Water flows through a pipe of area \(3.0\times10^{-4}\,\mathrm{m^2}\) at speed \(2.0\,\mathrm{m\,s^{-1}}\). Find the volume flow rate.

Oil of density \(850\,\mathrm{kg\,m^{-3}}\) flows through area \(4.0\times10^{-4}\,\mathrm{m^2}\) at speed \(1.5\,\mathrm{m\,s^{-1}}\). Find the mass flow rate.

Incompressible fluid flows from area \(8.0\times10^{-4}\,\mathrm{m^2}\) at speed \(1.2\,\mathrm{m\,s^{-1}}\) into area \(2.0\times10^{-4}\,\mathrm{m^2}\). Find the new speed.

A flow rate of \(1.5\times10^{-3}\,\mathrm{m^3\,s^{-1}}\) passes through a pipe at average speed \(0.75\,\mathrm{m\,s^{-1}}\). Find the pipe area.

An incompressible fluid moves from a pipe of radius \(4.0\,\mathrm{cm}\) into a nozzle of radius \(2.0\,\mathrm{cm}\). If the pipe speed is \(0.60\,\mathrm{m\,s^{-1}}\), find the nozzle speed.

A tank of volume \(0.30\,\mathrm{m^3}\) is filled by a steady inflow of \(2.5\times10^{-3}\,\mathrm{m^3\,s^{-1}}\). How long does it take to fill?

A steady incompressible flow of \(9.0\times10^{-4}\,\mathrm{m^3\,s^{-1}}\) splits into two branches. One branch carries \(3.5\times10^{-4}\,\mathrm{m^3\,s^{-1}}\). The other branch has area \(2.0\times10^{-4}\,\mathrm{m^2}\). Find the speed in the other branch.

A compressible gas flows steadily with \(\rho_1=1.2\,\mathrm{kg\,m^{-3}}\), \(A_1=0.080\,\mathrm{m^2}\), and \(v_1=25\,\mathrm{m\,s^{-1}}\). Downstream, \(\rho_2=0.90\,\mathrm{kg\,m^{-3}}\) and \(A_2=0.12\,\mathrm{m^2}\). Find \(v_2\).

A nozzle delivers \(12\,\mathrm{L}\) of water in \(30\,\mathrm{s}\). The nozzle area is \(5.0\times10^{-5}\,\mathrm{m^2}\). Find the average exit speed.

Two sections of the same steady gas flow have equal area. If the density at section 2 is \(80\%\) of the density at section 1, find \(v_2/v_1\).

Water has flow rate \(4.0\times10^{-3}\,\mathrm{m^3\,s^{-1}}\) in a circular pipe of radius \(2.5\,\mathrm{cm}\). Find the average speed.

An incompressible flow enters one pipe of area \(7.0\times10^{-4}\,\mathrm{m^2}\) at \(1.8\,\mathrm{m\,s^{-1}}\) and leaves through two outlets. Outlet A has area \(2.0\times10^{-4}\,\mathrm{m^2}\) and speed \(3.0\,\mathrm{m\,s^{-1}}\). Outlet B has area \(3.0\times10^{-4}\,\mathrm{m^2}\). Find the speed in outlet B.

In a tapered tube, the radius doubles between section 1 and section 2. For incompressible steady flow, find \(v_2/v_1\) and explain the physical reason.

A storage tank receives water at \(3.2\times10^{-3}\,\mathrm{m^3\,s^{-1}}\) and drains at \(2.5\times10^{-3}\,\mathrm{m^3\,s^{-1}}\). Find the rate at which the water volume in the tank changes and state whether the tank volume is steady.

A steady flow has mass flow rate \(0.72\,\mathrm{kg\,s^{-1}}\). At one section the fluid density is \(900\,\mathrm{kg\,m^{-3}}\) and the area is \(4.0\times10^{-4}\,\mathrm{m^2}\). Find the speed and decide whether treating \(Q\) as constant would require constant density.

A pipe radius varies as \(r(x)=r_0(1+x/L)\) for \(0\le x\le L\). For incompressible steady flow with speed \(v_0\) at \(x=0\), derive \(v(x)\).

A channel of width \(w\) and height \(H\) has velocity profile \(v(y)=v_0y/H\) for \(0\le y\le H\). Derive the volume flow rate and the average speed across the section.

A circular pipe has velocity profile \(v(r)=v_{\max}(1-r^2/R^2)\). Derive the volume flow rate \(Q\) and average speed \(Q/(\pi R^2)\).

For an incompressible steady flow in a smoothly varying tube, \(A(x)v(x)=Q\). Derive \(dv/dx\) in terms of \(v\), \(A\), and \(dA/dx\), and interpret the sign when the tube narrows.

A steady stream of liquid enters a mixing junction with \(\rho_1,A_1,v_1\), while a second stream enters with \(\rho_2,A_2,v_2\). The mixed stream exits through area \(A_3\) with density \(\rho_3\). Derive \(v_3\) from mass conservation and state the condition under which you could instead conserve volume flow rate.