1 Introduction

In the current societies, knowledge is an important aspect to ensure the competitiveness of enterprises. This is crucial to enable the satisfaction of social needs up to a level the regions can increase prosperity and wellbeing [1]. The proper relation among Universities, Enterprises and Government has been identified as key to promote the context of knowledge innovation [2]. The formulation of complex problems, required by society, leads to think on a global position in which the three actors’ interplay shall be efficiently executed to ensure social commitment towards the problem’s resolution [3]. One of the key questions to be asked in this article is related with the adequacy of the University contents to the demanded knowledge by companies and society within the theoretical framework introduced by the ATD. Recently, [4] aims to answer key questions on the evolution and structure of university contents related with mathematics in European and Canadian undergraduate programs making use of the ATD. Further in [5], Bosch and Winslow aims to understand how the didactic transpositions works from the wise knowledge to the academic contents for university mathematics education. Such transposition is far from being fully systematically conceived, leading to introduce concepts that may not be fully required by the noosphere. Indeed, this will be further highlighted along the presented article in relation with the Fluid Mechanical subject.

Fluid mechanics is a basic subject of notable relevance in technical or scientific degrees. During the training of an engineer, it is essential to successfully face future professional issues such as those related to the management and storage of the energy contained in a continuous fluid medium. Fluid mechanics is a subject taught in the middle courses of technical or scientific degrees, which can be related to an increasingly established scientific maturity of students. Fluid mechanics begins, in the form of a requirement to be known, in the physical and mathematical sciences touch in the degree first courses. Based on this, a meeting point is configured that allows the extension of scientific reasoning. Key aspects are the understanding of the object to be studied as a continuous element existing in reality, the understanding of mathematics and physics, which support as pillars the exercise of conceptualization and modeling, and the formation of a critical spirit in the interpretation of the result whose validity requires an exercise of calibration with the reality under study. From the scientific maturity provided by a fluid mechanics course, students acquire competences to study branches applied to their future field of specialization within the degree: hydraulics, thermal engines, plasmas, aerodynamics, aeroelasticity, etc.

The teaching and learning of fluid mechanics in tertiary studies has been linked to curricular criteria in relation to the institutional and degree levels to be taught. From the theory of didactic transposition, we start from the premise that any knowledge constituted as a discipline (such as fluid mechanics) is subject to an institutional superstructure. Let us think of two specific institutions, the University, as a reference center of knowledge and know-how, and the Company, as a competent area of applied knowledge and a place of transformation of knowledge into technology demanded by society. In both institutions, fluid mechanics develops and transforms over time. Certain notions about fluid mechanics, certain techniques or theories may remain unknown to one institution while they may be widely developed in another. This aspect may lead to an increase in the distance between the two institutions. We will refer to this concept as institutional distance or norm. The institutional distance between the University and the Company should be set at moderate values to ensure optimal functioning of the noosphere associated with fluid mechanics. This would ensure an optimal understanding between institutions as generators of knowledge and potential applications in the discipline in question. However, an analysis of the current situation is necessary to address and understand the state of the institutional distance, with the intention of identifying potential areas for improvement in the contents of fluid mechanics taught in technical schools within the Spanish context constituted of universities and enterprises.

The relevance of the mathematical models of fluid mechanics is explained from their fields of application to the real phenomena they try to model. Freudenthal [6] introduced the concept of inversion and conversion to justify how a mathematical knowledge arising from real and tangible experience ends up becoming an increasingly condensed and abstract form, becoming a universal mathematical content. Fluid mechanics can be understood as a science arising from observation and experimentation (we remember the studies of Da Vinci, Galileo or Torricelli on experimental hydraulics) that at a given moment in time, begins an escalation towards the economy of the concept to form cognitive superstructures close to mathematics. In fact, one of the great mathematical challenges of our time, the solution of the Navier–Stokes equations, arises from the application of patterns of knowledge and reasoning typical of fluid mechanics. In this line of thought, the rigorous character of mathematics and the application bases of fluid mechanics are conceived as a unique symbiosis that allows the segregation between the theoretical and the applied fields. The conception of fluid mechanics as an applied and experimental mathematics is appropriate, since it makes possible to give meaning to subjects such as differential calculus, algebra, differential equations, or numerical calculus. Authors such as Harris, Black, Hernandez-Martínez, Pepin and Williams [7] emphasize how the integration of mathematics in the applied sciences prevents mathematics from being perceived as isolated from the scientific or technical field. The fact that fluid mechanics and mathematics share the principle of inversion and conversion as well as that both constitute a symbiotic front in engineering and science degrees, suggests the possibility of applying Chevallard’s [8] Anthropological Theory of the Didactic which has been extensively treated for the field of mathematics didactics (also in its applied forms). Therefore, we have selected those authors and references that have emphasized the development of the ATD for applied mathematics and we have looked for common points with the possibilities of encounter offered by fluid mechanics.

