A shear force of \(18\,\mathrm{N}\) acts over area \(0.060\,\mathrm{m^2}\). Find the shear stress.

A Newtonian fluid has viscosity \(0.45\,\mathrm{Pa\,s}\) and velocity gradient \(120\,\mathrm{s^{-1}}\). Find the shear stress.

A plate of area \(0.15\,\mathrm{m^2}\) slides at \(0.40\,\mathrm{m\,s^{-1}}\) over a stationary plate with a \(2.0\,\mathrm{mm}\) fluid gap. The fluid viscosity is \(0.80\,\mathrm{Pa\,s}\). Assuming a linear velocity profile, find the viscous force.

Water with \(\rho=1000\,\mathrm{kg\,m^{-3}}\) and \(\eta=1.0\times10^{-3}\,\mathrm{Pa\,s}\) flows at \(1.2\,\mathrm{m\,s^{-1}}\) past a length scale \(0.030\,\mathrm{m}\). Find the Reynolds number.

Two flows have Reynolds numbers \(1200\) and \(2.0\times10^5\). Which is more likely to be turbulent, and why?

A Newtonian fluid has shear stress \(36\,\mathrm{Pa}\) when \(dv/dy=90\,\mathrm{s^{-1}}\). Find the viscosity.

A viscous force of \(10\,\mathrm{N}\) acts on area \(0.20\,\mathrm{m^2}\) in a fluid of viscosity \(0.50\,\mathrm{Pa\,s}\). Find the velocity gradient.

For oil with \(\rho=900\,\mathrm{kg\,m^{-3}}\), \(\eta=0.18\,\mathrm{Pa\,s}\), and \(L=0.10\,\mathrm{m}\), find the speed corresponding to \(\mathrm{Re}=2000\).

A flow's speed is doubled while its characteristic length is halved. Density and viscosity are unchanged. By what factor does the Reynolds number change?

In a plate-flow setup with linear velocity profile, the gap between plates is doubled while area, viscosity, and plate speed are unchanged. By what factor does the viscous force change?

A viscous force of \(30\,\mathrm{N}\) opposes a plate moving steadily at \(0.50\,\mathrm{m\,s^{-1}}\). Find the mechanical power needed to maintain the motion.

Water flows in a pipe of diameter \(2.0\,\mathrm{cm}\). Estimate the Reynolds number at speeds \(0.10\,\mathrm{m\,s^{-1}}\) and \(0.30\,\mathrm{m\,s^{-1}}\), using \(\eta=1.0\times10^{-3}\,\mathrm{Pa\,s}\). Interpret the change.

Air with \(\rho=1.2\,\mathrm{kg\,m^{-3}}\) and \(\eta=1.8\times10^{-5}\,\mathrm{Pa\,s}\) flows at \(40\,\mathrm{m\,s^{-1}}\) past a model of length \(0.25\,\mathrm{m}\). Find \(\mathrm{Re}\) and state whether inertial or viscous effects dominate the ratio.

A model is \(1/5\) the size of a full object but is tested at \(3\) times the full-object speed in a fluid of the same density. By what factor must the viscosity differ for the Reynolds number to match?

A fluid is tested at three velocity gradients: \(50\,\mathrm{s^{-1}}\), \(100\,\mathrm{s^{-1}}\), and \(150\,\mathrm{s^{-1}}\). The measured shear stresses are \(20\,\mathrm{Pa}\), \(40\,\mathrm{Pa}\), and \(65\,\mathrm{Pa}\). Decide whether the data fit a Newtonian model with constant viscosity.

A plate of area \(A\) moves at speed \(U\) over a stationary plate separated by fluid thickness \(d\). For a Newtonian fluid with linear velocity profile, derive the viscous force and the power required to maintain the motion.

A Newtonian fluid has velocity profile \(v(y)=ay+by^2\), where \(a>0\) and \(b>0\), between \(0\le y\le H\). Derive the shear stress \(\tau(y)\), and identify where the shear stress is largest.

A flow should remain below critical Reynolds number \(\mathrm{Re}_c\). Derive the maximum allowed characteristic length \(L_{\max}\) in terms of \(\rho\), \(v\), \(\eta\), and \(\mathrm{Re}_c\). Then calculate it for \(\rho=1000\,\mathrm{kg\,m^{-3}}\), \(v=0.40\,\mathrm{m\,s^{-1}}\), \(\eta=1.0\times10^{-3}\,\mathrm{Pa\,s}\), and \(\mathrm{Re}_c=2000\).

A channel has velocity profile \(v(y)=v_0(y/H)^n\) for \(0\le y\le H\), with \(n>0\). Derive the average speed and the wall shear stress at \(y=H\) for a Newtonian fluid.

An ideal Bernoulli calculation for a horizontal constriction predicts a pressure drop \(\Delta p_{\mathrm{ideal}}\). In the real flow, the Reynolds number is very large and turbulent losses are significant. Explain which pressure difference is larger, \(\Delta p_{\mathrm{actual}}\) or \(\Delta p_{\mathrm{ideal}}\), and justify the conclusion with an energy argument.