Pressure distribution in a squeeze film of a Shulman fluid between porous surfaces of revolution

International Journal of Engineering Science 69 (2013) 33–48

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Pressure distribution in a squeeze film of a Shulman fluid between porous surfaces of revolution A. Walicka ⇑ 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 27 June 2012 Received in revised form 21 March 2013 Accepted 24 March 2013 Available online 22 April 2013 Keywords: Shulman fluid Porous wall Squeeze film Curvilinear clearance Surfaces of revolution

a b s t r a c t The influence of a porosity of one porous wall limiting the narrow clearance between two surfaces of revolution on the pressure distribution in a squeeze flow of a Shulman fluid is considered. After general considerations on the flow of a Shulman fluid in a clearance and in a porous layer a modified Reynolds equation for the curvilinear squeeze film with a Shulman fluid 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 a squeeze film between two parallel disks is discussed in detail. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Recent trends in polymer, metal and food processing, thermal reactors, and biomechanics focus immensely on the application of fluids with variable viscosity. Further, there has been an increasing interest in the usage of non-Newtonian fluids with yield stress (called also viscoplastic fluids) as working media in different industrial mechanisms, like Bingham plastic, Casson, Vocˇadlo and Herschel-Bulkley fluids; the model of Shulman fluid is a generalized model for these fluids and from this model one can obtain simpler models by parameters reduction. Squeeze flow, 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 Adams and Edmondson (1987), Covey and Stanmore (1981), Dai and Bird (1981), Lipscomb and Denn (1984), Rodin (1996) and Smyrnaios and Tsamopoulos (2001) 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 (Covey & Stanmore, 1981; Engmann, Servais, & Burbidge, 2005; Xu, Yuan, Xu, & Hang, 2010). 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 between impermeable surfaces have been examined theoretically. The clearance walls have been modelled as two disks, two conical or spherical surfaces. The more general case is established by the clearance formed by two surfaces of revolution (Walicka, 1994). The flows in the clearances bounded by porous walls were generally applicated in porous bearings. These bearings have been widely used in industry for a long time (Bujurke, Jagadee, & Hiremath, 1987, 2007; Etsion & Michael, 1994; Morgan & ⇑ Tel.: +48 683282545; fax: +48 683247446. E-mail address: [email protected] 0020-7225/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2013.03.013

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A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

Nomenclature C im ; C in

Newton binomial coefficients

ðmÞ ðmÞ C ðnÞ ðx; tÞ; C ðnÞk ðx; tÞ

H e H h

auxiliary functions given by formulae Eq. (3.16)1 and (3.18)1, respectively

thickness of a porous layer non-dimensional thickness of a porous layer bearing clearance thickness

ðmÞ ðmÞ

F ðnÞ F ðnÞk auxiliary function given by Eq. (3.16)2 or (3.18)2, respectively K m, n p p Q R,R(x) rc SV V tx,ty Kyx

non-dimensional capillary radius of a porous matrix exponents in a Shulman fluid pressure in the bearing clearance pressure in the porous layer flow rate local radius of the upper boundary of a porous layer capillary radius of a porous matrix Saint–Venant plasticity number squeeze velocity components of the flow velocity component of shear stress

UðmÞ ðnÞ

auxiliary function in the plane flow of a Shulman fluid given by Eqs. (2.14)–(2.20)

WðmÞ ðnÞ

auxiliary function in the capillary flow of a Shulman fluid given by Eqs. (3.3)–(3.9) shear stress yield shear stress shear stress on the bearing clearance wall non-dimensional step bearing clearance thickness ratio s0 to sw for the clearance flow ratio s0 to sw for the capillary flow

s s0 sw e v !

