Consider the system consisting in a U shaped tube partially filled with liquid of density ρ, as presented Fig XXXXXXXXXXThe branches of the tube are cylindrical and of equal cross-section S. When a...

Consider the system consisting in a U shaped tube partially filled with liquid of density ρ, as presented Fig. 2.29. The branches of the tube are cylindrical and of equal cross-section S.

When a differential pressure Δp = p₁−p₂
is suddenly (step shaped) applied on the two branches of the U tube, the liquid level in the two branches will change, according to the simplified equation (momentum balance):

where: λ is the friction coefficient, L is the length of the column occupied by the liquid in the U tube, and D is the U tube diameter (cross-section area of the tube S = π D2/4).

(a) Compute the general solution of equation (2.123), taking into account the constant initial conditions h(0) = 0 and (dh/dt)(0) = 0.

(b) Verify whether the system is BIBO stable.