As it has been exposed, the fact that mathematics is constituted as a substantial part of fluid mechanics, leads us to consider aspects such as the perception of students towards mathematics or the ability of teachers to value it in a context of application to an object of study of interest in the fields of technical and scientific training. Baquero, Bosch and Gascón [9] argue that, in the university institution, mathematics is constructed from a perspective far removed from the model or application potential. In this way, they argue that mathematics constitutes an autonomous formation erected as a self-sufficient science in spite of being institutionalized in the field of technical schools or science faculties. In the present analysis, we try to build a framework of understanding about the relationship between fluid mechanics, which is taught in tertiary education, and the social environment, the institution and the teachers. From the theoretical scope provided by ATD, we employ tools and key elements of this theory with emphasis on the personal relationship of the teacher with fluid mechanics, the relationship of the institution with society and the teacher, and the identification of lines of didactic content demanded by society on the knowledge to be taught in fluid mechanics. Note that instead of running a complete massive data collection through questionnaires, our approach conception consists of selecting remarkable experts in the field of fluid mechanics. Consequently, the approximation shall be understood as a case study to explore the relations between several academic and professional profiles, so that the institutional distance can be initially formulated. Indeed, we will observe how fluid mechanics is perceived from several points of view: professional engineers without contacts with the University, an engineer working as professional in a company of the sector and as a part-time professor at the University, professors and researchers with experience in companies and professors and researchers with unique experience in the institution where they work. For each of the selected profiles, members of the group of experts in possession of the knowledge, a common questionnaire was carried out with the aim of knowing their thought as a source to study the relationships and restrictions existing at the levels of society, university, pedagogy, and discipline within the co-determination between fluid mechanics and didactics.

The paper outline is based on providing firstly a theoretical background in relation with the Anthropological Theory of Didactics and its relationship with the fluid mechanics. After having explored such relation, the interplay between Institutions is analyzed making use of advances in the ATD techniques. Afterwards, the methodologies are introduced together with the experts’ questionnaires and responses. Note that a discussion is introduced to make a comparative analysis among the different answers to show the institutional distance between Universities and Enterprises in relation with the Fluid Mechanics.

2 Theoretical framework

Our objective requires a multidimensional approach that highlights all the agents that make up the implementation of the teaching exercise in fluid mechanics. Consequently, we apply the ideas of Chevallard’s Anthropological Theory of Didactics [10]. The ATD has been widely used for the analysis in the didactics of mathematics, from different prisms, one of them focused on the elaboration of typical models of science [9]. Thus, we accept fluid mechanics as a knowledge with a high mathematical content oriented to the modeling of continuous mass systems. Gascón [11] established, as a constructivist genesis of mathematical knowledge, an effort focused on the modeling of a reality on which such knowledge operates. In this way, we provide the didactics of mathematics and fluid mechanics with a common thread where the general ideas of ATD have a place, allowing a controlled extension from the field of didactics of mathematics to the field of didactics of fluid mechanics. Along the same lines, it is necessary to emphasize that throughout this paper the problem of mathematical modeling is presented as an essential element of fluid mechanics. Modeling is the transforming attitude that allows the rigorous expression of human thought in order to broaden the vision of the observed reality. The approach to the reality of fluid mechanics, as a science, emanates from the current paradigm of the physical sciences, so that its implementation must be systematically supported by mathematical laws. Fluid mechanics enables mathematics with a functional vision, beyond the formal aspects, allowing a vision close to the questioning of the world, away from the monumentalism study of concepts.

The vision of the different paths of study and research [12] that emerge from fluid mechanics constitute a corpus of action to develop the perspective of analytical thinking, operation and change that properly configure the engineer and his work [13].

The ATD formulates that an Institution I is a social organization constituted by members over which I establishes a concrete way of doing and thinking [14]. Each member of an Institution I carries out its activity in terms of one or more praxeologies. These praxeologies refer to the type of tasks to be performed by a member of a given institution using specific techniques. In addition, the technological discourse associated with each specific technique makes it possible to justify why a particular technique is used and not another. The type of tasks, as well as their techniques, based on a technological discourse specific to Institution I, has a direct impact on the behaviour of each member of Institution I. It is relevant to highlight that, according to ATD, a change in the technological paradigm of a given institution entails a change in the discourse justifying its techniques, and therefore, a potential modification of these, impacting on the praxeology of each member of the institution.

