nGeneHS — Hemodynamic Simulation
Modified de Laval Nozzle Model

Peak velocities (model) – subvalvular: – m/s |  valvular: – m/s |  supravalvular: – m/s

Model equations

\[ A(x,t)=\begin{cases} A_{\text{sub}}\;-\;\dfrac{A_{\text{sub}}-A^{*}(t)}{L_{\text{conv}}}\,\bigl(x+L_{\text{conv}}\bigr), & x<0,\\[6pt] A^{*}(t)\;+\;\dfrac{A_{\text{sup}}-A^{*}(t)}{L_{\text{div}}}\,x, & x\ge 0. \end{cases} \]

\[ Q(t)=A(x,t)V(x,t)\quad\Longrightarrow\quad \frac{\partial V}{\partial x}= -\frac{1}{A}\frac{\partial A}{\partial x}\,V. \]

\[ \frac{dP}{dx} = -\rho\,V\frac{dV}{dx} - f\,\frac{\rho\,V|V|}{2D_h}, \qquad D_h(x)=2\sqrt{\frac{A(x,t)}{\pi}}. \]

These equations treat the right-ventricular outflow tract, pulmonary valve throat, and proximal pulmonary artery as a single converging–throat–diverging conduit. Given an instantaneous peak flow \(Q(t)\) (set here by the slider Q_peak), the model returns:

• continuous velocity profile \(V(x)\) from RVOT to main PA,
• spatial pressure distribution \(P(x)\) including viscous/turbulent losses (friction factor \(f\)),
• peak segmental velocities at the three anatomical levels, automatically satisfying continuity.

Unlike classical gas-dynamic nozzles, blood is incompressible; there is no sonic limit. The pressure-recovery effect downstream of the valve is reproduced naturally by the positive area gradient \(\partial A/\partial x > 0\) in the diverging section.