Effect of MHD with Micropolar Fluid Between Conical Rough Bearings

Figure 1 indicates the configuration of rough conical bearings in the existence of a transverse magnetic field. In this figure the thickness of fluid film between the plates is h. The magnetic field $B_0$ is applied along the z-direction. In the film area, micropolar fluid is taken into consideration as the lubricant. where u and v are the velocity components in r and z direction, $v_1$ is the micro-rotational velocity and $\sigma$ represents the electrical conductivity of the fluid. The parameter $\mu$ is the viscosity parameter of the base fluid as in the case of the Newtonian fluids, $\chi$ is the spin viscosity and $\gamma$ is the viscosity coefficient for micropolar fluid.

1.jpg

Figure 1. Configuration of the problem

The micropolar fluid equations are governed by

$\frac{1}{r} \frac{\partial(r u)}{\partial r}+\frac{\partial v}{\partial z}=0$       (1)

$\left(\mu+\frac{\chi}{2}\right) \frac{\partial^2 u}{\partial z^2}+\chi \frac{\partial v_1}{\partial z}-\sigma B_o^2 u=\frac{\partial p}{\partial r}$    (2)

$\gamma \frac{\partial^2 v_1}{\partial z^2}-\chi \frac{\partial u}{\partial z}-2 \chi v_1=0$     (3)

The relevant boundary conditions for the velocity and micro rotational velocity components are

(i) At the upper surface $z=h \sin \theta$

$u=0, v_1=0, v=\frac{d(2 h) \sin \theta}{d t}$    (4)

(ii) At the lower surface $z=-h \sin \theta$

$u=0, v_1=0, v=0$   (5)

These are the usual no-slip condition on velocity components for micropolar fluid. where the film height is h in the direction of the cone axis, the cone angle is $2 \theta$ and radius units a. The inner cone moves toward the housing with a squeezing velocity, $v_{\mathrm{sq}}=\frac{-d(2 h)}{d t} \sin \theta$ .

Eliminating $v_1$ from Eq. (2). and (3) gives

$\frac{\partial^4 u}{\partial z^4}-\xi_1 \frac{\partial^2 u}{\partial z^2}+\xi_2 u=f_1$   (6)

$\xi_1=\frac{4 \mu \chi+2 \sigma B_o^2}{\gamma(2 \mu+\chi)}, \xi_2=\frac{4 \chi \sigma B_o^2}{\gamma(2 \mu+\chi)} f_1=\frac{-4 \chi}{\gamma(2 \mu+\chi)} \frac{\partial p}{\partial r}$

To derive expression for u, integrate Eq. (6) and apply relevant velocity boundary conditions Eqns. (4)-(5) which gives,

$u=-\frac{\partial p}{\partial r} \frac{g_1-g_2}{\sigma B_o^2 g_3}$     (7)

$g_1=\varphi_2 \sinh \left(k_2 h \sin \theta\right)\left[\cosh \left(k_1 h \sin \theta\right)-\cosh \left(k_2 h \sin \theta\right)\right]$$g_2=\varphi_1 \sinh \left(k_1 h \sin \theta\right)\left[\cosh \left(k_2 h \sin \theta\right)-\cosh \left(k_2 h \sin \theta\right)\right]$$g_3=\left[\begin{array}{l}\varphi_2 \sinh \left(k_2 h \sin \theta\right) \cosh \left(k_1 h \sin \theta\right) \\ -\varphi_1 \sinh \left(k_1 h \sin \theta\right) \cosh \left(k_2 h \sin \theta\right)\end{array}\right]$

$k_1=\frac{\sqrt{\xi_1+\sqrt{\xi_1^2-4 \xi_2}}}{2}, k_2=\frac{\sqrt{\xi_1-\sqrt{\xi_1^2-4 \xi_2}}}{2}, \varphi_1=\frac{2 \sigma B_o^2-(2 \mu+\chi) k_1^2}{2 \chi k_1},\varphi_2=\frac{2 \sigma B_o^2-(2 \mu+\chi) k_2^2}{2 \chi k_2}$

The Reynolds equation for conical bearings derived by integrating the Eq. (1) and using the respective boundary conditions Eqns. (4)-(5)

$\frac{1}{r} \frac{d}{d r}\left\{f(H, N, L, M, \theta) r \frac{d p}{d r}\right\}=\frac{d H \sin \theta}{d t}$      (8)

where,

$f(H, N, L, M, \theta)=\frac{G_1-G_2}{\sigma B_0^2 k_1 k_2 G_3}$$G_1=k_1 \varphi_2 \sinh \left(k_2 h \sin \theta\right)\left[\begin{array}{l}k_1 h \sin \theta \cosh \left(k_1 h \sin \theta\right) \\ -\sinh \left(k_1 h \sin \theta\right)\end{array}\right]$$G_2=k_1 \varphi_1 \sinh \left(k_1 h \sin \theta\right)\left[\begin{array}{l}k_2 h \cosh \left(k_2 h \sin \theta\right) \\ -\sinh \left(k_2 h \sin \theta\right)\end{array}\right]$$G_3=\left[\begin{array}{l}\varphi_2 \sinh \left(k_2 h \sin \theta\right) \cosh \left(k_1 h \sin \theta\right) \\ -\varphi_1 \sinh \left(k_1 h \sin \theta\right) \cosh \left(k_2 h \sin \theta\right)\end{array}\right]$

For mathematical modeling of surface roughness, the stochastic film thickness H  which consist of two parts represented as $H=h(t)+h_s(r, \theta, \xi)$ is considered.