Fluid mechanics is a science in its own right, with its own characteristic lines of research, specialized journals and areas of training and doctoral research. Along the lines proposed by Chevallard [15], we can distinguish three fundamental activities that make it possible to explain, at a high level, the interrelationships between the learned knowledge of fluid mechanics and the knowledge to be taught in institutions, namely: Study process, thematic organization and didactic organization. Particularly and when referring to the mathematical activity, Chevallard [16] distinguishes two types of praxeologies: the Mathematical Organization that defines the thematics and the Didactic Praxeologies or Didactic Organization. Basically, Chevallard intention was to answer the following questions: what mathematical reality and thematics can be constructed in a didactic situation? While the latter, how is that reality studied? That is, a Mathematical Organization corresponds to the mathematical "reality" that is intended to be studied and, a Didactic Organization, to "the way" of studying that reality. The latter generates an isomorphism between a Mathematical Organization and a Didactic Organization, called a didactic-mathematical isomorphism. There is no Didactics without a Mathematical Organization to build or study, and Mathematical Organization cannot be built or studied without a Didactic one developed by the members of the community. Thus, the border between the didactic and the mathematical is blurred. The isomorphism postulated by Chevallard occurs at different levels at the same time, from the very top Civilization to the downstream Discipline. All such levels defined by Chevallard are provided in Fig. 1.

Fig. 1
figure 1

Source: Own elaboration

Levels of isomorphism to carry out the association between thematics and didactics.

Full size image

Focusing attention on the social hierarchy, it becomes relevant to understand how the stakeholders that are part of society, as demanders of knowledge, influence the thematic and didactic organization. Moreover, this demand permeates lower hierarchies, giving them meaning and structure.

The research presented aims at analyzing the opinions of a group of experts representative of the noosphere on fluid mechanics. Each expert is defined as a person e occupying a position p within an institution Ii (where the index i indicates each of the different institutions to which the surveyed expert belongs or has belonged). In our case, we will consider two key institutions: institution I1 which represents the aerospace engineering training school and institution I2 which represents an aerospace company of recognized prestige where future graduates can work and which is based in different European countries. Particularly, I1 is the School of Aerospace Engineers ascribed to the Polytechnics University of Madrid and the Institution I2 is the well-known company Airbus. Note that both institutions are geographically close one to another, thus it is natural that many early-graduates at I1 start their professional career at I2. Within the institution I1 there are different positions related to teaching and research, typical of any university institution. Within the I2 institution, a sample of fluid dynamic engineers have been selected to design, integrate and certify aeronautical systems based on the use of a fluid: specifically, fuel, hydraulic and air systems.

The type of tasks to be carried out, as well as their techniques and their justification or technological discourse are different for each of the positions available in each of the institutions. Moreover, it may be the case that the same expert holds a position in institution I2 and holds another position in institution I1 as an associate professor or has left his position in I2 to become a full-time professor in I1. The interrelationships between the different positions of the same expert are interesting as they define hybrid praxeologies of their own between two different institutions. We postulate that the techniques and justificatory discourses among hybrid experts with experience at both institutions (I1 and I2) will be different with respect to an expert who only knows I1 or I2. These possible interrelationships lead us to the concept of personal relationship. To represent this concept, let us first define an object as a delimited entity in the environment that constitutes a target of knowledge and/or attention for the expert. Contextually, in I1 we can consider as an object the preparation of the contents of a class or the teaching methodology. Similarly, in I2 we can define an object as the design and sizing of a fluid network or the preparation of a technical document. For whatever the object is in institution I1 (which we will call O1) and in institution I2 (O2), the personal relationship (Pr) of each expert with the object in question is defined.

