# 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).

© BrainMass Inc. brainmass.com October 10, 2019, 8:08 am ad1c9bdddfhttps://brainmass.com/math/functional-analysis/equation-motion-liquid-609265

#### Solution Preview

Hello, and thank you for posting your question to Brainmass.

The solution is attached below in two files. the files are identical in content, only differ in format. The first ...

#### 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.