Equation of motion is liquid
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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.
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