7. Hydrodynamic Lubrication: Reynolds ... Boundary lubrication Mixed lubrication Hydrodynamic...

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Transcript of 7. Hydrodynamic Lubrication: Reynolds ... Boundary lubrication Mixed lubrication Hydrodynamic...

  • IMAC-U course 6th semester

    Tribology 7. Hydrodynamic Lubrication: Reynolds Equation

    Assoc. Prof. Takeshi YAMAGUCHI [email protected]

  • Stribeck curve Fr

    ic ti

    o n

    c o

    ef fi

    ci e

    n t

    (Viscosity×velocity)/load (equivalent to fluid film thickness)

    Dry friction

    Boundary lubrication

    Mixed lubrication

    Hydrodynamic lubrication

    Bearing

    Shaft

    Lubricating oil

  • Upper fixed surface

    Lower moving surface

    U

    x

    y

    dy

    dx

    t

    t+dt

    p+dp p

    Microelement in fluid

    h

    dx

    dh U

    dx

    dph

    dx

    d 6

    3

     

      

     

    h (7.1)

    If h=h(x) is known, a distribution of fluid pressure can be obtained.

    Reynolds equation

  • Assumptions

    1. Fluid is incompressible Newtonian fluid. 2. Flow in the gap is a laminar flow and viscosity of the

    fluid is constant. 3. Inertia force of fluid is negligibly small compared to its

    viscous force. 4. Pressure change in the gap direction can be neglected

    because the thickness of fluid film is very small.

  • Derivation of Reynolds Equation (1)

    0 

      

     

      

      dxdy

    dy

    d dxdydx

    dx

    dp ppdy

    t tt (7.2)

    (7.3)

  • dy

    d

    dx

    dp t  (7.3)

    dy

    du ht  (7.4)

    Where h is viscosity of fluid, and u is the flow velocity in the x direction.

    (7.5)

    Derivation of Reynolds Equation (2)

  • Derivation of Reynolds Equation (3)

    2

    2

    dy

    ud

    dx

    dp h (7.5) (7.6)

    2nd integration

    Boundary condition: y=0, u= U; y=h, u=0

    (7.7)

  • Couette flow and Piseuille flow

      dx

    dp yhy

    h

    yh Uu 

     

    h2

    1

    ( ) ( )

    ( ) ( )

    (7.7)

  • Derivation of Reynolds Equation (4)

    Flow rate per unit depth length Q:

      h

    dx

    dphUh udyQ

    0

    2

    122 h (7.8) (7.9)

    Mass conservation raw

    (7.1)

  • Generalization of Reynolds Equation

    Upper moving surface

    Lower moving surface U1

    x

    y

    h z

    U2

    V

    W2

    W1

    u

    v

    w h

    2

    2

    2

    2

    y

    w

    z

    p

    y

    u

    x

    p

     

     

    h

    h (7.10)

    (7.11)

    Motion equation of fluid Boundary conditions

    y = 0: u = U1, w = W1 = 0

    y = h: u = U2, w = W2 = 0 (7.12)

  •    

    t

    h h

    x

    UU h

    x

    h UU

    z

    ph

    zx

    ph

    x

     

     

     

     

      

     

     

      

     

    1266 2121

    33

    hh

    Generalization of Reynolds Equation

    Upper moving surface

    Lower moving surface U1

    x

    y

    h z

    U2

    V

    W2

    W1

    u

    v

    w h

    (7.13)

    Wedge film action Stretch film action Squeeze film action

  •    

    t

    h h

    x

    UU h

    x

    h UU

    z

    ph

    zx

    ph

    x

     

     

     

     

      

     

     

      

     

    1266 2121

    33

    hh

    Wedge film action

    Wedge film action

    U

    0 

    x

    h It requires inclined surfaces to generate a fluid film wedge action that results in a pressure wave. Positive pressure is generated if the film thickness reduces in the x direction.

  •    

    t

    h h

    x

    UU h

    x

    h UU

    z

    ph

    zx

    ph

    x

     

     

     

     

      

     

     

      

     

    1266 2121

    33

    hh

    Wedge film action

    When the wall velocity reduces in the x direction, negative fluid pressure is generated. This is not usually occurred in normal sliding surfaces.

    Stretch film action

    0 

    x

    U

  •    

    t

    h h

    x

    UU h

    x

    h UU

    z

    ph

    zx

    ph

    x

     

     

     

     

      

     

     

      

     

    1266 2121

    33

    hh

    Wedge film action

    C

    A positive pressure is generated if the surfaces are approaching each other.

    Squeeze film action

    -V

    0 

    t

    h