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)d²z/dt² = (1-r)g - 3v/a²dz/dt

for two limiting cases:

i) assuming that the density ratio particle to fluid is negligibly small, r ≡ ρ_p/ρ_f << 1, and

ii) assuming that the particle density is much greater than the fluid density, r ≡ ρ_p/ρ_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).

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 → ∞ ) and the relaxation time Tr (the characteristic time required to reach approximately the terminal velocity) for both cases of light and heavy particle.

Question 2:

Consider the periodic function given on the interval -Π ≤ x ≤ Π by the equation:

a) Plot this function on the interval -10 ≤ x ≤ 10.

b) Find the Fourier series for that function.

c) Plot the Fourier spectrum A_n = √a²_n + b²_n against n in log-log scale where a_n and b_n are the amplitudes of sine and cosine Fourier harmonics and n is the harmonic number.

d) Show that the Fourier spectrum is the decaying function of harmonic number n and find the character of decay (exponential, power type, etc).

e) Plot again function y (x) together with its two approximations based on the Fourier series

i) with only first harmonic and ii) with three first harmonics.

f) Estimate the relative accuracy of approximations in percents at the function maximum.

Question 3:

a) Define which differential equation is called linear and which is called nonlinear?

b) Define which system of differential equations is called homogeneous and which is called nonhomogeneous?

c) What is a general structure of solution of a linear nonhomogeneous differential equation?

d) What is the principle of superposition in application to the solution of ordinary differential equations (ODEs)? To which kind of ODEs it is applicable?

Question 4:

Find the currents I1 and I2 in the circuits shown in Fig. 4.1 when R = 2.5 Ohms, L = 1 Henry, C = 0.04 Farad, and E (t) = 169 sint Volts. Assume that there are no currents in the circuits at the initial instant of time, i.e. I₁(0) = 0 and I₂(0) = 0.

Present graphics of the currents as functions of time in the same plot on the interval 0 ≤ t ≤ 20.