The probability density function given by $f\left(h_s\right)$, where h(t) represents nominal smooth part of the film thickness, $h_s$ denote the random part resulting from the surface roughness asperities measured from the nominal level and $\xi$  is an index describing the definite roughness arrangements.

Using Christensen [5]

$f\left(h_s\right)=\left\{\begin{array}{cc}\frac{35}{32 c^7}\left(c^2-h_s^2\right)^3 & -c \leq h_s \leq c \\ 0 & \text { otherwise }\end{array}\right.$   (9)

where, $\bar{\sigma}=\frac{c}{3}$ is the standard deviation.

The expectancy operator is denoted as $E(*)$ and defined by

$E(*)=\int_{-\infty}^{\infty}(*) f\left(h_s\right) d h_s$   (10)

According to the stochastic theory of Christensen, two types of one-dimensional surface roughness patterns are analyzed respectively radial roughness pattern and azimuthal roughness pattern. We obtain the stochastic modified Reynolds equation on taking the average of Eq. (8) with respect to $f\left(h_s\right)$.

$\frac{1}{r} \frac{d}{d r}\left\{F\left(H^*, N, L, M, \theta\right) r \frac{d E(p)}{d r}\right\}=\frac{d H \sin \theta}{d t}$

Radial Roughness Pattern

According to the stochastic theory of Christensen, two types of one-dimensional surface roughness patterns are analyzed, respectively, radial roughness pattern and azimuthal roughness pattern. The roughness structure has the form of long, narrow ridges and valleys running in the r-direction (i.e., they are straight ridges and valley passing through z = 0, r = 0 to form star pattern), in the case the film thickness takes the form: $H=h(t)+h_s(\theta, \xi)$.

The stochastic modified Reynolds equation becomes

$\frac{1}{r} \frac{d}{d r}\left\{E[f(H, N, L, M, \theta)] r \frac{d E(p)}{d r}\right\}=\frac{d h}{d t} \sin \theta$   (12)

Azimuthal Roughness Pattern

The structure for the one-dimensional azimuthal roughness pattern on the bearing surface has the roughness structure in the form of narrow ridges and valley running in $\theta$  -direction (i.e., they are circular ridges and valley on the flat plates that are concentric on $z=0, r=0$). In this case the film thickness takes the form $H=h(t)+h_s(r, \xi)$.

$\frac{1}{r} \frac{d}{d r}\left\{\frac{1}{E[f(H, N, L, M, \theta)]^{-1}} r \frac{d E(p)}{d r}\right\}=\frac{d h}{d t} \sin \theta$   (13)

Eq. (12) and (13) together can be expressed as

$\frac{1}{r} \frac{d}{d r}\left\{r \frac{d E(p)}{d r}\right\}=\frac{-12}{F\left(H^*, N, L, M, \theta\right)} \frac{d H \sin \theta}{d t}$    (14)

where,

$F\left(H^*, N, L, M, \theta\right)=\left\{\begin{array}{l}E[f(H, N, L, M, \theta)] \text { for radial } \\ E[1 / f(H, N, L, M, \theta)]^{-1} \text { for azimuthal }\end{array}\right.$$E[f(H, N, L, M, \theta)]=\frac{35}{32 c^7} \int_{-c}^c\left(c^2-h_s^2\right)^3 f(H, N, L, M, \theta) d h_s$$E\left[\frac{1}{f(H, N, L, M, \theta)}\right]=\frac{35}{32 c^7} \int_{-c}^c \frac{\left(c^2-h_s^2\right)^3}{f(H, N, L, M, \theta)} d h_s$

The pressure boundary conditions are:

$\frac{d p}{d r}=0$ at $r=0$    (15）

$p=0$ at $r=a \operatorname{cosec} \theta$   (16)

Introducing the non-dimensionless variables and parameters

$F\left(H^*, N, L, M, \theta\right)=\frac{24\left(G_1-G_2\right)}{M^2 k_1^* k_2^* G_3}$

