Dimensional analysis

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Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving a mix of different kinds of physical quantities. It is routinely used by physical scientists and engineers to check the correctness of derived equations and computations. It is also used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena.

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Introduction

The dimensions of a physical quantity are associated with symbols, such as M, L, T which represent mass, length and time, and each raised to rational powers. For instance, the dimension of the physical quantity, speed, is distance/time (L/T) and the dimension of a force is mass × distance/time² or ML/T2. In mechanics, every dimension of physical quantity can be expressed in terms of distance (which physicists often call "length"), time, and mass, or alternatively in terms of force, length and mass. Depending on the problem, it may be advantageous to choose one or another other set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of M, L, T, and Q, where Q represents quantity of electric charge.

The units of a physical variable and its dimensions should be clearly differentiated. The units of a physical quantity are defined by convention, related to some standard; e.g. length may have units of meters, feet, inches, miles or micrometres; but a length always has a dimension of L. Two different units of a variable have conversion factors between them. For example: 1 m = 39.37 in; the 39.37 is called a conversion factor. There are no conversion factors between dimensional symbols.

Dimensional symbols, such as L, form a group: there is an identity, <math>L^0=1</math>; there is an inverse to L, which is 1/L, and L raised to any rational power p is a member of the group, having an inverse of 1/L raised to the power p. The operation of the group is multiplication, with the usual rules for handling exponents.

In the most primitive form, dimensional analysis may be used to check the correctness of physical equations: the two sides of any equation must have the same dimensions, i.e., the equation must be dimensionally homogeneous. As a corollary of this requirement, it follows that in a physically meaningful expression, only quantities of the same dimension can be added or subtracted. For example, the mass of a rat and the mass of a flea may be added, but the mass of a flea and the length of a rat cannot be added. As a further corollary, scalar arguments to exponential, trigonometric and logarithmic functions must be dimensionless numbers. The logarithm of 3 kg is undefined, but the logarithm of 3 is nearly 0.477. This is essentially due to the requirement for the Taylor expansion of these functions to be dimensionally homogeneous, which means that the square of the argument must be of the same dimension as the argument itself. For scalar arguments, this means the argument must be dimensionless, but certain dimensioned tensors are dimensionally self-square (Hart, 1995) and may be used as arguments to these functions. The magnitudes of variables having different dimensions cannot be compared to one another either, or used in inequalities: 3 m > 1 g is not correct, nor is it a meaningful expression.

The value of a dimensional physical quantity is written as the product of a unit within the dimension and a dimensionless numerical factor. Strictly, when like dimensioned quantities are added or subtracted or compared, these dimensioned quantities must be expressed in consistent units so that the numerical values of these quantities may be directly added or subtracted. There is no problem adding quantities of the same dimension expressed in different units as long as a conversion factor is used to bring them into the same units.

Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time in this way in 1872 by Lord Rayleigh, who was trying to understand why the sky is blue.

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A simple example

What is the period of oscillation <math>T</math> of a mass <math>m</math> attached to an ideal linear spring with spring constant <math>k</math> suspended in gravity of strength <math>g</math>? The four quantities have the following dimensions: <math>T</math> [T]; <math>m</math> [M]; <math>k</math> [M/T^2]; and <math>g</math> [L/T^2]. From these we can form only one dimensionless product of powers of our chosen variables, <math>G_1</math> = <math>T^2 k/m</math>. The dimensionless product of powers of variables is sometimes referred to as a dimensionless group of variables, but the group, <math>G_1</math>, referred to means "collection" rather than mathematical group. They are often called dimensionless numbers as well.

Note that no other dimensionless product of powers involving <math>g</math> with k,m, T, and g alone can be formed, because only g involves L . Dimensional analysis can sometimes yield strong statements about the irrelevance of some quantities in a problem, or the need for additional parameters. If we have chosen enough variables to properly describe the problem, then from this argument we can conlude that the period of the mass on the spring is independent of g: it is the same on the earth or the moon. The equation demonstrating the existence of a product of powers for our problem can be written in an entirely equivalent way: <math>T = \kappa \sqrt{m/k}</math>, for some dimensionless constant <math>\kappa</math>.

