Continuum Mechanics · Stress measures and local balance laws
A silicone damping fluid (ρ=970 kg/m^3, μ=0.290 Pa· s) is driven by the constant…
Problem
A silicone damping fluid (\(\rho=970\,\mathrm{kg/m^3}\), \(\mu=0.290\,\mathrm{Pa\cdot s}\)) is driven by the constant pressure gradient \(\mathrm{d}p/\mathrm{d}x=-6.80\,\mathrm{kPa/m}\) between stationary parallel plates at \(y=0\) and \(y=h=2.75\,\mathrm{mm}\). The motion is steady, fully developed, and unidirectional, \(v=u(y)\,e_x\), with no body force in the \(x\)-direction. Reduce the \(x\)-component of incompressible Navier–Stokes to an ODE for \(u(y)\), apply the no-slip conditions, and find \(u(y)\). Determine the centreline speed, the volume flow rate per unit width, the wall shear stress, \(D\), and \(\Phi\). For a channel width of \(38.0\,\mathrm{mm}\), compute the mass flow rate. Recover \(p(x)\) given \(p(0)=175\,\mathrm{kPa}\), and report \(p\) and the extra-stress tensor \(2\mu D\) at \(x=0.65\,\mathrm{m}\), \(y=h/2\).
Hint
Fully developed parallel flow kills inertia; the \(x\)-momentum equation is \(\mu u''=\mathrm{d}p/\mathrm{d}x\).
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