Bingham Plastic Fluid Film Lubrication of Asymmetric Rollers

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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 8, ISSUE 11, NOVEMBER 2019 ISSN 2277-8616 Bingham Plastic Fluid Film Lubrication Of Asymmetric Rollers

Gadamsetty Revathi, Venkata Subrahmanyam Sajja, Dhaneshwar Prasad

Abstract: Hydrodynamic lubrication of asymmetric roller bearings under usual boundary conditions for heavily loaded rigid system is analyzed for incompressible Bingham plastic fluid in the operating behavior of line contact. The viscosity of the lubricant is assumed to vary with the hydrodynamic pressure. The fluid flow governing equations such as momentum and continuity equations are solved analytically first and then numerically using MATLAB. The lubricant velocity distributions are obtained and the results, particularly, pressure, load and traction forces are in good agreement with previous findings.

Index Terms: Hydrodynamic lubrication, Non-Newtonian, Bingham plastic, Asymmetric Rollers, Incompressible, Viscosity. ————————————————————

1 INTRODUCTION al. [9] analyzed theoretically the fluid lubrication in order to The non-Newtonian fluid flow behavior can be described by obtain the general expression so that it can be applied for both Bingham plastic fluid model, in particular greases, the flow of Newtonian and non-Newtonian lubricants. Using this melts and slurries in molds and also in chemical processing. expression the effect of sliding in contact and the influence of The Bingham model is characterized by two parameters: a unsteady load is also discussed. yield shear stress and a viscosity. If the magnitude of the The axisymmetric squeeze flow of viscoplastic materials are deviatoric stress tensor is less than the yield stress, then the examined by Smyrnaios and Tsamopoulos [10] using either material acts as rigid; when the yield stress is exceeded, the the original Bingham constitutive equation or the approximate material flows in a quasi-Newtonian manner [1]. It is required model suggested by Papanastasiou. The results are primarily to study the flow of Bingham fluid to discuss the applications of affected by the Bingham number that measures the magnitude lubricant transport in contact with a Newtonian fluid. In fact the of the yield stress with respect to the viscous stresses. As this Bingham fluid behaves like a viscous fluid if the shear stress number increases, large departures from the corresponding exceeds a yield value, and like a rigid body otherwise [2]. Jang Newtonian solution are obtained and limited flow and and Khonsari [3] analyzed finite slider bearings theoretically deformation of the material is predicted. In this paper, an with the Bingham rheological model including thermal effects attempt has been made to analyze the performance and presented a full thermo hydrodynamics solution for wide characteristics of Bingham plastic fluid film lubrication of two range of operating conditions. The squeeze flow of Bingham heavily loaded rigid cylindrical asymmetric roller bearings material is analyzed theoretically by Shi-Pu Yang, and Ke-Qin under usual boundary conditions. The viscosity variation of the Zhu [4] using the bi-viscosity model in the small gap between Bingham plastic fluid is considered in such a way that it varies parallel disks with the Navier slip condition. Bingham fluid is with the hydrodynamic pressure. described by using the bi-viscosity model avoiding the yield- surface paradox and the explicit expressions of the radial 2 Theoretical model velocity, pressure gradient, pressure and squeeze force are Reynolds, in 1886, had established the governing equations obtained. The performance characteristics and the core for inertialess flow of thin film of Newtonian fluid. However, in formation in a hydrodynamic journal bearing lubricated with a many applications, the Newtonian behaviour of the lubricant Bingham fluid are examined by Gertzos at al. [5] and the does not exist for a long time, i.e. the linear relationship FLUENT package is used to solve the Navier–Stokes between shear rate and shear stress collapse in a short period equations. The results of the developed 3-D CFD model were of time. Hence, the non-Newtonian effect of the lubricant has compared with theoretical and experimental results of previous to be incorporated along with hydrodynamic pressure. In this investigations for both Newtonian and Bingham lubricants and study a generalized Reynolds equation for the case of non- found to be in very good agreement. Osman Turan et al. [6] Newtonian incompressible Bingham plastic fluid lubrication of extended the analysis of Tarun et al. [7] by modifying the asymmetric rollers without inertia effect is considered under vertical sidewall boundary condition to one of constant wall usual boundary conditions. The thin film geometry for the heat flux rather than constant wall temperature. Laurent Talon lubrication of asymmetric rollers is presented in the following et al. [8] investigated that the flow rate in a 1D rough channel Fig. 1. for any aperture distribution and paid the particular attention to the case of aperture fields that are self affine. Tokio Sasaki et ______

