. Modified Reynolds equation 3.1. General considerations Integrating the equation of continuity (2.1) with respect to y and determining the arbitrary constants from the boundary conditions (2.7) we obtain [24,19]
^x Þ 1 @ðRht @h ¼ þ V; R @x @t or
" # 1 2nþ1 1 @ @p n @h n UðnÞ ðvÞ ¼ 12l V ; Rh R @x @x @t
ð3:1Þ
which is the modified Reynolds equation. Considering the porous matrix as a system of capillaries (each of radius rc) we may assume that the velocity components for the Vocˇadlo fluid flow in this matrix are given [26] as:
tx ¼ Up
1 n @p WðnÞ ð!Þ; @x
WðmÞ ð!Þ ¼
n 3n þ 1 1n 1 3nþ1 1 ! þ ! n ; ð3n þ 1Þ 3n 3n
ty ¼ Up
@p WðnÞ ð!Þ; @y
ð3:2Þ
where
ð3:3Þ
nþ1
r c n up s 2s0 !¼ 0 ¼ ; Up ¼ 1 ; @p sw rc ð @xÞ 2n l
up – is the coefficient of porosity of the porous matrix. Since the cross velocity component must be continuous at the porous wall-fluid film interface, one obtains from Eqs. (3.1) and (3.2)2 the modified Reynolds equation for the pressure distribution in the bearing with porous wall:
2 " # 1n 1 2nþ1 n 1 @ @p @h @p UðnÞ ðvÞ ¼ 12l4 WðnÞ ð!ÞUp Rh n R @x @x @t @y
3
5:
ð3:4Þ
y¼0
To determine the second term in the brackets on the right hand side of Eq. (3.4) let us consider the lubricant flow in the porous region. The equation of continuity for the flow in this region has the same form as Eq. (2.1). By substituting Eq. (3.2) into Eq. (2.1) one obtains the following equation for pressure distribution in the porous region
" " 1 # 1 # n n 1 @ @p @ @p þ ¼ 0: RWðnÞ ð!ÞUp WðnÞ ð!ÞUp R @x @y @x @y
ð3:5Þ
Integrating this equation with respect to y across the porous layer and using the Morgan–Cameron approximation [12] we have
1 n @p WðnÞ ð!ÞUp @y
y¼0
" 1 # n H @ @p : RWðnÞ ð!ÞUp ¼ R @x @x
ð3:6Þ
Note that on the porous wall-fluid film interface we have also:
@p @p h ¼ and ! ¼ v : rc @x @x When Eqs. (3.6) and (3.7) are substituted into Eq. (3.4) the modified Reynolds equation takes the form:
ð3:7Þ
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( ) h 2nþ1 i @p1n 1 @ @h n R h UðnÞ ðvÞ þ 12lHUp WðnÞ ð!Þ ¼ 12l : R @x @x @t
ð3:8Þ
The final shape of solution to the Reynolds equation (3.8) is formally given as follows:
pðx; tÞ ¼ po þ C ðnÞ ðxo ; tÞ C ðnÞ ðx; tÞ ;
ð3:9Þ
where
n Z 1 @h dx dx; 12l R F ðnÞ ðxÞ @t n h 2nþ1 io n : F ðnÞ ðxÞ ¼ R h n UðnÞ ðvÞ þ 12lHUp WðnÞ ð!Þ
C ðnÞ ðx; tÞ ¼
Z
ð3:10Þ
In reality the Eq. (3.8) is non-linear with respect to the pressure gradient P ¼ @p because it is also included into the @x functions U(n)(v) and W(n)(Y). One may propose the following simple iterative scheme to solve Eq. (3.8): 1. 2. 3. 4.
