Engineering Tribology Episode 1 Part 10 potx

200 ENGINEERING TRIBOLOGY 51 H. Hashimoto, S. Wada and M. Sumitomo, The Effects of Fluid Inertia Forces on the Dynamic Behaviour of Short Journal Bearings in Superlaminar Flow Regime, Transactions ASME, Journal of Tribology, Vol. 110, 1988, pp. 539-547. 52 J.A. Tichy, Measurements of Squeeze-Film Bearing Forces and Pressures, Including the Effect of Fluid Inertia, ASLE Transactions, Vol. 28, 1985, pp. 520-526. 53 L.A. San Andres and J.M. Vance, Effect of Fluid Inertia on Squeeze-Film Damper Forces for Small-Amplitude Circular-Centered Motions, ASLE Transactions, Vol. 30, 1987, pp. 63-68. 54 W.J. Harrison, The Hydrodynamical Theory of Lubrication with Special Reference to Air as a Lubricant, Trans. Cambridge Phil. Soc., Vol. 22, 1913, pp. 39-54. 55 W.A. Gross, Fluid Film Lubrication, John Wiley, New York, 1980. 56 A. Burgdorfer, The Influence of the Molecular Mean Free Path on the Performance of Hydrodynamic Gas Lubricated Bearings, Transactions ASME, Journal of Basic Engineering, Vol. 81, 1959, pp. 94-100. 57 B. Bhushan and K. Tonder, Roughness-Induced Shear- and Squeeze-Film Effects in Magnetic Recording, Parts 1 and 2, Transactions ASME, Journal of Tribology, Vol. 111, 1989, Part 1, pp. 220-227, Part 2, pp. 228-237. 58 Y. Mitsuya, Stokes Roughness Effects on Hydrodynamic Lubrication, Part 2 - Effects under Slip Flow Boundary Conditions, Transactions ASME, Journal of Tribology, Vol. 108, 1986, pp. 159-166. 59 S. Haber and I. Etsion, Analysis of an Oscillatory Oil Squeeze Film Containing a Central Gas Bubble, ASLE Transactions, Vol. 28, 1985, pp. 253-260. 60 D.W. Parkins and W.T. Stanley, Characteristics of an Oil Squeeze Film, Transactions ASME, Journal of Lubrication Technology, Vol. 104, 1982, pp. 497-503. 61 V.T. Morgan and A. Cameron, Mechanism of Lubrication in Porous Metal Bearings, Proceedings Conf. on Lubrication and Wear, Inst. Mech. Engrs., London, 1957, pp. 151-157. 62 F.E. Cardullo, Some Practical Deductions from the Theory of the Lubrication of Cylindrical Bearings, Transactions ASME, Vol. 52, 1930, pp. 143-153. 63 Cz. M. Rodkiewicz, K.W. Kim and J.S. Kennedy, On the Significance of the Inlet Pressure Build-up in the Design of Tilting-Pad Bearings, Transactions ASME, Journal of Tribology, Vol. 112, 1990, pp. 17-22. 64 J.A. Cole, Experimental Investigation of Power Loss in High Speed Plain Thrust Bearings, Proceedings Conf. on Lubrication and Wear, Inst. Mech. Engrs., London, 1957, pp. 158-163. 65 D. Dowson, Non-Steady State Effects in EHL, New Directions in Lubrication, Materials, Wear and Surface Interactions, Tribology in the 80's, edited by W.R. Loomis, Noyes Publications, Park Ridge, New Jersey, USA, 1985. 66 J.A. Cole and C.J. Hughes, Visual Study of Film Extent in Dynamically Loaded Complete Journal Bearings, Lubrication and Wear Conf., Proc. Inst. Mech. Engrs., 1957, pp. 147-149. 67 E.R. Booser, CRC Handbook of Lubrication, Volume II, CRC Press, Boca Raton, Florida, 1984. TEAM LRN COMPUTATIONAL 5 HYDRODYNAMICS 5.1 INTRODUCTION The differential equations which arose from the theories of Reynolds and later workers rapidly exceeded the capacity of analytical solution. For many years some heroic attempts were made to solve these equations using specialized and obscure mathematical functions but this process was tedious and the range of solutions was limited. A gap or discrepancy always existed between what was required in the engineering solutions to hydrodynamic problems and the