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Effect of couple stresses on static and dynamic characteristics of MHD wide tapered land slider bearing

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Abstract

This paper presents an analytical study on the effect of couple stress on static and dynamic characteristics of wide tapered land slider bearing in the presence of applied magnetic field. A non-dimensional modified Reynolds equation is derived for the bearing under consideration. Closed-form expressions for the steady fluid film pressure, steady load carrying capacity and dynamic characteristics, viz., dynamic stiffness and dynamic damping coefficients are obtained. Numerical computations of the results revealed that magnetohydrodynamics tapered land slider bearing lubricated with couple stress fluid provides higher steady load carrying capacity, dynamic stiffness coefficient and dynamic damping coefficient as compared with the corresponding Newtonian case. It is observed that the presence of applied magnetic field characterized by the Hartmann number in the couple stress lubrication of tapered land slider bearing provides the improved static and dynamic characteristics as compared with the non-magnetic case.

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Abbreviations

\( B_{0} \) :

applied magnetic field (Wb/m2)

\( \bar{C}_{d} \) :

non-dimensional dynamic damping coefficient

\( d \) :

difference between the inlet and outlet film thickness (m)

\( F \) :

frictional force (N)

\( \bar{F} \) :

non-dimensional frictional force, \( \bar{F} = \frac{{Fh_{ms}^{2} }}{{\mu UL^{2} B_{0} }} \)

\( h(x,t) \) :

film thickness (m)

\( \bar{h} \) :

non-dimensional film thickness, \( \bar{h}(\bar{x},\,\bar{t}) = h(x,\,t)/h_{ms} \)

\( h_{m} (t) \) :

minimum squeezing film thickness (m)

\( \bar{h}_{m} (\bar{t}) \) :

non-dimensional minimum squeezing film thickness \( \bar{h}_{m} (\bar{t}) = h_{m} (t)/h_{ms} \)

\( h_{ms} \) :

steady-state reference minimum film thickness at outlet (m)

\( h_{1} \) :

inlet film thickness (m)

L :

length of the bearing (m)

\( l \) :

couple stress parameter (m)

\( \bar{l} \) :

non-dimensional couple stress parameter, \( \bar{l} = 2l/h_{ms} \)

\( M \) :

Hartmann number, \( M = B_{0} h_{ms} \left( {\sigma /\mu } \right)^{1/2} \)

\( p \) :

film pressure (Pa)

\( p_{s} \) :

steady film pressure (Pa)

\( \bar{p} \) :

non-dimensional film pressure, \( \bar{p} = ph_{ms}^{2} /\mu UL \)

\( \bar{p}_{s} \) :

non-dimensional steady film pressure

\( \bar{S}_{d} \) :

non-dimensional dynamic damping coefficient

\( t,\,\bar{t} \) :

time, \( \bar{t} = Ut /L \) (s)

U:

sliding velocity of lower part (m/s)

\( \bar{V} \) :

non-dimensional squeezing velocity, \( \bar{V} = d\bar{h}_{m} /d\bar{t} \)

u, w :

velocity components in x and z directions

\( W_{s} \) :

steady load carrying capacity (N)

\( \bar{W}_{s} \) :

non-dimensional steady load carrying capacity,

x, z :

Cartesian coordinates

\( \bar{x} \) :

non-dimensional coordinate \( \bar{x} = x /L \)

\( \delta \) :

profile parameter \( \delta = d /h_{ms} \)

\( \eta \) :

material constant responsible for couple stress parameter (Ns)

\( \mu \) :

lubricant viscosity (Pa s)

\( \sigma \) :

conductivity of the lubricant (mho/m)

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Author information

Authors and Affiliations

  1. Department of Mathematics, Gulbarga University, Kalaburagi, 585106, India

    N B Naduvinamani & S Shridevi

  2. Department of Mathematics, Appa Institute of Engineering and Technology, Kalaburagi, 585103, India

    A Siddangouda

  3. Department of Science, Government Polytechnic, Shorapur, 585224, India

    Siddharama Patil

Authors
  1. N B Naduvinamani
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  2. A Siddangouda
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  3. Siddharama Patil
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  4. S Shridevi
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Corresponding author

Correspondence to N B Naduvinamani.