at first calculate P0 from Eq. (3.8) assuming that v = Y = 0, secondly using P0 calculate v(0) and Y0 from Eq. (2.14)5 and (3.7)2, at third calculate p1(x, t) from Eq. (3.9) using P0, v(0) and Y0 in F(n)(x), using P1 repeat the steps 2 and 3 to obtain p2(x, t). Finally the kth approximation for pressure distribution will be given as follows
pk ðx; tÞ ¼ po þ C ðnÞk ðxo ; tÞ C ðnÞk ðx; tÞ ;
ð3:11Þ
where
C ðnÞk ðx; tÞ ¼
Z
n Z 1 @h dx dx; 12l R F ðnÞk ðxÞ @t
ð3:12Þ
n h 2nþ1 io n F ðnÞk ðxÞ ¼ R h n UðnÞ ðvk1 Þ þ 12lHUp WðnÞ ð!k1 Þ : 3.2. Special forms of the Reynolds equation Let us discuss some special forms of the modified Reynolds equation – frequently met in practice – allowing to apply the general solution presented by formulae (3.11) and (3.12). The substitutions
R ¼ ax;
0 < a 6 1;
h ¼ hðx; tÞ;
ð3:13Þ
made in Eq. (3.8) lead the Reynolds equation for conical bearing (a < 1) or axial bearing (a = 1) with parallel surfaces:
( 1 ) 1 @ @p n @h ¼ 12l : x½GH x @x @x @t
ð3:14Þ
The iterative solution of this equation is generally described by formula (3.11) in which C(n)k and F(n)k are given as follows
C ðnÞk ðx; tÞ ¼
Z
n Z 1 @h dx dx; 12l x F ðnÞk ðxÞ @t
ð3:15Þ
n F ðnÞk ðxÞ ¼ x GHðk1Þ and:
h 2nþ1 i GH ¼ h n UðnÞ ðvÞ þ 12lHUp WðnÞ ð!Þ ;
ð3:16Þ
h 2nþ1 i GHðk1Þ ¼ h n UðnÞ ðvk1 Þ þ 12lHUp WðnÞ ð!k1 Þ :
ð3:17Þ
On the other hand the substitutions
R ¼ Rs sin u;
u¼
x ; Rs
h ¼ hðu; tÞ;
lead to the Reynolds equation for spherical bearing:
ð3:18Þ
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Fig. 2. Axial squeeze film bearing.
Fig. 3. Dimensionless pressure distribution in the axial bearing for n = 0.5, K = 1.0 and e = 1.0.
Fig. 4. Dimensionless pressure distribution in the axial bearing for n = 0.5, K = 0.5 and e = 1.0.
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( 1 ) nþ1 @h 1 @ @p n sin u½GH ; ¼ 12lRs n sin u @ u @u @t
9301
ð3:19Þ
here Rs is the radius of a spherical bush. In this case there are:
C ðnÞk ðu; tÞ ¼
Z
n Z nþ1 1 @h sin udu du; 12lRs n F ðnÞk ðuÞ @t
n F ðnÞk ðuÞ ¼ sin u GHðk1Þ :
ð3:20Þ ð3:21Þ
The substitutions
R ¼ Rb ¼ constant; x ¼ Rb u;
h ¼ hðu; tÞ;
lead to the Reynolds equation for journal bearing:
Fig. 5. Dimensionless pressure distribution in the axial bearing for n = 0.5, K = 1.0 and e = 0.5.
Fig. 6. Dimensionless pressure distribution in the axial bearing for n = 0.5, K = 0.5 and e = 0.5.
ð3:22Þ
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( 1 ) nþ1 @h @ @p n ½GH ; ¼ 12lRbn @u @u @t
ð3:23Þ
where Rb is the radius of a cylindrical bush and
C ðnÞk ðu; tÞ ¼
Z
n Z nþ1 1 @h du du; 12lRbn F ðnÞk ðuÞ @t
n F ðnÞk ðuÞ ¼ GHðk1Þ :
ð3:25Þ
In the last two cases it may introduce the clearance thickness in a following form:
hðu; tÞ ¼ Cð1 e cos uÞ;
ð3:24Þ
e ¼ eðtÞ;
Fig. 7. Dimensionless pressure distribution in the axial bearing for n = 1.0, K = 1.0 and e = 1.0.
Fig. 8. Dimensionless pressure distribution in the axial bearing for n = 1.0, K = 0.5 and e = 1.0.
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where
e C
e ¼ ; C ¼ Rs Re or C ¼ Rb Rc ; Rc – is a shaft radius, e – is an eccentricity. 4. Example of flow Let us consider the pressure distribution in a squeeze film bearing presented in Fig. 2 lubricated by the Vocˇadlo fluid. Assume also that the lubricant flow coincides with a small core, then v 1; ! 1 and the functions U(n)(v) and W(n)(Y) become quasi-linear:
UðnÞ ðvÞ
6n 1 n
2 ð2n þ 1Þ
1
2n þ 1 1n v ; 2n
Fig. 9. Dimensionless pressure distribution in the axial bearing for n = 1.0, K = 1.0 and e = 0.5.
Fig. 10. Dimensionless pressure distribution in the axial bearing for n = 1.0, K = 0.5 and e = 0.5.
