Intrinsic Viscosity - Formulae For Rigid Spheroids

Formulae For Rigid Spheroids

Generalizing from spheres to spheroids with an axial semiaxis (i.e., the semiaxis of revolution) and equatorial semiaxes, the intrinsic viscosity can be written

\left = \left( \frac{4}{15} \right) (J + K - L) + \left( \frac{2}{3} \right) L + \left( \frac{1}{3} \right) M + \left( \frac{1}{15} \right) N

where the constants are defined

M \ \stackrel{\mathrm{def}}{=}\ \frac{1}{a b^{4}} \frac{1}{J_{\alpha}^{\prime}}
K \ \stackrel{\mathrm{def}}{=}\ \frac{M}{2}
J \ \stackrel{\mathrm{def}}{=}\ K \frac{J_{\alpha}^{\prime\prime}}{J_{\beta}^{\prime\prime}}
L \ \stackrel{\mathrm{def}}{=}\ \frac{2}{a b^{2} \left( a^{2} + b^{2} \right)} \frac{1}{J_{\beta}^{\prime}}
N \ \stackrel{\mathrm{def}}{=}\ \frac{6}{a b^{2}} \frac{\left( a^{2} - b^{2} \right)}{a^{2} J_{\alpha} + b^{2} J_{\beta}}

The coefficients are the Jeffery functions

J_{\alpha} = \int_{0}^{\infty} \frac{dx}{\left( x + b^{2} \right) \sqrt{\left( x + a^{2} \right)^{3}}}
J_{\beta} = \int_{0}^{\infty} \frac{dx}{\left( x + b^{2} \right)^{2} \sqrt{\left( x + a^{2} \right)}}
J_{\alpha}^{\prime} = \int_{0}^{\infty} \frac{dx}{\left( x + b^{2} \right)^{3} \sqrt{\left( x + a^{2} \right)}}
J_{\beta}^{\prime} = \int_{0}^{\infty} \frac{dx}{\left( x + b^{2} \right)^{2} \sqrt{\left( x + a^{2} \right)^{3}}}
J_{\alpha}^{\prime\prime} = \int_{0}^{\infty} \frac{x\ dx}{\left( x + b^{2} \right)^{3} \sqrt{\left( x + a^{2} \right)}}
J_{\beta}^{\prime\prime} = \int_{0}^{\infty} \frac{x\ dx}{\left( x + b^{2} \right)^{2} \sqrt{\left( x + a^{2} \right)^{3}}}

Read more about this topic:  Intrinsic Viscosity

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