## Dimensionless Variables

The original seven variables in this problem can now be replaced by an equivalent set of four dimensionless groups of variables. For example, if it is desired to determine the driving force required to transport a given fluid at a given rate through a given pipe, the relation could be represented as

AP = fn(V; D , L,e,p,p) or, in terms of the equivalent dimensionless variables (groups),

Note that the number of variables has been reduced from the original seven to four (groups). Furthermore, the relationship between these dimensionless variables or groups is independent of scale. That is, any two similar systems will be exactly equivalent, regardless of size or scale, if the values of all dimensionless variables or groups are the same in each. By "similar" we mean that both systems must have the same geometry or shape (which is actually another dimensionless variable), and both must be operating under comparable dynamic conditions (e.g., either laminar or turbulentâ€”this will be expanded on later). Also, the fluids must be rheologically similar (e.g., Newtonian). The difference between Newtonian and non-Newtonian fluids will be discussed in Chapter 3. For the present, a Newtonian fluid is one that requires only one rheological property, the viscosity (p), to determine flow behavior, whereas a non-Newtonian fluid requires a rheological "function" that contains two or more parameters. Each of these parameters is a rheological property, so in place of the viscosity for a Newtonian fluid, the non-Newtonian fluid would require two or more ''rheological properties,'' depending upon the specific model that describes the fluid, with a corresponding increase in the number of dimensionless groups.

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