ð4:1Þ
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WðnÞ ð!Þ ¼
n 3n þ 1 1n ! : 1 3n þ 1 3n
Using the iterative procedure to Eq. (3.14), applying Eqs. (3.15), (3.16), (3.17) and introducing the following dimensionless variables and parameters
~x ¼
x ; Ro
~¼ R; R Ro
~¼ h; h ho
e_ ¼
de ; dt
V o ¼ ho ðe_ Þ;
K¼
~ ¼ H up ; H ho
rc ; ho
2nþ1
~¼ p
ðp po Þho
ln V no Rnþ1 o
;
SV ¼
h ¼ ho eðtÞ;
s0 h2n o ; ln V no Rno
ð4:2Þ
one may present formula (3.11) for the pressure distribution in an axial squeeze film bearing in the simple non-dimensional form
~ð~x; ~tÞ ¼ DðnÞ ð1 ~xnþ1 Þ þ EðnÞ ð1 ~xn Þ; p
Fig. 11. Dimensionless pressure distribution in the axial bearing for n = 1.5, K = 1.0 and e = 1.0.
Fig. 12. Dimensionless pressure distribution in the axial bearing for n = 1.5, K = 0.5 and e = 1.0.
ð4:3Þ
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where
DðnÞ ¼ ðn þ 1Þnn
1
2 e
2nþ1 n
2nþ1
~
; nþ1 n
HK n þ 23nþ1
EðnÞ ¼
~ e2 þ 43 HK ðSVÞn ; 2nþ1 nþ1 n ~ nn e n þ 2HK n 2nþ1 3nþ1
ð4:4Þ
SV – is the Saint–Venant plasticity number (plasticity index). The graphical presentation of the pressure distribution given by formula (4.3) is shown in Figs. 3–14; the graphs presented in Figs. 3, 4, 7, 8, 11 and 12 are made for e = 1 (the squeezing start) but the graphs presented in Figs. 5, 6, 9, 10, 13 and 14 are made for e = 0.5. The boundary values of power-law exponent (n = 0.5, 1.0, 1.5) and plasticity index (SV = 0, 0.5, 1.0) used to graphical presentation are near to the real values of these parameters for Vocˇadlo lubricants applied in thrust bearings [18]. The values SV = 0.5, 1.0 described the pressure distribution, in a squeeze film of a Vocˇadlo fluid which flows in the bearing clearance with a consistent core.
Fig. 13. Dimensionless pressure distribution in the axial bearing for n = 1.5, K = 1.0 and e = 0.5.
Fig. 14. Dimensionless pressure distribution in the axial bearing for n = 1.5, K = 0.5 and e = 0.5.
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5. Discussion and conclusions The above analysis has shown that the Reynolds approximation may be used to model the flow of the Vocˇadlo fluid as lubricant in the clearance of a curvilinear squeeze film bearing. The obtained results for the lubricant flow in an axial bearing with parallel surfaces are relatively simple and may be used in practical applications. The formula for pressure distribution consists of two terms: the first of them is connected with the power-law character of lubricant flow but the second one is connected with the plastic character of this flow and it is described – among other factors – by the plasticity index SV. Note that for SV = 0, and in effect E(n) = 0, formula (4.3) represents the pressure distribution only for the flow of a powerlaw lubricant; it is identical with that one obtained by Walicki et al. [27]. In the case when n ¼ 1:0; SV ¼ 0 there is a Newtonian lubricant flow and the results obtained here are the same as in [28]. If n = 1.0 and SV > 0, then one has a flow of a Bingham fluid lubricant and the results are identical with those presented in [23,24] for non-porous clearance walls ~ ¼ K ¼ 0Þ and identical with those presented in [16] for porous clearance walls. ðH For the axial bearing the results have been presented for three values of the Vocˇadlo exponent, namely: for n = 0.5, n = 1.0 (Bingham fluid) and for n = 1.5, for three values of the plasticity index SV and also for different parameters of the porous ~ and K). matrix (different values of H Analyzing the graphs presented in Figs. 3–14 it may approach to some conclusions. ~ For n = 1.0 all values of For n = 0.5 all values of the pressure are not great and generally decrease with increase of K and H. ~ the pressure are greater than those in previous case and also they decrease with the increase of K and H. ~ The decrease of the values of For n = 1.5 all values of the pressure are great and they decrease with the increase of K and H. e and the increase of the values of SV evoke the increase of all values of the pressure. ~ decrease with the Analysis of the formula (4.3) and its graphical presentations may conclude that the pressure values p ~ increase of the porous matrix parameters K and H. These values increase with the increase of the values of power exponent n and plasticity index SV. For SV > 0 the flows in a squeeze film exhibit the existence of a consistent core which considerably influences the pressure distribution. This influence is very substantial in the bearings with non-porous walls. In conclusion it may suggest that the case n = 1.5 is the best case for the Vocˇadlo fluid to be applicated as a lubricant in the squeeze film bearing.
