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Prompt: Please describe the value associated with momentum and heat transfer analogies and the role of the Prandtl number.

Discussion: The role of the material quantities ${\alpha,\nu,D}$ serve as coefficients for first order differential equations. As we have seen, at the surface interface between a fluid and solid, different properties decay rapidly from the solid. The region directly adjacent the the body is where the highest values are: temperature, velocity, shear stresses, etc. In order to assign a number and quantify both fluid boundary layer and surface gradients in general, the Nusselt and Prandtl are created. The Prandtl number is the ratio of two of these material quantities: $\text{Pr } =\frac{\nu}{\alpha}= \frac{\nu\rho c_p}{k}$, all of which are material properties. This is not to say that they are constant, but dependent on the material being analyzed. What when the Prandtl number is small, then the denominator or thermal diffusivity dominates the system. When the value is large, then the reverse is true and momentum transport dominates. This is because oftentimes both share the same “quantity carrier”: individual particles which have their own momentum and thermal energy. When the value is close to unity, then heat or momentum are being transferred using different mechanisms.

Response to : Joe Bostick

Hey Joe, thanks for the clarification on the mechanisms for these analogies! Since the transport mechanism for heat and momentum are oftentimes the same, their profiles are also similar in how their gradients are steepest at the interface.