The construction of the personal relations with the object depends on the institution in which the expert is located. For the same object, say calculation of head loss in a facility, the institution I1 imposes a strategy related to teaching and/or research, being, consequently, the personal relationship of the expert centered on both aspects. However, an expert in institution I2 will face the pressure drop problem using an approach that allows him to reduce the execution time and costs that act as constraints on his work. The construction of the personal relationship is particularly relevant in the case of the expert who has a synchronous or asynchronous relationship with institutions I1 and I2. During the research process we will note how this personal relationship is constructed and how it influences the praxeological conceptions of each institution. Trying to generalize, each expert e develops a personal experience in relation to the object O that can be summarized in an expression of the form:

$$ {\text{Re}} \left( {p1, \, p2, \, O} \right), $$
(1)

where e = 1, 2… indicates the numbered expert, p1 is the position of expert e in I1 and p2 indicates the position of expert e in I2. In the same way, the following personal relationships can be defined:

$$ {\text{Re}} \left( {p1, \, O} \right), $$
(2)
$$ {\text{Re}} \left( {p2, \, O} \right), $$
(3)

Throughout our research, it is observed how once the same object (i.e. present and future of the contents of fluid mechanics at university level) is identified, the approach towards it differs among the experts in relation to the positions (p1, p2) they occupy or have occupied in the institutions I1, I2.

3 Methodology and background

The objective of our research focuses on the present and emerging contents of a fluid mechanics course at university level. In addition, we delve into the knowledge and current state of the representative noosphere of the learning-teaching binomial for fluid mechanics. To this end, different profiles of experts with different personal relationships are analyzed. The methodology process is as follows: Firstly, the signing authors conveyed a dedicated campaign making use of social networks and online tools to find proper members of I1 and I2 to form part of the study. To this end, researchers’ profiles at Google Scholar and ResearchGate were considered together with professional profiles making use of LinkedIn network. To select and appropriate level of sampling, it shall be mentioned that the intention is to perform a questionnaire to high experts on the field, therefore the main premise was focused on the high quality of researchers and professionals rather than the quantity of both. Thus, the searching criteria was focused on:

  • Number of scientific publications in relevant journal under the Journal Citation Report scale in Web of Science.

  • Competitive Projects promoted by the involved researchers.

  • Years of experience as professional in a Company

  • Positions of responsibility hold

Some of the selected candidates to the next phase were competitive in the four criteria mentioned. When this happened, the researcher was classified as a hybrid with remarkable competencies in both the Institution I1 and I2.

Note that comparative studies between different reduced sourcing data have been already explored in the frame of the ATD in [17] and in relation of the ATD with the curriculum in [18]. Following these approaches, a total of ten representative experts were interviewed. Five of the experts are currently working as professors and researchers in fluid mechanics at a public university technical college. All five experts have more than 400 h of teaching experience in fluid mechanics and engineering. In addition, they have published more than twenty articles in specialized journals, with an average publication rate of 36 articles in journals of impact. It is worth mentioning the notoriety of one of the experts, who has been awarded the Prince of Asturias Prize for Scientific and Technical Research. It is, therefore, a group of extraordinary expertise in fluid mechanics at university level. On the other hand, the remaining five of the ten experts interviewed are engineers who carry out their professional work in the company where they have developed fluid-related systems in military and civil aircraft. One of these experts related to the company has research and teaching activities with the university as an associate professor in fluid mechanics. Our objective is, then, to establish a dialogue between University and Enterprises in the form of the commented expert profiles to measure the institutional distance for the Spanish context.

In addition, the questionnaire was designed in accordance with the following logical sequence as discussed in Stone [19].

  • Selection of Data and Items to be included: Data to be included was selected based on the objectives of the questionnaire, i.e. to deep into the knowledge of the noosphere associated to Fluid Mechanics in a local area of Spain. The Data to be included was, firstly, based on the searching criteria for experts described above. Then the data and items contained in the questionnaire was based in the expert academic and professional experience and their visions of what is fluid mechanics and its future development.

  • Definition and design of main questions for inclusion: The questions were formulated to permit the expert open opinions, views, potentials and suggestions.

  • Potential answers layout and coding: The definition of the questionnaire was intended to allow open answers; therefore, no specific coding has been accounted for. Typically, the coding is used in closed answers to correctly replicate the expert opinion in a gradual way. Nonetheless, in this case, the intention is to check emerging ideas on the fluid mechanics noosphere.

  • Survey launching process: The survey was sent to the experts via e-mail.

  • Questionnaire presentation and layout: The wording and layout is based on simple and open questions. The experts were contacted after the sending of questionnaire to check on potential doubts on the wording of questions. Based on the experts’ answers, some emerging patterns were identified as described later.

The relation between ATD and applied mathematics to physics has been explored based on the pedagogy known as Research and Questioning the World [20]. In addition, the relation between physics, ATD and teacher education has been tackled in [21]. In these cited references, the ATD methodologies were applied and successfully implemented and discussed. Alternatively, and along this work, the intention is to make the ATD participant in a particular area of physics, as it is the Fluid Mechanics, and to assess the co-determination levels involved, including experts and teachers, together with the ATD dedicated methodologies. Then, the introduction of the experts in the field configures an important aspect not previously done in the field of ATD and fluid mechanics.

