Dynamic characteristics for wide magneto-hydrodynamic slider bearings with a power-law film profile

Applied Mathematical Modelling 36 (2012) 4521–4528

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Dynamic characteristics for wide magneto-hydrodynamic slider bearings with a power-law film profile Jaw-Ren Lin Department of Mechanical Engineering, Nanya Institute of Technology, P.O. Box 324-22-59, Jhongli 320, Taiwan

a r t i c l e

i n f o

Article history: Received 23 March 2011 Received in revised form 8 November 2011 Accepted 15 November 2011 Available online 23 November 2011 Keywords: Power-law film profile Slider bearings MHD characteristics Dynamic coefficients

a b s t r a c t The magneto-hydrodynamic (MHD) dynamic characteristics of a wide power-law film-profile slider bearing lubricated with an electrically conducting fluid under the application of transverse magnetic fields has been proposed. A closed-form solution is obtained for the MHD power-law film-shape slider bearings, in which special bearing characteristics of the inclined-plane shape and the parabolic-film profile can also be included. Comparing with the non-conducting-fluid power-law film-shape bearing, the MHD bearing provides an increase in the load capacity, and the stiffness and damping coefficients. Comparing with the MHD inclined-plane slider bearing, the MHD parabolic-film bearing signifies an improvement in the steady performances and the dynamic characteristics. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction Fundamental analyses on steady slider bearings concentrate upon the use of a non-conducting fluid (NCF), such as the slider bearings with different film shapes [1–4]. Further investigations have been made by considering different operating situations [5–10]. On the other hand, the use of electrically conducting liquid metals as lubricants in fluid film lubrication has gained a lot of attention for a number of years. These kinds of conducting liquid-metal lubricants present high electrical-conductivity thermal-conductivity features. The lubrication performance of thin-film hydrodynamic bearings can be improved by the use of electrically conducting lubricants together with the application of externally magnetic fields. The use of magnetohydrodynamic (MHD) principle is also important for many areas of industrial engineering and applied science, such as the magneto-hydrodynamic braking [11], the nuclear reactor sodium cooling systems [12], the electric power generation [13], the microfluidic devices [14], and the biomechanics [15]. In the area of thin-film lubrication, the MHD characteristics of slider bearings lubricated with an electrically conducting fluid in the presence of externally magnetic fields have been carried out, such as the steady parallel-plate slider bearing [16–18]; the MHD steady slider bearing of different film profiles [19–25]. Recently, Lin et al. [26] and Lin [27] have investigated the MHD dynamic characteristics for slider bearings with inclined-plane film shape and tapered-land film shape, respectively. However, the inclined-plane film shape is actually a specific case of the power-law film profile. To provide more information in the variation of bearing performances with the film shape, a further analysis of the dynamic characteristics of MHD slider bearings with a power-law film profile is then motivated. 2. Analysis of power-law film slider bearing characteristics Fig. 1 describes physical geometry of a wide magneto-hydrodynamic power-law film-profile slider bearing of length L with a sliding velocity U in the x-direction and a squeezing velocity oh/ot in the z-direction. The inlet film thickness is h1(t) and the outlet film thickness is h2(t), and the power-law film profile for the bearing can be described by: E-mail address: [email protected] 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.11.052

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J.-R. Lin / Applied Mathematical Modelling 36 (2012) 4521–4528

Nomenclature B0 D, L Dd ; Dd ey f, f⁄ f0 F, F⁄ F f ; F f F f 0 ; F f 0 h h⁄ h1, h2 h10, h20  h2  h20 H n p, p⁄ r Sd ; Sd t, t⁄ u U V⁄ W0, W 0 x, z x⁄

