What is Non-Dimensionalising and How Can It Help Solve General Linear Equations?

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In summary, the term 'non-dimensionalising' refers to removing the units from a quantity to make it applicable to a wider range of situations. To non-dimensionalise a general linear equation, one can start by dividing by a fixed quantity with the same units as the original expression. The radians unit is an example of a non-dimensional quantity and can be used in algebraic equations without mentioning angles. The Buckingham π theorem provides a systematic method for obtaining powerful results using non-dimensional variables.
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newstudent
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Hi All,

I have often heard the term 'non-dimensionalising', and am unsure as to what it really means. I gather that it literally means non dimensionalising the units such that it may be applied to a wider range of situations. My question is, if i have a general linear equation and wish to non dimensionalise it, where should be my starting point? I would appreciate if someone could point me in the right direction. Thank you.

=)
cheers,
newstudent
 
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newstudent said:
Hi All,

I have often heard the term 'non-dimensionalising', and am unsure as to what it really means. I gather that it literally means non dimensionalising the units such that it may be applied to a wider range of situations. My question is, if i have a general linear equation and wish to non dimensionalise it, where should be my starting point? I would appreciate if someone could point me in the right direction. Thank you.

=)
cheers,
newstudent

I'm curious too. Would radians be a undimentionalising unit?
 
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"Radians", not "radiants". Yes, radians are an example of a non-dimensional quantity. Given a circle of any radius, the radian measure of an angle is the length of the arc it cuts from the circle, divided by the radius of the circle. Since those are both lengths, any units of length will cancel out. For example, a 60 degree angle, in a 40 inch in radius circle, would cut an arc length of [itex](60/360)(2\pi 40)= 41.9[/itex] inches long. The angle, in radians, is 251.3/40= 6.28. It is because the "radian" is really "dimensionless" that we can use it in purely algebraic equations with no mention of angles:
f(x)= cos(x) assumes x is "in radians".

In general, one forms dimensionless expressions by dividing by a fixed quantity having the same units as the original expression.

You might want to look at this:
http://astro.nmsu.edu/~aklypin/PM/pmcode/node2.html
 
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  • #4

FAQ: What is Non-Dimensionalising and How Can It Help Solve General Linear Equations?

What is "non-dimensionalising"?

"Non-dimensionalising" is a process in which physical quantities are expressed in terms of dimensionless numbers. This is done by dividing the original quantity by a characteristic value, often a scaling factor, to remove the units and make the quantity independent of the system's size or units of measurement.

Why is "non-dimensionalising" important in scientific research?

"Non-dimensionalising" is important in scientific research because it helps simplify complex equations and relationships between variables. This allows scientists to focus on the fundamental principles and behaviors of a system without getting bogged down by specific units of measurement. It also makes it easier to compare and analyze data from different systems or experiments.

What are some common examples of "non-dimensionalising" in science?

Some common examples of "non-dimensionalising" in science include the Reynolds number in fluid mechanics, the Mach number in aerodynamics, and the Froude number in ship hydrodynamics. These dimensionless numbers allow scientists to describe and compare the behavior of fluids without having to consider the specific units of measurement for velocity, density, and other variables.

Are there any limitations to "non-dimensionalising"?

While "non-dimensionalising" is a useful tool in many scientific fields, it does have limitations. It may not be applicable in systems with highly nonlinear relationships between variables, or in cases where the scaling factor or characteristic value is not clearly defined. Additionally, some physical quantities may not have a meaningful dimensionless representation.

Can "non-dimensionalising" be used in all scientific disciplines?

"Non-dimensionalising" can be used in many scientific disciplines, such as physics, engineering, and chemistry. However, it may not be as relevant in fields such as biology or social sciences, where the focus is on complex systems and behaviors rather than fundamental physical principles. Ultimately, the usefulness of "non-dimensionalising" will depend on the specific research question and the nature of the system being studied.

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