$G_1=\left(\begin{array}{l}0.5 k_1^* H^* \sin \theta \cosh \left(0.5 k_1^* H^* \sin \theta\right) k_2^* \varphi_2^* \sinh \left(0.5 k_2^* H^* \sin \theta\right) \\ -\sinh \left(0.5 k_1^* H^* \sin \theta\right) k_2^* \varphi_2^* \sinh \left(0.5 k_2^* H^* \sin \theta\right)\end{array}\right)$

$G_2=\left(\begin{array}{l}0.5 k_2^* H^* \sin \theta \cosh \left(0.5 k_2^* H^* \sin \theta\right) k_1^* \varphi_1^* \sinh \left(0.5 k_1^* H^* \sin \theta\right) \\ -\sinh \left(0.5 k_2^* H^* \sin \theta\right) k_1^* \varphi_1^* \sinh \left(0.5 k_1^* H^* \sin \theta\right)\end{array}\right)$

$\begin{aligned} G_3= & \varphi_2^* \sinh \left(0.5 k_2^* H^* \sin \theta\right) \cosh \left(0.5 k_1^* H^* \sin \theta\right) \\ & -\varphi_1^* \sinh \left(0.5 k_1^* H^* \sin \theta\right) \cosh \left(0.5 k_2^* H^* \sin \theta\right)\end{aligned}$

$\varphi_1^*=\varphi_1 H_0=\frac{M^2\left(1-N^2\right)-k_1^{* 2}}{2 N^2 k_1^{* 2}}$,

$\varphi_2^*=\varphi_2 H_o=\frac{M^2\left(1-N^2\right)-k_2^{* 2}}{2 N^2 k_2^{* 2}}$,

$k_1^*=k_1 H_0=\frac{\sqrt{\xi_1^*+\sqrt{\xi_1^{* 2}-4 \xi_2^*}}}{2}$,

$k_2^*=k_2 H_o=\frac{\sqrt{\xi_1^*-\sqrt{\xi_1^{* 2}-4 \xi_2^*}}}{2}$,

$\xi_1^*=\xi_1 H_o=\frac{N^2+M^2\left(1-N^2\right) L^2}{L^2}, \xi_2^*=\xi_2 H_o=\frac{N^2 M^2}{L^2}$

$H^*=\frac{H+h_s}{H_o}, N=\left(\frac{\chi}{2 \mu+\chi}\right)^{1 / 2}, L=\frac{(\gamma / 4 \mu)^{1 / 2}}{H_o}$,

$M=B_o H_o\left(\frac{\sigma}{\mu}\right)^{1 / 2}, H=\frac{2 h}{H_o}, \quad C=\frac{c}{H_o}$ and $r^*=\frac{r}{a \operatorname{cosec} \theta}$

Solution of Eq. (14) using boundary conditions

$\frac{d p^*}{d r^*}=0 \quad$ and $p^*=0, r^*=1$   (17)

gives the non-dimensional pressure equation as

$P^*=\frac{H_o^3 E(p)}{\mu\left(-\frac{d H}{d t}\right) a \operatorname{cosec} \theta}=\frac{-3\left(r^2-1\right)}{F\left(H^*, N, L, M, \theta\right)}$   (18)

The load carrying capacity equation considering the roughness effect is obtained as

$E(w)=2 \pi \int_{a \operatorname{cosec} \theta}^0 E(p) r d r$   (19)

The non-dimensional rough load carrying capacity equation is given by

$W^*=\frac{-H_o^3 E(w)}{\mu \frac{d H}{d t} a^4 \operatorname{cosec}^2 \theta}=\frac{3 \pi}{2 F\left(H^*, N, L, M, \theta\right)}$    (20)

The dimensionless time equation is given by

$T^*=\frac{-H_o^2 E(w)}{\mu a^4 \operatorname{cosec}^2 \theta}=\int_{h_j}^1 \frac{3 \pi}{2 F\left(H^*, N, L, M, \theta\right)} d H^*$    (21)

In this paper, the impact of roughness with MHD between the conical bearings using micropolar fluid is analyzed. The conclusions are obtained according to the numerical computations.

• The existence of micropolar fluid with MHD enhances the dimensionless pressure and the dimensionless load carrying capacity relative to the Newtonian case. Also, the roughness must be accounted for while designing this type of bearing system, even if a suitable magnetic effect is taken into consideration.
• When $(N, L$ and $M=0)$ the equation reduces to Newtonian and non- electrically conducting case.
• As the roughness parameter $(C=0)$ the result reduced to a smooth surface case for both roughness patterns for one-dimensional conical bearings.
• The impact of non-Newtonian micropolar fluid leads to an increase in the load carrying capacity and time, further it provides better performance for conical bearings operating with magnetic parameter values. As the magnetic parameter increases, the squeeze film time also increases. The applied magnetic field strongly opposes the fluid flow in the film region which results accumulation of fluid in the fluid region.
• An increase in the values of Hartmann number M makes the lubricant to acquire more magnetization which in turn interacts with microrotional properties of the micropolar fluid leading to an increased load bearing capacity.