Consider incompressible laminar flow of a constant property Newtonian fluid in an isothermal circular tube. The flow is steady with fully-developed temperature and velocity profiles. The Nusselt number for this flow depends on

The correct option is: neither the Reynolds number nor the Prandtl number

To understand why this is correct, let us analyze the conditions given in the problem statement:
1. The flow is incompressible and laminar.
2. The fluid is a Newtonian fluid with constant properties.
3. The tube is a circular tube and is isothermal (constant surface temperature).
4. The flow is steady and fully-developed both hydrodynamically (velocity profile) and thermally (temperature profile).

For a hydrodynamically and thermally fully-developed laminar flow in a circular tube, the governing energy equation simplifies significantly because the dimensionless temperature profile remains invariant in the flow direction.

When the boundary condition is constant surface temperature (isothermal wall), solving the energy equation analytically yields a constant value for the Nusselt number ($\mathrm{Nu}$). Specifically, the local and average Nusselt numbers are constant and given by:

${\mathrm{Nu}}_D=\frac{hD}{k}=3.66$

Similarly, if the boundary condition were constant surface heat flux, the Nusselt number would be:

${\mathrm{Nu}}_D=\frac{hD}{k}=\frac{48}{11}\approx 4.36$

In both fully-developed laminar cases, the Nusselt number is a constant numerical value. It is completely independent of flow velocity, viscosity, thermal diffusivity, or fluid type. Therefore, it depends on neither the Reynolds number ($\mathrm{Re}$) nor the Prandtl number ($\mathrm{Pr}$).