solutions available. Before numerical methods were developed, analogue methods, such as electrically conductive paper, were experimented with as a means of determining hydrodynamic pressure fields. These methods became largely obsolete with the advancement of numerical methods to solve differential equations. This change radically affected the general understanding and approach to hydrodynamic lubrication and other subjects, e.g. heat transfer. It is now possible to incorporate in the numerical analysis of the bearing common features such as heat transfer from a bearing to its housing. The application of traditional, analytical methods would require to assume that the bearing is either isothermal or adiabatic. Numerical solutions to hydrodynamic lubrication problems can now satisfy most engineering requirements for prediction of bearing characteristics and improvements in the quality of prediction continue to be found. In engineering practice problems like: what is the maximum size of the groove to reduce friction before lubricant leakage becomes excessive, or how does bending of the pad affect the load capacity of a bearing?, need to be solved. In this chapter the application of numerical analysis to problems encountered in hydrodynamic lubrication is described. A popular numerical technique, the ‘finite difference method’ is introduced and its application to the analysis of hydrodynamic lubrication is demonstrated. The steps necessary to obtain solutions for different bearing geometries and operating conditions are discussed. Based on the example of the finite journal bearing it is shown how fundamental characteristics of the bearing, e.g. the rigidity of the bearing, the intensity of frictional heat dissipation and its lubrication regime, control its load capacity. 5.2 NON-DIMENSIONALIZATION OF THE REYNOLDS EQUATION Non-dimensionalization is the substitution of all real variables in an equation, e.g. pressure, film thickness, etc., by dimensionless fractions of two or more real parameters. This process TEAM LRN 202 ENGINEERING TRIBOLOGY extends the generality of a numerical solution. A basic disadvantage of a numerical solution is that data is only provided for specific values of controlling variables, e.g. one value of friction force for a particular combination of sliding speed, lubricant viscosity, film thickness and bearing dimensions. Analytical expressions, on the other hand, are not limited to any specific values and are suited for providing data for general use, for example, they can be incorporated in an optimization process to determine the optimum lubricant viscosity. A computer program would have to be executed for literally thousands of cases to provide a comprehensive coverage of all the controlling parameters. The benefit of non- dimensionalization is that the number of controlling parameters is reduced and a relatively limited data set provides the required information on any bearing. The Reynolds equation (4.24) is expressed in terms of film thickness ‘h’, pressure ‘p’, entraining velocity ‘U’ and dynamic viscosity ‘η’. Non-dimensional forms of the equation's variables are following: h* = h c x* = x R y* = y L p* = pc 2 6UηR (5.1) where: h is the hydrodynamic film thickness [m]; c is the bearing radial clearance [m]; R is the bearing radius [m]; L is the bearing axial length [m]; p is the pressure [Pa]; U is the bearing entraining velocity [m/s], i.e. U = (U 1 + U 2 )/2; η is the dynamic viscosity of the bearing [Pas]; x, y are hydrodynamic film co-ordinates [m]. The Reynolds equation in its non-dimensional form is: = ∂ ∂x* ( ( ∂p* ∂x* h* 3 + ( ( R L 2 ( ( ∂p* ∂y* h* 3 ∂h* ∂x* ∂ ∂y* (5.2) All terms in equation (5.2) are non-dimensional apart from ‘R’ and ‘L’ which are only present as a non-dimensional ratio. Although any other scheme of non-dimensionalization can be used this particular scheme is the most popular and convenient. For planar pads, ‘R’ is substituted by the pad width ‘B’ in the direction of sliding. 