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I trying to understand the following coordinate transformation of the equation of continuity (spherical coordinates) for a vaporizing liquid droplet[tex]\frac{\partial \rho}{\partial t} + \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 \rho v) = 0[/tex] into [tex]\epsilon \sigma \frac{\partial \gamma}{\partial \sigma} + \frac{\partial \gamma}{\partial \tau} + \frac{1}{\sigma^2} \frac{\partial \mu}{\partial \sigma} = -\gamma \frac{\partial ln\rho_0}{\partial \tau}[/tex] with [tex]\sigma=r/r_0,\\ \tau=\int \rho D/\rho_0 r_0^2\, dt,\\ \gamma=\rho/\rho_0,\\ \mu=r^2\rho v/r_0\rho D \\ \epsilon=-(r_0\rho_0/\rho D)dr_0/dt[/tex]where [tex]\rho D=const.[/tex] and subscript 0 means condition at the droplet surface.

This transformation comes from F.A. Williams "On the Assumptions Underlying Droplet Vaporization and Combustion Theories" J. Chem. Phys. 33, 133 (1960); doi: 10.1063/1.1731068

Actually I only have problems reproducing the first term, so I appreciate if someone can guide me through this transformation.

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# Coordinate Transformation of the equation of continuity for a vaporizing droplet

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