Reynolds number effects in the flow of an electrorheological fluid of a Casson type between fixed surfaces of revolution

Applied Mathematics and Computation 250 (2015) 636–649

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Reynolds number effects in the flow of an electrorheological fluid of a Casson type between fixed surfaces of revolution A. Walicka ⇑, J. Falicki University of Zielona Góra, Department of Mechanics and Design of Machines, ul. Szafrana 2, P.O. Box 47, 65-516 Zielona Góra, Poland

a r t i c l e

i n f o

a b s t r a c t Many electrorheological fluids (ERFs) as fluids with micro-structure demonstrate viscoplastic behaviours. Rheological measurements indicate that the flows of these fluids may be modelled as the flows of a Casson fluid. Our concern in the paper is to examine the pressurized laminar flow of an ERF of a Casson type in a narrow clearance between to fixed surfaces of revolution. In order to solve this problem the boundary layer equations are used. The Reynolds number effects (the effects of inertia forces) on pressure distribution are examined by using the averaged inertia method. Numerical examples of externally flows in the clearance between parallel disks and concentric spherical surfaces are presented. Ó 2014 Elsevier Inc. All rights reserved.

Keywords: Inertia effects Fixed surfaces of revolution Casson ER fluid Pressurized flow

1. Introduction Electrorheological fluids (abbreviated to ERFs) are a kind of novel intelligent soft matter as a two-phase suspension system formed by dielectric solid particles dispersing in the insulating medium oil [1–7]. Electrorheological fluids have long held promise for use in vibration control and torque transmission devices, based on the characteristic dependence of their viscosity on the applied electric field strength. Since their discovery by Winslow [1] many kinds of organic or inorganic particles and their composites have been used and reported as the ER materials. When the external electric field is imposed to an ERF, it behaves as a viscoplastic fluid [2–11], displaying a field-dependent yield shear stress which is widely variable. Without the electric field, the ERF has a reversible and a constant viscosity so that it flows as a Newtonian fluid. Another salient feature of the ERF is that the time required for the variation is very short (<0.001 s). These attractive characteristics of the ERF provide the possibility of the appearance of new engineering technology (e.g.: nuclear and space engineering, mechatronics, etc.). Recently, the application of the ERF to rotor-bearing systems has been initiated Dimarogonas and Kollias [4] and Jung and Choi [6] and developed by Basaravaja et al. [12] and El Wahed [13], but its application to vibration control has been initiated by Lee et al. [14], Hong et al. [15,16] and Choi et al. [17]. To describe the rheological behaviour of viscoplastic fluids in complex geometries the Bingham model is used [2,18]. Recently, the non-linear model of Shulman [19] has been successfully applied. The constitutive equation of this model is given as follows:

h

in

1=m s ¼ s1=n þ ðlc_ Þ : 0

⇑ Corresponding author. E-mail addresses: [email protected] (A. Walicka), [email protected] (J. Falicki). http://dx.doi.org/10.1016/j.amc.2014.10.112 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

ð1Þ

A. Walicka, J. Falicki / Applied Mathematics and Computation 250 (2015) 636–649

637

Symbols A Aðm; bÞ ðxÞ D(x) h ðj; bÞ IðmÞ

auxiliary function defined by formula auxiliary function defined by formula auxiliary function defined by formula half of the fluid film thickness auxiliary function defined by formula

K KSV n p Q R; RðxÞ Ri Ro Rk S S(x) Tyx tx ; ty U

plasticity number de Saint-Venant plasticity number power-law index pressure flow rate local radius inlet radius to the clearance outlet radius from the clearance modified Reynolds number signum function auxiliary function defined by formula (40) or (43), respectively component of the shear stress tensor velocity components applied voltage central angle of spherical surface fluid density shear stress yield shear stress

u q s s0

(42)1 (42)3 (41) or (44), respectively (42)4

where s is the shear stress, s0 is the yield shear stress, l is the coefficient of plastic viscosity, c_ is the shear strain rate, n and m are the non-linearity indices. By reducing the coefficients in the Shulman equation one can obtain simpler models describing the flow of a viscoplastic fluid. One of these models is a Casson model [20]:

h

in

s ¼ s01=n þ ðlc_ Þ1=n :

ð2Þ

The most popular model of the ERF is the Bingham model for which m = n = 1 in Eq. (1). An analysis of many investigations [13,21–26] indicates that other models may be used – such as Casson (m = n), Vocˇadlo (m = 1) and Herschel–Bulkley (n = 1) in Eq. (1) – to describe ERFs flows.

