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Modeling and Analysis of the Slip Conditions in Hydrodynamic Lubrication

  • Research Article - Mechanical Engineering
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Abstract

In hydrodynamic lubrication, the slip conditions have been studied previously from the model of the slip length and the wall shear stress, but recently, in the literature on the slip at the interface fluid–solid, it has been shown that many questions will remain unsolved for long time to come. In fact, the modeling of the back-flow near the wall was not really taken into account. In this paper, we propose a more rigorous writing of the Reynolds equation (modified), taking into account the phenomena of slip and back-flow at wall. Subsequently, this equation is applied to the case of an inclined pad bearing and the results show that there is a significant influence of those phenomena on the operating characteristics.

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Abbreviations

A :

Dimensionless slip coefficient (A = α η/ h 2)

B (m):

Width of the inclined pad in the z-direction

b (μ m):

Slip length

\({\overline {F_m } }\) :

Dimensionless frictional force (\({\overline {F_m } =Fm.h_2 /ULB\eta )}\)

Fm (N):

Frictional force

H :

Dimensionless fluid-film thickness (H = h/ h 1)

h (μm):

Fluid-film thickness

h 1 (μm):

Fluid-film thickness at the entry

h 2 (μm):

Fluid-film thickness at the exit

L(m):

Length of the inclined pad in the x-direction

L xz :

Report of L/B

Oxyz:

Cartesian coordinate system

P :

Dimensionless pressure (\({P=ph_2^2 /\eta UL)}\)

p (Pa):

Pressure

Pu (W):

Dissipated power

\({\overline Q }\) :

Dimensionless carried load (\({\overline Q = Qh_2^2 /BL^2\eta U)}\)

Q (N):

Carried load

\({\overline {q_x }}\) :

Dimensionless flow rate (\({\overline {q_x } =q_x /UBh_2 )}\)

q x (m2/s):

Flow rate per length unit in the x direction

q z (m2/s):

Flow rate per length unit in the z direction

T c :

Dimensionless critical surface shear stress (\({T_c =\tau _c h_2 /\eta U)}\)

U (m/s):

Velocity of the bottom wall of the inclined pad

U(y):

Dimensionless velocity (U(y)=u(y)/U)

u p (m/s):

Fluid velocity at the wall

u, v, w (m/s):

Fluid velocities in the x, y, z directions

U 1, U 2 (m/s):

Velocities of the walls 1 and 2 in the x-direction

V 1, V 2 (m/s):

Velocities of the walls 1 and 2 in the y−direction

W 1, W 2 (m/s):

Velocities of the walls 1 and 2 in the z-direction

X :

Dimensionless length of the inclined pad in the x-direction (X = x/L)

Y :

Dimensionless height of the inclined pad in the y-direction (Y = y/ h 2)

Z :

Dimensionless width of the inclined pad in the z-direction (Z = z/B)

α (m/Pa s):

Slip coefficient

\({\dot {\gamma }_p }\) (s−1):

Wall shear rate

η (Pa s):

Dynamic viscosity of the lubricant

ρ(kg/m3):

Density of the lubricant

τ (Pa):

Shear stress in the fluid

τ c (Pa):

Critical surface shear stress

τ p (Pa):

Wall shear stress

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Authors and Affiliations

  1. Industrial Fluids, Measurements and Applications Laboratory, Faculty of Sciences and Technology, Khemis Miliana University, Route de Theniet El-Had Khemis, 44225, Aïn-Defla, Algeria

    Ibrahim Zidane

  2. Rheology and Mechanics Laboratory, Faculty of Technology, Hassiba Benbouali University, Chlef, Algeria

    Hamou Zahloul & Mohammed Abbadeni

  3. Mechanical Engineering Department and complex systems, Pprime Institute, UPR3346, CNRS-Poitiers University-ENSMA, Poitiers, France

    Mohamed Hajjam

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  1. Ibrahim Zidane
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  2. Hamou Zahloul
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  4. Mohammed Abbadeni
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Correspondence to Ibrahim Zidane.

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Zidane, I., Zahloul, H., Hajjam, M. et al. Modeling and Analysis of the Slip Conditions in Hydrodynamic Lubrication. Arab J Sci Eng 39, 7199–7210 (2014). https://doi.org/10.1007/s13369-014-1214-4

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  • DOI: https://doi.org/10.1007/s13369-014-1214-4

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