g lp r

externally magnetic field width and length of the bearing 3 dynamic damping coefficient, Dd ¼ Dd hm0 =gL3 D ¼ ð@F  =@V  Þ0 induced electric field  3 function in Reynolds equation, f  ðH; h Þ ¼ f ðH; hÞ=h20  non-dimensional function under steady state, f0 ¼ ðf  Þ0 2  dynamic film force, F  ¼ Fhm0 =gUL2 D ¼ F  ðhm ; V  Þ  friction force, F f ¼ F f h20 =gULD steady friction force, F f 0 ¼ ðF f Þ0 ; F f 0 ¼ ðF f Þ0 power-law film thickness, hðx; tÞ ¼ ðh10  h20 Þ  ð1  x=LÞn þ h2 ðtÞ   non-dimensional film thickness, h ðx ; t  Þ ¼ ðr  1Þ  ð1  x Þn þ h2 ðt  Þ inlet and outlet film thicknesses, h1 ðtÞ ¼ ðh10  h20 Þ þ h2 ðtÞ steady inlet and outlet film thicknesses, h10 ¼ ½h1 ðtÞ0 ; h20 ¼ ½h2 ðtÞ0 ,  non-dimensional outlet film thickness, h2 ðt  Þ ¼ h2 ðtÞ=h20   non-dimensional steady outlet film thickness, h20 ¼ ðh2 Þ0 non-dimensional Hartmann number, H ¼ B0 h20 ðr=gÞ1=2 power-law film-thickness index 2 dynamic film pressure, p ¼ ph20 =gUL steady inlet-outlet film-thickness ratio, r ¼ h10 =h20 2  dynamic stiffness coefficient, Sd ¼ Sd h20 =gUL2 D ¼ ð@F  =@h2 Þ0  time, t ¼ Ut=L velocity component in the x-direction sliding velocity of the lower part   non-dimensional squeezing velocity, V  ¼ dh2 =dt 2  2 steady load-carrying capacity, W 0 ¼ W 0 h20 =gUL D horizontal and vertical coordinates non-dimensional coordinate, x ¼ x=L lubricant viscosity steady friction parameter, lp ¼ ðF f 0 =W 0 Þ  ðL=h20 Þ ¼ F f 0 =W 0 electrical conductivity

Superscript ⁄ the non-dimensional quantity Subscript 0 the steady state

 xn hðx; tÞ ¼ ðh10  h20 Þ  1  þ h2 ðtÞ; L

ð1Þ

where the subscript ‘‘0’’ denotes the steady state, and n represents the power-law film-thickness index. For the bearing analyzed, the lubricant is taken to be an electrically conducting incompressible fluid with an electrical conductivity r, and in the z-direction an uniform transverse magnetic field B0 is externally applied. Assume that the thin-film lubrication theory [1–3] is applicable to the system, but the Lorentz body force is further considered, and the induced magnetic field is negligible as comparing with the externally applied magnetic field. Based on these MHD thin-film lubrication assumptions as Hughes [19], the velocity component of lubricant in the x-direction can be obtained (Appendix A).

u¼

  2 1 sinhðH  z=h20 Þ  sinh½H  ðh  zÞ=h20  hh @p   U  20 2 2 sinhðH  h=h20 Þ 2gH @x 

sinhðH  h=h20 Þ  sinhðH  z=h20 Þ  sinh½H  ðh  zÞ=h20  ; coshðH  h=h20 Þ  1

ð2Þ

In addition, the non-dimensional MHD dynamic Reynolds-type equation can be written as:

    @ @h  @p  ¼ 6 f ðH; h Þ þ 12V  ; @x @x @x 



ð3Þ

where V  ¼ dh2 =dt denotes the non-dimensional squeezing velocity. In addition, the non-dimensional function f⁄, the nondimensional film profile h⁄, the steady inlet-outlet film-thickness ratio r, and the non-dimensional Hartmann number H are defined by:

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Fig. 1. Physical geometry of the one-dimensional magneto-hydrodynamic power-law film-profile slider bearing.



Fig. 2. Film profiles of magneto-hydrodynamic slider bearings for different values of the power-law index n = 1, 2, 3, 4 and 5 under h20 ¼ 1. 







f  ðH; h Þ ¼ 6h ½H  h cothð0:5H  h Þ  2=H2 ;   h ðx ; t  Þ ¼ ðr  1Þ  ð1  x Þn þ h2 ðt  Þ;

ð4Þ ð5Þ

r ¼ h10 =h20 ;

ð6Þ

H ¼ B0 h20 ðr=gÞ

1=2

:

ð7Þ

It is noted that for n = 1 the film profile reduces to the inclined-plane shape; and for n = 2 the parabolic film bearing are recovered. The non-dimensional film thickness of MHD slider bearing for specific values of the power-law film-thickness index n is displayed in Fig. 2. Integrating the dynamic Reynolds-type equation with respect to x⁄ with the zero boundary conditions at x⁄ = 0 and at x⁄ = 1, the dynamic film pressure can be obtained. Integrating the film pressure over the film region, the dynamic film force F⁄ can also be derived. Taking both values of the outlet film thickness and the squeezing velocity under steady state, the steady load capacity W 0 can be obtained. Taking the partial derivative of u with respect to z, the shear-strain rate and then the shearing stress can be obtained. Integrating the shearing stress at the lower plate, we can obtain the friction force. Letting both the outlet film thickness be constant and the squeezing velocity be zero, the steady friction force can be derived.

