In laminar, fully developed flow within a tube, what is the typical shape of the velocity profile?

In laminar, fully developed flow within a tube, what is the typical shape of the velocity profile?

• A. Parabolic
• B. Uniform
• C. Linear
• D. Exponential

Answer: (A)

In this situation the velocity depends only on the radial position and the balance between viscous forces and the pressure-driven flow yields a parabolic distribution. Solving the steady Navier–Stokes equations for a circular tube with a constant pressure gradient and the no-slip condition at the wall shows that the velocity varies as u(r) = U_max[1 − (r/R)^2]. This means the centerline velocity is highest and the speed tapers to zero at the wall, producing a characteristic parabola when plotted across the radius. The other shapes don’t satisfy the boundary conditions for steady, fully developed flow in a tube: uniform would ignore shear, linear would not go to zero at the wall, and exponential isn’t the solution to this steady Poiseuille-type flow.

In this situation the velocity depends only on the radial position and the balance between viscous forces and the pressure-driven flow yields a parabolic distribution. Solving the steady Navier–Stokes equations for a circular tube with a constant pressure gradient and the no-slip condition at the wall shows that the velocity varies as u(r) = U_max[1 − (r/R)^2]. This means the centerline velocity is highest and the speed tapers to zero at the wall, producing a characteristic parabola when plotted across the radius. The other shapes don’t satisfy the boundary conditions for steady, fully developed flow in a tube: uniform would ignore shear, linear would not go to zero at the wall, and exponential isn’t the solution to this steady Poiseuille-type flow.