Fig. 1. The geometry of a curvilinear clearance between two surfaces of revolution.

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A. Walicka, J. Falicki / Applied Mathematics and Computation 250 (2015) 636–649

This paper deals with a laminar flow of ERFs modelled as a Casson fluid in the clearance between two surfaces of revolution shown in Fig. 1. Using the method of averaged inertia [27,28] the influence of inertia terms of the equations of motion and viscoplastic behaviour on the pressure distribution is analysed. The final results are presented for a ‘‘simple’’ Casson fluid (n = 2) and are also presented for a Newtonian fluid because for some physical conditions the insulating oil (or other solvents for ERFs) may manifest Newtonian behaviours. 2. Equations of motion of the ERF The yield shear stress for the ERF varies with respect to the electric field. According to the experimental results reported in [2,3,5,29,8,30,31] the relation between the yield shear stress s0 and the electric field strength E is given as follows:



s0 ðEÞ ¼ a

U 2h

b ð3Þ

where U and 2h are the applied voltage and the film thickness, respectively. Both parameters a and b are the experimental constants of which the range of the exponent b is 1–2.4. Other experimental data [32] suggest that the yield shear stress for some ERFs follows:

s0 ðEÞ ¼ a1



  2 U U þ a2 2h 2h

ð4Þ

where a1 and a2 are constants; for relatively high field strengths a simpler formula may be used



s0 ðEÞ ¼ a

U 2h

2 ð5Þ

where a is an experimental constant. The three-dimensional constitutive form of the equation of a Casson model follows [33,34]:

T ¼ p1 þ MA1 ;

M¼

h

in

s1=n þ ðlAÞ1=n A1 A ¼ 0

 1=2 1 trðA21 Þ 2

ð6Þ

where 1 is the unity tensor, p is the pressure, A is second invariant of the stretching tensor A1 (the first Rivlin–Ericksen tensor) defined by formulae:

A1 ¼ L þ LT ;

L ¼ grad

v

ð7Þ

where v is the velocity vector, but the symbol T denotes the matrix transpose. The general equations of motion of a Casson fluid have the following forms: – equation of continuity:

div

v¼0

ð8Þ

– equation of momentum

q

dv ¼ div T dt

ð9Þ

or

q

dv ¼ rp þ div K dt

ð10Þ

where:

K ¼ MA1 :

ð11Þ

The flow configuration is shown in Fig. 1. The surfaces of revolution are defined by the function R(x) which denotes the radius of the median between the surfaces, plus the function h(x) which denotes the distance to each surfaces from the median, measured along a normal to the median. An intrinsic curvilinear orthogonal coordinate system x; #; y is also depicted in Fig. 1. The vector E of an electric field strength is normal to the median. The physical parameters of the ERF flow are the velocity components tx ; ty and pressure p. With regard to the axial symmetry of the flow these parameters are not dependent on the angle #. Let us make the assumption that:

hðxÞ  RðxÞ;



tx ¼ 0ðV m Þ; ty ¼ 0 V m

 hm : Rm

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Here Vm is the mean velocity of longitudinal fluid flow and hm ; Rm are, respectively, mean values of h(x) and R(x) in the clearance. This assumption can be used to make order-of-magnitude arguments for Eqs. (8)–(11). If some asymptotic transformations are made, the same as in [23,33,34] these equations – for a simple Casson fluid – can be reduced to a simpler form:

1 @ðRtx Þ @ ty þ ¼ 0; R @x @y 

@ tx @ tx þ ty @x @y

q tx

ð12Þ

 ¼

dp @T yx þ ; dx @y

ð13Þ

where

"

T yx

  1n #n @ tx  ¼ S s0 þ l  ; @y

and S ¼ sgn



dtx dy

1 n



s0 ðxÞ ¼ a

UðxÞ 2hðxÞ

b ;

ð14Þ

 . The signum (sgn) function takes on the value +1 for a positive argument and 1 for a negative argument;

U E ¼ 2h is the value of the vector E.

