# Effect of Turbulence on Stability of Journal Bearing with Micropolar Lubrication: Linear and Non-linear Analysis

## Abstract

The object of the present article is to theoretically study the effect of turbulence on the stability of finite hydrodynamic journal bearing lubricated with micropolar fluid. Both linear and non-linear stability analyses have been carried out. In linear stability analysis, first-order perturbation method has been applied to obtain the steady-state and dynamic pressure equations. These equations have been solved to obtain the steady-state and dynamic pressures. These pressures were used to compute the dynamic film forces along with the stiffness and damping coefficients. These force components and dynamic response coefficients have been used to obtain the critical mass parameter and whirl ratio. In non-linear analysis, the transient Reynolds equation was solved using successive over relaxation scheme to obtain dynamic pressure field which in turn used to obtain the film forces and steady-state load capacity. The equations of motion have been solved using fourth-order Runge–Kutta method to obtain the threshold of stability. It is observed that the stability of the journal bearing system decreases with increase in turbulence.

## Keywords

Journal bearing Micropolar Non-linear Stability Turbulence## Nomenclature

*A*_{θ}Constant parameter of turbulent shear coefficient for circumferential flow (dimensionless)

*B*_{θ}Exponential constant parameter of turbulent shear coefficient for circumferential flow (dimensionless)

*C*Radial clearance (m)

- \(C_{{\overline{z} }}\)
Constant parameter of turbulent shear coefficient for axial flow (dimensionless)

*D*Journal diameter, (m)

- \(D_{ij}\)
Damping coefficients of micropolar fluid film, \(i = r,\phi\) and \(j = r,\phi\) (N s/m)

- \(\overline{D}_{ij}\)
Dimensionless damping coefficients of micropolar fluid film, \(\overline{D}_{ij} = \frac{{2D_{ij} C^{3} }}{{\mu\Omega R^{3} L}}\), \(i = r,\phi\) and \(j = r,\phi\) (dimensionless)

- \(D_{{\bar{z}}}\)
Exponential constant parameter of turbulent shear coefficient for axial flow (dimensionless)

- \(F_{i}\)
Fluid film force components along \(r\)- and \(\phi\)-directions, \(i = r\,{\text{and}}\,\phi\) (N)

- \(\overline{F}_{i}\)
Non-dimensional force components along

*r*- and \(\phi\)-directions, \(\bar{F}_{i} = F_{i} C^{2} /\left( {\mu\Omega ^{2} R^{3} L} \right)\), \(i = r{\text{ and }}\varphi\) (dimensionless)*h*Local film thickness (m)

- \(\overline{h}\)
Non-dimensional film thickness, \(\overline{h} = h/C\) (dimensionless)

- \(k_{\theta } ,k_{{\overline{z} }}\)
Turbulent shear coefficient along circumferential direction and axial directions, respectively, (dimensionless)

*L*Length of bearing (m)

*l*_{m}Non-dimensional characteristics length of micropolar fluid (dimensionless)

*M*Rotor mass per bearing (kg)

- \(\overline{M}\)
Non-dimensional mass parameter, \(\overline{M} = MC\Omega ^{2} /W_{0}\) (dimensionless)

- \(\overline{M}_{\text{cr}}\)
Critical value of non-dimensional mass parameter (dimensionless)

*N*Coupling number (dimensionless)

*p*Local micropolar film pressure in the film region, (N/m

^{2})- \(\overline{p}\)
Non-dimensional local micropolar film pressure in the film region, \(\overline{p} = pC^{2} /\left( {\mu\Omega R^{2} } \right)\) (dimensionless)

*P*_{i}Local micropolar film pressure in the film region,

*i*= 0, 1 and 2 for the steady-state and first-order perturbed film pressures along*r*- and \(\phi\)-directions, (N/m^{2})- \(\overline{p}_{i}\)
Local micropolar film pressure in the film region, \(\overline{p}_{i} = \frac{{p_{i} C^{2} }}{{\mu\Omega R^{2} }}\),

*i*= 0, 1 and 2 for the steady-state and first-order perturbed film pressures along*r*- and \(\phi\)-directions (dimensionless)*R*Radius of the journal (m)

*Re*Mean or average Reynolds number defined by radial clearance,

*C*, \(\text{Re} = \rho\Omega RC/\mu\) (dimensionless)- \(S_{ij}\)
Stiffness coefficients of micropolar fluid film, \(i = r,\phi\) and \(j = r,\phi\) (N/m)

- \(\overline{S}_{ij}\)
Stiffness damping coefficients of micropolar fluid film, \(\overline{S}_{ij} = \frac{{2S_{ij} C^{3} }}{{\mu\Omega R^{3} L}}\), \(i = r,\phi\) and \(j = r,\phi\) (dimensionless)

*t*Time (s)

*U**U*velocity of journal,*U*= Ω*R*, (m/s)*W*Load in bearing, (N)

*W*_{0}Steady-state load in bearing (N)

- \(\overline{W}_{0}\)
Non-dimensional steady-state load in bearing, \(\overline{W}_{0} = W_{0} /\left( {\mu\Omega ^{2} R^{3} L} \right)\) (dimensionless)

*x*Cartesian coordinate axis in the circumferential direction,

*x*=*Rθ*, (m)*z*Cartesian coordinate axis along the bearing axis (m)

- \(\overline{z}\)
Non-dimensional Cartesian coordinate axis along the bearing axis, \(\overline{z} = 2 {\text{z/}}L\) (dimensionless)

## Greek Symbols

*ε*Eccentricity ratio (dimensionless)

*ε*_{0}Steady-state eccentricity ratio (dimensionless)

*ε*_{1}Perturbed eccentricity ratio (dimensionless)

*λ*Whirl ratio,

*λ*=*ω*_{p}*/ Ω*(dimensionless)- \(\phi\)
Attitude angle (rad)

- \(\phi_{0}\)
Steady-state attitude angle, (rad)

- \(\Phi _{x,z}\)
Micropolar fluid functions along circumferential and axial directions

- \(\Phi _{{\theta ,\overline{z} }}\)
Non-dimensional micropolar fluid functions along circumferential and axial directions (dimensionless)

- Λ
Characteristics length of the micropolar fluid,

*Λ*=*(γ/*4*μ)*^{1/2}*γ*Viscosity coefficient of micropolar fluid, (N s)

*μ*Newtonian viscosity coefficient, (Pa s)

*ω*_{p}Angular velocity of the orbital motion of the journal centre, (rad/s)

- Ω
Angular velocity of journal (rad/s)

*ω*_{p}Angular velocity of the orbital motion of the journal centre (rad/s)

*θ*Circumferential coordinate (rad)

*θ*_{c}Circumferential coordinate where the film cavitates, (rad)

*τ*Non-dimensional time, \(\tau =\Omega t\) (dimensionless)

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