Analytic Approximate Solutions for Unsteady Two-Dimensional and Axisymmetric Squeezing Flows between Parallel Plates

<table>The <svg height="12.4375" id="M73" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 59.275002 12.4375" width="59.275002" 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="M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l112 522h-84l11 39h81l7 33q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-25 -117h165
l-12 -39h-161l-55 -261h2q77 102 195 167q44 24 79 24q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z" id="x210F"></path></g><g transform="matrix(.017,-0,0,-.017,8.834,12.162)"><path d="M300 253l-14 -55h-229l14 55h229z" id="x2D"></path></g><g transform="matrix(.017,-0,0,-.017,14.885,12.162)"><path d="M390 111l17 -21q-34 -45 -80 -73.5t-89 -28.5q-91 0 -146 62t-55 147q0 118 101 195q74 57 149 57h1q59 0 90 -27q16 -14 16 -30q0 -15 -12 -29t-21 -14q-8 0 -19 11q-44 41 -101 41q-52 0 -87.5 -42.5t-35.5 -117.5q0 -49 15 -87t39 -58t49 -30t48 -10q33 0 60.5 12
t60.5 43z" id="x63"></path></g><g transform="matrix(.017,-0,0,-.017,22.178,12.162)"><path d="M518 50v-26q-32 -4 -77 -16l-37 -11t-33 -9l-6 6v71q-56 -42 -74 -52q-43 -25 -77 -25q-55 0 -89.5 36.5t-34.5 112.5v196q0 38 -6.5 48.5t-26.5 15.5l-28 6v23q19 1 72 7q59 9 71 11q-3 -31 -3 -145v-135q0 -113 88 -113q59 0 108 48v234q0 38 -8 49t-34 15l-36 6v23
q44 2 90 8q37 4 67 10v-334q0 -37 8.5 -47t38.5 -12z" id="x75"></path></g><g transform="matrix(.017,-0,0,-.017,31.204,12.162)"><path d="M181 342h2q63 107 121 107q24 0 41 -16t17 -34q0 -34 -32 -49q-17 -7 -26 2q-21 20 -43 20q-20 0 -42 -22t-38 -63v-183q0 -50 12.5 -62t69.5 -16v-26h-230v26q47 5 58 17t11 61v207q0 49 -9 60.5t-53 16.5v23q74 12 141 40v-109z" id="x72"></path></g><g transform="matrix(.017,-0,0,-.017,38.004,12.162)"><path d="M478 437v-26q-39 -5 -53 -17.5t-35 -57.5q-68 -149 -142 -347h-28q-47 128 -138 350q-17 43 -30.5 54.5t-53.5 17.5v26h205v-26q-40 -4 -46.5 -14.5t6.5 -42.5q41 -115 95 -247q43 101 89 238q12 36 3.5 48.5t-50.5 17.5v26h178z" id="x76"></path></g><g transform="matrix(.017,-0,0,-.017,45.755,12.162)"><path d="M380 106l15 -22q-42 -52 -87 -74t-76 -22q-89 0 -142 62t-53 151q0 103 66 175q67 73 146 73h1q61 0 105.5 -46.5t44.5 -105.5q0 -16 -4.5 -22.5t-18.5 -9.5q-54 -10 -262 -17q1 -95 46 -143.5t104 -48.5q61 0 115 50zM225 412q-35 0 -64.5 -32.5t-40.5 -90.5q83 0 170 3
q14 0 19 4.5t5 15.5q1 41 -23.5 70.5t-65.5 29.5z" id="x65"></path></g><g transform="matrix(.017,-0,0,-.017,52.98,12.162)"><path d="M319 325l-25 -7q-33 99 -103 99q-29 0 -47 -19.5t-18 -49.5t22 -49.5t62 -36.5q63 -26 95 -57t32 -79q0 -64 -50 -101t-115 -37q-35 0 -67.5 10.5t-46.5 23.5q-5 11 -11 51t-6 67l27 5q14 -53 46.5 -88.5t75.5 -35.5q29 0 50 19.5t21 50.5t-19.5 51.5t-59.5 39.5
q-28 12 -46 22.5t-38.5 27t-30.5 38.5t-10 49q0 54 42.5 92t109.5 38q48 0 88 -18q6 -15 13 -50.5t9 -55.5z" id="x73"></path></g>
</svg> of <svg height="22.15" id="M74" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 35.112499 22.15" width="35.112499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,17.65)"><path d="M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19
q34 34 71 253l54 315h-90l-3 12l31 30h70q28 138 90 204q35 37 75 57.5t70 20.5q26 0 45 -14t19 -28z" id="x1D453"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.975,9.488)"><path d="M227 744l-123 -338l-31 15l73 368q12 3 41.5 -8t36.5 -20z" id="x2032"></path></g>
<g transform="matrix(.017,-0,0,-.017,15.138,17.65)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,21.019,17.65)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g><g transform="matrix(.017,-0,0,-.017,29.178,17.65)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg> given by the 15th-order approximation for the
axisymmetric case for the different values of the squeeze number <i>S</i>.</table>

Mathematical Problems in Engineering

fig2

Figure 2

Figure 2: Analytic Approximate Solutions for Unsteady Two-Dimensional and Axisymmetric Squeezing Flows between Parallel Plates 