The order-of-magnitude arguments shown that

p ¼ pðxÞ is a function of x only. For many ERFs the values of a yield shear stress are contained in the limits: 0ð1Þ 6 s0 6 0ð3Þ, where 0(m)  10m and denotes the order of magnitude [2,8,23]. Typical experimental values of the material parameters s0 and l for slit flow of the Casson ERF may find in Wunderlich [30] and Abu-Jdayil et al. [31]. In the flow of a fluid with the yield shear stress there exists a quasi-solid core flow (Fig. 1) bounded by surfaces lying at

jyj ¼ h0

for which the shear stresses are jT yx j ¼ s0 :

ð15Þ

Combining the expressions (10) and (11) one obtains the boundary conditions on liquid surfaces of the core flow as [35]:

@ tx ¼ 0 for y ¼ h0 : @y

ð16Þ

The boundary conditions on solid surfaces are stated as follows:

tx ¼ ty ¼ 0 for y ¼ h:

ð17Þ

On the median line there is also:

@ tx ¼ 0 for y ¼ 0 and hence jT yx j ¼ 0 for y ¼ 0: @y

ð18Þ

Moreover, in the inlet and the outlet of the clearance conditions for the pressure can be written in the form:

pðxi Þ ¼ pi ;

pðxo Þ ¼ po :

ð19Þ

where xi denotes the inlet coordinate and xo outlet coordinate. 3. Solution to the equation of motion 3.1. Solution for the flow of a Casson ER fluid Taking into account Eq. (12) one can rearrange Eq. (13) writing it in the form:

"   1n #n !   @ tx  1 1 @ðRt2x Þ @ dp @ n S s0 þ l  q þ ðtx ty Þ ¼  þ : R @x @y dx @y @y

ð20Þ

Then, averaging the left-hand side of Eq. (20) across the clearance thickness and taking into account boundary conditions (17) we obtain the following equation:

@ S @y

"

 

1n #n

@ t  1 sn0 þ l x  @y

where f(x) is defined as:

¼ f ðxÞ

ð21Þ

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A. Walicka, J. Falicki / Applied Mathematics and Computation 250 (2015) 636–649

f ðxÞ ¼

dp q @ R þ dx hR @x

Z

h

0

t2x dy:

ð22Þ

Integrating Eq. (21) we get [23,33–35]: – for shear flow:

h

txs ðyÞ ¼

2

l

  n y2ni X n ðf Þ F i 1  h i¼0

ð23Þ

where |Tyx| > s0 or |y| > h0; – for core flow 2

txc ¼ txc ðh0 Þ ¼

h

l

" #  2ni n X h0 n ðf Þ F i 1  h i¼0

ð24Þ

where jT yx j 6 s0 or jyj 6 h0 . Here

 ni n h0 C in ; 2n  i h

F i ¼ ð1Þi

C in ¼

n! : i!ðn  iÞ!

ð25Þ

The flow rate Q is defined as:

Q ¼ 2p R

Z

þh

tx dy ¼ 4pR

Z

h0

txc dy þ

Z

0

h

!

h

txs dy :

ð26Þ

h0

Using here the expressions (23) and (24) we obtain: 2

4pRh

Q¼

l

" #  3ni n X h0 n ; ðhf Þ F 2i 1  h i¼0

ð27Þ

 ni n h0 C in 3n  i h

ð28Þ

where

F 2i ¼ ð1Þi

the equation with an unknown quantity f. Introducing the following notations:



K¼



lQ hf h ¼ ; ; X¼  2 h0 s0 4pRh s0

ð29Þ

it may present Eq. (27) in the form: n X ð1Þi i¼0

 3ni  n C in X n  1  X 2 K ¼ 0: 3n  i

ð30Þ

This is an equation characteristic to the flow of the Casson ER fluid in a clearance between two surfaces of revolution. Its analytical solution Xs exists only for the large K ðP 5Þ and for small K ð6 0:05Þ and it is given as follows [23,33–35]: – for large K:

Xs ¼

  1 !n 3n ð1Þn n þ 3 K ; 3n  1 2

– for small K:

Xs ¼ 1 þ



ðn þ 1Þ! K UðnÞ

ð31Þ

1 nþ1

ð32Þ

where

UðnÞ ¼

n1 X ð1Þi i¼1

 n  n Y n i 1 Y C in 1 k þ n ð2 þ kÞ: 3n  i k¼0 n 2C 3n k¼0

ð33Þ

For the simple Casson ER fluid (n = 2) the characteristic Eq. (30) takes the form: 5

10X 3  24X 2 þ 15X 2 ð1  2KÞ  1 ¼ 0;