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The dynamic stiffness and damping coefficients can be obtained by performing the partial derivative of MHD dynamic film force with respect to the outlet film thickness and the squeezing velocity, and thereafter taking the results under steady state. In terms of a non-dimensional form, the MHD steady friction force, the steady load capacity, the steady friction parameter, the dynamic stiffness coefficient and the dynamic damping coefficient are written as:

F f 0 ¼ 0:5H

Z

þ1

x ¼0





cothð0:5Mh0 Þdx  3ðr  1Þ 







Z



þ1

x ¼0







h0  ½ð1  x Þn  fC ðh20 Þ  fA ðh20 Þ  dx ;  fC ðh20 Þ  f0



W 0 ¼ 6  ½F A ðh20 Þ  fC ðh20 Þ  fA ðh20 Þ  F C ðh20 Þ=fC ðh20 Þ;

ð9Þ

lp ¼ F f 0 =W 0 ; Sd ¼ 6 

ð8Þ

ð10Þ

    @F A ðGA Þ0  ðF C Þ0 g A ðh20 Þ @GC ; þ      6 @h2 0 g C ðh20 Þ @h2 0 





ð11Þ



Dd ¼ 12  ½F B ðh20 Þ  F C ðh20 Þ  fB ðh20 Þ=fC ðh20 Þ:

ð12Þ

The associated non-dimensional functions and relationships are described in Appendix B. 3. Results and discussion 

In order to present the bearing characteristics, representative data are: h20 ¼ 1; n = 1,2,3,4,5; h10 = (3, 4, 5, 6, 7)  104m; h20 = 2 104m; g = 1.55  103N  ts/m2; r = 1.07  106mho/m; B0 = (0, 3.7, 7.4, 11.1, 14.8, 18.5)  103Gauss. Accordingly, one can obtain: r = 1.5, 2.0, 2.5, 3.0, 3.5; H = 0, 1.94, 3.89, 5.83, 7.78, 9.72. For n = 1, H = 0, V⁄ = 0: the steady NCF inclined-plane bearing by Williams [3]. The predicted steady loads under r = 1.5, 2, 2.5, 3, 3.5 are: W 0 ¼ 0.13, 0.159, 0.158, 0.148 (Williams [3]); W 0 ¼ 0.131163, 0.158882, 0.157719, 0.147885 (present study). 3.1. The effects of n on the steady MHD characteristics Fig. 3 shows the load capacity as a function of n for different H. Comparing with the NCF case, higher loads are obtained for the MHD bearing. It is also observed that the MHD power-law film-shape bearing within the range of medium n is found to provide a higher load capacity as compared to the MHD inclined-plane bearing by Lin et al. [26]. Fig. 4 presents the friction parameter as a function of n. Comparing with the NCF case, the MHD bearing yields higher values of the friction parameter.

0.6

r=2 Present Inclined-plane: Lin et al. [26]

0.5

0.4

H=9.72

W0*

H=7.78 0.3

H=5.83 H=3.89

0.2

H=1.94 NCF 0.1 1

1.5

2

2.5

3

3.5

4

4.5

5

n Fig. 3. MHD steady load-carrying capacity W 0 as a function of n for different values of H under r = 2.

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J.-R. Lin / Applied Mathematical Modelling 36 (2012) 4521–4528

15 14

H=9.72

13

H=7.78

12 11

H=5.83

10 9

H=3.89

8 p

7

H=1.94

6

NCF

5 4 3 2

r=2

1 0 1

1.5

2

2.5

3

3.5

4

4.5

5

n Fig. 4. MHD steady friction parameter lp as a function of n for different values of H under r = 2.

0.9

r=2 Present Inclined-plane: Lin et al. [26]

0.8

0.7

H=9.72 0.6

H=7.78

S d*

H=5.83 0.5

H=3.89 H=1.94

0.4

NCF 0.3

0.2 1

1.5

2

2.5

3

3.5

4

4.5

5

n Fig. 5. MHD dynamic stiffness coefficient Sd as a function of n for different values of H under r = 2.