Cameron, 1957; Prakash & Vij, 1973; Shukla & Isa, 1978). Basing on the Darcy model Morgan and Cameron (1957) first presented theoretical research on that bearing. Recently the problem of slide bearings with porous walls lubricated by Bingham fluid was taken up by Walicki, Walicka, and Makhaniok (2000) but the problem of a porous squeeze film bearing lubricated by the same lubricant was presented by Walicka (2011). From various models of fluids with a yield shear stress the Shulman fluid flow began appear in many industrial branches: food processing, metal processing, petroleum industry (Falicki, 2007; Walicka, 2002a, 2002b; Walicki, 2005). The purpose of this study is to investigate the pressure distribution in the clearance with a squeeze film, formed by two surfaces of revolution, having common axis of symmetry, as shown in Fig. 1; the lower one of these surfaces is connected 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 fluid flow through the matrix in only one direction. To consider the porosity influence of a porous matrix on the fluid flow in a clearance the Morgan-Cameron approximation will be used. The constitutive equation for a Shulman fluid is given as follows (Shulman, 1975; Walicka, 2002a, 2002b):

T ¼ p1 þ K;

K ¼ MA1

ð1:1Þ

where

M¼

h

1

1

in

sn0 þ ðlAÞm A1 ; A ¼

 1 1  2 2 tr A1 2

ð1:2Þ

T is the stress tensor, p is the pressure, 1 is the unit tensor, K is the extra-stress tensor, M is the viscosity function, s0 is the yield stress, l is the plastic viscosity, A1 is the first Rivlin–Ericksen kinematic tensor, m and n are the power-law exponents. The values of m and n are included in the interval 1 6 m,n 6 3 for paints and lacquers (Shulman, 1975) and also for semiliquid lubricant (Falicki, 2007).

2. Analysis of the Shulman fluid in clearance between surfaces of revolution 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.

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A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

Fig. 1. Configuration of a clearance between surfaces of revolution with one porous wall.

By using the assumptions typical for the flows in a thin layer the equations of motion for a Shulman fluid for axial symmetry one can present in the form (Walicki & Walicka, 1997; Walicka, 2002a, 2002b; Walicka, Jurczak, & Walicki, 2011) (see also Appendix A):

1 @ðRtx Þ @ ty þ ¼ 0; R @x @y @p @ Kyx ¼ ; @x @y @p ¼ 0; @y

ð2:1Þ ð2:2Þ ð2:3Þ

where the non-zero component of stress tensor is

"

  1 #n  @ tx m   Kyx ¼ S s0 þ l @y  and S ¼ sgn



@ tx @y

1 n

ð2:4Þ



; the signum function (sgn) takes the value +1 for a positive argument   tx ative argument @@y < 0 for y P h  h0 .



@ tx @y

 > 0 for y 6 h0 and 1 for a neg-

In the flow of fluid with yield shear stress there exists a quasi-solid core bounded (Walicka et al., 2011) by surfaces laying at

y ¼ h0 or y ¼ h  h0

for which

@ tx ¼ 0 and the shear stress is : jKyx j ¼ s0 : @y

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 ty ðx; h; tÞ ¼ ; @t

ty ðx; 0; tÞ ¼ V;  @p @x 

¼ 0 pðxo Þ ¼ po :

here V is the fluid velocity on the upper boundary of a porous matrix, po is the outlet pressure. Putting Eq. (2.4) into Eq. (2.2) one will obtain two equations: – for the lower part of shear flow (y 6 ho):

"

1



s0n þ l

@ tx @y

ð2:7Þ ð2:8Þ

x¼0

@p @ ¼ @x @y

ð2:6Þ

m1 #n ;

– for the upper part of shear flow (y P h  ho):

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A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

@p @ ¼ @x @y

"

  1 #n @ tx m s0 þ  l : @y 1 n

Integrating these 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), the same manner as it was done in the paper (Walicka, 2002a), we obtain: – for shear flow (y 6 h0 or y P h  h0)



1

txs ¼

2

mþn n

l

" # m mþni  mþni m n n X mþn @p n n h h  2y i i n  ðh  2h0 Þ  ð1Þ  C @x m þ n  i m h  2h0 h  2h0 i¼0

ð2:9aÞ

" # m mþni  mþni m n n X mþn @p n n h 2y  h C im ðh  2h0 Þ n  ð1Þi  @x mþni h  2h0 h  2h0 i¼0

ð2:9bÞ

for y 6 h0 and



1

txs ¼

2

mþn n



l

for y P h  h0; – for core flow (h0 6 y 6 h  h0)



1

txc ¼

2

mþn n

l

@p  @x

mn

ðh  2h0 Þ

mþn n

m X

n Ci  ð1Þ mþni m i¼0 i

"

h h  2h0

mþni n

# 1 ;

ð2:9cÞ

Here the Newtonian binomial coefficient is as follows:

m! : i!ðm  iÞ!