ð34Þ

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641

Its analytical solutions are respectively:

X s  ð1:2 þ

pffiffiffiffiffiffiffi 2 3K Þ ;

Xs ¼ 1 þ

p ffiffiffiffiffiffiffiffiffi 3 12K :

ð35Þ

To determine the pressure distribution let us turn to the expression (22). Rearranging it we get:

dp q @ R ¼ f ðxÞ  dx hR @x

Z

h

0

t2x dy:

ð36Þ

Taking into account the expressions (35) and integrating Eq. (36) we will find:

p ¼ po þ ½SðxÞ  So  þ ½DðxÞ  Do ;

So ¼ Sðxo Þ;

Do ¼ Dðxo Þ

ð37Þ

where

SðxÞ ¼

V¼

Z o

Z

f s dx ¼ 

h

2

½tðsÞ x  dy;

Z

Z Xs 1 @ ðRVÞdx; dx; DðxÞ ¼ q hR @x h 8 h i2 9 ðsÞ > < txs = for jyj > h0 > ðsÞ 2 ½ t x  ¼ h i2 : > > : tðsÞ for jyj 6 h0 ; xc

s0

ð38Þ

Here:



V¼



s20 h3 2 2 88 32 73 36 12 1 1 4 32 1 2 31 3 1 ; Xs  Xs þ Xs  Xs  Xs þ Xs  X þ X þ 2 135 60 35 30 45 15 s 3780 s 3 l 15

ð39Þ

tðsÞ x is the velocity tx for f = fs. Note that the final forms of S(x) and D(x) – for U(x) = Uo = const – depend on the values of K: – for large values of K there are:

 b   Z Uo dx 1=2 ð2;bÞ ð1;bÞ ; SðxÞ ¼ a 1:44 þ 2:4ð3AÞ A ðxÞ þ 3AA ðxÞ 1þb 2 h # pffiffiffi   " qa2 U o 2b 6A2 ð2;bÞ 8 3 3=2 ð3;bÞ 33A ð1;bÞ DðxÞ ¼  2 I ðxÞ  A Ið2Þ ðxÞ þ I ðxÞ ; 500 ð1Þ l 2 5 ð1Þ 225

ð40aÞ

ð41bÞ

but for s0(x) = s0 = const there is b = 0 in Eq. (14)2, and a = s0 and then the formulae (40a), (41a) take the forms [23]:

  Z dx SðxÞ ¼ s0 1:44 þ 2:4ð3AÞ1=2 Að2;0Þ ðxÞ þ ð3AÞAð1;0Þ ðxÞ ; h "

ð40bÞ

pffiffiffi

qs2 6 2 ð2;0Þ 8 3 33 ð1;0Þ ð3;0Þ DðxÞ ¼  20 ðAÞ Ið1Þ ðxÞ  ðAÞ3=2 Ið2Þ ðxÞ þ AI ðxÞ l 5 500 ð1Þ 225

# ð41bÞ

Where

A¼ ðj;bÞ

lQ lQ ; or A ¼ ; Aðm;bÞ ðxÞ ¼ 4ps0 22b paU bo

IðmÞ ðxÞ ¼

Z

1 h 1mj 32jmð2mj Þb i0 R h dx; Rh

Z

dx 1

Rm h

mþ2 m1 m þ m b

; ð42Þ

– for small values of K there are:

 b Z  Uo dx 1=3 ð3;bÞ ; SðxÞ ¼ a þ ð12AÞ A ðxÞ 1þb 2 h

b h ð0; bÞ 2 ð1; bÞ ð2; bÞ DðxÞ ¼  10qa7 l2 U2o 5Ið3Þ ðxÞ  9ð12AÞ1=3 Ið3Þ ðxÞ þ 5ð12AÞ2=3 Ið3Þ ðxÞþ i ð3; bÞ ð4; bÞ ð12AÞ3=3 Ið3Þ ðxÞ þ 0; 08ð12AÞ4=3 Ið3Þ ðxÞ ;

ð43aÞ

ð44aÞ

but for s0(x) = s0 there are:

SðxÞ ¼ s0

Z

 dx þ ð12AÞ1=3 Að3Þ ðxÞ ; h

ð43bÞ

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A. Walicka, J. Falicki / Applied Mathematics and Computation 250 (2015) 636–649

h qs2 ð0; 0Þ ð1; 0Þ ð2; 0Þ DðxÞ ¼  107 l0 2 5Ið3Þ ðxÞ  9ð12AÞ1=3 Ið3Þ ðxÞ þ 5ð12AÞ2=3 Ið3Þ ðxÞþ i ð3; 0Þ ð4; 0Þ ð12AÞ3=3 Ið3Þ ðxÞ þ 0; 08ð12AÞ4=3 Ið3Þ ðxÞ ;

ð43bÞ

Here and throughout the paper the prime denotes differentiation with respect to x. 3.2. Solution for a Newtonian fluid Eq. (20) for a Newtonian fluid flow takes the form [33]:



q

 1 @ðRt2x Þ @ dp @ 2 tx þ ðtx ty Þ ¼  þ l 2 R @x @y dx @y

ð45Þ

After averaging the left-hand side it will be:

@ 2 tx ¼ f ðxÞ @y2

ð46Þ

where:

f ðxÞ ¼

1 dp q @ R þ l dx lhR @x

Z

h

ðt2x Þ dy:

0

ð47Þ

Integrating Eq. (40) we get [33]:

f 2

tx ¼ ðy2  h2 Þ

ð48Þ

and: 3

Q ¼

4pRh f; 3

tx ¼

  3Q 1 y2 1 2 : 8p Rh h

ð49Þ

The pressure distribution may be determined from the equation: 0

dp 3lQ 1 3qQ 2 ðRhÞ ¼ þ : dx 4p Rh3 40p2 ðRhÞ3

ð50Þ

Its solution has also the form given by Eq. (37), but now:

SðxÞ ¼

Z

fdx ¼ 

3l Q 4p

Z

dx Rh

3

;

DðxÞ ¼

3qQ 2 40p2

Z

0

ðRhÞ dx ðRhÞ

3

¼

3qQ 2 1 : 80p2 ðRhÞ2

ð51Þ

4. Graphic presentation of some results Taking into account the results obtained in the previous section, we will present the pressure distribution in the clearance of constant thickness between two parallel disks shown in Fig. 2 and between two concentric spherical surfaces shown in Fig. 3; to this aim we shall introduce the following dimensionless parameters:  for the clearance between parallel disks:

~x ¼

x ; xo

~ ¼ R ¼ ~x; R Ro

ð52Þ

Fig. 2. Clearance between two parallel disks.

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643

Fig. 3. Clearance between concentric spherical surfaces.

~ for the Newtonian flow in the clearance between parallel disks. Fig. 4. Pressure distributions p

 for the clearance between concentric spherical surfaces:

~x ¼

x ¼ u; Rs

~ ¼ R ¼ sin u R Rs

ð53Þ

and:

Rk ¼

b2 qQho ~ qðp  po Þð2ho Þ4 2b2 lQho ; p¼ ; K SV ¼ ; b 2 2 2 paU o Ro plRo l Ro Rk

ð54Þ

if b = 0 then a = s0 = const and K SV ¼ 4pRlQh2 s . o 0 For spherical surfaces: Ro = Rs in all dimensionless parameters. Rk is the modified Reynolds number and KSV is the ER plasticity number (de Saint–Venant number). Note that large values of the plasticity number correspond to the flow with a small core (small h0/h). The nondimensional formula for pressure distributions has the form:

~ ~xÞ  D ~ o: ~ ¼ ~Sð~xÞ  ~So þ Dð p

ð55Þ

For the Newtonian flows there are: – for parallel disks:

  3 1 ~ ¼ 12 ln ~x þ Rk 1  ; p ~x2 5

ð56Þ

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A. Walicka, J. Falicki / Applied Mathematics and Computation 250 (2015) 636–649

~ for the Newtonian flow in the clearance between concentric spherical surfaces. Fig. 5. Pressure distributions p

– for concentric spherical surfaces:

~ ¼ 12 ln tg p

! 3 1 þ Rk 1  : 2 2 5 sin u

u

ð57Þ

For the flows of the Casson ER fluid with a large value of K there are: – for parallel disks:

pffiffiffi ~ ð2Þ ð~xÞ  12A ~ ð1Þ ð~xÞ; ~Sð~xÞ ¼  5:76 ~x  9:6 3 A 1 K SV ðK SV Þ2

ð58Þ

( ) pffiffiffi 6 ~ð2Þ 8 3 1 ~ð3Þ ~ Dð~xÞ ¼ Rk I ð~xÞ  I ð~xÞ ; 5 ð1Þ 225 ðK SV Þ12 ð2Þ

