The theory of scaling

Scaled experimentation is an important approach for the investigation of complex systems but for centuries has been impeded by the want of a scaling theory that can accommodate scale effects. The present definition of a scale effect is founded on the violation of an invariance principle arising out of dimensional analysis, i.e. dimensionless equations do not change with scale. However, apart from all but the most rudimentary of systems, most dimensionless governing equations invariably do change with scale, thus providing a very severe constraint on the reach of scaled experimentation. This paper introduces the theory of scaling that in principle applies to all physics and quantifies either implicitly or explicitly all scale dependencies. It is shown here how the route offered by dimensional analysis is nothing more than a particular similitude condition among a countable infinite number of alternative possibilities provided by the new theory. The theory of scaling is founded on a metaphysical concept where space is scaled and the mathematical consequences of this are reflected in the governing equations in transport form. The theory is trialled for known problems in continuum mechanics, electromagnetism and heat transfer to illustrate the breath of the approach and additionally demonstrate the advantages offered by additional forms of similitude.


Introduction
For the most part of the twentieth century, scaled experimentation has been dominated by a single ubiquitous theory called dimensional analysis [1]. The theory, which is underpinned by an invariance principle that dimensionless governing equations remain invariant with scale, is relatively easy to apply and is standard fare on most physical sciences and engineering-related undergraduate degree programmes [2]. A particular advantage of dimensional analysis arises from its ability to characterise dominant physics with the identification of a subset of dimensionless parameters (the dominant Pi groups) [1,3]. A drawback in scaling however is its reliance on an invariance principle that seldom applies to all but the most rudimentary of systems [4,5]. If scale effects are pronounced, then the predictive capability of dimensional analysis combined with scaled experimentation is significantly diminished [2]. This is presently the situation and additionally the rise of computational modelling over recent decades, with ever greater sophistication in simulation approaches [6], has to a large extent significantly diminished the need for scaled experimentation.
Scaled experimentation has many advantages being accessible, time-saving and cost-effective but suffering principally from scale effects, where the behaviour at scale can be markedly different [4]. The change in behaviour at scale can significantly undermine the approach, and the literature is replete with ad hoc adaptations to scaled experiments in an attempt to more accurately capture the behaviour of the full-scale system. An example of this is in the field of earthquake engineering, where additional mass is a common feature in scaled models in an attempt to ameliorate the changes in behaviour that invariably occur with scale [7]. Unfortunately, such solutions bring with them their own problems, where mass added in the form of slabs being able to shift and cause irregular behaviour that is not a feature of full-scale buildings. Examples of shake-table experimentation with the application of ad hoc scaling rules applied to scaled models can be found in references [8][9][10][11], with reference [12] looking at the use of a centrifuge to augment gravity to negate scale effects. Another well-known area afflicted by scale effects is the process of indentation, which was known to the Swedish engineer Brinell in his early work (see reference [13]), where he observed that hardness was sensitive to ball diameter. This effect can be predominantly attributed to a size effect related to the size of the deformation zone under the influence of the indenter. Other scale effects are known to play a part however, which include surface contamination, roughness and energy, and indenter tip rounding, and friction (see reference [14]). Granular materials such as concrete are particularly afflicted by size effects as is revealed by the work of Bažant [15]. Since scale effects are presently characterized by a breakdown in similarity as defined by dimensional analysis, the approach is invariably required to investigate them. An example of this is a recent study concerned with scaling and its effects on fatigue and lifetimes [16], with dimensional analysis being applied to determine whether scale effects are present or not. Other studies concerned with fatigue [17,18] again confirm the presence of scale effects through dimensional analysis, through changes with scale in the dominant governing dimensionless equation. Another area where scale effects pose problems is the field of system dynamics. It is recognised that modern methods for the analysis of dynamic systems invariably involve computer simulation to analyse system kinematics, stability, long-term behaviour and control [19][20][21][22]. Computer simulation facilitates sensitivity analysis and the investigation of system responses to changes in boundary and initial conditions, but additional the effects of changes in system properties can be investigated [23,24]. However, for discrete systems that represent some physical system, experimental validation is necessary to support the use of what is undoubtedly a simplified model for practical modelling purposes [4,25,26]. Experimental studies can take many forms, but focus here is on those that involve a reduction in scale.
A question that has hardly been considered over the past century is whether the invariance principle that dimensional analysis provides [27] is the only one. It is only very recently that work has appeared in the open literature that addresses this question by viewing the scaling problem in an entirely new light. The new approach is called finite similitude and is founded on the metaphysical concept of space scaling and has been applied to a range of practical problems; see the references [28][29][30][31][32][33]. The underpinning concept being "metaphysical" is somewhat unusual in that it cannot be achieved practically but nevertheless can be imagined and most importantly can be mathematically defined. Through the mathematical formulation it becomes possible to assess the effect of metaphysical scaling on the governing constraining equations presented in an integral transport form. The principal step in the new scaling theory is the projection of the governing physics described in a trial space (where the scaled experiment sits) onto the physical space (where the full-scale experiment sits). This projection is key to the whole concept and has the effect of qualifying all scale dependencies. Some dependencies such as geometrical measures like volume and area are revealed explicitly by this transformation, but others including scalar, vector and tensor fields are revealed in implicit form only. This approach transforms the problem of scaling into a problem where the objective is the quantification of implicit scale dependencies. This paper examines the similitude rules that can be applied to reveal all hidden dependencies provided by the theory. It is proved here that there exists an infinite number of rules satisfying a particular recursive identity. It is confirmed that there exists no others and as such the theory of scaling is established as the sole procedure for the analysis of scaled experiments. The theory is applied to a broad range of physics to demonstrate the practical worth of the similitude rules. Problems are purposely kept relatively simple to illustrate the theory, but it is recognised of course that the real benefit of scaled experimentation is for the analysis of complex systems and physics. The work builds on the work presented in references [34][35][36][37][38], where for first time a new first-order finite similitude identity is applied involving experiments at two distinct scales.
Metaphysical space scaling is presented in brief in Sect. 2, where the principal focus is on isotropic scaling although anisotropic scaling [39,40] is noted. The starting point of the theory is the identification of inertial frames in the physical space where the full-scale experiment resides and in the trial space where the scaled experiment resides. The expansion and contraction of space to achieve scaling leads naturally to control volume formulations being impacted immediately by the space in which they reside. Scaled transport equations are introduced in Sect. 2.1 for Newtonian, Maxwellian and heat-related physics, where the critical step of projecting trial-space transport equations onto the physical space is presented. This step is important because it exposes all possible scale dependencies that arise with the scaling of physics. Some of the dependencies are hidden however and presented in Sect. 2.2 is a procedure for revealing these dependencies, i.e. the similitude rules. The uniqueness of these rules is discussed in Sect. 3, where it is shown that any alternative rule available is representable by the rules conveniently defined by a recursive differential identity. In Sect. 3.1 the rules pertaining to integration of the these identities are presented. It is shown how the nested form of the similitude rules leads to a generic theory, with a unified approach to scaled experimentation. Product terms are the focus of Sect. 4 which appear in transport physics and have the effect of limiting the order of the fields involved. This is followed in Sect. 4.1 with a focus on second-order finite similitude, which normally requires three scaled experiments but reduced to two by careful representation of the product terms. The finite similitude theory is showcased in Sect. 5 considering examples in electromagnetism, fluid mechanics, and heat transfer. The paper ends with a list of conclusions.

