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FIGURE 3 Fluid control volume.

FIGURE 4 Force equilibrium on fluid element.

FIGURE 3 Fluid control volume.

FIGURE 4 Force equilibrium on fluid element.

FIGURE 5 Laminar velocity distribution across film.

FIGURE 5 Laminar velocity distribution across film.

p dy dz — a P + dx] dy dz — t dx dz + [ t + dyb dx dz = 0 (1)

0P dt

For a Newtonian fluid in a laminar flow, the shear stress is directly related to the velocity gradient with the proportionality constant being the absolute viscosity m (see Figure 5):

dt d2v

where n is the fluid velocity. Substituting Equation 3 into Equation 2, we obtain d2v dP

dx m 0y2

Integrating with respect to y twice produces the following equation:

1 dp y2

where C1 and C2 are constants of integration.The boundary conditions are v = 0, y = 0 and (6)

Substituting the boundary conditions of Equation 6 into Equation 5 results in the following expression for y:

The velocity in the z direction would be similar, except that the surface velocity term would be omitted because no surface velocity exists in the z direction. In addition, the pressure gradient would be with respect to z.

Now let us consider the flow across the film due to this velocity. Note that Equation 7 is the velocity computed in the x direction, which is in the direction of rotation of the journal:

Qx vx dy

After integrating,

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