C im ¼

ð2:10Þ

The flow rate across the clearance is given by

^x ¼ 2 Q ¼ ht

Z

h0

txs dy þ 2

Z

h=2

txc dy ¼ 2

h0

0

Z

h0

txs dy þ txc ðh  2h0 Þ;

ð2:11Þ

0

^x is the mean velocity in the clearance; after calculations one has where t

^x ¼ ht

mþ2n  m h n @p n ðmÞ  UðnÞ ðvÞ: 12l @x

ð2:12Þ

Here

h0 ¼

sw  s0 ð @p Þ @x

¼

h ð1  vÞ; 2

sw ¼

  h @p ;  2 @x

s0 ¼



h  h0 2

  @p  ; @x

v¼

s0 2s0

; ¼ sw h  @p @x

ð2:13Þ

sw – is the shear stress on the clearance wall. ðmÞ

The forms of UðnÞ ðvÞ for different viscoplastic fluids are given in the Table 1 drawn from the paper (Walicka, 2011).

Table 1 ðmÞ The forms of UðnÞ ðvÞ for different models of viscoplastic fluids derivative from the Shulman model. Model of fluid (2.14) Shulman

ðmÞ

Form of UðnÞ ðvÞ 3

v

mn n

2

(2.15) Casson m = n (2.16) Casson ‘‘simple’’ m = n = 2 (2.17) Vocˇadlo m = 1 (2.18) Herschel-Bulkley n = 1

3v3

(2.20) Newton m = n = 1,s0 = 0

Pm

i i n i¼0 ð1Þ mþ2ni C m

Pn

i n i i¼0 ð1Þ 3ni C n 1 2

h

n v3ni

h

v

i 1

1  12 v þ 32 v  101 v3  5  2nþ1 1 3 2n 1  vn þ 2nþ1 v n 1 2n 2nþ1   3 1 ð1  vÞmþ1 1 þ mþ1 v m1 2

(2.19) Bingham m = n = 1

mþ2n n

ðmþ2Þ

1  32 v þ 12 v3 1

mþ2ni n

i 1

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A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

3. Modified Reynolds equation Integrating the equation of continuity (2.1) with respect to y and determining the arbitrary constants from the boundary conditions (2.7) we obtain (Walicki, 2005)

^x Þ 1 @ðRht @h ¼ þV R @x @t or

" #  m   mþ2n 1 @ @p n ðmÞ @h UðnÞ ðvÞ ¼ 12l V ; Rh n  R @x @x @t

ð3:1Þ

this is the modified Reynolds equation. To find the velocity V let us consider the Shulman fluid flow in a porous matrix. Considering the porous matrix as a system of capillaries (each of radius rc) we may find that the velocity components for the Shulman fluid in this matrix are given (Walicka, 2011; Walicka & Walicki, 2011) as



ðmÞ tx ¼ WðnÞ ð!ÞUp 

m @p n ; @x



ðmÞ ty ¼ WðnÞ ð!ÞUp 

m @p n ; @y

2s0

!¼ 

r c  @p @y

;

ð3:2Þ

ðmÞ

where the functions WðnÞ ðvÞ and coefficients Up are given in the Table 2 also drawn from the paper (Walicka, 2011; see Appendix B). 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 clearance

2 " #  m  mn  mþ2n 1 @ @p n ðmÞ @h @p  ðmÞ n Rh  UðnÞ ðvÞ ¼ 12l4  WðnÞ ð!ÞUp   R @x @x @t @y 