A brief overview of the finite similitude theory
The finite similitude theory was first introduced by Davey et al. in reference [28] and has subsequently been applied to a range of practical problems, which include: impact mechanics [31,32,34], biomechanics [30,41], powder compaction [29] and metal forming [42]. Although the theory is discussed in these references, it is convenient nonetheless to provide here an overview of the founding ideas prior to examining the mathematical foundation of the new similitude rules. The starting point for the theory of finite similitude is the imagined concept of space scaling, which transpires to be relatively easy to define mathematically. It is simply a matter of relating coordinate coefficients in the physical space x i ps (where the full-scale process sits) to those in the trial space x i ts (where the scaled process sits). This is achieved by the means of an affine mapping of the form x ps → x ts . The map is linear to reflect the homogeneity of space but additionally is required to be temporally invariant to reflect the homogeneity of time. The map is readily described in differential terms and takes the form dx ts = F · dx ps or equivalently dx i , where F is both temporally and spatially invariant. The basic idea is presented in a 2-D frame in Fig. 1, where the inclusion of clocks in the two spaces is to reflect the fact that processes can run at different rates in the two spaces. The times in the two spaces are related by a temporally and spatially invariant map t ps → t ts to reflect again the assumed homogeneity of space and time common to Newtonian physics. In differential terms the map takes the form dt ts = gdt ps , where g is both temporally and spatially invariant. It is assumed here for convenience that the inertial frames for the two spaces {G i } and {g i } shown in the figure are orthonormal. With this restriction in place the identity F = β I provides isotropic space scaling with β > 0 dictating the extent of linear scaling and I is a unit matrix. Contraction of space is associated with β < 1, with β = 1 providing no change in scale, and β > 1 associated with the expansion of space. Although in principle anisotropic space scaling is possible with an appropriate definition of F, in practice issues with the space metrics limit its range of applicability as discussed in reference [39]. Moreover, more recent work on localised anisotropic scaling [40] is noted here in passing only and not considered further. The focus on space as the backdrop to the scaling process suggests that physical descriptions founded on the control-volume approach are apposite. The reasons being that changes in geometric measures (e.g. volume, area) are readily reflected in the control volume and also evidently absent from point-based descriptions. Although it is apparent that control volumes formulations are not universally popular in many branches of physics, the laws of nature can invariably be described with them. Moreover, the new scaling theory founded on this approach ultimately leads to point-based identities and to those more familiar physical descriptions in common usage.
In scaling, the motion of control volumes in the trial and physical spaces is connected in order to describe the related physics between the two spaces The details of this can be found in references [34][35][36] ts and x * ps are points attached to their respective control volumes and velocities v * ts and v * ts are specified along with initial conditions; Frobenius's theorem guarantees unique solutions to these differential equations. Control volumes provide the means to focus on particular regions of space, but since space scaling is involved, the identity dx * ts = βdx * ps must apply. This relationship is similar to the space identity dx ts = βdx ps , but the former unlike the latter relates moving points (see Fig. 2). Recalling the temporal identity dt ts = gdt ps , the similitude identity between the moving control volumes velocities in the two spaces is the relationship v * ts = g −1 βv * ps , conferring synchronous motion for the two control volumes.

