Concept:
For a fully developed laminar flow,
\(u = - \frac{1}{{4\mu }}\frac{{\partial P}}{{\partial x}}\left( {{R^2} - {r^2}} \right)\)
where, R = radius of pipe, r = radial distance from centre (r = 0), \(\frac{{\partial P}}{{\partial x}}\)= pressure gradient, μ is dynamic viscosity.
\(u = - \frac{1}{{4\mu }}\frac{{\partial P}}{{\partial x}} \times {R^2}\left( {1 - \frac{{\;{r^2}}}{{{R^2}}}} \right)\)
At, r = 0, \(\;u\; = \;{u_{max}} = - \frac{1}{{4\mu }}\frac{{\partial P}}{{\partial x}} \times {R^2}\)
\(u = {u_{max}}\left( {1 - \frac{{{r^2}}}{{{R^2}}}} \right)\)
Calculation:
Given,
\(\frac{{\partial P}}{{\partial x}} = - 10\;Pa/m\) ,radius of pipe R = 5 cm = 0.05 m, μ = 0.001 Pa-s.
From,
\(u = - \frac{1}{{4\mu }}\frac{{\partial P}}{{\partial x}} \times {R^2}\left( {1 - \frac{{\;{r^2}}}{{{R^2}}}} \right)\)
at r = 0.2 cm = 0.002 m,
\(u = - \frac{1}{{4 \times 0.001}} \times \left( { - 10} \right) \times {0.05^2} \times \left( {1 - \frac{{{{0.002}^2}}}{{{{0.05}^2}}}} \right)\)
u = 6.24 m/s