The background for the convenance of using dimensionless parameters is the Buckingham π theorem. It states that if the physical problem can be described by n physical variables expressed by k independent physical units, the problem can be defined by p=n - k dimensionless variables. This reduces the number of parameters needed to determine the physical problem.
Problems that can be described by the same set of dimensionless parameters as well as share the same numerical values of these parameters are equivalent. For example, the behavior of the mathematical pendulum is described by the period of vibration, mass, gravity, and length of the rope, which consist of kilograms, meters, and seconds as physical units. According to the Buckingham π theorem, the problem can be described by p = 4 - 3 = 1 dimensionless parameter. In fact, the pendulum is completely determined by its natural frequency.
The physical significance is associated with the original differential equation only i.e. one in dimensional form. Once it is transformed through non-dimensionalization process, it is easier to analyse mathematically. Because this analysis can be mapped to any domain at later stage.
Following this, the governing equations in fluid and heat transfer problems, produce many important non-dimensioanlized numbers like Reynolds's , Nusselt, Prandtl etc. By knowing the value of these numbers itself can help in analyzing the characteristic of fluid and heat. In general, all these number help us in comparative study, i.e. for varying values associated with these number, decide the physics of fluid and heat.
The background for the convenance of using dimensionless parameters is the Buckingham π theorem. It states that if the physical problem can be described by n physical variables expressed by k independent physical units, the problem can be defined by p=n - k dimensionless variables. This reduces the number of parameters needed to determine the physical problem.
Problems that can be described by the same set of dimensionless parameters as well as share the same numerical values of these parameters are equivalent. For example, the behavior of the mathematical pendulum is described by the period of vibration, mass, gravity, and length of the rope, which consist of kilograms, meters, and seconds as physical units. According to the Buckingham π theorem, the problem can be described by p = 4 - 3 = 1 dimensionless parameter. In fact, the pendulum is completely determined by its natural frequency.
Dynamic, kinematic and geometric similarity. If the governing equations for apparently different problems (e.g. flow through a domestic water pipe, sewage flow through a sewage pipe, blood flow through a large artery, gas flow through a gas pipe - I am assuming these are all Newtonian and incompressible!) have the same nondimensional form, then mathematically they are identical. On a practical note, nondimensionalisation reduces the number of parameters one needs to vary as Prof Rozov has said.
Horimek Abderrahmane We can say the the system information's (Physical ) transferred to the non-dimensional numbers ?
One physical significance of non-dimensional equations is when you want to study a system, e.g. water flow behind a big dam. Then you can build a similar physical model, e.g. a small mock-up of dam in laboratory, and the non-dimensional equations describe both the model and the system, with the same numbers, and with the same solutions!
Another significance is that you can investigate the effects of parameters, with less number of experimentation, i.e. reducing the cost of experiments!
Another significance is that the order of magnitude of numbers will be normalized for different physical variables, like velocity, temperature, density, pressure, etc. That is important because; we use a same precision for all variables in our computer, e.g. 0.000000001, and this is the cut-off error for all variables. Then, put dimensions on this number, e.g. (K) for temperature, (m3) for volume, and (J) for energy, and you will sea that how much error produced and propagated in calculations. The numbers should be physically meaningful for all variables with a same magnitude, and this can be easily done using non-dimensional equations.
There are two methods to derive an equation in a non dimensional form (i). Rayleigh's method and (ii). Buckingham's Pi- theorem. Rayleigh's method is used to determine the expression of variable which depends upon a maximum of three or four variables only and expressed as X= K X1a X1b X1c, where K is constant and is rarely used. Whereas Buckingham's Pi- theorem is frequently used to express the function in terms of more than three or more variables also and expressed in (n-m) Pi equations where n is total variables and m is fundamental dimensions and all the Pi functions are combined to get a single equation. The physics behind is
(1). Using non dimensional parameters, it is quite easy to solve the equation and get the solution.
(2). The dependence of the variable can be split into a number of variables.
(3). The more accurate result is obtained when we go to more minute dependent parameters.
(4). Overall this does not give the accurate solution but gives the list of parameters dependence in non dimensional form.
(5). Performing the experiments, the actual equation can be framed by getting multiplication constants and powers of individual parameters.
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