3

5:

ð3:10Þ

y¼0

The equation of continuity for the flow in the porous 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

" "  m #  m # 1 @ @p n @ @p n ðmÞ ðmÞ þ ¼ 0: RWðnÞ ð!ÞUp  WðnÞ ð!ÞUp  R @x @x @y @y

ð3:11Þ

Integrating this equation with respect to y over the porous layer and using the Morgan–Cameron approximation we have

 m  @p n  !ÞUp   @y 

ðmÞ ðnÞ ð

W

y¼0

"  m # H @ @p n ðmÞ : ¼ RWðnÞ ð!ÞUp  R @x @x

ð3:12Þ

Note that on the porous wall-fluid film interface we have

@p @p h ¼ and ! ¼ v @x @x rc

ð3:13Þ

when Eqs. (3.12) and (3.13)1 are substituted into Eq. (3.10) the modified Reynolds equation takes the form

( ) h mþ2n i @pmn 1 @ @h ðmÞ ðmÞ n ¼ 12l : R h UðnÞ ðvÞ þ 12lHUp WðnÞ ð!Þ  R @x @x @t

ð3:14Þ

The final form of solution to the Reynolds Eq. (3.14) has the form

h i ðmÞ ðmÞ pðx; tÞ ¼ po þ C ðnÞ ðxo ; tÞ  C ðnÞ ðx; tÞ ;

ð3:15Þ

Table 2 Auxiliary functions and coefficients for the flow of the Shulman fluid and derivative fluids through a porous layer. Model of fluid (3.3) Shulman (3.4) Casson m = n (3.5) Casson ‘‘simple’’ m = n = 2 (3.6) Vocˇadlo m = 1 (3.7) Herschel–Bulkley n = 1 (3.8) Bingham m = n = 1 (3.9) Newton m = n = 1,s0 = 0

Up

ðmÞ

Form of WðnÞ ð!Þ

!

mþ3n n

Pm

4 Pn

i i n i¼0 ð1Þ mþ3ni C m





mþ3ni n

!

4ni n

 1

i n i i¼0 ð1Þ 4ni C n !   1 16 2 4 1 1  7 ! þ 3 !  21 !4   3nþ1 1 n 3nþ1 n 1 n 3nþ1 1  3n ! þ 3n !

! 1 4

h i ð1!Þ 2! 2!2 1 þ mþ2 þ ðmþ1Þðmþ2Þ ðmþ3Þ   1 4 1 4 4 1  3! þ 3! mþ1

1 4

 1

mþn

rc n

m

up

2n l r2c

up

2l r2c up 2l 1þ1

rcn

up

1

2n l rmþ1 c m 2

up l

r2c up 2l r2c up 2l

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A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

where

  mn @h dx dx; R  ðmÞ @t F ðnÞ ðxÞ n h mþ2n iomn ðmÞ ðmÞ ðmÞ : F ðnÞ ðxÞ ¼ R h n UðnÞ ðvÞ þ 12lHUp WðnÞ ð!Þ ðmÞ

C ðnÞ ðx; tÞ ¼

Z

1



12l

Z

ð3:16Þ



In reality the Eq. (3.14) is non-linear with respect to the pressure gradient P ¼  @p because it is also included into the @x ðmÞ ðmÞ functions UðnÞ ðvÞ and WðnÞ ð!Þ. One may propose the following simple iterative scheme to solve Eq. (3.14): 1. 2. 3. 4.

first calculate P0 from Eq. (3.14) assuming that v = ! = 0, secondly using P0 calculate v0 and !0 from Eqs. (2.13) and (3.2)3 or (3.13)2, ðmÞ thirdly calculate p1(x,t) from Eq. (3.15) using P0, v0 and !0 in F ðnÞ ðxÞ, 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

h i ðmÞ ðmÞ pk ðx; tÞ ¼ po þ C ðnÞk ðxo ; tÞ  C ðnÞk ðx; tÞ

ð3:17Þ

where ðmÞ

C ðnÞk ðx; tÞ ¼

Z

  mn Z  @h dx dx; R  12 l ðmÞ @t F ðxÞ 1

ð3:18Þ

ðnÞk

ðmÞ F ðnÞk ðxÞ

n h mþ2n iomn ðmÞ ðmÞ ¼ R h n UðnÞ ðvk1 Þ þ 12lHUp WðnÞ ð!k1 Þ :