Appendix A

Appendix A

$$ f_{11} = \frac{{\beta^{2} }}{{\left( {\alpha^{2} - \beta^{2} } \right)}}\left\{ {\frac{{\sinh \left( {\frac{\alpha h}{l}} \right) - \sinh \left( {\frac{\alpha z}{l}} \right) + \sinh \left( {\frac{{\alpha \left( {h - z} \right)}}{l}} \right)}}{{\sinh \left( {\frac{\alpha h}{l}} \right)}}} \right\} $$
(A1a)
$$ f_{12} = \frac{{\alpha^{2} }}{{\left( {\alpha^{2} - \beta^{2} } \right)}}\left\{ {\frac{{\sinh \left( {\frac{\beta h}{l}} \right) - \sinh \left( {\frac{\beta z}{l}} \right) + \sinh \frac{{\beta \left( {h - z} \right)}}{l}}}{{\sinh \left( {\frac{\beta h}{l}} \right)}}} \right\} $$
(A1b)
$$ f_{13} = \frac{{\beta^{2} }}{{\frac{{\beta^{2} }}{\alpha }\tanh \left( {\frac{\alpha h}{2l}} \right) - \frac{{\alpha^{2} }}{\beta }\tanh \left( {\frac{\beta h}{2l}} \right)}}\left\{ {\frac{{\sinh \left( {\frac{\alpha h}{l}} \right) - \sinh \left( {\frac{\alpha z}{l}} \right) - \sinh \frac{{\alpha \left( {h - z} \right)}}{l}}}{{\sinh \left( {\frac{\alpha h}{l}} \right)}}} \right\} $$
(A1c)
$$ f_{14} = \frac{{\alpha^{2} }}{{\frac{{\beta^{2} }}{\alpha }\tanh \left( {\frac{\alpha h}{2l}} \right) - \frac{{\alpha^{2} }}{\beta }\tanh \left( {\frac{\beta h}{2l}} \right)}}\left\{ {\frac{{\sinh \left( {\frac{\beta h}{l}} \right) - \sinh \left( {\frac{\beta z}{l}} \right) - \sinh \frac{{\beta \left( {h - z} \right)}}{l}}}{{\sinh \left( {\frac{\beta h}{l}} \right)}}} \right\} $$
(A1d)
$$ f_{21} = \frac{{{\sin}h\left( {\frac{z - h}{\sqrt 2 l}} \right) + {\sin}h\left( {\frac{z}{\sqrt 2 l}} \right) - {\sin}h\left( {\frac{h}{\sqrt 2 l}} \right)}}{{{\sin}h\left( {\frac{h}{\sqrt 2 l}} \right)}} $$
(A1e)
$$ f_{22} = \frac{{z{\cos}h\left( {\frac{z - h}{\sqrt 2 l}} \right) + y{\cos}h\left( {\frac{z}{\sqrt 2 l}} \right) - h\coth \left( {\frac{h}{2\sqrt 2 l}} \right)\sinh \left( {\frac{z}{\sqrt 2 l}} \right)}}{{2\sqrt 2 l\,{\sin}h\left( {\frac{h}{\sqrt 2 l}} \right)}} $$
(A1f)
$$ f_{23} = \frac{{z{\sin}h\left( {\frac{z - h}{\sqrt 2 l}} \right) + z\sinh \left( {\frac{z}{\sqrt 2 l}} \right) - h\sinh \left( {\frac{z}{\sqrt 2 l}} \right)}}{{\left( {6l\,{\sin}h\left( {\frac{h}{\sqrt 2 l}} \right) - \sqrt 2 h} \right)}} $$
(A1g)
$$ f_{24} = \frac{{2{\cos}{h}\left( {\frac{z - h}{\sqrt 2 l}} \right) + 2{\cos}{h}\left( {\frac{z}{\sqrt 2 l}} \right) - 2{\cos}{h}\left( {\frac{h}{\sqrt 2 l}} \right) - 2}}{{\left( {3\sqrt 2 {\sin}{h}\left( {\frac{h}{\sqrt 2 l}} \right) - \frac{h}{l}} \right)}} $$
(A1h)
$$ f_{31} = \frac{{{{{\cosh}}} \alpha_{1} z{\cos}\beta_{1} \left( {z - h} \right) - {\cos} \beta_{1} z{\cos}h\alpha_{1} \left( {z - h} \right)}}{{\left( {{\cos}h\alpha_{1} h - {\cos}\beta_{1} h} \right)}} $$
(A1i)
$$ f_{32} = \frac{{\cot\varphi \left\{ {{{{\sinh}}} \alpha_{1} z{\sin}\beta_{1} \left( {z - h} \right) - {\sin} \beta_{1} z{\sin}h\alpha_{1} \left( {z - h} \right)} \right\} + \left( {{\cos}h\alpha_{1} h - {\cos}\beta_{1} h} \right)}}{{\left( {{\cos}h\alpha_{1} h - {\cos}\beta_{1} h} \right)}} $$