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[26] A. Walicka, E. Walicki, Non-Newtonian fluids flows in porous media, Rheology – Theory and Application, vol. 2, EKMA Press, Warsaw, 2011, pp. 337– 367. [27] E. Walicki, A. Walicka, D. Michalski, The influence of a wall porosity on the pressure distribution in a curvilinear bearing lubricated by a power-law fluid, in: Problems of Non-Conventional Bearing Systems, 3rd Nat. Conf., Lodz, 1997, pp. 102–109. [28] E. Walicki, A. Walicka, T. Karpin´ski, Load capacity of a curvilinear thrust bearing with porous wall, Tribologia 5 (149) (1996) 610–620.
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Pressure distribution in a porous squeeze film bearing lubricated by a Vocˇadlo fluid A. Walicka ⇑, P. Jurczak University of Zielona Góra, Faculty of Mechanical Engineering, Ul. Szafrana 2, P.O. Box 47, 65-516 Zielona Góra, Poland
a r t i c l e
i n f o
Article history: Received 24 May 2012 Received in revised form 12 March 2013 Accepted 19 April 2013 Available online 10 May 2013 Keywords: Vocˇadlo lubricant Porous bearing Squeeze film Thrust bearing
a b s t r a c t The influence of a wall porosity on the pressure distribution in a curvilinear squeeze film bearing lubricated by a lubricant being a viscoplastic fluid of a Vocˇadlo type is considered. After general considerations on the flow of viscoplastic fluid (lubricant) in a bearing clearance and in a porous layer a modified Reynolds equation for the curvilinear squeeze film bearing with a Vocˇadlo lubricant is given. The solution of this equation is obtained by a method of successive approximation. As a result one obtains a formula expressing the pressure distribution. The example of squeeze film in an axial bearing (modeled by two parallel disks) is discussed in detail. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Squeeze flow, a simplified model based on the Reynolds’ lubrication theory, has been widely used in many applications, such as petroleum and chemical technology, food engineering, pharmaceutical manufacture, etc., involving a variety of fluids, including non-Newtonian fluids. Many investigators [1–6] have researched on the subject of squeeze flow. These flows are found in fabrication operations such as stamping, injection molding, and sheet forming. Also, material properties of highly viscous fluids are measured with a device called the ‘‘plastometer’’ which incorporates a parallel-disk squeeze flow geometry [2,7,8]. In addition, such flows are encountered in lubrication systems, and there is a considerable interest as to the degree which viscoplastic additives enhance the load-bearing capacity of lubricant. The flows of Newtonian fluids in the clearance of a thrust bearing with impermeable surfaces have been examined theoretically. The bearing walls have been modeled as two disks, two conical or spherical surfaces. The more general case is established by the bearing formed by two surfaces of revolution [9]. Porous bearings have been widely used in industry for a long time [10–14]. Basing on the Darcy model Morgan and Cameron [12] first presented theoretical research on these bearings. Recently the problem of slide bearings with porous walls lubricated by Bingham fluid was taken up by Walicki et al. [15] but the problem of a porous squeeze film bearing lubricated by the same lubricant was presented by Walicka [16]. From various models of fluids with a yield shear stress the Vocˇadlo fluid (sometimes called Robertson and Stiff [17] fluid) flow frequently appears in many industrial branches: food processing, metal processing, petroleum industry. It also appears in tribology modeling semi-fluid lubricants [18,19]. The purpose of this study is to investigate the pressure distribution in a clearance of squeeze film bearing formed by two surfaces of revolution, having common axis of symmetry, as shown in Fig. 1; the lower one of these surfaces is connected
⇑ Corresponding author. E-mail addresses: [email protected] (A. Walicka), [email protected] (P. Jurczak). 0307-904X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2013.04.046
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Nomenclature C ðnÞ ðx; tÞ; C kðnÞ ðx; tÞ auxiliary functions given by formulae (3.10)1 and (3.12)1, respectively H thickness of a porous layer ~ non-dimensional thickness of a porous layer H h bearing clearance thickness K non-dimensional capillary radius of a porous matrix n exponents in a Vocˇadlo fluid p pressure Q flow rate R; RðxÞ local bearing radius rc capillary radius of a porous matrix SV Saint–Venant plasticity number V squeeze velocity tx ; ty components of the flow velocity e non-dimensional axial bearing clearance thickness Kyx component of shear stress s shear stress s0 yield shear stress sw shear stress on the bearing clearance wall UðnÞ auxiliary function in the plane flow of a Vocˇadlo fluid given by Eq. (2.13) WðnÞ auxiliary function in the capillary flow of a Vocˇadlo fluid given by Eq. (3.3)1
with a porous layer. The analysis is based on the assumption that the porous matrix consists of a system of capillaries of very small radii restricting the lubricant flow through the matrix in only one direction. The lubricant is assumed to be the Vocˇadlo type for which the constitutive equation has a following form [20,21]:
1
n
s ¼ s0n þ lc_ ;
ð1:1Þ
where s is the shear stress, s0 is the yield shear stress, l is the coefficient of plastic viscosity, c_ is the rate of deformation, m is the exponent in the Vocˇadlo fluid.