Once the expert have been selected as per the searching mechanisms described, they were contacted via e-mail and were kindly asked to submit a summarized curriculum vitae, paying attention to the following aspects:

  • Bachelor’s Degree or Engineering

  • Master's degree (if any)

  • Doctorate (if any)

  • Main Occupation

  • Collaborations with public or private companies

  • Collaborations with the University

with the aim of pointing out those aspects related to the institutions I1, I2 and the respective positions p1, p2 that allow us to highlight the differences in the personal relationships of each of the experts with fluid mechanics and in the frame of the ATD.

The information extracted from the curriculum vitae of each of the experts is summarized below:

3.1 Expert 1 (E1)

Expert E1 has an in-depth training in fluid mechanics. His lines of research are focused on the fields of combustion, fluid mechanics and numerical methods. He has also collaborated with companies in which he has carried out studies of different nature related to fluid mechanics. He is the manager of a company dedicated to thermal engineering modeling. His current position is as a Full Time Professor at the University. In brief:

  • Aeronautical Engineer

  • PhD in Aeronautical Engineering

  • University Professor

  • He has carried out studies with public and private sector companies in thermal modeling. In addition, he supports a company dedicated to fluid modeling.

Therefore, the personal relationship of the expert E1 is built from his historical profile and from the positions p1 (Full Time University Professor) and p2 (Manager of a company).

3.2 Experts 2 (E2), 3 (E3), 4 (E4), 5 (E5)

Experts E2, E3, E4 and E5 can be considered as a scientific reference in fluid mechanics. Their lines of research focus on the fields of combustion, jet dynamics and semiconductor materials. These profiles are almost exclusively oriented to the academic field with some sporadic collaboration with the company in the form of subsidized projects. All three experts are university professors. In brief:

  • Aeronautical engineers

  • Doctors Aeronautical engineers

  • University professors

  • Their studies are almost exclusively related to the academic field, with the exception of some minor collaboration with companies.

The personal relationship of experts can be constructed from the academic perspective and from position p1 (University Professor).

3.3 Expert 6 (E6)

Expert E6 is an aeronautical engineer, he also has a PhD in science and technology. He is currently working in the private sector as a fluid modeling and simulation engineer. He has worked in the design of aeronautical systems in a leading company in the aerospace sector. In addition, he practices as an associate professor of fluid mechanics and mathematics. In brief:

  • Aeronautical engineer

  • Doctorate of Science in Technology

  • Fluid modeling and simulation engineer

  • He carries out his professional work in the company while working as an associate professor at the University.

The personal relationship of expert E6 is particularly interesting. We can observe that he presents a vision of fluid mechanics from the professional and applied perspective in the company and from the University, where he develops his work collaborating with the fluid mechanics department. His personal relationship is built, therefore, from the academic focus with position p1 (University associate professor) and from position p2 (Fluid modeling engineer).

3.4 Experts 7 (E7), 8 (E8), 9 (E9), 10 (E10)

This group of experts have aeronautical engineering as their basic training. Since the completion of their studies, they have not been in contact with the academic field, developing their work exclusively in the company as fluid systems design engineers in aircraft. Their trajectory can be summarized as follows:

  • Aeronautical engineers

  • Fluid systems design engineers

  • They work exclusively in-house

The personal relationship of experts towards fluid mechanics is built from position p2 (Fluid systems design engineers).

Each of the aforementioned experts was surveyed through an interview with the aim of bringing out the personal relationships of each of them towards the contents of fluid mechanics by emphasizing the existence of divergent elements among the experts that may explain their different conceptions towards what should be considered as knowledge to be taught in a fluid mechanics course. The questions followed the following sequence:

Question 1 (Q1): Which topics do you think should be studied in a Fluid Mechanics course?

Question 2 (Q2): Why those specific topics?

Question 3 (Q3): If you had the opportunity to add any content to a Fluid Mechanics course, what would it be and why?

Question 4 (Q4): Which teaching methodology do you consider as the most appropriate for the development of a Fluid Mechanics course?

Question 5 (Q5): What difficulties have you encountered with students coming to a fluid mechanics course? In your opinion, what are they due to?

