Oil (SG = 0.9, Dynamic viscosity = 1 Poise) is flowing with a mean velocity of 1 m/s between two fixed parallel plates which are 1 cm apart. What will be shear stress at the surface of the plate? 

Concept:

Pressure difference b/w two points in Laminar flow b/w two fixed parallel plate

\({P_2} - {P_1} = - \frac{{12\mu vl}}{{{t^2}}}\)

Maximum shear stress of Laminar flow b/w two fixed parallel plate

\({τ _{max}} = - \frac{1}{2}\left( {\frac{{dp}}{{dx}}} \right)t\)

Calculation:

Given:

Specific Gravity = 0.9 ⇒ Density (ρ) = 900 kg/m³

μ = 1 poise ⇒ 0.1 Ns/m²

v = 1 m/s, t = 1 cm ⇒ 0.01 m

The pressure difference is:

\({P_2} - {P_1} = - \frac{{12\mu vl}}{{{t^2}}}\)

\(\frac{{{P_2} - {P_1}}}{l} = - \frac{{12\mu v}}{{{t^2}}}\)

\(\left( {\frac{{dp}}{{dx}}} \right) = \; - \frac{{12\mu v}}{{{t^2}}}\)

\(\left( {\frac{{dp}}{{dx}}} \right) = - \frac{{12 \times .1 \times 1}}{{{{\left( {.01} \right)}^2}}} \Rightarrow - 12000\frac{N}{{{m^2}}}\)

Maximum shear stress is:

\({τ _{max}} = - \frac{1}{2}\left( {\frac{{dp}}{{dx}}} \right)t\)

\({τ _{max}} = - \frac{1}{2} \times \left( { - 12000} \right) \times 0.01\)

τ_max = 60 N/m²