# Equation of motion is liquid

Question 1:

Consider the equation of motion of a very light spherical solid particle in the creaping flow regime when the Reynolds number Re >> 1

a) Neglecting Bosinesq-Basset drag force find a solution to the equation of particle motion:

(r + 1/2)(d^2*z/d*t^2) = (1 - r)g - (3v/a^2)(dz/dt) (1.1)

for two limiting cases:

i) assuming that the density ratio particle to fluid is negligibly small, r = p_p/p_f <<1, and

ii) assuming that the particle density is much greater than the fluid density, r = p_p/p_f >>1

(assume that it is a platinum particle with the density 21.45 g/cm3).

In both case the particle commences its motion from the rest being at the point z = 0. For other parameters put g = 10 m/s2 (acceleration due to gravity), v = 1 cm2/s (kinematic water viscosity), a = 1 mm (particle radius).

Hint 1: Convert all dimensional quantities to SI system.

b) Plot your solution for the traversed path z (t) in mm against time in millisecond (up to 10 msec) and velocity v(t) in cm/s in the same time interval.

c) Find the terminal velocity (the asymptotic velocity v_t when t --> infinity) and the relaxation time Tr (the characteristic time required to reach approximately the terminal velocity) for both cases of light and heavy particle.

Hint 2: Introduce the particle velocity = dz/dt and solve the equation in terms of velocity, then find z (t).

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#### Solution Summary

The solution shows in detail how to solve the equation of motion (in two methods), applying initial conditions and analyze the limiting cases of a very dense and very non-dense cases. Includes graphs.