Hydrology without dimensions

By rigorously accounting for dimensional homogeneity in physical laws, the Pi theorem and the related self-similarity hypotheses allow us to achieve a dimensionless reformulation of scientific hypotheses in a lower dimensional context. This paper presents applications of these concepts to the partitioning of water and soil on terrestrial landscapes, for which the process complexity and lack of first principle formulation make dimensional analysis an excellent tool to formulate theories that are amenable to empirical testing and analytical developments. The resulting scaling laws help reveal the dominant environmental controls for these partitionings. In particular, we discuss how the dryness index and the storage index affect the long term rainfall partitioning, the key nonlinear control of the dryness index in global datasets of weathering rates, and the existence of new macroscopic relations among average variables in landscape evolution statistics. The scaling laws for the partitioning of sediments, the elevation profile, and the spectral scaling of self-similar topographies also unveil tantalizing analogies with turbulent fluctuations.

A critical appraisal of modern engineering science, and the changes required by the appraisal conclusions.

Until the nineteenth century, engineering science was founded on a view of dimensional homogeneity that required the following:· Parameters must not be multiplied or divided.· Dimensions must not be assigned to numbers.· Equations must be dimensionless. This view made it impossible to create equations such as the laws of modern engineering science. Modern engineering science is founded on Fourier’s view of dimensional homogeneity. His view allows the following, and makes it possible to create equations such as the laws of modern engineering science:· Parameters may be multiplied or divided.· Dimensions may be assigned to numbers.· Equations may or may not be dimensionless.Fourier did not prove the validity of his view of dimensional homogeneity. He merely stated that his view of dimensional homogeneity is equivalent to unspecified axioms left behind by the ancient Greeks. Presumably, his colleagues accepted his unproven view because it enabled him to solve problems they were unable to solve. A critical appraisal of Fourier’s unproven view of dimensional homogeneity results in the following conclusions:· Parameters cannot rationally be multiplied or divided. Only the numerical values of parameters can rationally be multiplied or divided.· Dimensions cannot rationally be assigned to numbers. If dimensions could be assigned to numbers, any equation could be regarded as dimensionally homogeneous.· Equations are inherently dimensionless because symbols in parametric equations can rationally represent only numerical value.The appraisal and the changes in modern engineering science required by the appraisal conclusions are presented in the text.

Generalized definition and application of tolerance

Tolerance with upper deviation less than lower deviation is defined as virtual tolerance; the dimension between two extreme values required with tolerance (virtual tolerance) is expressed by a set. The changed characteristics of the set range when the tolerance value is continuously reduced from positive to negative are explored. The nature of virtual tolerance is that the absolute value of virtual tolerance is the error compensation amount, and the dimension between two extreme values is the error compensation range. On the basis of the concept of positive and negative number, the theory of dimensional homogeneity, the existing conditions of the general formula of the dimensional chain, the accuracy of the calculation results of tolerance, and so on, the concept of virtual tolerance and its relationship of unity of opposites with tolerance are proposed. Based on the concept of virtual tolerance, analysis and calculation processes of various assembly dimensional chains are unified, and general formulation of calculating the range of false waste is established, and the method of determining the range of false waste by using probabilistic method in machining process is deduced.

Dimensional Analysis in Economics: A Study of the Neoclassical Economic Growth Model

The fundamental purpose of the present research article is to introduce the basic principles of dimensional analysis in the context of the neoclassical economic theory, in order to apply such principles to the fundamental relations that underlay most models of economic growth. In particular, basic instruments from dimensional analysis are used to evaluate the analytical consistency of the neoclassical economic growth model. The analysis shows that an adjustment to the model is required in such a way that the principle of dimensional homogeneity is satisfied. JEL: A12, C02, C65, O40

An Alternate View of Dimensional Homogeneity, and Its Impact on Engineering Science

Aims:This article proposes an alternate view of dimensional homogeneity that greatly simplifies the solution of nonlinear engineering problems.Background:The conventional view of dimensional homogeneity is generally credited to Fourier (1822).Objectives:The objectives of this article are to describe the alternate view of dimensional homogeneity and to demonstrate its application to practical engineering problems.Methods:By presenting the solution of several nonlinear engineering problems, this article compares solutions based on the alternate view of dimensional homogeneity with solutions based on the conventional view.Results:Example problems demonstrate that nonlinear engineering problems are much easier to solve if the solutions are based on the alternate view of dimensional homogeneity rather than the conventional view. The relative simplicity results because the alternate view of dimensional homogeneity reduces the number of variables in nonlinear problems.Conclusion:The widely accepted view of dimensional homogeneity should be replaced by the alternate view because the solution of nonlinear engineering problems is greatly simplified.

2. Do you want to know a secret? Turkeys, giants, and atomic bombs

When observing and trying to quantify the world around us, two simple facts can be appreciated and readily agreed upon. First, different physical quantities are characterized by different quantifiers, such as length and time. Second, objects come in different sizes. These two facts have far-reaching consequences that can be appreciated when we understand their mathematical implications and study the constraints that dimensions and size impose on physical processes. Obtaining information in this way is known as dimensional or scaling analysis. ‘Do you want to know a secret? Turkeys, giants, and atomic bombs’ considers several examples of scaling analysis and explains that dimensional analysis relies on a simple universal principle, the principle of dimensional homogeneity.

Units of measurement, dimensions, and dimensional analysis

In this chapter the crucial role of units and dimensions in the analysis of any problem involving physical quantities is explained. The International System of Units (SI) is introduced. The major advantage of collecting the physical quantities, which are included in either a theoretical analysis or an experiment, into non-dimensional groups is shown to be a reduction in the number of quantities which need to be considered separately. This process, known as dimensional analysis, is based upon the principle of dimensional homogeneity. Buckingham’s Π‎ theorem is introduced as a method for determining the number of non-dimensional groups (the Π‎’s) corresponding with a set of dimensional quantities and their dimensions. A systematic and simple procedure for identifying these groups is the sequential elimination of dimensions. The scale-up from a model to a geometrically similar full-size version is shown to require dynamic similarity. The definitions and names of the non-dimensional groups most frequently encountered in fluid mechanics have been introduced and their physical significance explained.

The Principle of Similitude in Biology: From Allometry to the Formulation of Dimensionally Homogenous ‘Laws’

ABSTRACTMeaningful laws of nature must be independent of the units employed to measure the variables. The principle of similitude (Rayleigh 1915) or dimensional homogeneity, states that only commensurable quantities (ones having the same dimension) may be compared, therefore, meaningful laws of nature must be homogeneous equations in their various units of measurement, a result which was formalized in the Π theorem (Vaschy 1892; Buckingham 1914). However, most relations in allometry do not satisfy this basic requirement, including the ‘3/4 Law’ (Kleiber 1932) that relates the basal metabolic rate and body mass, besides it is sometimes claimed to be the most fundamental biological rate (Brown et al. 2004) and the closest to a law in life sciences (Brown et al. 2004). Using the Π theorem, here we show that it is possible to construct an unique homogeneous equation for the metabolic rates, in agreement with data in the literature. We find that the variations in the dependence of the metabolic rates on body mass are secondary, coming from variations in the allometric dependence of the heart frequencies. This includes not only different classes of animals (mammals, birds, invertebrates) but also different aerobic conditions (basal and maximal). Our results demonstrate that most of the differences found in the allometric exponents (White et al. 2007) are due to compare incommensurable quantities and that our dimensionally homogenous formula, unify these differences into a single formulation. We discuss the ecological implications of this new formulation in the context of the Malthusian’s, Fenchel’s and Calder’s relations.