 Gadamsetty Revathi, Research Scholar, Department of Mathematics, Koneru Lakshmaiah Education Foundation, Guntur-522502, India, E-mail: [email protected]  Venkata Subrahmanyam Sajja, Department of Mathematics, Koneru Lakshmaiah Education Foundation, Guntur-522502, India, E-mail: [email protected]  Dhaneshwar Prasad, Department of Mathematics, K M Centre for Post Graduate Studies, Puducherry-605008, India, E-mail: [email protected]

2 h  1  x (10) 1 1

2.3 Reynolds equation Integrating the equation (2) using the boundary conditions, the pressure Reynolds equation is established and can be written as dp 3  U  U h  h   1 2 1 (11) dx 2 h 3

2.4 Dimensionless scheme 2 x , p   p , m  2 mc  , h  1  x x  n , R n n 1  2 n  1   U   2 R  p c      ,    e    / h n       0 0 4 n h h ,    0   0  2  R U h  h / h , U  U / U ,    , 0 1 2 0 2 0 h Fig.1: Lubrication of Asymmetric Rollers 0 The velocity expression and pressure Reynolds equation are 2.1 Fluid flow governing equations written in dimensionless form using the above mentioned The governing equations can be written by considering the dimensionless scheme case in which the deviatoric stress is related to the 1 1 2 2 (12) u  h  y  U h  y  (1  U )( y  h )( h  h 1 ) deformation rate by a viscosity function . Using the 3    ( p ) 2 h 4 h terminology of Fig. 1, the following governing equations such d p 3  1  U h  h 1  (13) as continuity and momentum equations are considered:  3 d x  u  v 4 h   0 (1)  x  y 2.5 Load and Traction dp    u  (2) The load-carrying capacity that the bearing will support is      0  found by integrating the pressure around the surface. The load dx  y  y   components W in the y-direction is calculated as where (3) x 2    0 exp ( p ) 2 W  p dx (14) x (4)  h  h 0    2 R w  R is the radius of the equivalent cylinder. The dimensionless load W  is given by 2 Rh 2.2 Boundary conditions 0 x 2 x 2 u  U at y   h ; and u  U at y  h (5) d p 1 2 W   p d x    x d x (15) dp d x p  0 at x       ; p  0 and  0 at x  x 2 (6) dx The surface traction force T is obtained from the integration F where are velocities of the rolling cylinders as of shear stress over the entire length and one may get U 1 and U 2 x x shown in Fig.(1). 2 2 Solving the equation (2) using the boundary conditions T    dx and T    dx (16) mentioned above, one can find the velocity expression as Fh   y   h Fh   y  h given below.     Dimensionless tractions are U  U y (U  U )  1 dp  1 2 1 2 2 2 (7) x 2 u       y  h   T F    0 2 2 h  2  dx  T Fh         d x and  h   y   h Now the volume flux ‗Q‘ for the fluid flow can be obtained by  0    integrating the velocity within the gap between the surfaces is x 2 presented below T Fh     d x (17)  y  h h 3  2 h dp    Q  u dy  h U  U     (8)  1 2   3  dx  h   3 RESULTS AND DISCUSSION: and the volume flux at the point of maximum pressure is The following representative values have been used for Q (  x )  (U  U ) h (9) 1 1 2 1 numerical computation: R=3 cm , 4 h0  4  10 cm 9 1 2 where the film thickness h at x   x 1 is considered to be , . 1 U 2  400 cm / s ,   1.6  10 dyne cm