2. Analysis of the Vocˇadlo fluid flow in a bearing clearance The flow configuration is shown in Fig. 1. The upper boundary of a porous layer is described by function R(x), which denotes the radius of this boundary. The fluid film thickness is given by function h(x,t). An intrinsic curvilinear orthogonal coordinate system (x, #, y) is also depicted in Fig. 1.
Fig. 1. Configuration of a thrust bearing with one porous wall.
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By using the assumptions typical for the flows in a thin layer the equations of motion for the Vocˇadlo fluid for axial symmetry one can present in the form [22–24]:
1 @ðRtx Þ @ ty þ ¼ 0; R @x @y
ð2:1Þ
@p @ Kyx ¼ ; @x @y
ð2:2Þ
@p ¼ 0; @y
ð2:3Þ
where the non-zero component of stress tensor is [23,24,18]
8 1 n > < s0n þ l @@ytx Kyx ¼ 1 n > : sn l @ tx 0 @y
y 6 h0 ;
for
ð2:4Þ
y P h h0 :
In the flow of a fluid with yield shear stress there exists a quasi-solid core bounded by surfaces laying at
y ¼ h0 or y ¼ h h0 for which the shear stress is : jKyx j ¼ s0 :
ð2:5Þ
The problem statement is complete after specification of boundary conditions which are
tx ðx; 0; tÞ ¼ 0; tx ðx; h; tÞ ¼ 0; @h ; @t
ty ðx; 0; tÞ ¼ V; ty ðx; h; tÞ ¼ @p ¼ 0; @x x¼0
ð2:6Þ ð2:7Þ
pðxo Þ ¼ po ;
ð2:8Þ
here V is the lubricant velocity on the upper boundary of a porous matrix, po is the outlet pressure. Putting Eq. (2.4) into Eq. (2.2) and integrating the resulted equations with respect to y in the interval 0 6 y 6 h and determining arbitrary constants from the boundary conditions (2.5) and (2.6) we obtain [23–25]: - for shear flow (y 6 h0 or y P h h0 )
txs ¼
1 1 @p n 1þ1n 2ðn þ 1Þ 1þ1n n yðh 2h h ðh 2yÞ Þ 0 nþ1 @x n 2 n lðn þ 1Þ n
ð2:9aÞ
for y 6 h0 and
txs ¼
1 1 @p n 1þ1n 2ðn þ 1Þ 1þ1n n h ð2y hÞ ðh yÞðh 2h Þ 0 nþ1 @x n 2 n lðn þ 1Þ n
ð2:9bÞ
for y P h h0 ; - for core flow (h0 6 y 6 h h0 )
txc ¼
1 1 1 @p n 1þ1n 2ðn þ 1Þ h0 ðh 2h0 Þn : h ðh 2h0 Þ1þn n lðn þ 1Þ @x n
nþ1 n
2
ð2:10Þ
The flow rate across the clearance is given by
^x ¼ 2 Q c ¼ ht
Z
h0
txs dy þ 2
0
Z
h=2
txc dy ¼ 2
Z
h0
txs dy þ txc ðh 2h0 Þ;
ð2:11Þ
0
h0
^x is the mean velocity in the clearance; after calculations there is where t
^x ¼ ht
2nþ1 1 h n @p n UðnÞ ðvÞ: @x 12l
ð2:12Þ
Here
UðnÞ ðvÞ ¼
3 2
1 n
2n 1 1 1 vn þ v2þn 2n þ 1 2n þ 1
ð2:13Þ
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and
s0 sw s0 h h @p ; ; h0 ¼ ¼ ð1 vÞ; sw ¼ 2 2 @x sw ð @p Þ @x h 2h0 @p 2s0 ; finally v ¼ @p ; s0 ¼
v¼
2
@x
ð2:14Þ
h @x