4. Example of flow Let us consider the pressure distribution in a squeeze film of the Shulman fluid between two parallel disks presented in Fig. 2. ðmÞ Assume also that the Shulman fluid flow coincides with a small core, then v  1,!  1 and the functions UðnÞ ðvÞ and WðmÞ ð ! Þ given by formulae (2.14) and (3.3), become simpler (and quasi-integrable): ðnÞ

  n mðm þ 2nÞ 1n 1 v ; m þ 2n  1 2 m þ2n  n mðm þ 3nÞ 1n ðmÞ WðnÞ ð!Þ  1 ! : m þ 3n m þ 3n  1

UðmÞ ðnÞ ðvÞ 

3

mn n

ð4:1Þ

Using the iterative procedure and introducing the following dimensionless variables and parameters

~x ¼

x ; Ro

e¼ R; R Ro

~¼ h; h ho

e_ ¼

de ; dt

V o ¼ ho ðe_ Þ;

K¼

e ¼ Hup ; H ho

rc ; ho

2nþ1

~¼ p

ðp  po Þhom n m

n m

n mþ1

l V o Ro

h ¼ ho eðtÞ; 2n

;

SV ¼

s0 hom n n ; lmn V mo Rmo

ð4:2Þ

one may present formula (3.17) for the pressure distribution in a step squeeze film bearing in the simple non-dimensional form

    mþn mþn1 ðmÞ ~ð~x; ~tÞ ¼ DðmÞ ~ n þ EðnÞ 1  ~x n : p ðnÞ 1  x

Fig. 2. Squeeze film between two parallel disks.

ð4:3Þ

A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

39

where ðmÞ DðnÞ

ðmÞ EðnÞ

" !#mn mþ2n e mþn n 2m e n 2 HK n ¼ þ ; mþn m þ 2n m þ 3n " !#1mn ! 1 mþ2n mþ2n1 m  m mþn1 e mþn e mþn1 n n ðSVÞn mn2 e n 2 HK e n 2 HK m n ¼ m1 2  n þ þ m þ 2n m þ 3n m þ 2n  1 m þ 3n  1 2 n ðm þ n  1Þ

ð4:4Þ

where 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 showing the pressure distribution are made for e = 1 (the squeezing start) and for e = 0.5 (full squeezing flow).

Fig. 3. Dimensionless pressure distribution in the clearance between two parallel disks for m = 3.0, n = 2.0, K = 0.5 and e = 0.5.

Fig. 4. Dimensionless pressure distribution in the clearance between two parallel disks for m = 3.0,n = 2.0, K = 1.0 and e = 0.5.

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A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

Fig. 5. Dimensionless pressure distribution in the clearance between two parallel disks for m = 3.0,n = 2.0, K = 0.5 and e = 1.0.

Fig. 6. Dimensionless pressure distribution in the clearance between two parallel disks for m = 3.0,n = 2.0, K = 1.0 and e = 1.0.

5. Discussion and conclusions The above analysis has shown that the Reynolds approximation may be used to model the flow of the Shulman fluid in the clearance of a curvilinear squeeze film between two surfaces of revolution. The obtained results are relatively simple and may be used in practical applications of different industrial processes. The formula for pressure distribution consist of two terms: the first of them is connected with the power law character of fluid flow but the second one is connected with the plastic character of this flow and it is described mainly by the Saint–VeðmÞ nant plasticity number SV. Note that for SV = 0, what involves the relation EðnÞ ¼ 0, formula (4.3) represents the pressure distribution for the flow of a power-law fluid and it is similar to that one obtained by Walicki, Walicka, and Michalski (1997). In the case when SV = 0 and m = n = 1, there is a Newtonian fluid flow and the results obtained here reduce to that ones presented in Walicki, Walicka, and Karpin´ski, 1996.

