The force balance presented in (9.35) describes the motion of a particle in a force field. As long as the particle is not moving in a vacuum, the drag force will always be present, so let us remove the drag force from the summation of forces m.
dx 3 up Dp dt Cr where F„ denotes external force i (those forces arising from external potential fields, such as gravity and electrical forces).
Situations in which a charged particle moves in an electric field are important in several gas-cleaning devices and aerosol measurements. If a particle has an electric charge q in an electric field of strength E, an electrostatic force
acts on the particle. The equation of motion for a particle of charge q moving at velocity v in a fluid with velocity u in the presence of an electric field of strength E is m, d\ 3 ji ]\.DP
At steady state in the absence of a background fluid velocity, the particle velocity is such that the electrical force is balanced by the drag force and where \e is termed the electrical migration velocity. Note that the characteristic time for relaxation of the particle velocity to its steady-state value is still given by x = mpCc/3n\iDp regardless of the external force influencing the particle. Thus, as long as x is small compared to the characteristic time of change in the electric force, the particle velocity is given by (9.49). Defining the electrical mobility of a charged particle B„ as then the electrical migration velocity is given by
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