Nanofluid Flow over a Permeable Surface with  Convective Boundary Conditions and Radiative Heat Transfer

<table>Effects of (a) thermal radiation <svg height="15.35" id="M170" style="vertical-align:-3.3525pt" version="1.1" viewbox="0 0 23.262501 15.35" width="23.262501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
l350 -516h5l48 257q25 131 25 171q0 34 -17.5 45t-77.5 15l7 28h240z" id="x1D441"></path></g>
<g transform="matrix(.012,-0,0,-.012,14.925,15.187)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
t-115.5 -72q40 -86 66 -133q39 -68 65 -101q28 -37 73 -51zM491 483q0 67 -33.5 101t-91.5 34q-35 0 -51 -10q-13 -8 -20 -48l-45 -245h49q71 0 113 28q79 52 79 140z" id="x1D445"></path></g>
</svg>, when <svg height="11.375" id="M171" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 50.275002 11.375" width="50.275002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M372 352v-2q176 -27 176 -159q0 -99 -95 -151q-73 -40 -196 -40h-221v28q65 5 78.5 19t13.5 76v404q0 63 -12 77t-72 18v28h257q102 0 150 -34q26 -18 41 -48.5t15 -65.5q0 -111 -135 -150zM213 362h47q80 0 117.5 32t37.5 96q0 56 -33 92t-100 36q-41 0 -56 -10
q-13 -8 -13 -50v-196zM213 328v-203q0 -54 17 -72.5t66 -18.5q67 1 110.5 37t43.5 110q0 147 -192 147h-45z" id="x42"></path></g><g transform="matrix(.017,-0,0,-.017,9.922,11.112)"><path d="M135 536q-20 0 -35 15.5t-15 35.5q0 22 15 37t36 15t35.5 -15t14.5 -37q0 -21 -15 -36t-36 -15zM252 0h-220v26q48 5 59 17t11 63v206q0 47 -9 58t-54 18v24q78 12 142 39v-345q0 -51 11.5 -63t59.5 -17v-26z" id="x69"></path></g><g transform="matrix(.017,-0,0,-.017,19.186,11.112)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,33.89,11.112)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,42.049,11.112)"><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>
</svg> and (b) effects of <svg height="11.175" id="M172" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 14.5375 11.175" width="14.5375" 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="M372 352v-2q176 -27 176 -159q0 -99 -95 -151q-73 -40 -196 -40h-221v28q65 5 78.5 19t13.5 76v404q0 63 -12 77t-72 18v28h257q102 0 150 -34q26 -18 41 -48.5t15 -65.5q0 -111 -135 -150zM213 362h47q80 0 117.5 32t37.5 96q0 56 -33 92t-100 36q-41 0 -56 -10
q-13 -8 -13 -50v-196zM213 328v-203q0 -54 17 -72.5t66 -18.5q67 1 110.5 37t43.5 110q0 147 -192 147h-45z" id="x42"></path></g><g transform="matrix(.017,-0,0,-.017,9.922,11.113)"><path d="M135 536q-20 0 -35 15.5t-15 35.5q0 22 15 37t36 15t35.5 -15t14.5 -37q0 -21 -15 -36t-36 -15zM252 0h-220v26q48 5 59 17t11 63v206q0 47 -9 58t-54 18v24q78 12 142 39v-345q0 -51 11.5 -63t59.5 -17v-26z" id="x69"></path></g>
</svg>,  when <svg height="15.35" id="M173" style="vertical-align:-3.3525pt" version="1.1" viewbox="0 0 50.849998 15.35" width="50.849998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M857 650l-6 -28q-44 -4 -61.5 -16.5t-29.5 -48.5q-11 -32 -37 -166l-78 -399h-29l-351 537h-4l-56 -276q-24 -120 -24 -164q0 -35 17.5 -46t75.5 -15l-6 -28h-245l7 28q41 2 62 14t31 44q10 30 41 171l53 245q8 44 6.5 60.5t-14.5 33.5q-10 15 -27 19.5t-64 6.5l6 28h153
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<g transform="matrix(.012,-0,0,-.012,14.925,15.187)"><path d="M627 18l-10 -26q-79 6 -116 27t-69 76q-41 71 -71 138q-13 29 -27.5 39t-42.5 10h-46l-27 -145q-13 -74 -2.5 -88.5t78.5 -20.5l-6 -28h-271l5 28q66 6 82.5 21.5t30.5 87.5l71 387q12 66 2 78.5t-77 19.5l8 28h233q102 0 147 -29q65 -43 65 -129q0 -69 -45.5 -117
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<g transform="matrix(.017,-0,0,-.017,27.925,11.112)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,42.612,11.112)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg> on temperature for various values of <svg height="16.3125" id="M174" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 49.349998 16.3125" width="49.349998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.813)"><path d="M542 242q0 -106 -80 -176.5t-192 -72.5l-55 -251l-6 -3q-18 7 -28 38t1 91l23 124q-85 16 -133.5 72.5t-48.5 137.5q0 95 62.5 159.5t166.5 97.5l14 -31q-77 -32 -115 -82.5t-38 -128.5q0 -63 31.5 -107t74.5 -59l125 597l56 43l16 -10l-40 -168q-9 -40 3 -45
q77 -33 120 -85.5t43 -140.5zM469 232q0 67 -37 115t-77 62l-76 -370q89 -4 139.5 52.5t50.5 140.5z" id="x1D719"></path></g><g transform="matrix(.017,-0,0,-.017,14.375,11.813)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,29.079,11.813)"><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,-.017,37.239,11.813)"><path d="M113 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16z" id="x2E"></path></g><g transform="matrix(.017,-0,0,-.017,41.114,11.813)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg>, <svg height="17.4" id="M175" style="vertical-align:-4.15506pt" version="1.1" viewbox="0 0 47.712502 17.4" width="47.712502" 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="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
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<g transform="matrix(.012,-0,0,-.012,10.975,17.2)"><path d="M689 332q0 -81 -40 -161.5t-103.5 -131.5t-127.5 -51q-32 0 -58.5 16t-40.5 46q-70 -62 -139 -62q-45 0 -77.5 30t-32.5 86q0 8 7 52q11 77 45 187q6 19 6 34q0 6 -7 6q-24 0 -78 -64l-20 23q32 48 73 77t78 29q32 0 32 -43q0 -30 -12 -73q-27 -95 -38 -152
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<g transform="matrix(.017,-0,0,-.017,24.787,12.162)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,39.491,12.162)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg>, and <svg height="11.6875" id="M176" style="vertical-align:-0.20473pt" version="1.1" viewbox="0 0 55.224998 11.6875" width="55.224998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.363)"><path d="M409 504l-29 -5q-16 60 -44.5 96t-86.5 36q-54 0 -82.5 -32t-28.5 -77q0 -51 30.5 -82.5t96.5 -65.5l35.5 -18t33 -19.5t33.5 -23.5l27 -25.5t24.5 -32t13.5 -36.5t6 -45q0 -80 -62 -134.5t-160 -54.5q-47 0 -98 15q-22 7 -50 21q-8 23 -27 155l30 7q7 -27 18 -52
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</svg>.</table>

Mathematical Problems in Engineering

fig7

Figure 7

Figure 7: Nanofluid Flow over a Permeable Surface with  Convective Boundary Conditions and Radiative Heat Transfer 