Gravity settling limiting conditions

When two immiscible fluids, which may be gas-liquid or liquid-liquid, are kept for a sufficiently long time, the denser droplets settle down, resulting in two separate phases. For a gas-liquid system, the terminal or free settling velocity can be mathematically defined as [1,2] 

The drag coefficient C is a function of the Reynolds number, which is defined as 

Depending on the Reynolds number, the terminal velocity can be defined further. 

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Now we re ready to tackle the full two-dimensional problem. As we claimed at the end of Section 4.6, for sufficiently weak damping the pendulum and the Josephson junction can exhibit intriguing hysteresis effects, thanks to the coexistence of a stable limit cycle and a stable fixed point. In physical terms, the pendulum can settle into either a rotating solution where it whirls over the top, or a stable rest state where gravity balances the applied torque. The final state depends on the initial conditions. Our goal now is to understand how this bistability comes about. 

Calculate the upper limit of particle diameter x ax as a function of particle density Pp for gravity sedimentation in the Stokes law regime. Plot the results as versus Pp over the range 0ambient conditions. Assume that the particles are spherical and that Stokes law holds for RCp < 0.3. 

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