27 x p z x t y p z y t p o x Remembering the formula for dynamic absolute

27 x p z x t y p z y t p o x remembering the formula

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27 x p z x t = y p z y t = p o x =
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Remembering the formula for dynamic (absolute) viscosity: [N.s/m 2 =Pa.s] (5.6) Substituting Eq. (5.6) into (5.5) (5.7) 28 du dy t m = x u u h z t m m = = y v v h z t m m = = p u x z z m æ ö = ç ÷ è ø p v y z z m æ ö = ç ÷ è ø
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The above equations can now be integrated. Since the viscosity of the fluid is constant throughout the film (Assumption 8) and is not a function of ‘z’, the process of integration is simple. (5.8) 29 p u z x z m æ ö ¶ =¶ ç ÷ è ø 1 p u z C x z m + = 1 1 p z C z C u x m æ ö + ¶ + = ç ÷ è ø 2 1 2 2 p z C z C u x m + + =
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Since there is no slip or velocity discontinuity between liquid and solid at the boundaries of the wedge (Assumption 3), the boundary conditions are: u = U 2 at z = 0 u = U 1 at z = h In the general case, there are two velocities corresponding to each of the surfaces ‘U 1 ’ and ‘U 2 ’ . By substituting these boundary conditions into (5.8) the constants ‘C 1 ’ and ‘C 2 are calculated: , (5.9) 30 1 1 2 ( ) 2 p h C U U h x m = - - 2 2 C U m =
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Substituting these into (5.8) yields (5.10) Dividing and simplifying gives the expressions for velocity in the ‘x’ and ‘y’ directions: (5.11) 31 2 1 2 2 ( ) 2 2 p z z p hz U U U u x h x m m m + - - + = 2 1 2 2 ( ) 2 2 z zh z p hz u U U U h x m æ ö - = + - - + ç ÷ è ø 2 1 2 2 ( ) 2 z zh p z v V V V y h m æ ö - = + - + ç ÷ è ø
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c) continuity of flow in a column Consider a column of lubricant as shown below: 32
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The principle of continuity of flow requires that the influx of a liquid must equal its efflux from a control volume under steady conditions. If the density of the lubricant is constant (Assumption 7) then the following relation applies: Simplifying: the equation of continuity of flow in a column (5.12) 33 ( ) y x x y o x y h q q q dy q dx w dxdy q dx dy q dy dx w dxdy x y æ ö + + = + + + + ç ÷ è ø ( ) 0 y x h o q q w w x y + + - = ( ) 0 y x h o q q dxdy dxdy w w dxdy x y + + - =
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Flow rates per unit length, ‘q x ’ and ‘q y ’ , can be found from integrating the lubricant velocity profile over the film thickness, i.e.: Substituting for ‘u’ from equation (5.11) yields: (5.13) 34 0 h x q udz = ò 0 h y q vdz = ò 3 2 2 1 2 2 0 ( ) 3 2 2 2 h x z z h p z q U U U z x h m æ ö = - + - + ç ÷ è ø 3 1 2 ( ) 12 2 x h p h q U U x m =- + + 3 1 2 ( ) 12 2 y h p h q V V x m =- + +
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Substituting now for flow rates into the continuity of flow equation (4.15) Defining U= U 1 + U 2 and V= V 1 + V 2 and assuming that there is no local variation in surface velocity in the x and y directions gives: Full Reynolds equation in three dimensions: (5.14) 35 3 3 1 2 1 2 0 ( ) ( ) ( ) 0 12 2 12 2 h h p h h p h U U V V w w x x y y m m é ù é ù - + + + - + + + - = ê ú ê ú ë û ë û 3 3 0 ( ) 0 12 2 12 2 h h p U dh h p V dh w w x x dx y y dy m m æ ö æ ö - + - + + - = ç ÷ ç ÷ è ø è ø 3 3 0 6 12( ) h h p h p dh dh U V w w x x y y dx dy m m é ù é ù é ù + = + + - ê ú ê ú ê ú ë û ë û ë û
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Simplifications to the Reynolds Equation
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  • Msc. Amar Yemim

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