A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

41

Fig. 7. Dimensionless pressure distribution in the clearance between two parallel disks for m = 2.0,n = 3.0, K = 0.5 and e = 0.5.

Fig. 8. Dimensionless pressure distribution in the clearance between two parallel disks for m = 2.0,n = 3.0, K = 1.0 and e = 0.5.

The particular results for the flow between two disks have been presented for three pairs of the Schulman exponents, namely: for m = 3, n = 2; m = 2,;n = 3; m = n = 2 and also for different parameters of the porous matrix (different values of e and K) and different values of the Saint–Venant plasticity number SV. H For the first pair of m and n the values of pressure are moderate; for the second pair these values are very great; for the third pair of m and n these values are intermediate. Generally the pressure values increase with the increase of exponent n and decrease with the increase of exponent m. These values also increase with the decrease of e, i.e., with the increase of squeezing flow phenomenon. e and K connected with permeability of the porous layer. The pressure also increases with the decrease of the parameters H This pressure increases with the increase of the Saint–Venant plasticity number SV.

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A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

Fig. 9. Dimensionless pressure distribution in the clearance between two parallel disks for m = 2.0,n = 3.0, K = 0.5 and e = 1.0.

Fig. 10. Dimensionless pressure distribution in the clearance between two parallel disks for m = 2.0,n = 3.0, K = 1.0 and e = 1.0.

Appendix A A.1. Equations of motion of the Shulman fluid Let us consider a flow of the Shulman incompressible fluid in a thin layer between two surfaces of revolution (Fig. 1) for which there is h  R (Walicka, 1994, 2002a, 2002b). The differential form of momentum equations in the curvilinear coordinates system (x,#,y) are written as:

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A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

Fig. 11. Dimensionless pressure distribution in the clearance between two parallel disks for m = n = 2.0, K = 0.5 and e = 0.5.

Fig. 12. Dimensionless pressure distribution in the clearance between two parallel disks for m = n = 2.0, K = 1.0 and e = 0.5.

@T xx 1 @T x# @T xy T xx  T ## 0 þ þ þ R; R @# @x @y R @T 1 @T ## @T #y 2T #x 0 qa# ¼ #x þ þ þ R; R @# @x @y R @T 1 @T y# @T yy T yx 0 qay ¼ yx þ þ þ R: R @# @x @y R

qax ¼

ðA:1Þ ðA:2Þ ðA:3Þ

The acceleration components are as follows:

ax ¼

dtx t2# 0  R; dt R

a# ¼

dt# tx t# 0 þ R; dt R

ay ¼

dty ; dt

The individual stress tensor components are written as:

d @ @ t# @ @ ¼ þ tx þ þ ty ; dt @t @x R @# @y

R0 ¼

dR : dx

ðA:4Þ

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A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

Fig. 13. Dimensionless pressure distribution in the clearance between two parallel disks for m = n = 2.0, K = 0.5 and e = 1.0.

Fig. 14. Dimensionless pressure distribution in the clearance between two parallel disks for m = n = 2.0, K = 1.0 and e = 1.0.

@ tx ; @x   1 @ t# tx 0 T ## ¼ p þ K## ¼ p þ 2M þ R ; R @# R @ ty T yy ¼ p þ Kyy ¼ p þ 2M ; @y   1 @ tx @ t# t# 0 þ  R ; T x# ¼ T #x ¼ Kx# ¼ K#x ¼ M R @# @x R   @ t# 1 @ ty ; þ T #y ¼ T y# ¼ K#y ¼ Ky# ¼ M @y R @#   @ ty @ tx þ ; T yx ¼ T xy ¼ Kyx ¼ Kxy ¼ M @x @y T xx ¼ p þ Kxx ¼ p þ 2M

ðA:5Þ

A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

here M is given by formula (1.2)

