The General Form of Reynolds Equation

THE GENERAL FORM  OF REYNOLDS  EQUATIONS • For a general (moving) boundary with h ( x,t ) • The flow can  be  unsteady  but is fully-­‐developed   locally.   1. Beginning from  the locally fully-­‐developed  flow derived in class, for which we required 2 2 ⎛ h⎞ ⎛ h⎞ that   ⎜ ⎟  1 , Re L ⎜ ⎟  1 , we  know that the  velocity  profile  in the  gap  is: ⎝ L⎠ ⎝ L⎠ 2 ⎛ y ⎞ ⎤ ⎛ y ⎞ h(x,t)2 ⎛ dp ⎞ ⎡ y + vx (x, y,t) = U ⎜ 1 − − ⎥ ⎜− ⎟ ⎢ ⎝ 2 µ ⎝ dx ⎠ ⎢⎣ h(x,t) ⎜⎝ h(x,t)⎠⎟ ⎥⎦ h(x,t)⎠⎟ Couette Poiseuille (wall driven) (pressure driven) (1) Integrating  this expression  gives an expression  for  the  local flow rate   q′ at any given slice: q′(x,t) ≡ h(x,t ) ∫0 vx dy = h(x,t)3 ⎛ ∂p ⎞ 1 ⎜ − ⎟ + Uh(x,t) 12 µ ⎝ ∂x ⎠ 2 (2) **Note that this may not be constant along the channel because of the squeezing flow induced by vertical motion of the channel boundary at h(x,t).   2. We combine with this with an integrated analysis of the flow in the vertical direction, and the use of what is commonly called the “kinematic boundary condition” at the upper plate. – This links the displacement in the boundary motion to the local  fluid  flow rate – Essentially  we apply conservation of mass to the thin strip of fluid h ( x,t ) dx a).   We start with ∂vx ∂vy + = 0 , integrate  in y to  obtain ∂x ∂y h( x ) ⌠ ⎮ ⌡0 h( x ) ∂vx ⌠ dx + ⎮ ∂x ⌡0 (? ) + Vy y= h ( x,t ) y= 0 ∂vy ∂y dy = 0 (3a) =0 (3b) The second integral introduces  the  boundary  conditions  for the   vertical rate of displacement of the upper and lower surfaces (which may be stationary  or moving BCs).   To find the first  term, labeled (? ) in eq. 3, we need to use the Leibnitz Theorem,  in the form: ∂f ∂ ⎧ ∫a ∂x dy = ∂x ∫a f dy − ⎨⎩ f b b 1 y=b db −f dx y= a da ⎫ ⎬ dx ⎭ recognizing that here f = vx , a = 0 and  b = h(x), we can  write that   h( x ) ⌠ ⎮ ⌡0 ∂vx d h ⎧ dh dy = ∫ vx dy − ⎨ vx ∂x dx 0 ⎩ dx y= h − d(0) U dx y= 0 ⎫ ⎬ ⎭ (4) The first term  on the right-hand  side is nothing  but the  variation  in the  flow rate  of fluid   along  the gap (which again  we re-emphasize might not be constant because of the squeezing  induced  by  the  top plate  etc).   • The terms  in { } again have to be evaluated  based  on the  actual boundary  conditions  of the specific  problem at the top and bottom  plates. Combining eq.(3) with 2(b) results in the expression: dq′ dh − U dx dx y= h +V y= h =0 (5)   [Alternately you can apply a control volume analysis directly to a thin slice of the form   shown  in the  sketch  above  to  arrive  at exactly the same final result]. b) To simplify this result, we can recognize that the equation for the location of the free surface is simply the solution of the equation: y − h(x,t) = 0 . Since material points on the   surface  stay  on the  surface as it moves and deforms, we can  take the convected derivative of this expression  so that D ( y − h(x,t)) = 0 Dt ⇒ − ∂h − vx ∂t y=h dh + vy dx y=h =0 (6) This is commonly known as the  kinematic boundary  condition and it is of exactly the same form  as the   expression  derived in eq. (5). Combining eq. (5) and (6)  thus  results  in the   following  much simpler expression: dq′ ∂h + =0 dx ∂t (7) 3) We can now combine this with the result obtained  in eq. (2) to  obtain  the following nonlinear coupled PDE: 1 ∂ ⎛ 3 ∂p ⎞ ∂h ∂h + 12 ⎜⎝ h ⎟⎠ = 6U ∂x µ ∂x ∂t ∂x (8) This  is referred to  as  the general form  of Reynolds Lubrication  Equation.   If the shape  of the boundary and it’s displacement are specified (i.e. h(x,t) is given) then  it  gives a second order differential equation  that can  be  solved  for the  pressure  field  p(x,t) (subject to appropriate boundary  conditions on  the pressure field at the edge of the fluid film). Alternatively it can be combined with a global force balance on a moving block or plate or other object that has a thin film  of viscous fluid underneath it to develop  a differential equation for the motion of the block. It is thus the starting  point for many different research  topics  as  well as  homework or qualifying problems! 2 There are many, many generalizations, e.g. – cylindrical coordinates – compressible fluids – ‘suction’ or ‘porous wall’  boundary conditions – stretching  surfaces,  collapsing,  bending  surfaces *All collapse down to the same basic physics, which is the key thing to internalize: (i) nondimensionalize governing  equations of motion. 2 ⎛ ⎛H⎞ ⎞ (ii) drop “small” terms i.e. 0 ⎜ Re L ⎜ ⎟ ⎟ ... ⎝ L⎠ ⎠ ⎝ (iii) solve  locally  fully-­‐developed  viscous  flow in slice dx (iv) integrate  across  the slice  (i.e. the   “fast” direction)  to find an integrated quantity   such  as  the  local flow rate  per unit depth,   q′(x) (v) use conservation of mass or other appropriate BCs to integrate  along  the “slow”   direction,  along  which  integrated  quantity   q′(x) varies Remember that effectively we have the following  two independent  constraints: L ty  and t y  t x V i.e. the time-­‐scale for diffusion of vorticity information across the  thin gap is always much shorter than the time required for the information to be convected along the gap by the fluid  flow (with convection time t conv ~ L / V ) or to  diffuse along  the  gap  (with  a characteristic time scale t x ~ L2 ν ). This is why  we  say  the  flow is locally  fully-­developed. • The second constraint is the same as saying: h 2 L2  1 • The first  constraint  can  be rearranged several  different  ways: 2 ⎛ ρVL ⎞ ⎛ h ⎞ ⎛ ρVh ⎞ ⎛ h ⎞ h2 L ρVh 2 ρ  ⇒ 1 or ⎜ ⎜ ⎟  1 or ⎜ ⎜ ⎟ 1 ⎝ µ ⎟⎠ ⎝ L ⎠ ⎝ µ ⎟⎠ ⎝ L ⎠ µ V µL 3