In the present research we will focus on an analysis of the contents of a fluid mechanics course, so that our objective will be the answers to questions Q1, Q2 and Q3, leaving the analysis of questions Q4 and Q5 for further research focused on the forms and methodologies of teaching and learning which might required further approaches beyond the ATD principles.

4 Analysis of the data obtained from the survey

Each of the questions and the answers of each of the experts surveyed are presented below with a comparative analysis. As the questionnaire was formulated with open questions subjected to open answers, the main intention is to compare them so that an institutional distance can be postulated. Comparison is executed based on the topics introduced for each question and based on the authors subjective analysis. Although subjectiveness cannot be fully discarded, it is possible to build a global knowledge based on an inductive approach to conclude on the factual objective, i.e., postulation of an institutional distance between university and companies in the way the fluid mechanics is exposed and treated.

Question 1 (Q1): Which topics do you think should be studied in a Fluid Mechanics course?

Answer E1: Those taught in the School (of Engineering), in principle. But I would put more emphasis on the estimation of orders of magnitude and dimensional analysis, since they are given too much on top and they are two very powerful tools.

The contents taught at the School (meaning engineering education) are detailed in the answer of the expert E2 with whom E1 shares a subject.

Answer E2: Macroscopic variables. Local thermodynamic equilibrium and hydrodynamic equations. Conservation equations. Equations of state. Transport phenomena. Fluid statics. Unidirectional and quasi-unidirectional motions. Dimensional analysis. Movements at low Reynolds numbers. Hydrodynamic lubrication. High Reynolds number motions. Euler equations and discontinuity layers. Boundary layer. Laminar boundary layer equations and characteristics. Detachment. Thermal boundary layer. Transition and turbulent boundary layer. Turbulence. Energy scales and cascade. Free turbulent flows with shear. Turbulent flows limited by walls.

The answers of experts E3, E4 and E5 are exactly the same as the answers of expert E2. Note that all these experts have configured the fluid mechanics subject during a joint work. We observe here the construction of a content that lasts over time and subject to the principle of teaching economy.

Answer E6: Equations of conservation of mass, energy and momentum. Numerical analysis of the most important equations. Fluid statics. Motion in pipes, pumps, turbines and compressors. Dimensional analysis. Fluid regimes. Transient motion. Turbulence.

Answer E7: Bernoulli's equation. Pressure drops in pipes. Pumps, turbines and compressors. Transient flow. Equation of motion. Energy equation. Dimensional analysis. Regimes of a fluid.

Answer E8: Types of currents. Reynolds number. Energy equation and conservation laws. Turbomachine's and design of pressure systems. Dimensional analysis. Liquid Hammering.

The answers from experts E9 and E10 are already contained in the answers provided by E7 and E8. First, we can compare the answers given by the experts with an academic background (experts E1, E2, E3, E4 and E5). As it can be seen, these experts share their opinion on what should be taught in a fluid mechanics course. Expert E1, in addition, stresses the importance of studying in some detail the elements related to dimensional analysis and orders of magnitude for their computational power. Dimensional analysis is considered by all the experts surveyed as an appropriate subject. Regarding the answers given by the experts with experience in companies of the sector (E6, E7, E8, E9 and E10) we observe that in all cases, they maintain as relevant the equations of conservation and motion, introduce the study of transient motion and give relevance to applications related to pressure drops, pumps, turbines and compressors. It is noteworthy that expert E6 also introduces a topic related to the numerical analysis of equations and introduces a topic on fluid statics. In turn, we can compare the answers of experts with mainly academic background with those offered by experts working in companies as prime occupation. This exercise allows us to relate and compare the vision of two institutions I1 (represented by E1, E2, E3, E4 and E5) and I2 (represented by E6, E7, E8, E9 and E10). The praxeology imposed on its experts by institution I2 suggests the need to have a solid base in those applied aspects related to calculations of fluid systems (pipes, pumps, compressors and turbines) based on simple numerical or analytical analysis (by means of dimensional analysis). In addition, the requirement of I2, not contemplated in I1, towards the study of transient phenomena in fluids is highlighted, since situations such as liquid hammering can cause irreversible damage to a hydraulic installation. At the same time, expert E8 suggests the need to introduce aspects related to the design of pressure systems and turbomachinery. We also note that expert E6, who has a relationship as an associate professor with the institution I1, establishes common thematic areas with the answers given by the members of I1 in terms of fluid statics, turbulence or dimensional analysis.

Question 2 (Q2): Why these specific topics?

The answers to question Q2 can be summarized as follows:

Answer E1: Because of the above.

