Effect of slip and percolation of polar additives of coupled-stress lubricant on the steady-state characteristics of double-layered porous journal bearings

Abstract

A theoretical study has been made into the steady-state solution of externally pressurized double-layered porous journal bearings lubricated with coupled-stress fluid with tangential velocity slip at the fine porous interface. The analysis takes into account the tangential velocity slip based on the Beavers–Joseph criterion. Moreover, the present study includes the effects of percolation of the polar additives (microstructures) into the coarse and fine layers of a porous medium. The most general modified Reynolds type equation has been derived for a porous journal bearing lubricated with coupled stress fluids. The governing equations are derived for flow in the coarse and fine layers of the porous medium incorporating the percolation of polar additives of the lubricant. The effects of slip, speed parameter, percolation factor and coupled-stress parameter on the static characteristics in terms of load capacity, attitude angle, side leakage and frictional parameter were investigated. The results are exhibited in the form of graphs.

Appendix

The summary of the derivation procedure of modified Reynold’s Eq. (5) in the clearance region of porous bearing is as follows.

Slip velocity along the x direction is:

$$U_{S} = U\xi_{0X} - \frac{{h^{2} \xi_{x} }}{{6\mu \left( {1 - \gamma_{xf} } \right)}}\frac{\partial p}{\partial x} + \frac{h}{2\mu }\xi_{0X} \frac{\partial p}{\partial x}2l\tanh \left( {\frac{h}{2l}} \right).$$

(9)

Slip velocity along the z direction is:

$$W_{\text{S}} = - \frac{{h^{2} \xi_{z} }}{{6\mu \left( {1 - \gamma_{zf} } \right)}}\frac{\partial p}{\partial z} + \frac{h}{2\mu }\xi_{0z} \frac{\partial p}{\partial z}2l\tanh \left( {\frac{h}{2l}} \right),$$

(10)

where

$$\xi_{0n} = \frac{{\sqrt {k_{nf} } }}{{\alpha \left( {h + \frac{{\sqrt {k_{nf} } }}{\alpha }} \right)}} \xi_{n} = \frac{{3\sqrt {k_{nf} } \left\{ {2\alpha \sqrt {k_{nf} } + h\left( {1 - \gamma_{nf} } \right)} \right\}}}{{h\alpha \left( {h + \frac{{\sqrt {k_{nf} } }}{\alpha }} \right)}} n = x,z.$$

The velocity component, $u\left( {x,y} \right)$, along the circumferential direction,

$$\begin{aligned} u\left( {x,y} \right) & = U\left( {\frac{y}{h}} \right) + U\xi_{0X} \left( {1 - \frac{y}{h}} \right) - \frac{{h^{2} \xi_{x} }}{{6\mu \left( {1 - \gamma_{xf} } \right)}}\frac{\partial p}{\partial x}\left( {1 - \frac{y}{h}} \right) \\ & \quad \quad \quad \quad + \frac{h}{2\mu }\xi_{0X} \frac{\partial p}{\partial x}2l\tanh \left( {\frac{h}{2l}} \right)\left( {1 - \frac{y}{h}} \right) \\ & \quad \quad \quad \quad + \frac{1}{2\mu }\frac{\partial p}{\partial x}\left[ {y\left( {y - h} \right) + 2l^{2} \left\{ {1 - \frac{{\cosh \left( {\frac{2y - h}{2l}} \right)}}{{\cosh \left( {\frac{h}{2l}} \right)}}} \right\}} \right]. \\ \end{aligned}$$

(11)

The velocity component, $w\left( {z,y} \right)$, along the axial direction,

$$w\left( {z,y} \right) = - \frac{{h^{2} \xi_{z} }}{{6\mu \left( {1 - \gamma_{zf} } \right)}}\frac{\partial p}{\partial z}\left( {1 - \frac{y}{h}} \right) + \frac{h}{2\mu }\xi_{0z} \frac{\partial p}{\partial z}2l\tanh \left( {\frac{h}{2l}} \right)\left( {1 - \frac{y}{h}} \right) + \frac{1}{2\mu }\frac{\partial p}{\partial z}\left[ {y\left( {y - h} \right) + 2l^{2} \left\{ {1 - \frac{{\cosh \left( {\frac{2y - h}{2l}} \right)}}{{\cosh \left( {\frac{h}{2l}} \right)}}} \right\}} \right].$$

(12)

The flow rate $q_{x}$ in the x direction per unit width of z is given by

$$q_{x} = \mathop \int \limits_{0}^{h} u.{\text{d}}y = \left( {\frac{1}{12\mu }} \right)\left[ {6\mu Uh\left( {1 + \xi_{0X} } \right) - f\left( {h,k_{xf} ,l,\gamma_{xf} } \right)\frac{\partial p}{\partial x}} \right],$$

(13)

where $f\left( {h,k_{xf} ,l,\gamma_{xf} } \right) = h^{3} \left( {1 + \frac{{\xi_{x} }}{{\left( {1 - \gamma_{xf} } \right)}}} \right) - 6h^{2} \xi_{0X} l\tanh \left( {\frac{h}{2l}} \right) - 12l^{2} \left\{ {h - 2l\tanh \left( {\frac{h}{2l}} \right)} \right\}.$Similarly,

$$q_{z} = \mathop \int \limits_{0}^{h} w.{\text{d}}y = \left( {\frac{1}{12\mu }} \right)\left[ { - f\left( {h,k_{zf} ,l,\gamma_{zf} } \right)\frac{\partial p}{\partial z}} \right],$$

(14)

where $f\left( {h,k_{zf} ,l,\gamma_{zf} } \right) = h^{3} \left( {1 + \frac{{\xi_{z} }}{{\left( {1 - \gamma_{zf} } \right)}}} \right) - 6h^{2} \xi_{0z} l\tanh \left( {\frac{h}{2l}} \right) - 12l^{2} \left\{ {h - 2l\tanh \left( {\frac{h}{2l}} \right)} \right\}.$The flow continuity equation is:

$$\frac{{\partial q_{x} }}{\partial x} + \frac{{\partial q_{z} }}{\partial z} + \left( {v_{h} - v_{0} } \right) = 0.$$

(15)

Here,$v_{h}$ = velocity of the journal surface at y = h

$$={\text{squeeze\;velocity}}= \frac{\partial h}{\partial t} {\text{(it\;is\;neglected\;in\;steady-state\;analysis)}}.$$

$v_{0}$ = velocity of fluid (filter velocity) at y = 0,

$$= v^{'} = - \frac{{k_{yf} }}{{\mu \left( {1 - \gamma_{yf} } \right)}}\left. {\frac{{\partial p^{'}_{f} }}{\partial y}} \right|_{y = 0} .$$

Modified Reynold’s Eq. (5) is established with the help of Eqs. (13), (14) and (15).