A liquid of dynamic viscosity Pa s flows steadily along a long horizontal channel…

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Viscous flow and dynamical similarity · Viscous flow and dynamical similarity

Fluids and Continuum Physics, Ch The Navier-Stokes equations and exact viscous solutions

Problem

A liquid of dynamic viscosity μ=1.5×10−3\mu = 1.5\times10^{-3} Pa s flows steadily along a long horizontal channel between two fixed parallel plates. The flow is unidirectional and fully developed, u=(u(y),0,0)\mathbf{u} = (u(y),0,0), with u(y)=u0(1−y2b2),u0=0.30 m s−1,b=2.0 mm,u(y) = u_0\left(1 - \frac{y^{2}}{b^{2}}\right), \qquad u_0 = 0.30\ \mathrm{m\,s^{-1}}, \quad b = 2.0\ \mathrm{mm}, where yy is measured from the mid-plane and the plates are at y=±by = \pm b. The fluid is incompressible and Newtonian. ((a)) Find the net viscous force per unit volume, μ d2u/dy2\mu\,\mathrm{d}^{2}u/\mathrm{d}y^{2}, on a fluid element. ((b)) Using the xx component of the Navier-Stokes equation for this steady unidirectional horizontal flow, find dp/dx\mathrm{d}p/\mathrm{d}x. ((c)) Find the magnitude of the viscous shear stress on the plate at y=+by = +b.

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