M. Cánovas*, I. Alhama and F. Alhama Mathematical Characterization of Bénard-Type Geothermal Scenarios Using Discriminated Non-dimensionalization of the Governing Equations Abstract: The classical non-dimensionalization process of governing equations is a low-cost method commonly applied for a first approximation to the dimensionless numbers that determine the solution patterns in many problems; however, this procedure fails in complex pro- blems, where it is not even possible to define reference quantities – because they are not established in the state- ment of the problem – to make the dependent or inde- pendent variables dimensionless. The application of discrimination corrects this obstacle and allows suitable dimensionless groups to be defined. These, in turn, have two interesting properties: (i) they are of order of magni- tude unity, and (ii) they have a clear meaning in terms of balance of physical quantities that counteract each other in a domain or sub-domain of the problem. In this paper, discriminated non-dimensionalization is applied to geothermal scenarios of Bénard-type convective flow, large horizontal boundary sides under a temperature gra- dient in porous media, to determine the dimensionless groups that control the steady temperature and stream patterns and, from these, the order of magnitude of the main unknowns of the problem. The results were checked numerically for many cases. Keywords: Bénard flow, heat convection, groundwater flow and heat transport, mathematical characterization, dimensionless groups MSC: 76S05 PACS. 35B36 DOI 10.1515/ijnsns-2014-0068 Received June 11, 2014; accepted November 14, 2014 1 Introduction To find the minimum number of dimensionless groups that characterize the resulting pattern –or solution – of problem is the first aim when a physical or engineering scenario is to be studied, since from these unknown parameters the more universal graphical patterns of the solution can be repre- sented. Although such investigation has often been carried out experimentally, its cost, when these parameters are obtained following a mathematical procedure, must be considered low in light of the advantages offered. Different techniques are currently used in the search for dimensionless numbers, as they are classically known, among them (i) the application of the theory of dimensional analysis (see Palacios [1] and Szirtes [2]), a technique that has been gradually abandoned due to the poor results provided for practical problems, and (ii) the reduction of governing equations to their non-dimen- sional form, a process still followed in many books and papers within the field of porous media (as in Holzbecher and Yusa [3], Holzbecher [4], Nield and Bejan [5] and Vafai [6]). The first is carried out by the looks for the dimensional equations of the variables and physical characteristics of the problem in a given dimensional basis; then, by applying the pi theorem to this table of quantities, dimensionless groups are deduced. A critical and formal application of dimensional analysis to a nat- ural convection problem can be read in Alhama and Madrid [7]; the main advantage of this method is that it is not necessary to know the governing equations of the process, although, on the other hand, the researcher needs to have a deep understanding of the phenomenon involved. The second technique, which we follow in this work, may be considered more formal since it starts from the mathematical model and provides the dimensionless groups following clear and well-known rules. Obviously, both dimensional analysis and non-dimensionalization lead to the same results when they are correctly applied. Let us now comment on an interesting aspect that often goes unnoticed as regards the order of magnitude of *Corresponding author: M. Cánovas, Mining, Geologic and Cartographic Engineering Department, Technical University of Cartagena, Murcia, Spain, E-mail: manuel.canovas@upct.es I. Alhama, Civil Engineering Department, Technical University of Cartagena, Murcia, Spain F. Alhama, Applied Physics Department, Technical University of Cartagena, Murcia, Spain http://orcid.org/0000-0001-6215-3115 IJNSNS 2015; 16(1): 23–34 Brought to you by | Western University Authenticated Download Date | 6/10/15 2:13 PM