Scaled transport equations
In this section transport equations are introduced for Newtonian mechanics, Maxwellian electromagnetism and thermodynamics along with their corresponding projections. No attempt is made to couple the distinct physics although such a coupling is not a barrier to the scaling theory presented. The starting point for all the physical descriptions is volume conservation described by the transport equation, where focus is initially directed to the trial space where the scaled experiment is assumed to reside.
The key step in the finite similitude theory is the projection of trial-space transport equations onto the physical space. This is achieved by first substituting dV * ts = β 3 dV * ps , n ts d * ts = β 2 n ps d * ps and dt ts = gdt ps , and on multiplication by g (and α 1 0 in this case) to provide where use is made here of the identity v * ts = g −1 βv * ps . It will be revealed that projected transport equations serve to enforce similitude relationships (such as v * ts = g −1 βv * ps ) and the scalars of the type α 1 0 aid in their enforcement but more on this in subsequent sections.

Newtonian physics
In addition to momentum, mass conservation is included here but also a transport equation for movement (first introduced in reference [43]) to bring the displacement field directly into play. The transport equations of interest to Newtonian mechanics are: where ρ ts is mass density, u ts is displacement, σ ts is Cauchy stress, b v ts is specific-body force (i.e. force per unit mass), and n ts is an outer pointing unit normal on the boundary * ts of control volume * ts . As mentioned above the key step in the finite similitude theory is the projection of these equations onto the physical space. Consider then the substitution of dV * ts = β 3 dV * ps , n ts d * ts = β 2 n ps d * ps and dt ts = gdt ps , and on multiplication by g and, respectively, α ρ 0 , α v 0 and α u 0 to provide ts and the allimportant scalars α 1 0 , α ρ 0 , α v 0 and α u 0 (whose role will become clear) along with g are assumed to be functions of β.
Equations (4) (and Eq. (1)) provide an alternative viewpoint for scaling, where the objective is to reveal β dependency. Geometrical measures such as volume and area are known immediately and are identified with β 3 and β 2 terms, but scalar, vector and tensor fields are only revealed implicitly. For example, the behaviour of the fields v β , u β , σ β are unknown although linked to β. Similitude rules provide the means to uncover hidden behaviours, but prior to examining this aspect electromagnetism is considered next.

Maxwellian physics
Maxwell equations can be represented in transport form with the help of the Levi-Civita tensorˆ i jk , which takes up the values of one for even permutations of {i jk}, minus one for odd permutations and zero otherwise. The tensorˆ i jk is useful because it provides a convenient means for the representation of cross-products and a particular example is where B is the magnetic field, ∂ j ≡ ∂ ∂ x j and B k = δ m k B m , and where the Kronecker delta symbol δ m k is either zero or one.
The advantage of Eq. (5) is that it equates to ∇ × B = ∇ · (ˆ · B), where the operators ∇·, ∇× signify divergence and curl. This notation enables Maxwell equations to be written in transport form, i.e. 0 = − * ts D ts · n ts d ts + * ts ρ f ts dV ts (6a) 0 = * ts B ts · n ts d ts (6b) where for linear stationary materials the magnetising field H ts = μ −1 ts B ts and the electric displacement field D ts = ts E ts , with permeability μ ts = μ 0 ts (1 + χ m ts ) and permittivity ts = 0 ts (1 + χ e ts ), and where χ m ts and χ e ts are magnetic and electric susceptibility, respectively; additionally, B ts , E ts are the magnetic and electric fields, μ 0 ts , 0 ts , are the magnetic and electric constants, ρ f ts and J f ts are the charge and current densities for free particles.
Although not independent of Eq. (6), it is sometimes convenient to include a transport equation for charge, which takes the form where current density J f ts for a stationary material is typically related to the electric field by J f ts = σ ts E ts , and where σ ts is electrical conductivity.
Following the procedure in Sect. 2.1.1 and on multiplication Eqs. (6) by g and, respectively, (7)) to provide in slightly different order:

Thermodynamics
The governing transport equations for in thermodynamics take the form where h ts and s ts are specific enthalpy and entropy, q ts is heat flux (i.e. rate of heat transfer per unit area), Q ts is a source representing specific heat transfer (i.e. per unit mass) rate, andṡ ts is entropy density production associated with irreversibility. Following the procedure above for projection with multiplication by g, α h 0 and α s 0 provide where the velocity field v β is included here for completeness but not involved in examples presented below, ts . Note here that all the physics considered in this section is described by transport equations of the form α ψ 0 T ψ 0 = 0, where ψ is a label indicating the equation under consideration.