1

45

and

 1         12 1  2 2 1 A¼ tr A1 ¼ A21 þ A21 þ A21 xx ## yy 2 2 "        2  2  2 #12 2 2 2 @ tx 1 @ t# tx 0 @ ty 1 @ tx @ t# t# 0 @ t# 1 @ ty @ ty @ tx ¼ 2 þ2 þ R þ2 þ þ  R þ þ þ þ : R @# R @# @x R @y @x R @y R @# @x @y

ðA:6Þ

The differential form of the continuity equation is written as

1 @ðRtx Þ 1 @ t# @ ty þ þ ¼ 0: R @x R @# @y

ðA:7Þ

Taking into account Eq. (1.1)2 and putting Eq. (A.5) into Eqs. A.1, A.2, A.3 we will obtain the equations of motion expressed in velocity components. The flow in the squeeze film under considerations is axially symmetric that:

@ ¼ 0; @#

t# ¼ 0; h ¼ hðx; tÞ:

ðA:8Þ

The assumption that h(x,t)  R(x), which was made earlier and assumptions (A.8) can be used for some considerations concerning the order-of-magnitude arguments for Eqs. A.1, (A.2)–(A.6) or equations of motion expressed in velocity components. Let us make the following assumptions on the velocity orders:



tx ¼ OðV m Þ; ty ¼ O V m

 hm : Rm

ðA:9Þ

Here Vm is the mean velocity of longitudinal fluid flow and hm, Rm are, respectively, mean values of h(x,t) and R(x) in the clearance. If some asymptotic transformations are made, the same as in Walicka (1994), we will obtain:

A¼

"

@ tx @y

2 #12

  @ tx  ¼  ; @y

@ tx Kyx ¼ M ¼S @y

"

"

 1 #n     @ tx m @ tx 1    ;  M ¼ s0 þ l @y   @y  1 n

  1 #n  @ t m s0 þ l x  @y

ðA:10Þ

1 n

and

  @ tx @ tx @ tx @p @ Kyx ¼ þ þ tx þ ty ; @x @t @x @y @y   @ ty @ ty @ ty @p ¼ : q þ tx þ ty @y @t @x @y

q

ðA:11Þ ðA:12Þ

Assuming that the inertia effects, given by left hand sides, are very small one may omit them and these equations will take the form presented by Eqs. (2.2) and (2.3). Appendix B B.1. Flows of the Shulman fluid in a porous bed The flow of the Newtonian fluids in a porous bed describes the experimental Darcy law usually noted as follows (Walicka, 2002a, 2002b):

t¼

Un

l

rp

ðB:1Þ

where the average velocity vector

t ¼ tp up ; up ¼

Vf : Vp

t is connected with the pore velocity vector tp by the relation: ðB:2Þ

Here Un is the permeability of a bed, up is the porosity of a bed, Vp is the total volume of a porous bed, Vf is the volume filled by a fluid (total volume of open pores), p is the pressure in a porous bed (the bar is usually used to note the flow in a porous bed).

46

A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

To find a generalization of the Darcy law, consider a porous layer of thickness H, shown in Fig. B.1, being a matrix of rectilinear parallel capillary tubes. The velocity of the Newtonian flow in a layer – according to the Darcy law – is equal to:

ty ¼ 

Un dp

l dy

ðB:3Þ

:

The flow velocity of the Newtonian fluid in a capillary tube of radius rc (Fig. B.1) is given by formula (Walicka, 2002a):



ty ¼ 1 

r2 r 2c

  dp  dy

ðB:4Þ

but the flow rate Q is defined as

Q ¼ 2p

Z

rc

ty rdr:

ðB:5Þ

0

Finally, the flow rate of the Newtonian fluid flowing in the capillary tube – by virtue of formulae (B.4) and (B.5) – is equal to:

Q ¼ 2p

Z

rc

ty rdy ¼ 2p

Z

0

0

rc

   r2 dp pr4 dp  rdy ¼  c 1 2 dy rc 8l dy

ðB:6Þ

and the average velocity in the capillary tube is:

ta ¼

Q r 2c dp ¼  : pr2c 8l dy

ðB:7Þ

Taking into account the porosity coefficient for the matrix of capillary tubes given by Eq. (B.2)2:

up ¼

Nc pr 2c Hp N c pr2c ¼ ; AHp A

ðB:8Þ

where Nc is the number of capillary tubes on the surface A of the matrix, we will obtain:

t ¼ t a up ¼ 

r 2c up dp : 8l dy

ðB:9Þ

Comparing Eqs. (B.3) and (B.9) one obtains:

Un ¼

r 2c up ; 8

ðB:10Þ

the formula defining the permeability of a porous matrix for the flow of the Newtonian fluid. Let us consider now the steady laminar flow of the Shulman fluid in a capillary tube of radius rc (Fig. B.2). The velocity field is given as follows:

Fig. B.1. Porous matrix composed from rectilinear parallel capillary tubes.

Fig. B.2. Coordinates system and velocity field in a capillary tube.

A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

tr ¼ 0; t# ¼ 0; ty ¼ ty ðrÞ;

47

ðB:11Þ

Changing the curvilinear coordinates system (x,#,y) in Eqs. (A.1)–(A.3) on the cylindrical coordinates system (r,#,y) we arrive to the replacement x and R by r. Thus the equations of motion (A.1)–(A.3) – after taking into account Eq. (B.11) – reduce to the one equation:

"   1 #n 1 dp 1 @ðrKry Þ dty m ¼ ; where Kry ¼  sn0 þ l : dy r @r dr

ðB:12Þ

The solution to Eq. (B.12) with boundary condition:

ty ¼ 0 for r ¼ rc ;

dty ¼ 0; dr

Kyx ¼ s0

for r ¼ r 0

takes the form: – for shear flow:

" #  m  mþni m n r0 r0 s0 n mn X r tys ðrÞ ¼ ! Ui 1  rc l rc i¼0

ðB:13Þ

– for core flow:

tyc ¼ tys ðr0 Þ;

ðB:14Þ

here

Ui ¼ ð1Þi

mþni n 2s0 : C i ! n and ! ¼  mþni m r c  dp

ðB:15Þ

dy

The flow rate Q is now defined as:

Q ¼ 2p

Z

rc

ty rdr ¼ 2p

Z

0

r0

tyc rdr þ

0

Z

rc



tys rdr :

0

Using expressions (B.13) and (B.14) one obtains: m

Q¼

m h mþ3ni i pr3c s0n 3 X n C im ! n  1 ! ð1Þi m þ 3n  i l i¼0

ðB:16Þ

and the average velocity in the capillary tube is

ta ¼

m h mþ3ni i Q rc mn 3 X n i  n i s ¼ ! ð1Þ !  1 : C m 0 m þ 3n  i pr2c l i¼0

  one obtains the analogue of the Darcy law for the Shulman fluid in the form: Introducing here the relation that s0 ¼ rc2!  dp dy



t ¼ ta up ¼ WðmÞ ðnÞ ð!ÞUp 

dp dy

mn ðB:17Þ

where mþ3n n

WðmÞ ðnÞ ð!Þ ¼ !

m X ð1Þi i¼0

mþn n

Up ¼

r c up m

2n l

h mþ3ni i n C i ! n  1 ; m þ 3n  i m ðB:18Þ

:

Eq. (B.18) is the same as Eq. (3.3) in the Table 2. It is easy to see that by a successive reduction of the exponents in Eq. (B.17) one obtains the auxiliary functions and coefficients for simpler models of the fluid flow through a porous bed given in the Table 2. References Adams, M. J., & Edmondson, B. (1987). Forces between particles in continuous and discrete liquid media. In B. J. Briscoe & M. J. Adams (Eds.), Tribology in particulate technology (pp. 154–172). New York: IOP Publishing. Bujurke, N. M., Jagadee, M., & Hiremath, P. S. (1987). Analysis of normal stress effects in squeeze film porous bearing. Wear, 116, 237–248.

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A. Walicka / International Journal of Engineering Science 69 (2013) 33–48

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