Physical Interpretation of the Dimensionless Numbers

The dimensionless parameters, such as the Nusselt and Reynolds numbers, can be thought of as measures of the relative importance of of certain aspects of the flow. For example, if the flow through an area dA is considered, as shown in Fig. 1.19, the rate momentum passes through this area is equal to the mass flow rate times the velocity, i.e., equal to mV dA, i.e., equal to pVVdA, i.e., equal to pV2dA. If, therefore, U is a measure of the velocity, the quantity pU1 is a measure of the magnitude of the momentum flux in the flow. This quantity is often termed the inertia force . Further, since the Newtonian viscosity law indicates that the viscous shear stresses 

Because dA is small the velocity is effectively constant over dA 

consider the Nusselt number. The convective heat transfer from a surface will depend on the magnitude of h(Tw - T/). Also, if there was no flow, i.e., if the heat transfer was purely by conduction, Fourier s law indicates that the quantity k(Tw - Tf)U would be a measure of the heat transfer rate. Now, the Nusselt number can be written as  

the Grashof number is a measure of the magnitude of the ratio of the inertia forces induced in the flow by the buoyancy forces to the viscous stresses, i.e., it is effectively1 a measure of the magnitude of the ratio of the buoyancy forces to the viscous forces in the flow. 

Attention will, lastly, be given to the Prandtl number. Consider a steady flow over a surface which is at a different temperature from the fluid flowing over the surface, the situation considered being shown in Fig. 1.20. 

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