A fluid with a density of \(2000 \mathrm{~kg} / \mathrm{m}^{3}\) flows steadily between two flat plates as shown in Fig. P6.23. The bottom plate is fixed and the top one moves at a constant speed in the \(x\) direction. The velocity is \(\boldsymbol{V}=0.20 y \hat{\mathbf{i}} \mathrm{m} / \mathrm{s}\) where \(y\) is in meters. The acceleration of gravity is \(g=-9.8 \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}^{2}\). The only nonzero shear stresses, \(\tau_{y x}=\tau_{x y}\), are constant throughout the flow with a value of \(5 \mathrm{~N} / \mathrm{m}^{2}\). The normal stress at the origin \((x=y=0)\) is \(\sigma_{x x}=-100 \mathrm{kPa}\). Use the \(x\) and \(y\) components of the equations of motion (Eqs. 6.50 \(a\) and \(b\) ) to determine the normal stress throughout the fluid. Assume that \(\sigma_{x x}=\sigma_{y y}\).

Figure P6.23

Eq. 6.50 (a, b)

Transcribed Image Text:

b 1 Moving plate x Fixed plate