5.3 THE VOGELPOHL PARAMETER The Vogelpohl parameter was developed to improve the accuracy of numerical solutions of the Reynolds equation and was introduced by Vogelpohl [1] in the 1930's. The Vogelpohl parameter ‘M v ’ is defined as follows: TEAM LRN COMPUTATIONAL HYDRODYNAMICS 203 M v = p*h* 1.5 (5.3) Substitution into the non-dimensional form of Reynolds equation (5.2) yields the ‘Vogelpohl equation’: = FM v + G ∂ 2 M v ∂x* 2 + ( ( R L 2 ∂ 2 M v ∂y* 2 (5.4) where parameters ‘F’ and ‘G’ for journal bearings are as follows: ∂h* ∂x* + ( ( R L 2 [( ( 2 ∂h* ∂y* ( ( 2 [ 0.75 h* 2 F = ∂ 2 h* ∂x* 2 + ( ( R L 2 [ ∂ 2 h* ∂y* 2 [ 1.5 h* + (5.5) ∂h* ( ( h* 1.5 G = ∂x* (5.6) The Vogelpohl parameter facilitates computing by simplifying the differential operators of the Reynolds equation, and furthermore it does not show high values of higher derivatives in the final solution, i.e. d n M v /dx * n where n > 2, unlike the dimensionless pressure ‘p*’. This is because, where there is a sharp increase in ‘p*’ close to the minimum of hydrodynamic film thickness ‘h*’, ‘M v ’ remains at moderate values. Large values of higher derivatives cause significant truncation error in numerical analysis. The characteristics of ‘M v ’ and ‘p*’ for a journal bearing at an eccentricity of 0.95 are shown in Figure 5.1. 0 100 200 Degrees around bearing Vogelpohl parameter M v Dimensionless pressure p* p*, M v L/D = 1 ε = 0.95 360° bearing FIGURE 5.1 Variation of dimensionless pressure and the Vogelpohl parameter along the centre plane of a journal bearing [4]. TEAM LRN 204 ENGINEERING TRIBOLOGY It can be seen from Figure 5.1 that the introduction of the Vogelpohl parameter does not complicate the boundary conditions in the Reynolds equation, since wherever p* = 0, also M v = 0 (zero values of ‘h’, i.e. solid to solid contact, are not included in the analysis). As discussed later in this chapter, wherever cavitation occurs, the gradient of ‘M v ’ adjacent and normal to the cavitation front is zero like that of ‘p*’. Numerical solutions of the Reynolds equation are obtained in terms of ‘M v ’ and values of ‘p*’ found from the definition M v /h * 1.5 = p*. 5.4 FINITE DIFFERENCE EQUIVALENT OF THE REYNOLDS EQUATION Journal and pad bearing problems are usually solved by ‘finite difference’ methods although ‘finite element’ methods have also been employed [2]. The finite difference method is based on approximating a differential quantity by the difference between function values at two or more adjacent nodes. For example, the finite difference approximation to ∂M v /∂x* is given by: ∂M v ∂x* ≈ M v,i+1 − M v,i−1 2δx* ( ( i M v,i−1 M v,i+1 M v x* x* i x* i+1 x* i−1 M v,i 2δx* ∂M v ∂x* ( ( i Approximation to (5.7) where the subscripts i- 1 and i+1 denote positions immediately behind and in front of the central position ‘i’ and ‘δx*’ is the step length between nodes. A similar expression results for the second differential ∂ 2 M v /∂x * 2 . This expression can be found according to the principle illustrated in