Flow in a Lubricated Bearing (COMSOL) Consider the situation in Fig. E8.5.1, in which one inclined plate (surface 3) moves at a velocity V relative to a stationary plate at y = 0, the intervening space being occupied by a Newtonian fluid of density 1,000 and viscosity 1.0. All units are mutually co

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Flow in a Lubricated Bearing (COMSOL)

Consider the situation in Fig. E8.5.1, in which one inclined plate (surface 3) moves at a velocity [latex]V[/latex] relative to a stationary plate at [latex]y=0[/latex], the intervening space being occupied by a Newtonian fluid of density 1,000 and viscosity [latex]1.0[/latex]. All units are mutually consistent (taken as SI by COMSOL unless we specify otherwise). The velocity components of the moving plate are [latex]u_{0}=0.05[/latex] in the [latex]x[/latex] direction and [latex]v_{0}=-0.01[/latex] in the [latex]y[/latex] direction. The other three boundary conditions (on surfaces 1, 2 , and 4) are shown in Fig. E8.5.1. The two "ends," at [latex]x=-1.4[/latex] and [latex]x=1.4[/latex], are exposed to a pressure of [latex]p=0[/latex]. In reality, a bearing would have a much smaller mean thickness/length ratio, and would generate higher internal pressures, but the present choice allows the main features to be displayed without "stretching" the [latex]y[/latex] coordinate.

Solve for the streamlines, the pressure distribution, and the variation of pressure along the straight line between midpoints of the opposite "ends"---between points with coordinates [latex](-1.4,0.3)[/latex] and [latex](1.4,0.1)[/latex].

Accepted Answer

See Chapter 14 for more details about COMSOL Multiphysics. All mouse-clicks are left-clicks (the same as Select) unless specifically denoted as right-clicks (R).

Select the Physics

Open COMSOL, L-click Model Wizard, 2D, Fluid Flow (the little rotating triangle is called a "glyph"), Single-Phase Flow, Laminar Flow, Add.
L-click Study, Stationary, Done.

Establish the Geometry

R-click Geometry, select Polygon. Under Coordinates in the Settings window enter the [latex]x[/latex] values [latex]-1.4,1.4,1.4,-1.4[/latex] and [latex]y[/latex] values [latex]0.0,0.0,0.2,0.6[/latex] (which automatically have units of meters). L-click Build Selected.

Define the Fluid Properties

R-click the Materials node within Component 1 and select Blank Material. Enter 1000 and 1 for the values of rho and [latex]m u[/latex], which automatically have units [latex]\mathrm{kg} / \mathrm{m}^{3}[/latex] and Pa [latex]\mathrm{s}[/latex].

Define the Boundary Conditions

R-click the Laminar Flow node and select Wall to define a velocity boundary condition for the upper wall. Hover over the top wall until it turns red, then select it with a L-click and note that it turns blue. L-click and change the Boundary Condition to Moving Wall and enter the values [latex]0.05[/latex] and [latex]-0.01[/latex] for the wall velocity components (assumed to be [latex]\mathrm{m} / \mathrm{s}[/latex] ).
R-click the Laminar Flow node and select Outlet. Add the left and right boundary segments to the Boundary Condition by hovering and L-clicking each segment and note that Pressure Conditions give [latex]p=0[/latex] for each segment. Deselect the Suppress backflow option. L-click the Laminar Flow glyph and select the Equation glyph under Settings, to see the equations being solved.

Create the Mesh and Solve the Problem

L-click the Mesh node and change the Element size to Fine; Build All. Then R-click Mesh and pull down to Statistics, noting that there are approximately 7,000 elements. Zoom in three times and note the elements next to the boundaries are rectangular, with all the interior elements being triangular.
As a reminder, save your work frequently.
R-click the Study node and select Compute. The result is a surface plot for the velocity magnitudes, which we don't particularly want.

Display the Results

R-click the Results node and select 2D Plot Group. R-click the new [latex]2 \mathrm{D}[/latex] Plot Group 3 and select Streamline. Under Streamline Positioning select the Positioning option and select Start point controlled. L-Click Plot. Change Points from 20 to 25 , Plot, giving Fig. E8.5.2.
R-click the Results node and select 2D Plot Group. R-click the new 2D Plot Group 4 and select Surface. Enter [latex]p[/latex] in the Expression field. L-Click Plot, giving Fig. E8.5.3, with a color bar at the right.
R-click Data Sets within the Results tree and select Cut Line 2D. In the Settings window enter the values [latex](-1.4,0.3)[/latex] and [latex](1.4,0.1)[/latex] for Points 1 and [latex]2 .[/latex] L-click Plot near the top of the Settings.
R-click Results and L-click 1D Plot Group. In the Settings window, L-click the dropdown dialog and change the data set to Cut Line 2D 1 .
R-click the 1D Plot Group and select Line Graph. Within the Settings window enter the variable for pressure, [latex]p[/latex], in the Expression field. Finally, L-Click Plot, giving E3.5.4.

Discussion of Results

The COMSOL results are shown in Figs. E8.5.2-E8.5.4. The first of these displays the streamlines. In the upper part of the left-hand side, and for almost all of the right-hand side, the motion is from left to right, driven by the moving upper plate. But much of the left-hand region is occupied by a clockwise-rotating vortex-type flow; in the bottom half of this vortex, the reason for the flow in the negative [latex]x[/latex] direction is readily seen by examining Fig. E8.5.3, which shows the following important features:

(a) A pressure that essentially depends only on [latex]x[/latex] and not on [latex]y[/latex]---a fact that is at the heart of simplified treatments of boundary-layer and lubrication-type flows.

(b) A high-pressure zone centered at about [latex]75 \%[/latex] of the distance from the left end. The maximum pressure is approximately [latex]p=2.15 \mathrm{~Pa}[/latex].

Fig. E8.5.4 clearly shows the pressure variation between the midpoints of the two ends --- between points [latex](-1.4,0.3)[/latex] and [latex](1.4,0.1)[/latex]. The variation is broadly parabolic but with a peak that is noticeably skewed toward the right of the center.

Question 8.5: Flow in a Lubricated Bearing (COMSOL) Consider the situation...

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