When faced with a case where our analysis rejects a variable (g, here) that we feel sure really belongs in a physical description of the situation, we might also consider the possibility that the rejected variable is in fact relevant, and that some other relevant variable has been omitted, which might combine with the rejected variable to form a dimensionless quantity. That is, however, not the case here.

When dimensional analysis yields a solution of problems where only one dimensionless product of powers is involved, as here, there are no unknown functions, and the solution is said to be "complete."

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A more complex example

Consider the case of a vibrating wire of length l [<math>L</math>] vibrating with an amplitude A [<math>L</math>]. The wire has a linear density of ρ [<math>M/L</math>] and is under tension s [<math>ML/T^2</math>], and we want to know the energy, E, in the wire. Now we can easily find that we can form two dimensionless products of powers of the variables chosen. <math>\pi_1 = E/As</math>, and <math>\pi_2 = l/A</math>. Perhaps surprisingly, like the g in the simple example given above, the linear density of the wire is not involved in either. The two groups found can be combined into an equivalent form as an equation

<math>F (E/As, l/A) = 0</math>,

where F is some unknown function, or, equivalently as

<math>E = A s f(l/A) </math>,

where f is some other unknown function. Here the unknown function implies that our solution is now incomplete, but dimensional analysis has given us something that may not have been obvious: The energy is proportional to the first power of the tension. Barring further analytical analysis, we might proceed to experiments to discover the form for the unknown function f. But our experiments are simpler than in the absence of dimensional analysis. We'd perform none to verify that the energy is proportional to the tension. Or perhaps we might guess that the Energy is proportional to l, and so infer that <math>E = l s</math>. The power of dimensional analysis as an aid to experiment and forming hypotheses becomes evident.

The power of dimensional analysis really becomes apparent when it is applied to situations, unlike those given above, that are more complicated, the set of variables involved are not apparent, and the underlying equations hopelessly complex. Consider for example, a small pebble sitting on the bed of a river. If the river flows fast enough, it will actually raise the pebble and cause it to flow along with the water. At what critical velocity will this occur? Sorting out the guessed variables is not so easy as before. But dimensional analysis can be a powerful aid in understanding problems like this, and is usually the very first tool to be applied to complex problems where the underlying equations and constraints are poorly understood.

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Huntley's addition

Huntley (Huntley, 1967) has claimed that it is sometimes productive to refine our concept of dimension. Two possible refinements are:

  • The magnitude of the components of a vector are to be considered dimensionally distinct. For example, rather than an undifferentiated length unit L, we may have <math>L_x</math> represent length in the x direction, and so forth. This requirement stems ultimately from the requirement that each component of a physically meaningful equation (scalar, vector, or tensor) must be dimensionally consistent.
  • Mass as a measure of quantity is to be considered dimensionally distinct from mass as a measure of inertia.

As an example of the usefulness of the first refinement, suppose we wish to calculate the distance a cannon ball travels when fired with a vertical velocity component <math>V_y</math> and a horizontal velocity component <math>V_x</math>, assuming it is fired on a flat surface. Assuming no use of directed lengths, the quantities of interest are then <math>V_x</math>, <math>V_y</math>, both dimensioned as <math>L/T</math>, R, the distance travelled, having dimension L, and g the downward acceleration of gravity, with dimension <math>L/T^2</math>

With these four quantities, we may conclude that the equation for the range R may be written:

<math>R \propto V_x^a\,V_y^b\,g^c\,</math>

Or dimensionally

<math>L = (L/T)^{a+b} (L/T^2)^c\,</math>

from which we may deduce that <math>a+b+c=1</math> and <math>a+b+2c=0</math> which leaves one exponent undetermined. This is to be expected since we have two fundamental quantities L and T and four parameters, with one equation.

If, however, we use directed length dimensions, then <math>V_x</math> will be dimensioned as <math>L_x/T</math>, <math>V_y</math> as <math>L_y/T</math>, R as <math>L_x</math> and g as <math>L_y/T^2</math>. The dimensional equation becomes:

<math>L_x = (L_x/T)^a\,(L_y/T)^b (L_y/T^2)^c\,</math>

and we may solve completely as <math>a=1</math>, <math>b=1</math> and <math>c=-1</math>. The increase in deductive power gained by the use of directed length dimensions seems apparent.

