Weakly Nonlinear Hydrodynamic Stability of the Thin Newtonian Fluid Flowing on a Rotating Circular Disk

<table>(a) Threshold amplitude in subcritical instability region for
three different <svg height="10.625" id="M142" style="vertical-align:-0.12538pt" version="1.1" viewbox="0 0 17.987499 10.625" width="17.987499" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.625)">
<g transform="translate(72,-63.5)">
<text transform="matrix(1,0,0,-1,-71.95,63.67)">
<tspan style="font-size: 12.50px; " x="0" y="0">R</tspan>
<tspan style="font-size: 12.50px; " x="8.0394287" y="0">o</tspan>
</text>
</g>
</g>
</svg> values at <svg height="11.0375" id="M143" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 47.075001 11.0375" width="47.075001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.0375)">
<g transform="translate(72,-63.17)">
<text transform="matrix(1,0,0,-1,-71.95,63.4)">
<tspan style="font-size: 12.50px; " x="0" y="0">R</tspan>
<tspan style="font-size: 12.50px; " x="8.1644592" y="0">e</tspan>
<tspan style="font-size: 12.50px; " x="19.267122" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.307512" y="0">6</tspan>
</text>
</g>
</g>
</svg> and <svg height="11.1" id="M144" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 41.512501 11.1" width="41.512501" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.1)">
<g transform="translate(72,-63.12)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑟</tspan>
<tspan style="font-size: 12.50px; " x="8.5645552" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="20.604944" y="0">5</tspan>
<tspan style="font-size: 12.50px; " x="26.856443" y="0">0</tspan>
</text>
</g>
</g>
</svg>. (b) Threshold amplitude in subcritical instability region for
three different <svg height="7.0124998" id="M145" style="vertical-align:-0.0pt" version="1.1" viewbox="0 0 6.5 7.0124998" width="6.5" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,7.0125)">
<g transform="translate(72,-66.39)">
<text transform="matrix(1,0,0,-1,-71.95,66.44)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑟</tspan>
</text>
</g>
</g>
</svg> values at <svg height="11.0375" id="M146" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 47.075001 11.0375" width="47.075001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.0375)">
<g transform="translate(72,-63.17)">
<text transform="matrix(1,0,0,-1,-71.95,63.4)">
<tspan style="font-size: 12.50px; " x="0" y="0">R</tspan>
<tspan style="font-size: 12.50px; " x="8.1644592" y="0">e</tspan>
<tspan style="font-size: 12.50px; " x="19.267122" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="31.307512" y="0">6</tspan>
</text>
</g>
</g>
</svg> and <svg height="10.9125" id="M147" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 56.912498 10.9125" width="56.912498" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
<g transform="translate(72,-63.27)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">R</tspan>
<tspan style="font-size: 12.50px; " x="8.0394287" y="0">o</tspan>
<tspan style="font-size: 12.50px; " x="17.754259" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="29.794649" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="36.04615" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="39.171898" y="0">1</tspan>
</text>
</g>
</g>
</svg>.</table>

Mathematical Problems in Engineering

fig4

Figure 4

Figure 4: Weakly Nonlinear Hydrodynamic Stability of the Thin Newtonian Fluid Flowing on a Rotating Circular Disk 