Figure 5.2. The second differential ∂ 2 M v /∂x * 2 is found by subtracting the expression for ∂M v /∂x* at the i- 0.5 nodal position from the i+0.5 nodal position and dividing by δx*, i.e.: δx* ≈ ∂ 2 M v ∂x* 2 ∂M v ∂x* ( ( i−0.5 ∂M v ∂x* ( ( i+0.5 − ( ( i (5.8) where: ∂M v ∂x* ≈ M v,i+1 − M v,i δx* ( ( i+0.5 ∂M v ∂x* ≈ M v,i − M v,i−1 δx* ( ( i−0.5 substituting into (5.8) yields: ∂ 2 M v ∂x* 2 ≈ M v,i+1 + M v,i−1 − 2M v,i (δx*) 2 ( ( i TEAM LRN COMPUTATIONAL HYDRODYNAMICS 205 M v,i−1 M v,i M v,i+1 Closest approximations to dM v dx* at (x i * − 1 2 δx*) and (x i * + 1 2 δx*) M v dM v dx* x* x* Closest approximation to d 2 M v dx* 2 at x i * x i * − 1 2 δx* x i * + 1 2 δx*x i *x i * −δx* x i * +δx* x i * − 1 2 δx* x i * + 1 2 δx*x i * FIGURE 5.2 Illustration of the principle for the derivation of the finite difference approximation of the second derivative of a function. The finite difference equivalent of (∂ 2 M v /∂x * 2 + ∂ 2 M v /∂y * 2 ) is found by considering the nodal variation of ‘M v ’ in two axes, i.e. the ‘x’ and ‘y’ axes. A second nodal position variable is introduced along the ‘y’ axis, the ‘j’ parameter. The expressions for ∂M v /∂y * and ∂ 2 M v /∂y * 2 are exactly the same as the expressions for the ‘x’ axis but with ‘i’ substituted by ‘j’. The coefficients of ‘M v ’ at the ‘i’-th node and adjacent nodes required by the Reynolds equation which form a ‘finite difference operator’ are usually conveniently illustrated as a ‘computing molecule’ as shown in Figure 5.3. The finite difference operator is convenient for computation and does not create any difficulties with boundary conditions. When the finite difference operator is located at the boundary of a solution domain, special arrangements may be required with imaginary nodes outside of the boundary. The solution domain is the range over which a solution is applicable, i.e. the dimensions of a bearing. There are more complex finite difference operators available based on longer strings of nodes but these are difficult to apply because of the requirement for nodes outside of the solution domain and are rarely used despite their greater accuracy. The terms ‘F’ and ‘G’ can be included with the finite difference operator to form a complete equivalent of the Reynolds equation. The equation can then be rearranged to provide an expression for ‘M v,i,j ’ i.e.: + ( ( R L 2 2C 1 + 2C 2 + F i,j M v,i,j = C 1 ( ( M v,i+1,j + M v,i−1,j C 2 ( ( M v,i,j+1 + M v,i,j−1 − G i,j (5.9) TEAM LRN 206 ENGINEERING TRIBOLOGY where: C 1 = 1 δx* 2 C 2 = 1 δy* 2 This expression forms the basis of the finite difference method for the solution of the Reynolds equation. Its solution gives the required nodal values of ‘M v ’. −2 δx* 2 −2 δy* 2 1 δx* 2 1 δx* 2 1 δy* 2 1 δy* 2 δy* δx* To boundary of solution domain j-1 j j+1 i-1 i i+1 FIGURE 5.3 Finite difference operator and nodal scheme for numerical analysis of the Reynolds equation. Definition of Solution Domain and Boundary Conditions After establishing the controlling equation, the next step in numerical analysis is to define the boundary conditions and range of values to be computed. For the journal or pad bearing, the boundary conditions require that ‘p*’ or ‘M v ’ are zero at the edges of the bearing and also that cavitation can occur to prevent negative pressures occurring within the bearing. The range of ‘x*’ is between 0 - 2π (360° angle) for a complete bearing or some smaller angle for a partial arc bearing. The range of ‘y*’ is from -0.5 to +0.5 if