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Flow Around A Corner • A particle at (x0,y) will descend at speed U until y=x0. – Ignore gravity • Then it finds itself in a horizontal fluid flow. • Drag force on particle takes more general form: – FD = -(v-u)/B • Solve for particle trajectory in x and y directions. t x x 0 vdt ' 0 t t x x 0 mBv 0 1exp ut umB1 exp mB mB dv v u dt mB mB v u u v t 0 exp mB mB mB mB mB t v u v 0 u exp mB t t v v 0 exp u1 exp mB mB Approaching Drift Velocity Here we consider the trajectory of a charged particle in a constant electric field • The force on a charged particle is the charge on the particle times the electric field at its location • e is the elementary unit of charge, and –e is the charge on a single electron. Assume the aerosol particle has a single extra electron. • The electric field is calculated as E = -V, where V is the electric potential (voltage) V2 Parallel plates at different voltages produce a nearly constant field between them. Let V2 > V1. The distance between them is h. E V2 V1 ˆ k h FE eE V1 dv m FD FE Now consider the generalized force equation for the particle, dt dv v m eE dt B Equations of this form have the solution, v e e v t E 0 E exp mB m mB mB m V V t v v 0 exp eB 1 2 h mB Memory of original velocity decays away t 1 exp mB A “drift velocity” takes over on same timescale Approaching Drift Velocity Now that we have solved for velocity, we need to integrate to get trajectory t t v v 0 exp v D 1 exp mB mB Assume particle starts at rest, vD = 1 cm/s, t = 10s t z z 0 vdt ' 0 t z z 0 v Dt v D v 0 mB1 exp mB zoffset The result is that the particle initially accelerates until it approaches a path parallel to the constant drift path. The offset between these paths asymptotes to z offset v D v 0 mB z=vDt z