Phase shift ellipses for pulsating flows

A pulsating laminar flow of a viscous, incompressible fluid through a pipe with an orifice has been studied at relatively low Reynolds numbers. The motion is caused by an imposed sinusoidally varying pressure difference, Δp(t). The induced flow rate, Q(t), has a phase shift, φ_Q, with respect to the imposed pressure oscillations. We have used that a phase (x,y)-plane instantaneous state plot of two phase-shifted trajectories: x(t =sin(ωt+φ_x) and y(t)=sin(ωl+φ_y , is an ellipse. The ellipses of the instantaneous states Q(t) vs Δp(t) during a cycle allow readily computing the phase shift, Δ_Q.