Chapter 11. Equation of continuity for steady flow of an imcompressible fluid (11-19)

Review

It takes some time interval \(\Delta{t}\) for the slug of fluid to enter the region at point 1. And since the volume of fluid between the points must remain constant, the slug of fluid at point 2 must exit the region in the same time interval. If we divide Equation 11-17 by the time interval \(\Delta{t}\) it takes for the slugs of fluid to enter or exit the region, we arrive at

\(A_1\frac{\Delta{x_1}}{\Delta{t}} = A_2\frac{\Delta{x_2}}{\Delta{t}}\)

To see why we divided through by \(\Delta{t}\), note that the fluid at point 1 moves a distance \(\Delta{x_1}\) during the time interval \(\Delta{t}\) and so has speed \(v_1 = \Delta{x_1}/\Delta{t}\). SImilarly, the fluid at point 2 (which moves a distance \(\Delta{x_2}\) during the same time interval) has speed \(v_2 = \Delta{x_2}/\Delta{t}\). So we can rewrite Equation 11-18 as the following relationship called the \(\textbf{equation of continuity}\):