The laws of physics do not depend on the choice of units of measurement. Dimensional analysis explores this immutability and its consequences and applications. The invariants under the unit transformation of the unconstomend, the laws of physics must eventually only be expressed as unconstomends. The number of unorthodored quantities that can be constructed from the physical variables in a problem is less than the number of original variables, resulting in simplification, and the constructed unorthodored quantities can more deeply reflect the internal relationship between physical quantities. The concept of dimensions is deep enough, but the method is simple enough to be an important part of physical training. The article expounds the basic concepts, principles and applications of dimensional analysis, most of which comes from the literature, focusing on the dimensional system and its relationship with the unit system, in an attempt to correct some confusion in the literature. In particular, for dimensional analysis operations alone, it is possible to use only the MLT dimension system.
Written by | Zheng Weimou (Institute of Theoretical Physics, Chinese Academy of Sciences)
Source | This article is from Physics, No. 12, 2021
The symmetry of the laws of physics means the invariance of the laws of physics under various transformations. A simple principle is that the laws of physics do not depend on the choice of units of measurement. A dimension is a character in which a physical quantity is not altered by a unit transformation, and dimensional analysis explores this invariance and its consequences and applications. A special class of physical quantities is an unorthodox, an invariant under the unit transformation, and the laws of physics must ultimately only be expressed as unorthodoxes. The number of unorthodored quantities that can be constructed from the physical variables in a problem is less than the number of original variables, resulting in simplification, and the unorthoded quantities can more deeply reflect the intrinsic relationship between physical quantities.
The dimensional analysis method is a universal tool for exploring scientific laws and solving scientific and engineering problems, which is very worth learning and mastering. Dimensional analysis can be used for experimental design and data collation, as well as to have a quantitative and qualitative grasp of the problem before solving the problem, and help to deepen the understanding of physical laws. When faced with a complex problem, it can be very difficult to build a mathematical model, or the equations are very complex and difficult to solve, or it is difficult to understand the meaning of the resulting solution. Sometimes it is necessary to do experiments, and the actual size is difficult to achieve in the experimental conditions, it is necessary to reduce the size to do model experiments, and certain similar conditions must be met, which must be based on dimensional analysis and similarity theory.
Dimensional analysis is hardly to say when it began. The term Dimension was first used by Poisson in 1833, and before that used homogeneity. In 1822 Fourier made it clear that the laws of physics should be independent of units, written in his famous book Analytical Theory of Heat.[1] This leads to an important conclusion: any law of significance must be a homogeneous equation for every unit of measurement. Many scientific masters such as Newton, Euler, and Maxwell used the concept of quantifiers to deal with problems. Rayleigh's explanation of sky blue in 1871 and later studies of the sound of strings in the wind,[2,3] and Raynaud's work on Reynolds numbers in 1883 were early examples of dimensional analysis. The normal form of dimensional analysis is the Π-theorem : if a physical relationship contains n independent variables and m fundamental dimensions , it can be expressed as n-m dimensionless quantities. Using the principle of dimensional equilibrium (homogeneous) of the laws of physics, the relationship between physical quantities can be determined, so that problems can be solved from qualitative to semi-quantitative to quantitative. The origin of the Chinese "dimensionality" was not found, which is called "dimensional analysis" in Japanese. There is a great deal of literature on dimensional analysis that cannot be enumerated, such as the Chinese Book[4,5,6] and more recently,[7] Bridgeman's English book[8] is a classic, and later, it is cited,[9] with a wealth of literature and exercises, and [10].