The answers of experts E2, E3, E4, E5 were agreed among the four experts and was sent as a single common answer.

Answer E2, E3, E4 and E5: The first two topics constitute the general formulation of fluid mechanics. The third topic and part of the fourth serve to become familiar with the handling of the equations in simple cases. The fourth topic also includes problems of practical interest, particularly if complemented with the hydraulic approximation for turbulent flows, although formally turbulence is studied in the last topic. The fifth topic is the study of the simplification of fluid mechanics problems by exploiting their invariance to changes in unit systems. The sixth topic is a study of some characteristics of flows with negligible inertial effects. The seventh and eighth topics are the study of flow characteristics at high Reynolds numbers, both for liquids and gases. They are essential for aeronautical applications and are the point of connection with aerodynamics. The eighth topic is also the point of connection with a more detailed study of heat transfer problems. The last topic is the study of turbulent flows. Some parts may preempt other topics.

Answer E6: In my opinion the contents are appropriate, although I consider that more relevance should be given to fluid statics to know how to calculate pressure distributions in a fluid, too dimensional analysis for its simplicity and power. On the other hand, I consider that every fluid or hydraulic engineer should know how to properly design pressure drops and diameters in pipes. I consider that this part is not treated too much in fluid mechanics curricula.

Answer E7: I believe that a fluid mechanics course should emphasise those concepts that will be useful for their future as engineers. Bernoulli's equation is basic and simple. It allows you to understand and calculate pressure drops and to analyze all types of turbomachinery. I think it is very important that this equation is treated well and with plenty of time to understand the concepts. Because of its simplicity, Bernoulli's equation is widely used in the company. There is no technical meeting that does not end up talking about Bernoulli's equation. In addition, I consider it very important to emphasize transient flow in order to properly calculate liquid hammering. I believe that this part is sometimes undervalued in fluid mechanics courses, but the truth is that knowing how to calculate transients in a pipe line is not easy. The equations of motion and energy should be studied for their possible use for numerical integration, but, above all, as a basis for dimensional analysis, which is also very important, since it allows preliminary analysis of solutions. In the early design phases, these preliminary analyses are of great value. Finally, I believe that time should be invested in explaining well the regimes of a fluid (laminar, turbulent and mixed) since for an engineer it can mean a saving or an important cost in the installation depending on the fluid regime.

Answer E8 and E9: Every engineer should know the most important applications of fluid mechanics.

Answer E10: The applications to real problems shall be a paramount content.

The answers provided by experts E8, E9 and E10 where very short and provide only some general remarks on introducing applied contents. On this occasion, the noosphere of the educational system in fluid mechanics suggests that there are two ways of dealing with the subject matter institutionalized through I1 and I2. In both cases, common areas are observed (conservation equations and dimensional analysis mainly), however, the I2 institution demands to deal with topics related to transient phenomena and greater applications to fluid systems. I2 experts justify their choice of contents by the fact that every professional engineer should have a deep knowledge of his profession and, therefore, attention should be paid to those basic contents whose reinforcement will have an impact on a more qualified professional. However, the experts of the institution I1 have established a reasonable and guided approach towards the broad problem of fluid mechanics, trying to find a common thread that generates and gives meaning to the contents.

Question 3 (Q3): If you had the opportunity to add any content to a fluid mechanics course, what would it be and why?

Let us look at each of the answers again:

Answer E1: I would add an extensive topic devoted to estimating orders of magnitude. It is something that students have a hard time understanding and using, and yet it is something very simple and logical that has very high analytical power.

Answer E2: Numerical calculation. Commercial codes and specific methods for potential flows, vorticity dynamics and boundary layer. The reason is the obvious practical importance of numerical computation in fluid mechanics. Microfluidics. This is a broad topic of current interest, although it may be somewhat removed from aeronautical applications.

The answers of experts E3, E4 and E5 were previously agreed between them so that they were sent jointly.

Responses E3, E4 and E5: I think it would be essential to dedicate an extensive topic to modeling. I think that students should know how to make a model and, immediately after, know the mathematical tools that fluid mechanics provides for its resolution. Students who come to the company do not fully understand that they are the ones who generate the model to be solved based on an understanding of reality. Normally, they think that in their professional work the models are given to them because they are used to being given them already made in engineering schools, when the truth is that they themselves are the ones who, in the future, must extract the model. For this reason, I think it is important to model more.

Answer E6: I would give them an introduction to fluid mechanics models as a basis for the mathematical activity they will develop in the course.