Similitude rules
The similitude rules are in effect about how the projected transport equations change with β. The simplest assumption is that no change with β takes place, which is expressed mathematically by the identity where the equality sign "≡" means "identically" making the derivative of the transport equations identically zero, i.e. the left-hand side of Eq. (11) disappears completely under the derivative. The satisfaction of this identity is termed zeroth-order finite similitude, and it is this rule that was first introduced in reference [39] and subsequently explored in references [28][29][30][31][32][33]. It might be anticipated that higher derivatives of the projected transport equations can capture behaviours that Eq. (11) fails to represent, i.e. scale effects as presently defined. It is recognised that the application of a scale-invariant identity to the projected transport equations in Sect. 2.1 is about assuming how things change with β. It can be anticipated therefore for an arbitrary problem that the applied rule will invariably be incorrect. The advantage of similitude however is that it can be used to design-scaled experiments, making the problem one of design-of-experiments with the similitude rule providing the vehicle for the design process.
Higher-order derivatives can therefore be expected to capture what are termed here zeroth-order scale effects, which are defined here to be those effects that lead to Eq. (11) not being satisfied. The following definition is critical to the whole approach: ∀β > 0, is the lowest value of k for which the identity holds (termed k th -order finite similitude), where smooth α ψ k : R + → R, α ψ k (1) = 1 and α ψ 0 T ψ 0 = 0 are projected transport equations (see Sect. 2.1). Definition 2.1 was formally introduced in reference [35] and is designed to capture the basic requirements of similitude rules. It is shown below that all possible similitude rules are captured by this definition having certain necessary prescribed features. The first of these is the nesting of similitude rules, where lower-order similitude conditions if satisfied infer the satisfaction of higher-order rules. This is a critical requirement that similitude conditions must satisfy for without it the approach would cease to have practical value. Note that zeroth-order finite similitude under this definition is T ψ 1 ≡ 0 and first-order finite similitude is therefore defined by T ψ 2 ≡ 0 and second-order finite similitude by T ψ 3 ≡ 0 and so on. Note also that first order involves two derivatives T and second order involves three derivatives 1 4 ) and so forth for higher order, where the number of derivatives involves matches the number of scaled experiments required. Note that Eqs. (11) to (14) are differential equations, which here are initiated at β = β 0 = 1. This is to ensure that projected equations of the form α ψ k T ψ k = 0 match the full-scale system at β = β 0 = 1 and is the reason why α ψ k (1) = 1 in Definition 2.1. Note also that g(1) = 1 since dt ts (1) = g(1)dt ps = dt ps and all fields must match at β = β 0 = 1, e.g. ρ β (1) = ρ ps , v β (1) = v ps , u β (1) = u ps , σ β (1) = σ ps , E ps (1) = E β , B β (1) = B ps and so on for other fields.
Note also that integration of Eq. (11) between the limits β 1 and This equality returns what dimensional analysis provides being that it confirms that the projected trialspace transport equations do not change with scale. Higher-order identities follow the exact same rule with the integration of Eq. (12) (1), but in this case the fields involved are derivatives with respect to β of the fields populating the original transport equations. The solution of Eqs. (11) to (14) can be achieved relatively easily to link scaled experiments. However, prior to examining this aspect it is important to establish that the countable infinite number of identities provided by Eq. (12) are the only ones that exist.

Uniqueness of similitude rules
The nesting of the similitude rules imposes constraints on their form although at first sight it appears that more general forms might exist. This transpires not to be the case but to show this first consider the expansion of identity in Eq. (12) to give which in principle could be replaced by a more general looking equation giving rise to the proposition: The proof is confirmed by Eqs. (15) to (17).
The following proposition is needed to proceed.

Proposition 3.2 For arbitrary natural numbers i and j the identity
holds.
provides an element in the right-hand span by Leibniz rule of differentiation. The existence of elements not in the span to the immediate left in Eq. (18) is evident.
This leads to the next proposition, which is needed for an inductive step required for the general proof.
, is true for natural numbers k and m with m < k, then it is also true that span T since all the derivatives appearing are present in this last relationship.
Note however this provides span T It is now a simple matter to provide the principal theorem of this paper which is: The matching of the spans in Theorem 3.1 is an equivalence relationship, so it transpires therefore, that each similitude rule is unique in the sense, that for a given order (see Definition 2.1) there exists no alternative rule of the same order that cannot be represented by Eq. (12). Thus, for a single scaled experiment there is no alternative to zeroth-order finite similitude and for two scaled experiments, first-order finite similitude is the similitude rule that uniquely applies. The advantage offered by this uniqueness is that the relatively simple form of Eq. (12) can be readily integrated with a generic procedure. It is through integration that scaled experiments are related and a generic approach has the advantage of giving rise to a standardised procedure for scaled investigations.

The rules of integration
The similitude identities defined by Eq. (12) are remarkably straightforward to integrate and involve divided differences but also involves the application of a mean-value theorem to ensure exact similitude rules are returned.

Zeroth-order integration
The simplest similitude rule is defined by Eq. (11), which readily integrates over the limits β 1 and β 0 but in divided difference terms takes the form where a mean-value theorem is utilised to provide exact identities although, with the identity being identically zero, this feature is not needed in this case.
Note that α confirms that projected transport equations do not depend on β and consequently their integrands do not change. Provided in Table 1 are the fields and scalar relationships for zeroth-order finite similitude arising from this identity.