(A1j)
$$ f_{33} = \frac{M}{{h_{ms} }}\left\{ {\frac{{\cot\varphi \left\{ {{{\sin}h}\alpha_{1} z\sin\beta_{1} \left( {z - h} \right) + \sin\beta_{1} z{\sin}h\alpha_{1} \left( {z - h} \right)} \right\} + \left( {\cos\beta_{1} h + {\cos}h\alpha_{1} h} \right)}}{{\left( {\beta_{1} - \alpha_{1} \cot\varphi } \right)\sin\beta_{1} h + \left( {\alpha_{1} + \beta_{1} \cot\varphi } \right){\sin}h\alpha_{1} h}}} \right\} $$
(A1k)
$$ f_{34} = \frac{M}{{h_{ms} }}\left\{ {\frac{{{\cos} \beta_{1} z{\cos}h\alpha_{1} \left( {z - h} \right) + {\cosh} \alpha_{1} z\cos\beta_{1} \left( {z - h} \right)}}{{\left( {\beta_{1} - \alpha_{1} \cot\varphi } \right)\sin\beta_{1} h + \left( {\alpha_{1} + \beta_{1} \cot\varphi } \right){\sin}h\alpha_{1} h}}} \right\} $$
(A1l)
$$ \xi_{A} \left( {\bar{x},\,\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 0}}^{{\bar{x}}} {\frac{{\bar{h}_{1} (\bar{x},t)}}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}} d\bar{x} $$
(A2a)
$$ \xi_{B} \left( {\bar{x},\,\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 0}}^{{\bar{x}}} {\frac{{\bar{x}}}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}} d\bar{x} $$
(A2b)
$$ \xi_{C} \left( {\bar{x},\,\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 0}}^{{\bar{x}}} {\frac{1}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}} d\bar{x} $$
(A2c)
$$ \xi_{D} \left( {\bar{x},\,\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 1}}^{{\bar{x}}} {\frac{{\bar{x}}}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}} d\bar{x} $$
(A2d)
$$ \xi_{E} \left( {\bar{x},\,\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 1}}^{{\bar{x}}} {\frac{1}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}} d\bar{x} $$
(A2e)
$$ \xi_{AK} \left( {\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 0}}^{K} {\frac{{\bar{h}_{1} (\bar{x},t)}}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}} d\bar{x} $$
(A2f)
$$ \xi_{BK} \left( {\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 0}}^{K} {\frac{{\bar{x}}}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}} d\bar{x} $$
(A2g)
$$ \xi_{CK} \left( {\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 0}}^{K} {\frac{1}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}} d\bar{x} $$
(A2h)
$$ \xi_{DK} \left( {\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 1}}^{K} {\frac{{\bar{x}}}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}} d\bar{x} $$
(A2i)
$$ \xi_{EK} \left( {\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 1}}^{K} {\frac{1}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}} d\bar{x} $$
(A2j)
$$ \chi_{AK} \left( {\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 0}}^{K} {\int\limits_{{\bar{x} = 0}}^{{\bar{x}}} {\frac{{\bar{h}_{1} (\bar{x},t)}}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}d\bar{x}\,} } d\bar{x} $$
