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Lubrication of rollers by power law fluids considering consistency-variation with pressure and temperature

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Summary

The steady state lubrication of two rigid cylindrical rollers under entraining and squeezing motion including cavitation is studied. The lubricant consistencym of the power law fluid is assumed to be pressure and temperature dependent. A numerical solution to the coupled Reynolds' and energy equation for pressure and temperature has been obtained using higher order predictor corrector method. Also several other bearing characteristics such as load, traction ratios, maximum pressure, location of pressure maximum, temperature ratios etc. are calculated and discussed in detail for different values of the power law index including the Newtonian fluid behaviour. A comparison of the present results with those under isothermal condition is also made. It is concluded that the effect of temperature is to shift the position of the pressure peak slightly away from the center line of contact, whereas the trend is opposite for the cavitation points. The variation in consistency is found to be very significant, especially in the pressure peak region.

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Abbreviations

c :

specific heat

c n :

\(\left( {\frac{{4n + 2}}{n}} \right)^n \left( {\frac{U}{{h_0 }}} \right)^n \left( {\frac{{2R}}{{h_0 }}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\)

F :

X 2-X 1 2+ 2q(X+X 1)

G :

−F

H :

1+X 2

h :

film thickness

h 0 :

minimum film thickness

h 1 :

film thickness at maximum pressure

K :

thermal conductivity of the lubricant

m :

consistency of the fluid

m 0 :

consistency at ambient pressure and temperature

\(\bar m\) :

non-dimensional consistency (=2mc n α)

\(\bar m_0\) :

2m o c n α

n :

flow behaviour index

p :

hydrodynamic pressure

\(\bar p\) :

non-dimensional pressure (=αp)

\(\bar p_{\max }\) :

maximum pressure

\(\bar p_{\max R}\) :

maximum pressure ratio

q :

velocity ratio parameter\(\left( {\frac{V}{U}\left( {\frac{R}{{2h_0 }}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right)\)

R :

equivalent radius

r :

radius of the cylinder

S :

1+X 1 2-2q(X+X 1)

T :

temperature

T 0 :

ambient temperature

\(\bar T\) :

β(T-T 0)

U :

rolling speed of cylinder

u :

velocity of lubricant inx-direction

V :

normal velocity of cylinder

v :

velocity of lubricant imy-direction

v h :

velocity aty=h/2

X :

x/(2Rh 0)1/2

X 1 :

x 1 2Rh 0 1/2

X 2 :

x 2 2Rh 0 1/2

x, y :

coordinate axes

W x +,W y + :

load components inx andy-directions, respectively

W x ,W y :

non-dimensional load components inx andy-directions respectively

\(\bar W_x ,\bar W_y\) :

load ratio components inx andy-directions respectively

\(\bar W_{Y_i }\) :

isothermal load ratio

T F :

traction force

\(\bar T_F\) :

non-dimensional traction,TF/(h 0/2α)

\(\bar T_{FR}\) :

traction ratio,\(\bar T_F (q \ne 0)/\bar T_F (q = 0)\)

\(\bar T_{FR_i }\) :

isothermal traction ratio

α:

pressure coefficient, Eq. (4)

β:

temperature coefficient, Eq. (4)

γ:

β(ϱcα)

ϱ:

density of the lubricant

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Author notes

  1. P. Sinha

    Present address: Department of Mathematics, Indian Institute of Technology, 208016, Kanpur, India

  2. D. Prasad

    Present address: Department of Mathematics, Dr. S. R. K. Govt. College, Yanam, 533464, Pondicherry, India

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    1. P. Sinha
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    2. D. Prasad
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    Additional information

    Subscripts 1 and 2 refer to inlet and outlet zones in case of pressure, temperature, and consistency.

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    Sinha, P., Prasad, D. Lubrication of rollers by power law fluids considering consistency-variation with pressure and temperature. Acta Mechanica 111, 223–239 (1995). https://doi.org/10.1007/BF01376932

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