s0 – is the yield shear stress, sw – is the shear stress on the clearance wall. 3. Modified Reynolds equation 3.1. General considerations Integrating the equation of continuity (2.1) with respect to y and determining the arbitrary constants from the boundary conditions (2.7) we obtain [24,19]
^x Þ 1 @ðRht @h ¼ þ V; R @x @t or
" # 1 2nþ1 1 @ @p n @h n UðnÞ ðvÞ ¼ 12l V ; Rh R @x @x @t
ð3:1Þ
which is the modified Reynolds equation. Considering the porous matrix as a system of capillaries (each of radius rc) we may assume that the velocity components for the Vocˇadlo fluid flow in this matrix are given [26] as:
tx ¼ Up
1 n @p WðnÞ ð!Þ; @x
WðmÞ ð!Þ ¼
n 3n þ 1 1n 1 3nþ1 1 ! þ ! n ; ð3n þ 1Þ 3n 3n
ty ¼ Up
@p WðnÞ ð!Þ; @y
ð3:2Þ
where
ð3:3Þ
nþ1
r c n up s 2s0 !¼ 0 ¼ ; Up ¼ 1 ; @p sw rc ð @xÞ 2n l
up – is the coefficient of porosity of the porous matrix. Since the cross velocity component must be continuous at the porous wall-fluid film interface, one obtains from Eqs. (3.1) and (3.2)2 the modified Reynolds equation for the pressure distribution in the bearing with porous wall:
2 " # 1n 1 2nþ1 n 1 @ @p @h @p UðnÞ ðvÞ ¼ 12l4 WðnÞ ð!ÞUp Rh n R @x @x @t @y
3
5:
ð3:4Þ
y¼0
To determine the second term in the brackets on the right hand side of Eq. (3.4) let us consider the lubricant flow in the porous region. The equation of continuity for the flow in this region has the same form as Eq. (2.1). By substituting Eq. (3.2) into Eq. (2.1) one obtains the following equation for pressure distribution in the porous region
" " 1 # 1 # n n 1 @ @p @ @p þ ¼ 0: RWðnÞ ð!ÞUp WðnÞ ð!ÞUp R @x @y @x @y
ð3:5Þ
Integrating this equation with respect to y across the porous layer and using the Morgan–Cameron approximation [12] we have
1 n @p WðnÞ ð!ÞUp @y
y¼0
" 1 # n H @ @p : RWðnÞ ð!ÞUp ¼ R @x @x
ð3:6Þ
Note that on the porous wall-fluid film interface we have also:
@p @p h ¼ and ! ¼ v : rc @x @x When Eqs. (3.6) and (3.7) are substituted into Eq. (3.4) the modified Reynolds equation takes the form:
ð3:7Þ
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( ) h 2nþ1 i @p1n 1 @ @h n R h UðnÞ ðvÞ þ 12lHUp WðnÞ ð!Þ ¼ 12l : R @x @x @t
ð3:8Þ
The final shape of solution to the Reynolds equation (3.8) is formally given as follows:
pðx; tÞ ¼ po þ C ðnÞ ðxo ; tÞ C ðnÞ ðx; tÞ ;
ð3:9Þ
where
n Z 1 @h dx dx; 12l R F ðnÞ ðxÞ @t n h 2nþ1 io n : F ðnÞ ðxÞ ¼ R h n UðnÞ ðvÞ þ 12lHUp WðnÞ ð!Þ
C ðnÞ ðx; tÞ ¼
Z
ð3:10Þ
In reality the Eq. (3.8) is non-linear with respect to the pressure gradient P ¼ @p because it is also included into the @x functions U(n)(v) and W(n)(Y). One may propose the following simple iterative scheme to solve Eq. (3.8): 1. 2. 3. 4.
