User:Biwas Mrinmoy/sandbox

Non-dimensionalization of Navier-Stokes equation refers to the conversion of the Navier-Stokes equation to a form which is easier to use and reducing the number of free parameters in the problem to be studied. The non-dimensionalized Navier-Stokes equation is beneficial to use when posed with similar physical situations that is problems where the only change are in the basic dimensions of the system. Scaling of Navier-Stokes equation refers to the process of selecting the proper scales for non-dimensionalization of the equation. Since the final equation needs to be dimensionless, using the parameters and constants of the equation a suitable combination of these is found having the same dimensions as that of the variables in the equation. As a result of this combination the number of parameters to be analyzed is reduced and the results are obtained in terms of scaled variables.

Need For Non-dimensionalization and Scaling
In addition to reducing the number of parameters, non-dimensionalized equation helps to gain a greater insight in to the relative size of various terms present in the equation. Following appropriate selecting of scales for the non-dimensionalization process, this leads to identification of small terms in the equation. Neglecting the smaller terms against the bigger ones allows for the simplification of the situation.

For the case of flow without heat transfer, the non-dimensionalized Navier Stokes equation depend only one the Reynolds Number and hence all physical realizations of the related experiment will have the same value of non-dimensionalized variables for the same Reynolds Number. Scaling helps provide better understanding of the physical situation, with the variation in dimensions of the parameters involved in the equation. This allows for experiments to be conducted on smaller scale prototypes provided that any physical effects which are not included in the non-dimensionalized equation are unimportant.