Answer E7: I think I would introduce turbulence phenomena in pipes and a chapter on design and modeling.

Answer E8: A topic on real experiences in the design and modeling of fluid systems would be useful, for example, they could learn how an oil-hydraulic network has been designed.

Answer E9: Much more applied modelling.

Answer E10: I would say the importance of costing in the design procedure associated to hydraulics

In the answers to question Q3 we observe that there is consensus on the need to introduce more content on modeling and design of fluid components. Even the expert E10 introduces an interesting emerging variable related with the design to cost in fluid lines. Of particular interest is the response of experts E3 and E4. Both experts argue that the future graduates undertake their professional activity without having any indication of how to model, in addition, the common value highlighted by the experts E6, E7, E8 and E9 members of the institution I2. We observe, once again, the existence of an institutional distance around phenomena related to the construction of fluid mechanics knowledge. On this occasion, the noosphere claims a construction of knowledge based on the model and its conception.

5 Main discussion

During the development of the present work, the existence of an institutional distance between the University or institution I1 and the Company or institution I2 regarding the contents that a fluid mechanics course should contain has become evident. Each of the experts surveyed have highlighted the different needs of the noosphere related with fluid mechanics. There are points of agreement between the experts of each institution regarding the contents of conservation equations and dimensional analysis due to their computational power and ability to approach highly complex problems. In addition, the survey reveals important differences between the institutions regarding the contents that should be taught in a fluid mechanics course. Specifically, I2 experts suggest the importance of teaching contents related to transient phenomena (or, informally, Liquid Hammering), turbomachinery and system design. Expert E5, who has a relationship with both institutions, suggests meeting points around fluid statics, turbulence or dimensional analysis. In conclusion, the pressure of the noosphere of fluid mechanics around the institution I1 as an agglutination of wise knowledge, suggests an update of contents in the subjects of fluid mechanics in engineering schools, especially, with the introduction of a subject on transient movements of fluids, pressure drops and relations between turbomachinery and the design of hydraulic systems. In addition, the need to introduce new concepts and contents about modeling in fluid mechanics and design phenomena that establish the guidelines for the future professional is highlighted. In the line of professional conception, a notable difference in the way of elaborating the contents has been observed among the experts. Thus, the experts of the I2 institution have given greater importance to contents related with the future engineering professional, while the experts of I1 elaborate the contents by means of a logical and reasonable thread based on a wise or scientific knowledge.

During the elaboration of these comparison principles, a remarkable limitation emerged, particularly when analyzing the possibility of a deeper content. To the understanding of the signing authors, further detailed questions on each of the contents compared may be necessary. For example, we have identified an institutional distance around turbomachinery and hydraulic system design, nonetheless these contents may be already briefly discussed at the Institution I1 under other lessons (conservation law for instance). It shall be stated that, even this may happen, the extent of the remarked limitation is reduced, otherwise, the institution I1 would have dedicated lessons with dedicated material and contents.

6 Conclusions and future lines

Along the presented case of study, It has been proposed an expert questionnaire to relevant professionals coming from the academia and companies. The case study has been performed within the Spanish institutional system. Particularly, our main hypothesis consisted of measuring the institutional distance between Academia (as knowledge generator) and Enterprises (as knowledge practitioner) in the field of Fluid Mechanics and making use of the systematic methodology introduced by the Anthropological Theory of Didactics. Once the answers from the different experts were received, it was revealed that, indeed, there exists an institutional distance on the conception of Fluid Mechanics knowledge. An efficient noosphere associated to Fluid Mechanics should have such institutional distance to small levels. In other words, the institution I1 would produce and teach the exact knowledge demanded by the institution I2. Along the presented analysis, it has been shown that this is not the case within the Spanish academia and companies. The reduction of such institutional distance would require the institution I1 to introduce contents related with turbomachinery, transient phenomena, and system design principles to fully comply with institution I2 expectation in those professionals demanded.

Among the future lines of work, it is particularly interesting to further develop each of the contents mentioned, i.e., to understand what are the particular granulated knowledges produce at I1 and demanded at I2 to optimize the expectations of both Institutions. In addition, an analysis to questions Q4 and Q5 can be further developed. Note that the answers to these questions are related with the teaching and learning techniques only applicable to the academic field (not related directly to enterprises and the noosphere associated to Fluid Mechanics). In addition, the analysis of questions Q5 and Q6 would lead to introduce didactics approaches related with techniques not influenced by the ATD principles as it has been exposed along this article.