3.1 Velocity profile: Velocity profile Lubricant velocity u verses y have been computed at 1.3 different locations of the fluid in the gap between the surfaces 1.2 and depicted in Figs. (2-7) respectively for the regions before, 1.1 after, and at the point of maximum pressure. The Figs. (2-4) have similar characteristics, i.e. they are like parabolas with 1 vertices downward in the regions before the point of maximum 0.9 pressure. Figs. (5-6) show that the velocity of the fluid after point of maximum pressure. It can be seen from Fig. (7) that 0.8 the velocity of the lubricant increases linearly as increases 0.7

Dimensionless velocity u at the point of maximum pressure. This is in good agreement 0.6 with the previous findings of Revathi et al. [11]. 0.5

Velocity profile 0.4 1.2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Dimensionless y

1 Fig.4: Velocity at x = -1.0 0.8

0.6 Velocity profile 0.4 1.5

0.2 1.45 1.4 0 Dimensionless velocity u 1.35 -0.2 1.3 -0.4 1.25 -0.6 -25 -20 -15 -10 -5 0 5 10 15 20 25 1.2 Dimensionless y

Dimensionless velocity u 1.15 Fig.2: Velocity at x = -4.5 1.1

1.05 Velocity profile 1.2 1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1 Dimensionless y

0.8 Fig.5: Velocity at x=0

0.6

Velocity profile 0.4 1.25

0.2

Dimensionless velocity u 1.2 0

-0.2 1.15 -0.4 -8 -6 -4 -2 0 2 4 6 8 Dimensionless y 1.1

Fig.3: Velocity at x = -2.5 Dimensionless velocity u 1.05

1 -1.5 -1 -0.5 0 0.5 1 1.5 Dimensionless y

Fig.6: Velocity at x=0.4

Velocity profile 1.2 0.14 1.18 0.12 1.16

1.14 0.1 1.12 0.08 1.1

1.08 0.06

Dimensionless velocity u 1.06 dimensionless p pressure 0.04 1.04

1.02 0.02

1 -1.5 -1 -0.5 0 0.5 1 1.5 0 Dimensionless y -7 -6 -5 -4 -3 -2 -1 0 1 dimensionless x

Fig.7: Velocity at point of maximum pressure Fig.9: Pressure profile for Ū=1.2

3.2 Pressure profile: The pressure distribution p trends for the lubricant are 0.16 depicted in Figs. (8-11). One can observe the pressure profile 0.14 in Fig. (8) for pure rolling case and in Figs. (9-10) for sliding cases. The profiles for different values of Ū are presented 0.12 in Fig. (11) and can be seen that the pressure increases as 0.1 rolling ratio Ū increases. This kind of the behavior was observed in [11-15]. 0.08

0.06

0.14 dimensionless p pressure 0.04

0.12 0.02

0.1 0 -7 -6 -5 -4 -3 -2 -1 0 1 dimensionless x 0.08

Fig.10: Pressure profile for Ū=1.4 0.06

dimensionless p pressure 0.04

0.02

0 -7 -6 -5 -4 -3 -2 -1 0 1 dimensionless x

Fig.8: Pressure profile for Ū=1.0

Fig.11: Pressure profile for different Ū

3.3 Viscosity (  ) Profile: -3 x 10 Load profile 2.5 The lubricant viscosity is computed for different values of x and is presented in Fig. (12). One can observe from the figure 2.4 that the lubricant viscosity profile is similar to pressure profile. 2.3 Since the viscosity considered in this problem is a function of hydrodynamic pressure, hence, the viscosity profile looks like 2.2 pressure profile. The same trend can be observed in 2.1 Dhaneshwar Prasad and Venkata Subrahmanyam Sajja [15]. 2

Viscosity profile 1.9 0.8014 Dimensionless Wx load 1.8

0.8012 1.7

1.6  0.801 1.5 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 0.8008 Rolling ratio

