Numerical Solution of the MHD Reynolds Equation for Squeeze-Film Lubrication between Porous and Rough Rectangular Plates

<table>The physical configuration of squeeze-film lubrication between two rectangular plates in which the lower plate has a porous material and the upper plate has a rough structure under uniform magnetic field <svg height="11.175" id="M15" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 17.475 11.175" width="17.475" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28
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</svg>. The film thickness <svg height="11.175" id="M16" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 15.0375 11.175" width="15.0375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28
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</svg> contains height of nominal smooth part and roughness. The upper plate approaches the lower one with velocity <svg height="15" id="M17" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 46.700001 15" width="46.700001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M530 686q-79 -330 -126 -595q-5 -24 4 -24q22 0 77 53l16 -24q-42 -49 -88 -78.5t-72 -29.5q-37 0 -19 83l21 99h-2q-62 -83 -153 -141q-65 -41 -97 -41q-28 0 -48 31.5t-20 91.5q0 71 32.5 144t88.5 118q36 30 94 52.5t94 22.5q38 0 68 -13l29 158q7 40 2 47.5t-38 7.5
h-35l1 26q39 4 76 13.5t58.5 17t27.5 7.5q16 0 9 -26zM387 375q-11 11 -38.5 20t-51.5 9q-45 0 -79 -23q-46 -32 -77.5 -107.5t-31.5 -143.5q0 -40 9.5 -58.5t23.5 -18.5q40 0 113.5 70t99.5 118z" id="x1D451"></path></g><g transform="matrix(.017,-0,0,-.017,9.497,12.162)"><path d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28
h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" id="x1D43B"></path></g><g transform="matrix(.017,-0,0,-.017,24.405,12.162)"><path d="M368 703l-264 -866h-60l265 866h59z" id="x2F"></path></g><g transform="matrix(.017,-0,0,-.017,31.408,12.162)"><path d="M530 686q-79 -330 -126 -595q-5 -24 4 -24q22 0 77 53l16 -24q-42 -49 -88 -78.5t-72 -29.5q-37 0 -19 83l21 99h-2q-62 -83 -153 -141q-65 -41 -97 -41q-28 0 -48 31.5t-20 91.5q0 71 32.5 144t88.5 118q36 30 94 52.5t94 22.5q38 0 68 -13l29 158q7 40 2 47.5t-38 7.5
h-35l1 26q39 4 76 13.5t58.5 17t27.5 7.5q16 0 9 -26zM387 375q-11 11 -38.5 20t-51.5 9q-45 0 -79 -23q-46 -32 -77.5 -107.5t-31.5 -143.5q0 -40 9.5 -58.5t23.5 -18.5q40 0 113.5 70t99.5 118z" id="x1D451"></path></g><g transform="matrix(.017,-0,0,-.017,40.843,12.162)"><path d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" id="x1D461"></path></g>
</svg><i>. </i></table>

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fig1

Figure 1

Figure 1: Numerical Solution of the MHD Reynolds Equation for Squeeze-Film Lubrication between Porous and Rough Rectangular Plates 