ð59Þ

because for a clearance of constant thickness there are formal dependences:

~ ðm; bÞ ð~xÞ ¼ A ~ ðm; 0Þ ð~xÞ ¼ A ~ ðmÞ ð~xÞ and ~Iðj; bÞ ð~xÞ ¼ ~Iðj; 0Þ ð~xÞ ¼ ~IðjÞ ð~xÞ: A ðmÞ ðmÞ ðmÞ

ð60Þ

Here:

~ ð1Þ ð~xÞ ¼ A

Z (

~IðjÞ ð~xÞ ¼ ðmÞ

d~x ¼ ln ~x; ~ R  m1 j ~ ln R

1 j ~m R

~ ð2Þ ð~xÞ ¼ A for j > 0

Z

d~x 1 ¼ 2~x2 ; ~ 12 R

)

for j ¼ 0

ð61Þ (

ðjÞ or ~IðmÞ ð~xÞ ¼

 m1 j ln ~x

1 j

for j > 0

~xm

for j ¼ 0

) ð62Þ

and

~Ið2Þ ð~xÞ ¼  1 ; ð1Þ 2~x2

~Ið3Þ ð~xÞ ¼  1 ; 3 ð2Þ 3~x2

ð63Þ

– for concentric spherical surfaces:

~x ¼ u;

  ~ ð1Þ ðuÞ ¼ ln tg u ; A 2

~ ð2Þ ðuÞ ¼ Ið12Þ ðuÞ; A

ð64Þ

A. Walicka, J. Falicki / Applied Mathematics and Computation 250 (2015) 636–649

( ~IðjÞ ðuÞ ¼ ðmÞ

 m1 j

1 j

for j > 0

)

ðsin uÞm

ln sin u

645

ð65Þ for j ¼ 0

and

~Ið2Þ ðuÞ ¼  ð1Þ

1 2

2 sin u

;

~Ið3Þ ðuÞ ¼  ð2Þ

1 3

3 sin2 u

:

ð66Þ

~ for the Casson ERF flow in the clearance between parallel disks for different values of Rk and a large value of KSV (KSV = 10). Fig. 6. Pressure distributions p

~ for the Casson ERF flow in the clearance between concentric spherical surfaces for different values of Rk and a large value of Fig. 7. Pressure distributions p K SV ðK SV ¼ 10Þ.

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A. Walicka, J. Falicki / Applied Mathematics and Computation 250 (2015) 636–649

~ for the Casson ERF flow in the clearance between parallel disks for different values of Rk and a small value of K SV ðK SV ¼ 0:05Þ. Fig. 8. Pressure distributions p

Fig. 9. Pressure distributions for the Casson ERF flow in the clearance between concentric spherical surfaces for different values of Rk and a small value of K SV ðK SV ¼ 0:05Þ.

For the flows of the Casson fluid with small values of K there are: – for parallel disks:

48 ~ ð3Þ ð~xÞ; ~Sð~xÞ ¼  4 x~  A 2 K SV ð12K SV Þ3 ~ ~xÞ ¼  Dð

n

Rk 7

2

10 ðK SV Þ

o 1 ð1Þ 2 ð2Þ 3 ð3Þ 4 ð4Þ ð0Þ 5~Ið3Þ ð~xÞ  9ð12K SV Þ3~Ið3Þ ð~xÞ þ 5ð12K SV Þ3~Ið3Þ ð~xÞ  ð12K SV Þ3~Ið3Þ ð~xÞ þ 0:08ð12K SV Þ3~Ið3Þ ð~xÞ ;

ð67Þ

ð68Þ

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647

~ for the Casson ERF flow in the clearance between parallel disks for large values of KSV and for Rk ¼ 0:5. Fig. 10. Pressure distributions p

where:

~ ð3Þ ð~xÞ ¼ 3 ~x23 ; ~Ið0Þ ð~xÞ ¼ ln x; ~Ið1Þ ð~xÞ ¼  2 ; A 1 ð3Þ ð3Þ 2 ~x3 ~Ið2Þ ð~xÞ ¼  1 ; ~Ið3Þ ð~xÞ ¼ 0; ~Ið4Þ ð~xÞ ¼ 1 ; 4 ð3Þ ð3Þ ð3Þ 2~x 4~x3

ð69Þ

– for concentric spherical surfaces:

