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Ferrofluid lubrication of porous-rough circular squeeze film bearings

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Abstract

In this paper, porous-roughness effects on different circular discs squeeze film bearing designs lubricated with ferrofluid (FF) are studied. Using FF flow model by Rosensweig and continuity equation for the film as well as porous region, modified stochastic Reynolds–Darcy equation is derived for variable magnetic field by considering effects of porosity together with roughness (circumferential and radial) at the lower disc and rotations of both the discs, under the assumption of validity of the Darcy’s law in the porous region. This equation is then applied to predict and compare performances of parallel, exponential, secant and mirror image of secant squeeze film bearing designs for load-carrying capacity \(\overline{W}\). According to results, higher values of circumferential roughness parameter and curvature of the upper discs at the centre in downward direction, and smaller values of radial permeability parameter and lateral surface area of the generated opening cylindrical surface gives better performance of the system. In addition, the performance is more pronounced in the case of exponential squeeze film bearing. The following two empirical relations are also established.

\(\overline{W} \propto Curvature\;of\;the\;upper\;disc\;at\;the\;centre\;in\;downward\;direction,\;\)

\( \overline{W} \propto \frac{1}{{Lateral\;Surface\;Area\;of\;the\;generated\;opening\;cylindrical\;surface}}. \)

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Abbreviations

a :

Radius of both the circular discs (upper and lower) (m)

c :

Maximum asperity deviation from nominal film height (m)

C :

Dimensionless surface roughness parameter defined in Eq. (40)

E :

Expectancy operator defined in Eq. (35)

E(p):

Mean film pressure (N/m2)

E(W):

Mean load-carrying capacity (N)

F :

Body force

FF:

Ferrofluid

f :

Probability density function

h 0 :

Central film thickness at time t = 0 (m)

h s :

Deviation of film height from nominal level (m)

h n :

Nominal film height (m)

h :

Film thickness (m)

\(\dot{h}_{0}\) :

Normal velocity (or Squeeze velocity), \(\frac{{dh_{0} }}{dt}\;\;(\text{m}/\text{s})\)

H :

Magnetic field vector

H :

Strength of H (A/m)

\(H_{r} ,H_{z}\) :

Components of the applied magnetic field vector H in r and z-directions

H * :

Thickness of the porous matrix (m)

K :

Quantity defined in Eq. (39) (A2/m4)

\(({\text{L.S.A.}})_{e}\) :

Lateral surface area of the generated opening cylindrical surface for exponential squeeze film bearing

\(({\text{L.S.A.}})_{is}\) :

Lateral surface area of the generated opening cylindrical surface for mirror image of secant squeeze film bearing

\(({\text{L.S.A.}})_{p}\) :

Lateral surface area of the generated opening cylindrical surface for parallel squeeze film bearing

\(({\text{L.S.A.}})_{s}\) :

Lateral surface area of the generated opening cylindrical surface for secant squeeze film bearing

M :

Magnetization vector

\(M_{r} ,M_{z}\) :

Components of the magnetization vector M in r and z-directions

MF:

Magnetic fluid

p :

Film pressure (N/m2)

P :

Fluid pressure in the porous matrix (N/m2)

q :

Fluid velocity vector

s :

Slip parameter (1/m)

t :

Time (s)

VMF:

Variable magnetic field

w.r.t.:

With respect to

W :

Load-carrying capacity (N)

\(\overline{W}\) :

Dimensionless load-carrying capacity defined in Eq. (45)

\(\overline{W}_{e}\) :

Dimensionless load-carrying capacity for exponential squeeze film bearing

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

Dimensionless load-carrying capacity for secant squeeze film bearing

\(\overline{W}_{is}\) :

Dimensionless load-carrying capacity for mirror image of secant squeeze film bearing

\(\overline{W}_{p} \;\) :

Dimensionless load-carrying capacity for parallel squeeze film bearing

r,θ, z :

Cylindrical polar co-ordinates

u, v, w :

Radial, tangential and axial (or transverse) velocity components of q

\(\overline{u}\,\), \(\overline{w}\) :

Radial and axial velocity components of fluid velocity in the porous matrix

α :

Inclination of H with the lower plate

\(\beta\) :

Curvature of the upper discs (1/m2)

\(\overline{\beta }\) :

\(\beta\) a2, Dimensionless curvature parameter of the upper disc

\(\xi\) :

Random variable

σ :

Standard deviation

\(\eta\) :

Viscosity of the FF suspension (N s/m2)

\(\eta_{r}\) :

Porosity of the porous matrix in r-direction

\(\mu_{{0}}\) :

Free space permeability (N/A2)

\(\rho\) :

Fluid density (N s2/m4)

\(\phi\) :

Scalar function

\(\phi_{r} {,}\;\phi_{z}\) :

Permeabilities of the porous matrix in r and z-directions, respectively (m2)

\(\Omega _{u}\) :

Angular (or rotational) velocity of the impermeable upper disc (rad./s)

\(\Omega _{l}\) :

Angular (or rotational) velocity of the porous-rough lower disc (rad./s)

\(\overline{\mu }\) :

Magnetic susceptibility

μ * :

Dimensionless magnetization parameter defined in Eq. (40)

ψ r :

Dimensionless radial permeability parameter defined in Eq. (40)

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  1. Department of Applied Mathematics, Faculty of Technology & Engineering, The Maharaja Sayajirao University of Baroda, Vadodara, Gujarat, 390 001, India

    Rajesh C. Shah

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Shah, R.C. Ferrofluid lubrication of porous-rough circular squeeze film bearings. Eur. Phys. J. Plus 137, 190 (2022). https://doi.org/10.1140/epjp/s13360-022-02344-z

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