However, the MHD power-law film-shape bearing with medium n depending up H is observed to provide a smaller friction parameter when comparing with the MHD inclined-plane bearing (n = 1). 3.2. The effects of n on the dynamic MHD characteristics Figs. 5 and 6 shows the stiffness coefficient as a function of n. Comparing with the NCF case, higher stiffness coefficients are predicted for the MHD bearing. Comparing with the MHD inclined-plane bearing by Lin et al. [26], the MHD power-law film-shape bearing results in higher stiffness coefficients within the range of medium n (e.g., 1< n 63 for H = 9.72). Fig. 6

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J.-R. Lin / Applied Mathematical Modelling 36 (2012) 4521–4528

1.4

r=2 1.3

H=9.72

Present Inclined-plane: Lin et al. [26]

1.2

H=7.78

1.1 1

H=5.83

0.9

Dd* 0.8

H=3.89

0.7 0.6

H=1.94

0.5

NCF 0.4 0.3 1

1.5

2

2.5

3

3.5

4

4.5

5

n Fig. 6. MHD dynamic damping coefficient Dd as a function of n for different values of H under r = 2.

shows the damping coefficient as a function of n. Comparing with the NCL power-law film-shape bearing, the MHD bearings signify an improvement in the dynamic damping coefficients. Generally speaking, the increments in the damping coefficients are more emphasized for MHD power-law film-shape bearings with larger values of H. 4. Conclusions A closed-form solution of dynamic characteristics is obtained for the MHD power-law film-shape slider bearings, in which bearing characteristics of the inclined-plane shape and the parabolic-film profile can also be included by setting the powerlaw film-thickness index n = 1 and n = 2 respectively. Comparing with the NCL power-law film-shape bearing, the MHD power-law film-shape bearing provides an increase in the steady load capacity, and the dynamic stiffness and damping coefficients. Comparing with the MHD inclined-plane slider bearing (n = 1), the MHD parabolic-film slider bearing (n = 2) signifies an improvement in the steady performances and the dynamic characteristics. Further numerical results of the MHD parabolic-film slider bearing (n = 2) are also included for engineering references. Appendix A. Steady performances and dynamic characteristics Following the procedure of Lin et al. [26], the MHD motion equation in the x-direction is

g

@2u @p þ rB0 ey :  rB20 ¼ @x2 @x

ðA:1Þ

Using the boundary conditions u(z = 0) = U and u(z = h) = 0, one can obtain

u ¼ ½coshðHz=h20 Þ  cothðHh=h20 Þ sinhðHz=h20 Þ  U      2  h @p Hz Hh Hz þ tanh sinh :  202  rB0 ey 1  cosh h20 2h20 h20 gH @x

ðA:2Þ

Assume the net current flow is zero across the film, then:

Z

h

ðey þ B0 uÞdz ¼ 0:

ðA:3Þ

z¼0

Then, u is obtained by solving equations (A.2) and (A.3), simultaneously.

u¼

  2 1 sinhðH  z=h20 Þ  sinh½H  ðh  zÞ=h20  hh @p sinhðH  h=h20 Þ  sinhðH  z=h20 Þ  sinh½H  ðh  zÞ=h20    U  20 : 2 2 sinhðH  h=h20 Þ coshðH  h=h20 Þ  1 2gH @x ðA:4Þ

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J.-R. Lin / Applied Mathematical Modelling 36 (2012) 4521–4528

The dynamic friction force is obtained by integrating the shearing stress.

Z

Ff ¼



L

g x¼0

  Z Z L @u gUDH L Hh D @p dx   Ddx ¼   coth h dx: @z z¼0 2h20 2h20 2 x¼0 @x x¼0

ðA:5Þ

The integrated continuity equation across the film is given by

Z

h

@u dz ¼  @x

z¼0

Z

h

z¼0

@w dz: @z

ðA:6Þ

Applying the conditions w(z = 0) = 0 and w(z = h) = oh/ot, one can derive

( 2  )   @ hh20 Hh Hh @p @h @h ¼ gU  2  þ 2g : coth @x H2 h20 2h20 @x @x @t

ðA:7Þ

Using in a non-dimensional form, the non-dimensional dynamic Reynolds equation is

    @ @h  @p f  ðH; h Þ  ¼ 6  þ 12V  ;  @x @x @x 

f  ðH; h Þ ¼

6h

ðA:8Þ

 

H2



½H  h cothð0:5H  h Þ  2:

ðA:9Þ

Integrating the Reynolds equation gives the dynamic film pressure.