First-order integration
First-order integration is concerned with the solution of Eq. (13), which is achieved by considering an additional scale β 2 and a similar divided-difference identity to Eq. (20), i.e.
with β 2 ≤ β 1 2 ≤ β 1 , where as above, a mean-value theorem is again utilised to provide an exact identity. , The first-order assumption means that α , which on substitution of Eqs. (20) and (21) provides after a little rearrangement where and since α ψ 1 is an indeterminate function of β, it is understood that R ψ 1 takes the form of a parameter. Provided in Table 2 are the fields and scalar relationships for first-order finite similitude arising from this identity. Note in the table that for a consistent field expressions it is required that R respectively, as indicated in the table.

Second-order integration
Second-order integration is concerned with the solution of Eq. (14), which follows on from first order by considering an additional scale β 3 and a similar divided-difference identity to Eqs. (20) and (21), i.e.
with β 3 ≤ β 2 3 ≤ β 2 , where as above, a mean-value theorem is again utilised to provide an exact identity. Given identities Eqs. (20), (21) and (24) and recalling that α where β 2 3 ≤ β 1 3 ≤ β 1 2 and β 1 2 ≤ β 0 2 ≤ β 0 1 . Given that second-order finite similitude requires α it follows substitution of Eqs. (20), (21) and (24) into this identity and after a significant degree of algebraic manipulation that, where and as above α ψ 2 is an indeterminate function of β, making R ψ 2 a parameter. Provided in Table 3 are the fields and scalar relationships for second-order finite similitude arising from this identity. Note that the exact same requirements imposed on α ψ 1 , apparent in Table 2, not too unexpectedly apply to α ψ 2 for consistent field expressions. It is important to appreciate also that the appearance of products of fields in the transport equations places constraints on the orders of the field expressions provided in Tables  2 and 3. An immediately apparent example arises from the entropy equation where the product of q β and T −1 β places limits on the orders involved. The role played by the nesting of similitude rules is critically important here as it enables mixed orders to be involved so that the product q β T −1 β is no greater than the order of the similitude rule being applied. The next section examines this issue in greater detail.

The constraint of field products
The similitude field relationships listed in Tables 1, 2 and 3 cannot all be simultaneously applied as a consequence of field products, should these appear in scaled transport Eqs. (4), (8), (10). Fortunately, the nesting of the similitude laws provides a solution to this issue by allowing field products of mixed order to be formed. The following proposition sheds some light on the result: Prior to examining the proof it is worth noting the manner in which products appear in the transport equations. In thermomechanics for example, mass density ρ ts appears as a multiplier of a number of fields having the effect of converting specific quantities (i.e. per unit mass) into density quantities (i.e. per unit volume), and additionally is central to Newtonian physics. In the problems of interest in this paper (not involving gases) it is sufficient to use a zeroth-order approximation for density as stipulated in Table 1, where ρ 1 = ρ β = α ρ 0 ρ ts β 3 , which means that d dβ (α ρ 0 ρ ts β 3 ) ≡ 0. Zeroth-order approximations impose no constraints on the order of any field product. Another product form, observed in mechanics, in the transport equation for momentum, is the appearance of the velocity field v β and its dyadic v β v β . In first-order theory the simplest solution is to replace one v β in the dyadic by the zeroth-order field v 1 making v β v 1 first order. This approximation is particularly convenient for solid mechanics, where convective terms are small and are typically neglected. In fluid mechanics however such an approximation may be insufficient and second-order theory can be applied, but additionally limiting the velocity field v β to be first order so that v β v β is second order. This issue also arises when kinetic energy features involving the product v β · v β , but again readily catered for by limiting the order of the velocity field v β . An alternative form is where a field is present in one transport equation but also appears in a product form in another. This situation arises in the energy and entropy equations for example, where q β features in the energy transport equation and the product q β T −1 β is in the entropy equation. Mixing orders is again the solution here, although it is appreciated that it is common practice to relate q β to temperature through a constitutive law, which can impose additional constraint. More is said about this below, but it is worth noting at this point that the finite similitude theory does not directly feature constitutive equations, although these can be conveniently used to set free parameters [34,[34][35][36]39,42].
Proof 4.1 Without loss of generality it can be assumed that the field features in a scaled-transport equation whose scalars are α θ n , (n = 1 : N ) and the product · features in a scaled-transport equation whose scalars are α θψ k , (k = 1 : K ). It can be assumed further that the field is only present as a product, but nevertheless had it featured in another transport equation it would have its own set of scalars α ψ m , (m = 1 : M). Note that the similitude rules presented in Sect. 2.2 apply directly to the fields appearing in the integrands of the scaled transport equations introduced in Sect. 3. This is a result of the projection of transport equations onto the physical space, with integration domains independent of β. Having a scaling theory underpinned by a calculus offers a particular advantage, in that the similitude rules in Definition 2.1 immediately apply to the fields of interest. Setting 0 = and n+1 = d dβ (α θ n n ) in accordance with Definition 2.1 and similarly for the field product · , i.e. 0 · 0 = · , and ( · ) k+1 = d dβ (α θψ k ( · ) k ), additionally it is convenient to assume the existence of m+1 = d dβ (α ψ m m ). It is evident that the order of the individual fields in the product ( · ) k has an effect on the order of the product. Without loss of generality let M ≤ N and consider then K = 1 (and M = 0) with ( · ) 1 which is identically zero if and only if α θψ 1 = α θ 1 . Consider further K = 2, which provides two possibilities, i.e. M = 0 and M = 1, with the former of these which is identically zero if and only if α θ 1 = α θ 1 and α θ 2 = α θ 2 . It can be readily deduced that for M = 0, that α θ k = α θ k , ∀k ≤ K = N ; as mentioned above a zeroth-order field in a product imposes no additional constraints. The more interesting situation is M = 1 (with K = 2) and note that which reveals an important requirement that α θψ 1 = α θ 1 = α ψ 1 in accordance with the finite-similitude approach where lower-order terms are required to be set prior to higher-order ones.
Satisfying this requirement provides as a consequence of the first-order condition (v β ) 2 . Turning now to the energy and entropy equations and the product term q β T −1 β in the entropy equation, which again is captured by Proposition 4.1, where α s 2 = α s 1 = α h 1 . This follows because

