Effects of Thermocapillarity and Thermal Radiation on Flow and Heat Transfer in a Thin Liquid Film on an Unsteady Stretching Sheet

<table>Effects of <svg height="11.175" id="M302" 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)"><use xlink:href="#x1D440"></use></g>
</svg> on (a) <svg height="18.025" id="M303" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 36.25 18.025" width="36.25" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,13.525)"><use xlink:href="#x1D453"></use></g><g transform="matrix(.017,-0,0,-.017,10.976,13.525)"><use xlink:href="#x2032"></use></g><g transform="matrix(.017,-0,0,-.017,16.024,13.525)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,21.906,13.525)"><use xlink:href="#x1D702"></use></g><g transform="matrix(.017,-0,0,-.017,30.303,13.525)"><use xlink:href="#x29"></use></g>
</svg> and (b) <svg height="16.6625" id="M304" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 28.75 16.6625" width="28.75" 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)"><use xlink:href="#x1D703"></use></g><g transform="matrix(.017,-0,0,-.017,8.528,12.162)"><use xlink:href="#x28"></use></g><g transform="matrix(.017,-0,0,-.017,14.409,12.162)"><use xlink:href="#x1D702"></use></g><g transform="matrix(.017,-0,0,-.017,22.807,12.162)"><use xlink:href="#x29"></use></g>
</svg> using 10th-order HAM approximation for the case of <svg height="19.4125" id="M305" style="vertical-align:-5.76984pt" version="1.1" viewbox="0 0 66.75 19.4125" width="66.75" 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)"><use xlink:href="#x210F"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.837,16.25)"><use xlink:href="#x1D453"></use></g>
<g transform="matrix(.017,-0,0,-.017,21.813,12.162)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,36.516,12.162)"><use xlink:href="#x2212"></use></g><g transform="matrix(.017,-0,0,-.017,46.495,12.162)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,54.654,12.162)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,58.53,12.162)"><use xlink:href="#x36"></use></g>
</svg>, <svg height="16.450001" id="M306" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 65.037498 16.450001" width="65.037498" 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)"><use xlink:href="#x210F"></use></g>
<g transform="matrix(.012,-0,0,-.012,8.837,16.25)"><use xlink:href="#x1D703"></use></g>
<g transform="matrix(.017,-0,0,-.017,20.1,12.162)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,34.804,12.162)"><use xlink:href="#x2212"></use></g><g transform="matrix(.017,-0,0,-.017,44.782,12.162)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,52.942,12.162)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,56.817,12.162)"><use xlink:href="#x38"></use></g>
</svg>, <svg height="11.75" id="M307" style="vertical-align:-0.23206pt" version="1.1" viewbox="0 0 47.900002 11.75" width="47.900002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.4)"><use xlink:href="#x1D446"></use></g><g transform="matrix(.017,-0,0,-.017,12.931,11.4)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,27.634,11.4)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,35.794,11.4)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,39.67,11.4)"><use xlink:href="#x34"></use></g>
</svg>, <svg height="15.35" id="M308" style="vertical-align:-3.35245pt" 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)"><use xlink:href="#x1D441"></use></g>
<g transform="matrix(.012,-0,0,-.012,14.925,15.187)"><use xlink:href="#x1D445"></use></g>
<g transform="matrix(.017,-0,0,-.017,27.925,11.112)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,42.612,11.112)"><use xlink:href="#x31"></use></g>
</svg>, <svg height="11.175" id="M309" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 43.424999 11.175" width="43.424999" 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)"><use xlink:href="#x50"></use></g><g transform="matrix(.017,-0,0,-.017,9.48,11.113)"><use xlink:href="#x72"></use></g><g transform="matrix(.017,-0,0,-.017,20.495,11.113)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,35.199,11.113)"><use xlink:href="#x31"></use></g>
</svg>, and <svg height="13.95" id="M310" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 66 13.95" width="66" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,13.688)"><use xlink:href="#x1D435"></use></g>
<g transform="matrix(.012,-0,0,-.012,10.55,5.525)"><use xlink:href="#x2217"></use></g>
<g transform="matrix(.017,-0,0,-.017,22.875,13.688)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,37.579,13.688)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,45.738,13.688)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,49.614,13.688)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,57.773,13.688)"><use xlink:href="#x35"></use></g>
</svg>.</table>

Mathematical Problems in Engineering

fig4

Figure 4

Figure 4: Effects of Thermocapillarity and Thermal Radiation on Flow and Heat Transfer in a Thin Liquid Film on an Unsteady Stretching Sheet 