01
Start with a story and an example
02
Basic concepts and methods of dimensional analysis
2.1 Dimensional systems, dimensional representation vectors, dimensional representation matrices, and the Π- theorem
Physics is a highly quantitative discipline. Physical quantities are inseparable from measurement, which gives the value of physical quantities. Each physical quantity is related by definition or physical law, and a few physical quantities can be selected as the basic quantity, and the basic measurement unit is specified for it, which constitutes the unit system. Conventional fundamental quantities cannot be derived from each other according to the laws of fundamental physics. In particular, the set of fundamental quantities is referred to here as a dimensional system, regardless of its units; the unit system determines the dimensional system, but vice versa. Other physical quantities are derived by definition or basic laws of physics. The attribute of a fundamental quantity that does not depend on its unit selection is its dimension, such as the length of one of the dimensions, and the dimension of the radius of the earth is the length. After selecting the basic quantity and the unit system, the units of the exported quantity can be tabled in the basic unit table, this expression is called the dimension of the exported quantity, and the attribute of the export quantity that is not dependent on the unit selection related to this is its dimension, which describes the minimalist relationship between the exported quantity and the basic quantity that is not related to the quantity value. For example, in the MLT dimension system there are three basic quantities: mass, length and time, the corresponding three basic dimensions are recorded as mass M, length L and time T, such as L can not be derived from M and T according to the basic laws of physics, and the velocity v = dx/dt is the derived quantity, which is the result of the length divided by time, and its dimension is recorded
2.2 Steps of dimensional analysis
The steps for dimensional analysis are as follows:
Step 1, select the dimensional system, list all the independent key parameters of the problem, and set a total of n;
Step 2, determine the dimensions of all n parameters;
Step 3, determine the rank m of the matrix representation of the dimension. Often m-dimensional independent reference dimension quantities are appropriately selected from n variables;
Step 4 constructs an unequal πj with l=n-m unequal valences. Often m reference dimension quantities are used to construct non-class quantities one by one for the remaining l parameters;
Step 5, again verify that all amorous quantities are indeed dimensionless, and appropriately organize the simplified combinations of some amortized quantities to make them common to the scientific community as named unorganized quantities, usually so that each original variable to be examined appears only in one inorganic quantity, or is convenient for the investigation of the limit situation;
Step 6, write out the final acontrolled relation of the problem. Ideally, it can be represented as a simple power form, often referred to as scaling law.
Dimensional analysis does have some skills, dimensional analysis to solve complex problems often give people a magical enjoyment of beauty, dimensional analysis of many examples are scientific masterpieces. How to determine the key physical quantities of the problem involves how to simplify the problem, what are the physical factors, which are the more important, and requires profound scientific accomplishment, rich physical knowledge and a deep understanding of the problem, especially for emerging scientific problems. Adding extra physical quantities that have little to do with each other increases the complexity of the analysis, but the loss of critical physical quantities leads to failure, which obviously does not solve the problem. The choice of basic and reference dimensions is arbitrary, one choice will be more convenient than the other, and the specific form of the non-program can be different, but different group of non-stationary quantities are equivalent to each other, and the final non-program quantity, that is, the degree of freedom of the physical problem, is unchanged.
03
Application examples
Qualitative thinking and semi-qualitative experiments, which seek to estimate the nature and harmony of the problem, require long-term thinking and cultivation by physical intuition and insight, experience and skill. "To learn without thinking is to be reckless, and to think without learning is to perish."
3.1 Proof of the Pythagorean string theorem
04
The number of basic quantities of the quantum system
There are only 3 basic units in the CGS or Gaussian system of units, 4 in the MKSA system of units, and 7 in the SI of the International System of Units. The number of basic units is the number of basic dimension quantities. Increasing the number of fundamental dimension quantities, such as the introduction of temperature or current, facilitates the distinction from mechanical processes, analyzes the role played by thermal or electrical processes, but increases the number of fundamental dimension quantities, according to the Π- theorem, it seems to reduce the number of inorganic quantities. The degree of freedom of the physical problem obviously does not change due to the choice of the unit system, and increasing the basic quantity generally also increases the relevant physical quantity, such as the introduction of the basic quantity temperature after the corresponding introduction of the classbolzmann constant, the degree of freedom of the problem is ultimately unchanged. For dimensional analysis operations alone, it is possible to use only MLT dimensions
05
Theoretical physics applications
06
There is compression and recovery of class physical constants
07
Similarity theory and model experiments
Dimensional analysis can be used for complex problems, often the equations that describe the problem are unknown, let alone solved, but dimensional analysis reduces the problem to a small number of arbitrary functions without an objective. In the field of engineering technology, simulation tests are commonly used instead of real tests such as small model wind tunnel tests, which are economical and convenient, and can exceed practical limitations. Model trials include medical trials with model animals. In 1868, W. Froude overhauled ship design by proposing the Freud number (v2/gl) to guide small model experiments. He was ridiculed that small models might be fun, but they didn't make any practical sense, but the engineering community eventually changed its attitude. The theoretical basis of model experiments is the theory of similarity based on dimensional analysis. Although the specific form of any function that appears in dimensional analysis is unknown, as long as the unorganized quantity remains unchanged, the behavior of the model is equivalent to that of the prototype. If the combinations of inordents are properly selected and fixed in the prototype, the number of arguments of the functions involved can be reduced and the model experiment can be simplified. Of course, the similarity here is not necessarily geometric similarity. Only homogeneous quantities can be compared, and the physical similarities between systems are characterized by unorganized quantities. In addition, relative to the equations that describe the problem, the similarity of the solution conditions such as the initial condition and the edge condition should be considered together.