A second-order two-experiment theory
It is clear from Proposition 4.1 that products limit the order of the fields involved, but additionally constrain the freedom to arbitrarily set some of the higher-order scaling parameters R k . The precise nature of this constraint is explored in the case of second-order finite similitude, but limiting fields to at most first order only (i.e. K = 2, M = N = 1). With the limitation that first-order fields are involved it is shown here that it is possible to limit the number of scaled experiments to two despite a second-order theory being required. This situation is of practical value, so is examined in detail here and the starting point is a corollary of Proposition 4.1. The condition α s 2 = α s 1 = α h 1 places a constraint on the parameter R s 2 and suggests the existence of a possible relationship with R s 1,1 and R s 1,2 . The restriction of q β and T −1 β to first order means that where the expectation is that the product of these equations should equate to for some value of the parameter R s 2 and leads to the following proposition.

Proof 4.3 Rearrangement of (34) gives
where terms have been added and subtracted so that differences appear but otherwise leaving Eq. 34 unchanged, and on bringing heat-flux differences to one side and temperature-inverse differences to the other yields which on substitution of Eqs. (33) gives which on substitution of Eqs. (33) again gives and in view of the arbitrariness of the fields this reduces to and given that by assumption R h 1,1 R h 1,2 = 0, this equation yields Eq. (35) in the proposition. Note that the conditions R h 1,1 = 0, R h 1,2 = 0, R h 1,1 = −1 and R h 1,2 = −1 all signify zeroth order, as seen on examination of Eqs. (33).
The final piece of information needed to remove the requirement for an experiment at scale β 3 is the determination of the parameter R h 1,2 , which is readily set equal to R h 1,1 to provide R s 2 = R h 1,1 according to Eq. (35). The justification for this follows from the β−translation invariance of the finite similitude conditions, which is readily apparent from their definition in differential form being absent of explicit β terms (e.g. Eq. (13)). Thus first-order identity Eq. (22) applies equally to {β 3 , β 2 , β 1 } as it does to {β 2 , β 1 , β 0 } and consequently setting R h 1,1 = R h 1,2 is permitted and convenient. Note that the term q β 2 T −1 β 2 − q β 3 T −1 β 3 in Eq. (34) can by means of algebraic manipulation be transformed into which removes any reference to fields at β 3 , and achieved with substitution of Eqs. (33). Similarly, the dyadic velocity expression v β v β has a second-order expansion of the form which reduces to the two-experiment form on setting R 2 = R 1,2 = R 1,1 and on elimination of the product v β 3 v β 3 (by similar means as Eq. (41)) to give which reduces further to confirming explicitly that the setting R 2 = R 1,2 = R 1,1 reduces the second-order dyadic to the product of two first-order velocity fields as required.

Case Studies
Relatively simple examples are presented here to showcase the finite similitude theory and confirm the theoretical results above. Three case studies are analysed with one study from each the fields of electromagnetism, fluid mechanics, and heat transfer. The electromagnetism study is restricted to an introductory problem that is captured by zeroth-and first-order theories, whilst the second-order, two-experiment approach is demonstrated for fluid mechanics and heat transfer examples.