the mid-line of the bearing is selected as a datum. A domain of the journal bearing where symmetry can be exploited to cover either half of the bearing area, i.e. from y* = 0 to y* = 0.5, or the whole bearing area is shown in Figure 5.4. Nodes on the edges of the bearing remain at a pre-determined zero value while all other nodes require solution by the finite difference method. When symmetry is exploited to solve for only a half domain, it should be noted that nodes on the mid-line of the bearing are also variable and the finite difference operator requires an extra column of nodes outside the solution domain, as zero values along the edge of the solution domain cannot be assumed. This extra column is generated by adopting node values from the column one step from the mid-line on the opposite side. In analytical terms this is achieved by setting: TEAM LRN COMPUTATIONAL HYDRODYNAMICS 207 M v,i,jnode+1 = M v,i,jnode−1 (5.10) where ‘jnode’ is the number of nodes in the ‘j’ or ‘y*’ direction. A split domain reduces the number of nodes but when analyzing a non-symmetric or misaligned bearing then a domain covering the complete bearing area is necessary. The domain in this case is the complete bearing with limits of ‘y*’ from -0.5 to +0.5 and the mid- line boundary condition vanishes. Sliding direction Load line i=1  x* = 0  x* = π i=INODE x* = 2π j=1 j=JNODE  half bearing j=JNODE full bearing Extra row for half bearing Extra row of nodes for overlap with x* = 0 FIGURE 5.4 Nodal pressure or Vogelpohl parameter domains for finite difference analysis of hydrodynamic bearings. Calculation of Pressure Field It is possible to apply the direct solution method to calculate the pressure field but this is quite complex. In practice the pressure field in a bearing is calculated by iteration procedure and this will be discussed in the next section. Calculation of Dimensionless Friction Force and Friction Coefficient Once the pressure field has been found, it is possible to calculate the friction force and friction coefficient from the film thickness and pressure gradient data. As discussed already in Chapter 4 the frictional force operating across the hydrodynamic film is calculated by integrating the shear stress ‘τ’ over the bearing area, i.e.: F = ⌠ ⌡ 0 L ⌠ ⌡ 0 2πR τdxdy (5.11) where the shear stress ‘τ’ is given by: τ = ηU h + h 2 dp dx (5.12) where: τ is the shear stress [Pa]; η is the dynamic viscosity of the lubricant [Pas]; U is the entraining velocity [m/s]; TEAM LRN 208 ENGINEERING TRIBOLOGY h is the hydrodynamic film thickness [m]; p is the hydrodynamic pressure [Pa]; F is the friction force [N]; x is the distance in the direction of sliding [m]; y is the distance normal to the direction of sliding [m]. In a manner similar to the computation of pressure, the equation for friction force can be expressed in terms of non-dimensional quantities. From (5.1) h = h*c, x = x*R and p = p*(6UηR)/c 2 and substituting into (5.12) yields: τ = ηU c 1 h* ) + ch* 2 6UηR c 2 1 R dp* dx* = Uη c 1 h* + 3h* ()( dp* dx* (5.13) Substituting for ‘x’ and ‘y’ from (5.1) the ‘τdxdy’ in terms of non-dimensional quantities is: τdxdy = τdx*dy*RL (5.14) Substituting (5.14) and (5.13) into (5.11) results in an expression for frictional force in terms of non-dimensional quantities: F = ⌠ ⌡ 0 L ⌠ ⌡ 0 2πR τdxdy = RL ⌠ ⌡ 0 1 ⌠ ⌡ 0 2π τdx*dy* = ⌠ ⌡ 0 1 ⌠ ⌡ 0 2π dx*dy* RLηU c ) 1 h* + 3h* dp* dx* ( (5.15) It can be seen from equation (5.15) that the non-dimensional shear stress ‘τ*’ is