0.8006 Fig. 14: Load in x-direction

dimensionless viscosity 0.8004 The traction forces T have been evaluated and presented in f the form of a tables for both the lower and upper surfaces for 0.8002

distinct values of  0 and . Here  0  0 represents 0.8 -7 -6 -5 -4 -3 -2 -1 0 1 Newtonian case and   0 represents non-Newtonian 0 dimensionless x case. The Traction forces at upper surface increase with rolling Fig.12: Viscosity versus ratio Ū and decrease as rolling ratio Ū increase in Newtonian

and non-Newtonian case as well. Further, it is observed that 3.5 Load and Traction: Numerically computed values of the normal load carrying traction forces are increasing as increasing for a fixed value of Ū. capacity W is presented graphically in Fig. (13) and it can be y seen that the increase in normal load with the increase in Table 1: Traction force T f values at upper surface rolling ratio U . It is in conformity with the previous findings of T for   0 T for   0 .5 T for   1 [11-15]. Further, the load in x-direction W is also computed f 0 f 0 f 0 x U and presented in Fig. (14). A trend similar to can be 1.0 0.7301 4.4634 8.1967 observed in Fig. (14). 1.1 0.7317 4.4651 8.1984 1.2 0.7334 4.4667 8.2001 Load profile 1.3 0.7351 4.4684 8.2018 0.68 1.4 0.7368 4.4701 8.2035 0.66 1.5 0.7385 4.4718 8.2052

0.64 Table 2: Traction force values at lower surface 0.62

0.6

0.58 1.0 -0.7301 3.0032 6.7365 0.56 1.1 -0.7318 3.0016 6.7349 Dimensionless load, Wy 0.54 1.2 -0.7334 2.9999 6.7332 1.3 -0.7351 2.9982 6.7315 0.52 1.4 -0.7368 2.9965 6.7298 0.5 1.5 -0.7386 2.9948 6.7281 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 Rolling ratio, U 4 CONCLUSION Fig.13: Load in y-direction The problem has been attempted to study the hydrodynamic lubrication analysis of rolling and sliding line contact by an incompressible Bingham Plastic fluid under the usual boundary conditions. The pressure Reynolds equation and is 2553 IJSTR©2019 www.ijstr.org INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 8, ISSUE 11, NOVEMBER 2019 ISSN 2277-8616 solved for pressure, load and traction forces for various values [12] Dhaneshwar Prasad, S.V. Subrahmanyam, S.S. Panda, of the sliding parameter U and the viscosity coefficient. The ―Thermal effects in Hydrodynamic Lubrication of Asymmetric Rollers using R-K Fehlberg Method‖, lubricant velocity distributions at different locations of the fluid International Journal of Engineering Science and are also discussed. The following facts may be drawn from the Advanced Technology, 2012, Vol. 2, Issue 3, pp. 422-437. results obtained here: [13] Venkata Subrahmanyam Sajja, Dhaneshwar Prasad, • Velocity of the lubricant at points of maximum ―Characterization of Lubrication of Asymmetric Rollers pressure increases with increase of viscosity including Thermal Effects‖, Industrial Lubrication and coefficient. Tribology, 2015, Vol. 67, Issue 3, pp. 246-255. • A notable increase in lubricant pressure is observed [14] Dhaneshwar Prasad, Venkata Subrahmanyam Sajja, for different values of rolling ratio Ū ―Thermal Effects in Non-Newtonian Lubrication of • The load in x-direction and as well as in y-direction is Asymmetric Rollers under adiabatic and Isothermal found to be increasing with sliding parameter Ū. Boundaries‖, International Journal of Chemical Sciences, • Traction forces at the upper surface are increasing 2016, Vol. 14, Issue 3, pp. 1641-1656. with  and Ū. Further, the traction forces at lower 0 [15] Dhaneshwar Prasad, Venkata Subrahmanyam Sajja, surface are found to be decreasing as sliding ―Non-Newtonian Lubrication of Asymmetric Rollers with parameter Ū increases, but increasing with  0. Thermal and Inertia Effects‖, Tribology Transactions,

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