Case study I: Electromagnetism
The particular problem under scrutiny is presented in Fig. 3 consisting of nested conductive square solid and hollow rods, encircled by a rectangular cross-sectional current carrying coil. The lengths of the rods are equal to 0.50m, with side lengths set to 0.035m and 0.05m for the inner and outer rods, respectively. Also, the inner and outer radius and depth of the encircling conductor are, respectively, equal to 0.09m, 0.11m and 0.02m. The medium surrounding the rod and coil being represented by a cylinder of radius and length of 0.20m and 0.50m is assumed to have the properties of air. Depicted in Fig. 3 is a the uniform mesh employed to discretize the model consisting of EMC3D8 elements [44] of size 0.0125m. Material properties assigned to the different sections of the full-scale model are tabulated in Table 4. The properties for the coil and air are considered identical since the current in the coil is directly specified as opposed to modelling the source responsible for the current. Furthermore, the current density in the tangential direction shown in Fig. 3 along with the lower and upper frequencies of the full-scale model is provided in Table 4. Scaling factors, frequencies and current  Table 5, where dimensional scaling factors for trial models 1 and 2 are, respectively, set equal to 0.50 and 0.25, which scales the dimensions of these models to half and one-fourth of the full-scale model. Trial models 1 and 2 are designed based on the zeroth-order theory, whilst the virtual models, which are a combination of trial models 1 and 2, are designed based on the first-order theory. The principal relationship of concern in this case study is which is a first-order improvement on α F 0 σ −1 ts1 = α ρ f 0 σ −1 ps , which is used for conductivity matching on setting α F 0 , with g set to satisfy α F 0 μ −1 ps = α ρ f 01 gβ −2 μ −1 ts for permeability matching, where throughout α ρ f 0 = 1 as indicated in Table 5.
Specifically, R F 1 is set to satisfy Eq. 44 and α F 01 and α F 02 set by α F 01 σ −1 ts1 = σ −1 ps and α F 02 σ −1 ts2 = σ −1 ps , respectively. The magnetic field along a longitudinal path for the full-scale outer rod (see Fig. 4), made from expensive gold, is predicted using trial models 1 and 2, respectively, made from aluminium and copper. Based on the presented results in Fig. 4, it is clear that the response of the full-scale outer rod is well represented by scaled down trial models 1 and 2 with all pertinent parameters captured to good accuracy. The errors in the magnetic field prediction of the prototype are 4.75% and 7.74% for the scaled down trial models 1 and 2, respectively. This reduces to 2.90% on employing the first-order theory combining the results of the small-scale models 1 and 2. Note here that the errors are calculated based on the area under the curves. Note however, from Fig. 5, the magnetic field of the full-scale inner rod made from expensive silver, is not been predicted to good accuracy using scaled down trial models 1 and 2, which are, respectively, made of iron and carbon steel.   Large errors are recorded for the full-scale inner rod, i.e. 20.88% and 37.71% for models 1 and 2, respectively. This, of course, is a consequence of the zeroth-order theory possessing insufficient degrees of freedom, akin to dimensional analysis. To deal with this problem, the extra degree of freedom (i.e. R F 1 ) provided by the first-order theory is used to counter the scale effects apparent in the inner rods. Application of Eq. 44 and on combining the results of models 1 and 2 using first-order expressions in Table 2 and the scaling factors for virtual model 1 in Table 5, the magnetic field of the full-scale inner rod is predicted with improved accuracy (i.e. 10.48%). Moreover, as apparent from Fig. 5 and Table 2, by setting R F 1 = 1.25 the magnetic field of the full-scale model is predicted with a zero error. In other words R F 1 can be used to completely annihilate the errors in scaled down trial models.  The focus in this section is mechanics in the form of a fluid-solid interaction problem involving the dropping of a hollow spherical metallic ball into a body of water. The main issue here is gravity with the recognition that conventional scaling methodologies (including zeroth-order finite similitude) are unable to account for scale effects that typically arise with the scaling of gravitational problems. This is readily apparent on consideration of the zeroth-order relationship for velocity v ts = g −1 βv ps (see Table 1) which provides acceleration a ts = g −2 βa ps and acceleration due to gravity G ts = g −2 βG ps . As observed from references [35,36,45] problems in mechanics invariably necessitate the condition g = β which infers the relationship G ts = β −1 G ps and the requirement that G ts varies with scale. A question of interest addressed in this case study is whether the extra degree of freedom afforded the first-order theory is sufficient to capture the zeroth-order scale effect presented by gravity.
The Abaqus finite element software [44] is applied to obtain numerical results for the simulation of an hollow spherical ball being dropped into water. The mid-surface radius and thickness of the full-scale ball is set equal to 0.40 m and 0.05 m, respectively. The ball is made from steel with density, elastic modulus and Poisson's ratio, respectively, equal to 7800 kg/m 3 , 200 GPa and 0.30. The details of the problem considered are depicted in Fig. 6 and feature an Eulerian component consists of water and a void above the water. The properties of water including density, dynamic viscosity and speed of sound are 1000 kg/m 3 , 0.001 Pa s and 1500 m/s, respectively. The mesh and a simulation result can be found in Fig. 7. A dynamic explicit solver is utilized to obtain the numerical results, where a general contact model is employed to simulate all contact interactions. Boundary and initial conditions for the full-scale model are provided in Fig. 6 and acceleration due to gravity is set equal to 9.81 m/s 2 . The elements shown in Fig. 6 are 0.05 m in size and are applied to both the ball and the Eulerian components of the model. A mesh sensitivity study confirmed that the mesh depicted in Fig. 6 makes the results independent of further mesh refinement. Element types S4R and EC3D8R [44] are  Table 6. Note that trial models 1 and 2 are, respectively, scaled to one-fifth and one-tenth of the dimensions of the full-scale model and the virtual model is a combination of the two trial models.
Given the first-order relationship for acceleration a 1 = a β 1 + R 1 (a β 1 − a β 2 ) with a β = g 2 β −1 a ts it follows that acceleration due to gravity can be represented as and with the requirement that G ps = G ts1 = G ts2 it follows that R 1 = (1 − β 1 )(β 1 − β 2 ) −1 and for β 1 = 0.2 and β 2 = 0.1 gives R 1 = 8 as recorded in Table 6. The velocity variation of the ball as it penetrates and moves through the water is shown in Fig. 8 with a visual image for the full-scale model depicted to Fig. 7. The results presented in Fig. 8 confirm as anticipated that trial models 1 and 2 (founded in zeroth-order theory) are unrepresentative of the behaviour of the full-scale with significant differences involved. Based on the area under the curves, the trial models 1 and 2 predict the response of the full-scale model with errors, respectively, equal to 3.90% and 4.39%. The final velocities are recorded for the trial models 1 and 2 give errors of 6.97% and 7.83%, respectively, on comparison with the full-scale model. The errors are significantly reduced on application of the first-order theory which combines the results of the trial models 1 and 2 based on the first-order velocity relationship provided in Table 2. Note additionally, the energy per mass variation of the ball has been presented for the full-scale and virtual models Fig. 9. This information is obtained using dyadic Eq. (43b) by recording the velocity product in the vertical direction. The left-hand side of Eq. (43b) represents the energy per mass of the full-scale ball, whilst the right-hand side provides the specific energy of the virtual ball. Apart from undulations in the results good replication is revealed in Fig. 9. The undulations are a result of the 10 and 5-fold magnification of the results from the scaled models.