expressed as: τ* = 1 h* + 3h* dp* dx* (5.16) and equation (5.15) can be re-written in the following form: F = ⌠ ⌡ 0 1 ⌠ ⌡ 0 2π τ*dx*dy* = F* RLηU c RLηU c )( (5.17) The coefficient of friction is calculated by dividing the friction force by the load. A similar quantity is also found when the dimensionless friction is divided by the dimensionless load. Thus in journal bearings the friction coefficient is the ratio of circumferential friction force divided by the load: µ = F W = ⌠ ⌡ 0 L ⌠ ⌡ 0 2πR τdxdy ⌠ ⌡ 0 L ⌠ ⌡ 0 2πR pdxdy (5.18) TEAM LRN COMPUTATIONAL HYDRODYNAMICS 209 where: µ is the coefficient of friction; W is the bearing load [N]. Load on a journal bearing is often expressed as: ⌠ ⌡ 0 L ⌠ ⌡ 0 2πR −cos(x*)pdx dyW = (5.19) where the term -cos(x*) arises from the fact that load supporting pressure is located close to x* = π or cos(x*) = -1. Any pressure close to x* = 0 merely imposes an extra load on the bearing since it acts in the direction of the load. The negative sign refers to the fact that the load vector does not coincide with the position of maximum film thickness. Expressing equation (5.19) in terms of non-dimensional quantities yields: 6UηR c 2 () RL ⌠ ⌡ 0 1 ⌠ ⌡ 0 2π −cos(x*)p*dx*dy* = W* 6R 2 LUη c 2 () W = (5.20) Substituting (5.17) and (5.20) into (5.18) gives the expression for coefficient of friction: RLηU c () = F W 6R 2 LUη c 2 () F* W* = ( c 6R )( F* W* ) µ = (5.21) hence: F* W* = ( ) 6R c µ (5.22) The presence of cavitation in the bearing adds some complication to the calculation of the coefficient of friction. Within the cavitated region, the proportion of clearance space between shaft and bush that is filled by lubricant is: h* cav h* where: h cav * is the dimensionless film thickness at the cavitation front; h* is the dimensionless film thickness at a specified position down-stream of the cavitation front. The average or ‘effective’ coefficient of friction is proportional to the lubricant filled fraction of the clearance space and within the cavitated region ‘p*’ and ‘dp*/dx*’ are equal to zero. The symbol for ‘effective’ dimensionless shear stress is ‘τ e * ’. Assuming a simple proportionality between fluid filled volume and total shear force, an average value of ‘τ e * ’ that allows for zero shear stress between streamers of lubricant, is given by: TEAM LRN [...]... program ‘PARTIAL’ and is arranged in a perspective view to show the entire profile rather than just a section through ‘x*’ or ‘y*’ All pressures are presented as percentages of the peak pressure P% 10 0% Pressure along the load line Outlet 12 0 96 8 84 10 72 L 0.75 L 0.5 L Inlet 0.25 L 12 0 FIGURE 5.7 24 36 60 48 es gre De Computed pressure profile for 12 0° perfectly aligned partial bearing 10 0% P% Pressure... Since the load ‘W’ and pressure ‘p’ expressed in non-dimensional terms are: W = W* p = p* 6R 2LUη c2 6RUη c2 substituting the bearing data yields: 2 10 W = 0.6029 6 × 0 .1 × 0.2 × 2 × 0.05 0.0002 × p max = 1. 2684 6 × 0 .1 × 10 2 0.05 0.0002 = 90.435 [kN] = 9. 513 [MPa] The difference in lubricant film thickness or shaft clearance along the load line from one side of the bearing to the other (i.e the distance... dimensions are R = 0 .1 [m], L = 0.2 [m], c = 0.0002 [m] and that the bearing entraining velocity is U = 10 [m/s] and the dynamic viscosity of the lubricant is η = 0.05 [Pas] · Solution Executing the program ‘PARTIAL’ with a mesh size of 11 columns in the ‘x*’ direction and 9 rows in the ‘y*’ direction yields: dimensionless load W* = 0.6029 and maximum dimensionless pressure p* = 1. 