Term information
The Lamm equation[1] describes the sedimentation and diffusion of a solute under ultracentrifugation in traditional sector-shaped cells. (Cells of other shapes require much more complex equations.) source: http://en.wikipedia.org/wiki/Lamm_equation
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <mrow> <mfrac> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>c</mi> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <mrow> <mi>D</mi> <mo>⁢</mo> <mfenced close="]" open="["> <mrow> <mfenced close=")" open="("> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <mo>⁢</mo> <mi>c</mi> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mfenced> <mo>+</mo> <mrow> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> <mo>⁢</mo> <mfenced close=")" open="("> <mfrac> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>c</mi> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>r</mi> </mrow> </mfrac> </mfenced> </mrow> </mrow> </mfenced> </mrow> <mo>-</mo> <mrow> <mi>s</mi> <mo>⁢</mo> <msup> <mi>ω</mi> <mn>2</mn> </msup> <mo>⁢</mo> <mfenced close="]" open="["> <mrow> <mrow> <mi>r</mi> <mo>⁢</mo> <mfenced close=")" open="("> <mfrac> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>c</mi> </mrow> <mrow> <mo>∂</mo> <mo>⁢</mo> <mi>r</mi> </mrow> </mfrac> </mfenced> </mrow> <mo>+</mo> <mrow> <mn>2</mn> <mo>⁢</mo> <mi>c</mi> </mrow> </mrow> </mfenced> </mrow> </mrow> </mrow> </math>
\frac{\partial c}{\partial t} = D \left[ \left( \frac{\partial^{2} c}{\partial r^2} \right) + \frac{1}{r} \left( \frac{\partial c}{\partial r} \right) \right] - s \omega^{2} \left[ r \left( \frac{\partial c}{\partial r} \right) + 2c \right]
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"> <semantics> <mrow> <mfrac> <mrow> <mo>∂</mo> <mi>c</mi> </mrow> <mrow> <mo>∂</mo> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mi>D</mi> <mfenced close="]" open="["> <mrow> <mfenced close=")" open="("> <mfrac> <mrow> <msup> <mo>∂</mo> <mn>2</mn> </msup> <mi>c</mi> </mrow> <mrow> <mo>∂</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> </mfrac> </mfenced> <mo>+</mo> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> <mfenced close=")" open="("> <mfrac> <mrow> <mo>∂</mo> <mi>c</mi> </mrow> <mrow> <mo>∂</mo> <mi>r</mi> </mrow> </mfrac> </mfenced> </mrow> </mfenced> <mo>-</mo> <mi>s</mi> <msup> <mi>ω</mi> <mn>2</mn> </msup> <mfenced close="]" open="["> <mrow> <mi>r</mi> <mfenced close=")" open="("> <mfrac> <mrow> <mo>∂</mo> <mi>c</mi> </mrow> <mrow> <mo>∂</mo> <mi>r</mi> </mrow> </mfrac> </mfenced> <mo>+</mo> <mn>2</mn> <mi>c</mi> </mrow> </mfenced> </mrow> <annotation encoding="SnuggleTeX">\[ \frac{\partial c}{\partial t} = D \left[ \left( \frac{\partial^{2} c}{\partial r^2} \right) + \frac{1}{r} \left( \frac{\partial c}{\partial r} \right) \right] - s \omega^{2} \left[ r \left( \frac{\partial c}{\partial r} \right) + 2c \right] \]</annotation> </semantics> </math>