(A3a)
$$ \chi_{BK} \left( {\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 0}}^{K} {\int\limits_{{\bar{x} = 0}}^{{\bar{x}}} {\frac{{\bar{x}}}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}d\bar{x}\,} } d\bar{x} $$
(A3b)
$$ \chi_{CK} \left( {\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = 0}}^{K} {\int\limits_{{\bar{x} = 0}}^{{\bar{x}}} {\frac{1}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}d\bar{x}\,} } d\bar{x} $$
(A3c)
$$ \chi_{DK} \left( {\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = K}}^{1} {\int\limits_{{\bar{x} = 1}}^{{\bar{x}}} {\frac{{\bar{x}}}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}d\bar{x}\,} } d\bar{x} $$
(A3d)
$$ \chi_{EK} \left( {\bar{h}_{m} } \right) = \int\limits_{{\bar{x} = K}}^{1} {\int\limits_{{\bar{x} = 1}}^{{\bar{x}}} {\frac{1}{{\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)}}d\bar{x}\,} } d\bar{x} $$
(A3e)
$$ \frac{{\partial \chi_{AK} }}{{\partial \bar{h}_{m} }} = - \int\limits_{{\bar{x} = 0}}^{K} {\int\limits_{{\bar{x} = 0}}^{{\bar{x}}} {\frac{{\bar{h}_{1} \left( {\bar{x}} \right)}}{{\left\{ {\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)} \right\}^{2} }}\frac{{\partial \bar{g}}}{{\partial \bar{h}_{m} }}} }\,d\bar{x}\,d\bar{x} $$
(A4a)
$$ \frac{{\partial \chi_{CK} }}{{\partial \bar{h}_{m} }} = - \int\limits_{{\bar{x} = 0}}^{K} {\int\limits_{{\bar{x} = 0}}^{{\bar{x}}} {\frac{1}{{\left\{ {\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)} \right\}^{2} }}\frac{{\partial \bar{g}}}{{\partial \bar{h}_{m} }}} }\,d\bar{x}\,d\bar{x} $$
(A4b)
$$ \frac{{\partial \chi_{EK} }}{{\partial \bar{h}_{m} }} = - \int\limits_{{\bar{x} = K}}^{1} {\int\limits_{{\bar{x} = 1}}^{{\bar{x}}} {\frac{1}{{\left\{ {\bar{g}\left( {\bar{h},\,\bar{l},\,M} \right)} \right\}^{2} }}\frac{{\partial \bar{g}}}{{\partial \bar{h}_{m} }}} }\,d\bar{x}\,d\bar{x} $$
(A4c)
$$ \left( {\frac{{\partial C_{1} }}{{\partial \bar{h}_{m} }}} \right)_{s} = - 6\left[ {\frac{{\left\{ {\xi_{CK} - \xi_{EK} } \right\}\frac{{\partial \xi_{AK} }}{{\partial \bar{h}_{m} }} - \xi_{AK} \left( {\frac{{\partial \xi_{CK} }}{{\partial \bar{h}_{m} }} - \frac{{\partial \xi_{EK} }}{{\partial \bar{h}_{m} }}} \right)}}{{\left\{ {\xi_{CK} - \xi_{EK} } \right\}^{2} }}} \right] $$
(A4d)
$$ \left( {\frac{{\partial C_{1} }}{{\partial \bar{V}}}} \right)_{s} = 12\left[ {\frac{{\xi_{DK} - \xi_{BK} }}{{\xi_{CK} - \xi_{EK} }}} \right] $$
(A4e)

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Naduvinamani, N.B., Siddangouda, A., Patil, S. et al. Effect of couple stresses on static and dynamic characteristics of MHD wide tapered land slider bearing. Sādhanā 43, 162 (2018). https://doi.org/10.1007/s12046-018-0940-9

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