2684 max Since the... the sum of eccentricity and half the misalignment must be less than 1 for no solid contact TEAM LRN 214 ENGINEERING TRIBOLOGY This effect can also be demonstrated easily by comparing the pressure fields for perfectly aligned and misaligned bearings The computed pressure field for a perfectly aligned 12 0° partial arc bearing, L/D = 1 at an eccentricity ratio of 0.7 is shown in Figure 5.7 and the computed... the bearing) is: m = tc Substituting gives: m = 0.4 × 0.0002 = 8 × 10 -5 [m] TEAM LRN 216 ENGINEERING TRIBOLOGY The minimum film thickness has two components: one due to eccentricity (as described in Chapter 4, i.e h min,ecc = c (1 − ε)) and one due to misalignment and can be estimated from the formula: h min = hmin,ecc − 0.5tc = c (1 − ε) − 0.5tc Misalignment is calculated from the centre of the bearing,... is TEAM LRN 224 ENGINEERING TRIBOLOGY denoted by a subscript ‘p' An example of the control volume mesh with this coding is shown in Figure 5 .12 The dimensions of the mesh are individually labelled to allow the asymmetry of the control volume to be incorporated in the controlling equations k=KNODE Inlet Pad δz Outlet z x δx k =1 i =1 Sliding surface i=INODE Sliding direction FIGURE 5 .11 Control volume... pronounced influence on maximum hydrodynamic pressure In Figure 5.6 this effect is illustrated for a 12 0° arc bearing, L/D = 1 and an eccentricity ratio of 0.7 Maximum dimensionless pressure p* max 2 1. 5 1 0.5 0 0 .1 0.2 0.3 0.4 0.5 0.6 t Effect of misalignment on maximum hydrodynamic pressure in a partial arc bearing Misalignment parameter FIGURE 5.6 It can be seen that the maximum pressure is more... 5.7 24 36 60 48 es gre De Computed pressure profile for 12 0° perfectly aligned partial bearing 10 0% P% Pressure along the load line Outlet 12 0 96 8 84 10 72 L 0.75 L 0.5 L Inlet 0.25 L 0 FIGURE 5.8 12 24 36 60 48 es gre De Computed pressure field for 12 0° misaligned partial bearing It can be seen from Figures 5.7 and 5.8 that the misaligned bearing shows a skewed pressure field with a relatively high... composed of trapezia with the dimensions normal to the plane of the lubricant film varying in proportion to the lubricant film thickness is illustrated in Figure 5 .11 The grid-scheme shown in Figure 5 .11 is two dimensional Other researchers [e.g 10 ] have developed three-dimensional schemes which are more accurate but far more demanding of computing time The edges or faces of each control volume (which is... 210 ENGINEERING TRIBOLOGY τ* = e h* cav h*2 (5.23) This value of dimensionless shear stress is included in the integral for dimensionless friction force (eq 5 .17 ) with no further modification Values of h*, ∂h*/∂x*, ∂h*/∂y* and ∂ 2 h*/∂x * 2 are also required in computation and the expressions for these are: h* = y*tcos(x*) + εcos(x* − β) + 1 (5.24) where: ε is the eccentricity . 94 -10 0. 57 B. Bhushan and K. Tonder, Roughness-Induced Shear- and Squeeze-Film Effects in Magnetic Recording, Parts 1 and 2, Transactions ASME, Journal of Tribology, Vol. 11 1, 19 89, Part 1, pp + ( ( R L 2 2C 1 + 2C 2 + F i,j M v,i,j = C 1 ( ( M v,i +1, j + M v,i 1, j C 2 ( ( M v,i,j +1 + M v,i,j 1 − G i,j (5.9) TEAM LRN 206 ENGINEERING TRIBOLOGY where: C 1 = 1 δx* 2 C 2 = 1 δy* 2 This. given by: ∂M v ∂x* ≈ M v,i +1 − M v,i 1 2δx* ( ( i M v,i 1 M v,i +1 M v x* x* i x* i +1 x* i 1 M v,i 2δx* ∂M v ∂x* ( ( i Approximation to (5.7) where the subscripts i- 1 and i +1 denote positions immediately