What is the physical significance of the Prandtl number? Does the value of the Prandtl number depend on the type of flow or the flow geometry? Does the Prandtl number of air change with pressure? Does it change with temperature?

The Prandtl number (Pr) is a dimensionless number that defines the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. It gives an indication of the relative importance of fluid flow and heat transfer in a specific problem. Mathematically, it is given by the formula: Pr = \frac{\text{kinematic viscosity}}{\text{thermal diffusivity}} = \frac{\nu}{\alpha} where \nu is the kinematic viscosity and \alpha is the thermal diffusivity. In practical applications, a low Prandtl number (Pr << 1) means that heat diffuses faster than momentum, which occurs in cases such as liquid metals. On the other hand, a high Prandtl number (Pr >> 1) indicates that momentum diffuses faster than heat, which is common in fluids like water and air at room temperature.

The Prandtl number itself does not depend on the type of flow (e.g., laminar or turbulent). However, the type of flow might influence the relative importance of fluid flow and heat transfer in certain problems. This influence is not directly related to the value of the Prandtl number but is instead represented by other dimensionless numbers, like Reynolds number or Grashof number, which describe the flow regime.

The Prandtl number does not depend on the flow geometry since it is a property of the fluid itself. However, flow geometry can affect the importance of fluid flow and heat transfer in a problem, which could influence how the Prandtl number is used to analyze the problem. For example, flow around a cylinder might have different considerations than flow inside a pipe, but the value of the Prandtl number remains the same for the same fluid in both cases.

The Prandtl number of air (or any other fluid) does not change significantly with pressure. This is because both kinematic viscosity and thermal diffusivity are mainly dependent on temperature, and their dependency on pressure is generally negligible. Therefore, changes in pressure have a minor impact on the Prandtl number itself.

The Prandtl number of air (or any other fluid) can change with temperature. Both kinematic viscosity and thermal diffusivity depend on temperature, and their dependencies can result in a variation of the Prandtl number with temperature. However, it's essential to note that, in practice, the change in Prandtl number with temperature might not be large for specific temperature ranges, especially for some fluids like air.