 Z p ¼ 6  ðr  1Þ 

x

ð1  x Þn  fA dx   f fC

x ¼0

Z

x

x ¼0

  Z x   Z x 1  x fB 1   dx þ 12V   dx   dx :    f fC x ¼0 f x ¼0 f

ðA:10Þ



Integrating the film pressure gives the dynamic film force F  ¼ Fðh2 ; V  Þ .

F  ¼ 6½F A  fA  F C =fC  þ 12V  ½F B  fB  F C =fC :

ðA:11Þ

The steady load and the steady friction force are obtained by taking under steady state. 









W 0 ¼ ðF  Þ0 ¼ 6  ½F A ðh20 Þ  fC ðh20 Þ  fA ðh20 Þ  F C ðh20 Þ=fC ðh20 Þ; F f 0 ¼ ðF f Þ ¼ 0:5H

Z

1





cothð0:5Mh0 Þdx  3ðr  1Þ 

x ¼0

Z

1

x ¼0

ðA:12Þ 





h0  ½ð1  x Þn  fC ðh20 Þ  fA ðh20 Þ  dx :  fC ðh20 Þ  f0

ðA:13Þ



Performing the derivative of F⁄ with respect to h2 and V⁄, one can achieve

Sd ¼ 



@F   @h2

¼ 6  0

    @F A ðGA Þ0  ðF C Þ0 fA ðh20 Þ @F C  ; þ     6 @h2 0 fC ðh20 Þ @h2 0

     @F fB ðh20 Þ   Dd ¼  ; ¼ 12  F B ðh20 Þ  F C ðh20 Þ    @V 0 fC ðh20 Þ

ðA:14Þ

ðA:15Þ

Appendix B. The associated functions and relationships



fA ðh2 Þ ¼ ðr  1Þ 



fB ðh2 Þ ¼



fC ðh2 Þ ¼

Z

1

x ¼0

Z

1

x ¼0

Z

x ¼0

Z

1

x ¼0

ðB:1aÞ

ðB:1bÞ

1   dx ; f  ðH; h Þ

ðB:1cÞ





ð1  x Þn   dx ; f  ðH; h Þ

x   dx ; f  ðH; h Þ

F A ðh2 Þ ¼ ðr  1Þ 

F B ðh2 Þ ¼

1

Z

Z

1

x ¼0

x

x ¼0

Z

x

x ¼0

ð1  x Þn    dx dx ; f  ðH; h Þ

x    dx dx ; f  ðH; h Þ

ðB:2aÞ

ðB:2bÞ

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J.-R. Lin / Applied Mathematical Modelling 36 (2012) 4521–4528

Z

1

Z

x

1    dx dx ; f  ðH; h Þ

ðB:2cÞ

  Z 1 Z x  @F A ð1  x Þn @f    ¼ ðr  1Þ  ;     dx dx 2 @h2 0 x ¼0 x ¼0 f ðH; h Þ @h2 0

ðB:3aÞ

  Z 1 Z x  @F C 1 @f    ¼ dx dx ;     2 @h2 0 x ¼0 x ¼0 f ðH; h Þ @h2 0

ðB:3bÞ



F C ðh2 Þ ¼

ðGA Þ0 ¼

x ¼0

x ¼0

    6 @fC @fA   ; ðh Þ   f ðh Þ   f A C    20 20 fC2 ðh20 Þ @h2 0 @h2 0

@f  3 2 2 2 2    ½H h  H2 h coth ð0:5Hh Þ þ 4Hh cothð0:5Mh Þ  4;  ¼ @h2 H2 @fA  ¼ ðr  1Þ  @h2 @fC  ¼  @h2

Z

1

x ¼0

Z

1

x ¼0

ð1  x Þn @f      dx ; f 2 ðH; h Þ @h2

1  f 2 ðH; h Þ



@f    dx : @h2

ðB:4Þ

ðB:5Þ

ðB:6aÞ

ðB:6bÞ

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