at first calculate P0 from Eq. (3.8) assuming that v = Y = 0, secondly using P0 calculate v(0) and Y0 from Eq. (2.14)5 and (3.7)2, at third calculate p1(x, t) from Eq. (3.9) using P0, v(0) and Y0 in F(n)(x), using P1 repeat the steps 2 and 3 to obtain p2(x, t). Finally the kth approximation for pressure distribution will be given as follows
pk ðx; tÞ ¼ po þ C ðnÞk ðxo ; tÞ C ðnÞk ðx; tÞ ;
ð3:11Þ
where
C ðnÞk ðx; tÞ ¼
Z
n Z 1 @h dx dx; 12l R F ðnÞk ðxÞ @t
ð3:12Þ
n h 2nþ1 io n F ðnÞk ðxÞ ¼ R h n UðnÞ ðvk1 Þ þ 12lHUp WðnÞ ð!k1 Þ : 3.2. Special forms of the Reynolds equation Let us discuss some special forms of the modified Reynolds equation – frequently met in practice – allowing to apply the general solution presented by formulae (3.11) and (3.12). The substitutions
R ¼ ax;
0 < a 6 1;
h ¼ hðx; tÞ;
ð3:13Þ
made in Eq. (3.8) lead the Reynolds equation for conical bearing (a < 1) or axial bearing (a = 1) with parallel surfaces:
( 1 ) 1 @ @p n @h ¼ 12l : x½GH x @x @x @t
ð3:14Þ
The iterative solution of this equation is generally described by formula (3.11) in which C(n)k and F(n)k are given as follows
C ðnÞk ðx; tÞ ¼
Z
n Z 1 @h dx dx; 12l x F ðnÞk ðxÞ @t
ð3:15Þ
n F ðnÞk ðxÞ ¼ x GHðk1Þ and:
h 2nþ1 i GH ¼ h n UðnÞ ðvÞ þ 12lHUp WðnÞ ð!Þ ;
ð3:16Þ
h 2nþ1 i GHðk1Þ ¼ h n UðnÞ ðvk1 Þ þ 12lHUp WðnÞ ð!k1 Þ :
ð3:17Þ
On the other hand the substitutions
R ¼ Rs sin u;
u¼
x ; Rs
h ¼ hðu; tÞ;
lead to the Reynolds equation for spherical bearing:
ð3:18Þ
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Fig. 2. Axial squeeze film bearing.
Fig. 3. Dimensionless pressure distribution in the axial bearing for n = 0.5, K = 1.0 and e = 1.0.
Fig. 4. Dimensionless pressure distribution in the axial bearing for n = 0.5, K = 0.5 and e = 1.0.
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( 1 ) nþ1 @h 1 @ @p n sin u½GH ; ¼ 12lRs n sin u @ u @u @t
9301
ð3:19Þ
here Rs is the radius of a spherical bush. In this case there are:
C ðnÞk ðu; tÞ ¼
Z
n Z nþ1 1 @h sin udu du; 12lRs n F ðnÞk ðuÞ @t
n F ðnÞk ðuÞ ¼ sin u GHðk1Þ :
ð3:20Þ ð3:21Þ
The substitutions
R ¼ Rb ¼ constant; x ¼ Rb u;
h ¼ hðu; tÞ;
lead to the Reynolds equation for journal bearing:
Fig. 5. Dimensionless pressure distribution in the axial bearing for n = 0.5, K = 1.0 and e = 0.5.
Fig. 6. Dimensionless pressure distribution in the axial bearing for n = 0.5, K = 0.5 and e = 0.5.
ð3:22Þ
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( 1 ) nþ1 @h @ @p n ½GH ; ¼ 12lRbn @u @u @t
ð3:23Þ
where Rb is the radius of a cylindrical bush and
C ðnÞk ðu; tÞ ¼
Z
n Z nþ1 1 @h du du; 12lRbn F ðnÞk ðuÞ @t
n F ðnÞk ðuÞ ¼ GHðk1Þ :
ð3:25Þ
In the last two cases it may introduce the clearance thickness in a following form:
hðu; tÞ ¼ Cð1 e cos uÞ;
ð3:24Þ
e ¼ eðtÞ;
Fig. 7. Dimensionless pressure distribution in the axial bearing for n = 1.0, K = 1.0 and e = 1.0.
Fig. 8. Dimensionless pressure distribution in the axial bearing for n = 1.0, K = 0.5 and e = 1.0.
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where
e C
e ¼ ; C ¼ Rs Re or C ¼ Rb Rc ; Rc – is a shaft radius, e – is an eccentricity. 4. Example of flow Let us consider the pressure distribution in a squeeze film bearing presented in Fig. 2 lubricated by the Vocˇadlo fluid. Assume also that the lubricant flow coincides with a small core, then v 1; ! 1 and the functions U(n)(v) and W(n)(Y) become quasi-linear:
UðnÞ ðvÞ
6n 1 n
2 ð2n þ 1Þ
1
2n þ 1 1n v ; 2n
Fig. 9. Dimensionless pressure distribution in the axial bearing for n = 1.0, K = 1.0 and e = 0.5.
Fig. 10. Dimensionless pressure distribution in the axial bearing for n = 1.0, K = 0.5 and e = 0.5.