In a similar manner, it is sometimes found useful (e.g., in fluid mechanics and thermodynamics) to distinguish between mass as a measure of inertia (inertial mass), and mass as a measure of quantity (substantial mass). For example, consider the derivation of Poiseuille's Law. We wish to find the rate of mass flow of a viscous fluid through a circular pipe. Without drawing distinctions between inertial and substantial mass we may choose as the relevant variables

  • <math>\dot{m}</math> the mass flow rate with dimensions <math>M/T</math>
  • <math>p_x</math> the pressure gradient along the pipe with dimensions <math>M/L^2T^2</math>
  • <math>\rho</math> the density with dimensions <math>M/L^3</math>
  • <math>\eta</math> the dynamic fluid viscosity with dimensions <math>M/LT</math>
  • <math>r</math> the radius of the pipe with dimensions <math>L</math>

There are three fundamental variables so the above five equations will yield two dimensionless variables which we may take to be <math>\pi_1=\dot{m}/\eta r</math> and <math>\pi_2=p_x\rho r^5/\dot{m}^2</math> and we may express the dimensional equation as

<math>C=\pi_1\pi_2^a=\left(\frac{\dot{m}}{\eta r}\right)\left(\frac{p_x\rho r^5}{\dot{m}^2}\right)^a</math>

where C and a are undetermined constants. If we draw a distinction between inertial mass with dimensions <math>M_i</math> and substantial mass with dimensions <math>M_s</math>, then mass flow rate and density will use substantial mass as the mass parameter, while the pressure gradient and coefficient of viscosity will use inertial mass. We now have four fundamental parameters, and one dimensionless constant, so that the dimensional equation may be written:

<math>C=\frac{p_x\rho r^4}{\eta \dot{m}}</math>

where now only C is an undetermined constant (found to be equal to <math>\pi/8</math> by methods outside of dimensional analysis). This equation may be solved for the mass flow rate to yield Poiseuille's law.

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Dimensionless constants

The dimensionless constants that arise in the results obtained, such as the C in the Poiseuille's Law problem and the <math>\kappa</math> in the spring problems discussed above come from a more detailed analysis of the underlying physics, and often arises from integrating some differential equation. Dimensional analysis itself has little to say about these constants, but it is useful to know that they very often have a magnitude of order unity. This observation can allow one to sometimes make "back of the envelope" calculations about the phenomenon of interest, and therefore be able to more efficiently design experiments to measure it, or to judge whether it is important, etc.

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Orientational analysis

Huntley's addition has some serious drawbacks. It does not deal well with vector equations involving the cross product, nor does it handle well the use of angles as physical variables. It also is often quite difficult to assign the L, <math>L_x</math>, <math>L_y</math>, <math>L_z</math> symbols to the physical variables involved in the problem of interest. He invokes a procedure that involves the "symmetry" of the physical problem. This is often very difficult to apply reliably: it is unclear as to what parts of the problem that the notion of "symmetry" is being invoked. Is it the symmetry of the physical body that forces are acting upon, or to the points, lines or areas at which forces are being applied? What if more than one body is involved with different symmetries? Consider the spherical bubble attached to a cylindrical tube, where one wants the flow rate of air as a function of the pressure difference in the two parts. What are the Huntley extended dimensions of the viscosity of the air contained in the connected parts? What are the extended dimensions of the pressure of the two parts? Are they the same or different? These difficulties are responsible for the limited application of Huntley's addition to real problems.

Angles are conventionally considered to be dimensionless variables, and so the use of angles as physical variables in dimensional analysis can give less meaningful results. As an example, consider the projectile problem mentioned above. Suppose that, instead of the x- and y-component of the initial velocity, we had chosen the magnitude of the velocity v and the angle <math>\theta</math> at which the projectile was fired. The angle is conventionally considered to be dimensionless, and the magnitude of a vector has no directional quality, so that no dimensionless variable can be composed of the four variables g, v, R, and θ. Conventional analysis will correctly give the powers of g and v, but will give no information concerning the dimensionless angle θ.