Case study III: Heat transfer
Heat transfer is the focus of this section in the form of a long rod illustrated to Fig. 10 and considered in reference [46]. The diameter of the rod with one end maintained at 100 • C is 5 mm, and its surface is exposed to ambient air at 25 o C with a convection heat transfer coefficient of 100W/m 2 K. The circular fin is made from copper with thermal conductivity equal to 398 W/mK [46]. Subject to the assumption of an infinitely long fin, the equations for the temperature and heat transfer distributions are [46] where T ∞ is ambient air temperature, T b is the base temperature, h is the convective heat transfer coefficient, P is the fin perimeter, k is thermal conductivity, A is cross-sectional area, and x is the longitudinal coordinate. Depicted in Fig. 11 is the profile for heat transfer distribution q for the full-scale model, trial models 1 and 2 and the virtual model. The scaling factors of the trial models 1 and 2 and the virtual model are listed in Table 7. The material properties of the full-scale and scaled down trial models 1 and 2 are identical, and the dimensions of the trial models 1 and 2 are scaled to the half and one-fourth of the dimensions of the full-scale model, respectively. The virtual model is the combination of the trial models 1 and 2, which are designed based on the zeroth-order finite similitude theory. As evident from Fig. 11, the small-scale trial models return a substantially different response than full-scale model with errors equal to 92.80% and 257.23% for trial models 1 and 2, respectively. The errors calculated based on the areas under the curves. Note however that the virtual model which combines the two trial models 1 and 2 predicts the response of the full-scale model to reasonable accuracy (10.59% error). The value for R 1 used in the virtual model is obtained from the first-order approximation which on setting h ps P ps = h ts1 P ts1 = h ts2 P ts2 it follows that R 1 = (1 − β 1 )(β 1 − β 2 ) −1 and for β 1 = 0.5 and β 2 = 0.25 gives R 1 = 0.5 as recorded in Table 7. Finally, examination of Eq. (34) provides the two curves in Fig. 12 for the left-and right-hand sides of Eq. (34) confirming reasonable accuracy for the approximation.

Conclusion
The focus of this paper is on the fundamentals of a new scaling theory called finite similitude with a particular focus on zeroth-, first-and second-order similitude rules but limiting to two scaled experiments at most. The following conclusions can be drawn from the work presented in the paper: • The theory of finite similitude has been further developed to capture all scale dependencies that arise in the fields describing electromagnetism, continuum mechanics and heat transfer. • Differential forms of similitude have been shown to be unique in the sense that all other similitude rules of identical order are representable by the high-order similitude rules presented in the paper. • Product terms that appear in the transport equations impact on the similitude rules and established is the precise nature of the constraint. • A second-order, two-experiment theory has been established that has practical value being applicable to fluid mechanics and heat transfer.
More specifically, from the case studies examined in electromagnetism, fluid mechanics and heat transfer, it has been shown that: • Magnetic field predictions for a Eddy current heating problem involving a metallic composite saw errors in a half-scale model of 4.75%, quarter-scale model of 7.74%, reduce to 2.90% for a virtual model that combines the results of the two scaled models. • Ball velocity predictions for a water penetration problem saw errors in a fifth-scale model of 3.90%, tenthscale model of 4.39%, reduce to 0% for a virtual model that combines the results of the two scaled models. Small undulations in the prediction from the virtual model were visible arising from the large magnification of differences due to the small models involved. • Estimation of the second-order dyadic v 1 v 1 from first-order field information provided good supporting evidence for the theory developed in Sect. 4. • Heat transfer predictions for a fin problem saw errors in a half-scale model of 92.80%, quarter-scale model of 257.23%, reduce to 10.59% for a virtual model that combines the results of the two scaled models. • Estimation of the second-order product q 1 T −1 1 from first-order field information provided good supporting evidence for the theory developed in Sect. 4.
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