ð4:1Þ
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WðnÞ ð!Þ ¼
n 3n þ 1 1n ! : 1 3n þ 1 3n
Using the iterative procedure to Eq. (3.14), applying Eqs. (3.15), (3.16), (3.17) and introducing the following dimensionless variables and parameters
~x ¼
x ; Ro
~¼ R; R Ro
~¼ h; h ho
e_ ¼
de ; dt
V o ¼ ho ðe_ Þ;
K¼
~ ¼ H up ; H ho
rc ; ho
2nþ1
~¼ p
ðp po Þho
ln V no Rnþ1 o
;
SV ¼
h ¼ ho eðtÞ;
s0 h2n o ; ln V no Rno
ð4:2Þ
one may present formula (3.11) for the pressure distribution in an axial squeeze film bearing in the simple non-dimensional form
~ð~x; ~tÞ ¼ DðnÞ ð1 ~xnþ1 Þ þ EðnÞ ð1 ~xn Þ; p
Fig. 11. Dimensionless pressure distribution in the axial bearing for n = 1.5, K = 1.0 and e = 1.0.
Fig. 12. Dimensionless pressure distribution in the axial bearing for n = 1.5, K = 0.5 and e = 1.0.
ð4:3Þ
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where
DðnÞ ¼ ðn þ 1Þnn
1
2 e
2nþ1 n
2nþ1
~
; nþ1 n
HK n þ 23nþ1
EðnÞ ¼
~ e2 þ 43 HK ðSVÞn ; 2nþ1 nþ1 n ~ nn e n þ 2HK n 2nþ1 3nþ1
ð4:4Þ
SV – is the Saint–Venant plasticity number (plasticity index). The graphical presentation of the pressure distribution given by formula (4.3) is shown in Figs. 3–14; the graphs presented in Figs. 3, 4, 7, 8, 11 and 12 are made for e = 1 (the squeezing start) but the graphs presented in Figs. 5, 6, 9, 10, 13 and 14 are made for e = 0.5. The boundary values of power-law exponent (n = 0.5, 1.0, 1.5) and plasticity index (SV = 0, 0.5, 1.0) used to graphical presentation are near to the real values of these parameters for Vocˇadlo lubricants applied in thrust bearings [18]. The values SV = 0.5, 1.0 described the pressure distribution, in a squeeze film of a Vocˇadlo fluid which flows in the bearing clearance with a consistent core.
Fig. 13. Dimensionless pressure distribution in the axial bearing for n = 1.5, K = 1.0 and e = 0.5.
Fig. 14. Dimensionless pressure distribution in the axial bearing for n = 1.5, K = 0.5 and e = 0.5.
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5. Discussion and conclusions The above analysis has shown that the Reynolds approximation may be used to model the flow of the Vocˇadlo fluid as lubricant in the clearance of a curvilinear squeeze film bearing. The obtained results for the lubricant flow in an axial bearing with parallel surfaces are relatively simple and may be used in practical applications. The formula for pressure distribution consists of two terms: the first of them is connected with the power-law character of lubricant flow but the second one is connected with the plastic character of this flow and it is described – among other factors – by the plasticity index SV. Note that for SV = 0, and in effect E(n) = 0, formula (4.3) represents the pressure distribution only for the flow of a powerlaw lubricant; it is identical with that one obtained by Walicki et al. [27]. In the case when n ¼ 1:0; SV ¼ 0 there is a Newtonian lubricant flow and the results obtained here are the same as in [28]. If n = 1.0 and SV > 0, then one has a flow of a Bingham fluid lubricant and the results are identical with those presented in [23,24] for non-porous clearance walls ~ ¼ K ¼ 0Þ and identical with those presented in [16] for porous clearance walls. ðH For the axial bearing the results have been presented for three values of the Vocˇadlo exponent, namely: for n = 0.5, n = 1.0 (Bingham fluid) and for n = 1.5, for three values of the plasticity index SV and also for different parameters of the porous ~ and K). matrix (different values of H Analyzing the graphs presented in Figs. 3–14 it may approach to some conclusions. ~ For n = 1.0 all values of For n = 0.5 all values of the pressure are not great and generally decrease with increase of K and H. ~ the pressure are greater than those in previous case and also they decrease with the increase of K and H. ~ The decrease of the values of For n = 1.5 all values of the pressure are great and they decrease with the increase of K and H. e and the increase of the values of SV evoke the increase of all values of the pressure. ~ decrease with the Analysis of the formula (4.3) and its graphical presentations may conclude that the pressure values p ~ increase of the porous matrix parameters K and H. These values increase with the increase of the values of power exponent n and plasticity index SV. For SV > 0 the flows in a squeeze film exhibit the existence of a consistent core which considerably influences the pressure distribution. This influence is very substantial in the bearings with non-porous walls. In conclusion it may suggest that the case n = 1.5 is the best case for the Vocˇadlo fluid to be applicated as a lubricant in the squeeze film bearing.
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