Siano (Siano, 1985-I, 1985-II) has suggested that the directed dimensions of Huntley be replaced by using orientational symbols <math>1_x,\;1_y,\;1_z</math>to denote vector directions, and an orientationless symbol <math>1_0\,</math>. Thus, Huntley's <math>L_x</math> becomes <math>L\,1_x</math> with L  specifying the dimension of length, and <math>1_x</math> specifying the orientation. Siano further shows that the orientational symbols have an algebra of their own. Along with the requirement that <math>1_i^{-1}=1_i</math>, the following multiplication table for the orientation symbols results:

<math>

\begin{matrix} 

  &\mathbf{1_0}&\mathbf{1_x}&\mathbf{1_y}&\mathbf{1_z}\\

\mathbf{1_0}&1_0&1_x&1_y&1_z\\ \mathbf{1_x}&1_x&1_0&1_z&1_y\\ \mathbf{1_y}&1_y&1_z&1_0&1_x\\ \mathbf{1_z}&1_z&1_y&1_x&1_0 \end{matrix} </math>

Note that the orientational symbols form a group (the Klein four-group or "viergruppe"). In this system, scalars always have the same orientation as the identity element, independent of the "symmetry of the problem." Physical quantities that are vectors have the orientation expected: a force or a velocity in the x-direction has the orientation of <math>1_x</math>. For angles, consider an angle θ that lies in the z plane. Form a right triangle in the z plane with θ being one of the acute angles. The side of the right triangle adjacent to the angle then has an orientation <math>1_x</math> and the side opposite has an orientation <math>1_y</math>. Then, since tan(θ) = ly/lx = θ + ... we conclude that an angle in the xy plane must have an orientation <math>1_y</math>/<math>1_x</math> = <math>1_z</math>, which is not unreasonable. Analogous reasoning forces the conclusion that sin(θ) has orientation <math>1_z</math> while cos(θ) has orientation <math>1_0</math>. These are different, so one concludes (correctly), for example, that there are no solutions of physical equations that are of the form a sin(θ) + b cos(θ), where a and b are scalars.

The assignment of orientational symbols to physical quantities and the requirement that physical equations be orientationally homogeneous can actually be used in a way that is similar to dimensional analysis to derive a little more information about acceptable solutions of physical problems. In this approach one sets up the dimensional equation and solves it as far as one can. If the lowest power of a physical variable is fractional, both sides of the solution is raised to a power such that all powers are integral. This puts it into "normal form". The orientational equation is then solved to give a more restrictive condition on the unknown powers of the orientational symbols, arriving at a solution that is more complete than the one that dimensional analysis alone gives. Often the added information is that one of the powers of a certain variable is even or odd.

As an example, for the projectile problem, using orientational symbols, θ, being in the x-y plane will thus have dimension <math>1_z</math> and the range of the projectile R will be of the form:

<math>R=g^a\,v^b\,\theta^c</math> which means <math>L\,1_x\sim

\left(\frac{L\,1_y}{T^2}\right)^a\left(\frac{L}{T}\right)^b\,1_z^c</math>

Dimensional homogeneity will now correctly yield a=-1 and b=2, and orientational homogeneity requires that c be an odd integer. In fact the required function of theta will be <math>\sin(\theta)\cos(\theta)</math> which is a series of odd powers of <math>\theta</math>.

It is seen that the Taylor series of <math>\sin(\theta)</math> and <math>\cos(\theta)</math> are orientationally homogeneous using the above multiplication table, while expressions like <math>\cos(\theta)+\sin(\theta)</math> and <math>\exp(\theta)</math> are not, and are (correctly) deemed unphysical.

It should be clear that the multiplication rule used for the orientational symbols is not the same as that for the cross product of two vectors. The cross product of two identical vectors is zero, while the product of two identical orientational symbols are the identity element.

Ultimately, it can be seen that dimensional analysis and the requirement for physical equations to be dimensionally homogeneous reflects the idea that the laws of physics are independent of the units employed to measure the physical variables. That is, F=ma, for example, is true whether the unit system used is SI, English, or cgs, or any other consistent system of units. Orientational analysis and the requirement for physical equations to be orientationally homogeneous reflects the idea that the equations of physics must be independent of the coordinate system used.

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Buckingham π theorem

The Buckingham π theorem forms the basis of the central tool of dimensional analysis. This theorem describes how every physically meaningful equation involving n variables can be equivalently rewritten as an equation of n–m dimensionless parameters, where m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables, even if the form of the equation is still unknown.

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See also

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References

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