08
Scale law
A narrow-sense scale law, usually manifested as a single term. The scale law, from the perspective of rigorous dimensional analysis, is necessary but not necessarily sufficient, equivalent to a simplified version of dimensional analysis. If the problem is simple enough and has the ability to grasp the key quantity of the problem, so that there is only one unorganized quantity, then the scale law is determined. Starting from the complete dimensional analysis, it is prudent to distinguish the degradation situation and obtain the scale law, which can refer to the previous example of the beverage plus ice cold and the resistance of moving objects in the fluid. Scale law, geometrically corresponding to fractals. Scale laws are used to describe complex phenomena such as the aforementioned Kleber power law of 3/4 power of metabolic rate versus animal body weight.
8.1 Scale law for load-bearing cylinders
09
epilogue
The laws of physics do not depend on the choice of units of measurement and are the most basic and simplest symmetry. In the end, the laws of physics must only be expressed in the form of unconstitutional quantities, which leads to the reduction of independent and effective physical quantities, that is, to reduce the degree of freedom of the problem, and to reflect more deeply the internal relationship between physical quantities. By simply analyzing the dimensions of each physical quantity in a problem, the number of valid variables and possible forms of the problem can be given, limiting some possible relationship between these physical quantities. Although the quantum method cannot predict explicit functional relations, its conclusions also exhibit universality unrelated to any particular functional form. The dimensional analysis method is a universal tool for exploring scientific laws and solving scientific and engineering problems, which is very worth learning and mastering. But it is very strange that although the principles of dimensional analysis are extremely simple, such an important method of dimensional analysis is generally not systematically taught in physics courses. Many examples of dimensional analysis are masterpieces of science. The dimensional method cannot determine in advance which key physical quantities are, and what determines the success or failure of the dimensional analysis is how to determine the key physical quantities of the problem, how to simplify the problem, what are the physical factors, which is more important, and it is necessary to have a profound scientific accomplishment, rich physical knowledge and a deep understanding of the problem, especially for emerging scientific problems, such as dealing with phenomena involving life. Once the system of units is determined, so is the system of dimensions, but vice versa. For dimensional analysis operations alone, it is possible to use only the MLT dimension system. The number and form of valid variables in a problem do not change substantially depending on the dimensional system. The dimensional method belongs to the qualitative and semi-quantitative methods, and many physical people understand that the qualitative method seems to be unavoidable, in fact, the mathematical qualitative method answers the question of whether there is or not, and it is often not comparable to the quantitative method. Dimensional analysis has applications in various fields, with fluid mechanics being a traditional field and biomedicine being an emerging field. Most of the content of this article is only a collation and summary of some literature, and it also attempts to correct some confusion in the literature.
bibliography
[1] Fourier J B J. Analytical Theory of Heat. New York：Dover Pub.，1955
Tan Qingming. Dimensional Analysis. Hefei: University of Science and Technology of China Press, 2007
SUN Bohua. Dimensional Analysis and Lie's Group. Beijing: Higher Education Press, 2016
ZHAO Kaihua. Qualitative and semi-quantitative physics. Beijing: Higher Education Press, 2007
Liang Canbin,Cao Zhoujian. Dimensional Theory and Applications. Beijing: Science Press, 2020
[6] Rayleigh J S W，Strutt B J W. The Theory of Sound. MacMillan，1877
[7] Rayleigh J S W. Nature，1915，95：66
[8] Bridgman P W. Dimensional Analysis，2nd ed. New Haven：Yale University Press，1931
[9] White F M. Fluid Mechanics，4th ed：Ch. 5 Dimensional Analysisand Similarity. McGraw-Hill College，1998
[10] Sonin A A. The Physical Basis of Dimensional Analysis，2nd Ed. Cambridge：MIT，2001. http://web.mit.edu/2.25/www/pdf/DA unified.pdf
[11] Buckingham E. Phys. Rev.，1914，4：345
[12] West S G, translated by Zhang Pei. Scale: Simple Laws for Complex Worlds. CITIC Publishing Group, 2018
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