~ ð3Þ ðuÞ ¼ Ið13Þ ðuÞ; A

~x ¼ u;

~Ið0Þ ðu ~ Þ ¼ ln sin u; ð3Þ ~Ið2Þ ðu ~Þ ¼  ð3Þ ~Ið4Þ ðu ~Þ ¼ ð3Þ

1 ; 2 sin u 1

~Ið1Þ ðu ~Þ ¼  ð3Þ

2 1

sin3 u

~Ið3Þ ðu ~ Þ ¼ 0; ð3Þ

; ð70Þ

4

4 sin3 u

The formulae for the functions ~IðnÞ ðuÞ are given in the Appendix A. The nondimensional pressure distributions for Newtonian flows are presented in Figs. 4 and 5. The nondimensional pressure distributions for the Casson ERF flows are presented in Figs. 6–10. All dimensionless values of KSV and Rk that were used to the graphic illustrations of the pressure distributions are taken from the analysis of the experimental viscometric data for ERFs of the Casson type in slit flows [30,31] and also from the analysis of the experimental results for the Casson lubricants flows in thrust slide bearings [23]. 1

5. Conclusions In this work the relative inertia effect as a function of the Reynolds number was investigated in a clearance flow of an ERF of the Casson type between two surfaces of revolution. It was also investigated the Newtonian fluid flow for the reason that under some physical condition ERFs may manifest Newtonian behaviour. All these theoretical results was obtained by using the method of averaged inertia. From the general considerations, formulae and graphs presented here for the flows in the narrow clearance of constant thickness between parallel disks and concentric spherical surfaces shown in Figs. 2 and 3 one may conclude that: – for the Newtonian flows: with the increase of the modified Reynolds number Rk the pressure decreases. The bigger losses of the pressure coincide with the fluid move to the clearance outlet. The maxima of the pressure distributions for both the clearances are similar between parallel disks;

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– for the Casson ERF flows: the pressure values decrease with the increase of the modified Reynolds number Rk for large values of the de Saint– Venant ER number KSV = 5 10 and these decreases are only considerable near to the clearance inlet between parallel disks. For small values of the de Saint–Venant ER number KSV the influence of the Reynolds number Rk on the pressure values is inconsiderable. Therefore, it may be considered that for these values of KSV the Casson ERF flows without inertia effects. The pressure values are larger for the flows between concentric spherical surfaces than these for flows between parallel disks. The theoretical results obtained for the Newtonian fluid flow between two parallel disks are nearly identical with the experimental results presented by Chen [36] and Patrat [37]. This accordance of the both results indicates on the great usability of the averaged inertia method to the investigation of Newtonian flow between surfaces of revolution [27,33]. Some general informations about experimental data concerning the considered here flows of ERFs of the Casson type may find in the works carried out by Korobko [38] and Abu-Jdayil [39]. Korobko present her experimental results in graphic form and these results are generally similar to the present results. Appendix A. The integral:

~Ið1nÞ ðuÞ ¼

Z

du 1

sinn u

may be expressed by the following series:

~Ið1nÞ ðuÞ ¼ b u11n þ b u31n þ b u51n þ b u71n þ b u91n þ b u111n þ b u131n 1 3 5 7 9 11 13 where 1

1



 1 þ 1n 1  ; 5 3

b1 ¼

1 ; 1  1n

b7 ¼

n

144 7  1n

b9 ¼





 1 þ 1n ð2 þ 1nÞ 3 þ 1n 2 þ 1n 1 1

:  þ þ 105 100 108 30 288 9  1n

b3 ¼ n 1 ; 6 3n

1

b11 ¼

b13

1 n



b5 ¼

n 24 5  1n



 1 þ 1n 2 þ 1n 1 þ 1n 1 ;  þ 9 35 5













  1 þ 1n ð2 þ 1nÞ 3 þ 1n 4 þ 1n 2 þ 1n 3 þ 1n 2 þ 1n 2 þ 1n 1

;  þ þ  35 21 10 54 9 17280 11  1n 1 n





















 1 þ 1n 2 þ 1n 3 þ 1n 4 þ 1n 5 þ 1n 2 þ 1n 3 þ 1n 4 þ 1n 2 þ 1n 3 þ 1n 2 þ 1n 3 þ 1n 2 þ 1n

¼  þ þ  1 324 36 53 20 35 103680 13  n  1 2þn 1 : þ  490 100 1 n



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