Foundations of Engineering Mathematics Applied for Fluid Flows

Abstract

Based on a brief historical excursion, a list of principles is formulated which substantiates the choice of axioms and methods for studying nature. The axiomatics of fluid flows are based on conservation laws in the frames of engineering mathematics and technical physics. In the theory of fluid flows within the continuous medium model, a key role for the total energy is distinguished. To describe a fluid flow, a system of fundamental equations is chosen, supplemented by the equations of the state for the Gibbs potential and the medium density. The system is supplemented by the physically based initial and boundary conditions and analyzed, taking into account the compatibility condition. The complete solutions constructed describe both the structure and dynamics of non-stationary flows. The classification of structural components, including waves, ligaments, and vortices, is given on the basis of the complete solutions of the linearized system. The results of compatible theoretical and experimental studies are compared for the cases of potential and actual homogeneous and stratified fluid flow past an arbitrarily oriented plate. The importance of studying the transfer and transformation processes of energy components is illustrated by the description of the fine structures of flows formed by a free-falling drop coalescing with a target fluid at rest.

Foundations of Engineering Mathematics for Fluid Flows

“…Connected equations (1) forming a parametrically and length scale-invariant system of coupled algebraic-differential equations is supplemented with physically based initial and boundary conditions. Consistency of the infinitesimal symmetries of the system (1) with the basic principles of physics [2] testifies to the validity of its choice as a universal basis for studying fluid flows. The complete solution of the system (1) is searched with the compatibility condition [2].…”

“…The total energy concept gives room to consider all the mechanisms for its transfer in a flow, including transfer with a local flow velocity and a group wave one, slow diffusion processes with a characteristic rate, and rather fast ones in the course of localized direct atomic-molecular interactions (for example, at the size of a molecular cluster upon free surface elimination in merging fluids [2]. )…”

“…In modern fluid mechanics, the main parameter of an equilibrium continuous medium at rest is the free enthalpy or the Gibbs potential G [1,2]. Derivatives of the thermodynamic potentials define density  …”

“…The axiomatically introduced system of equations for fluid motion, taking into account the general principles for the choice of physical quantities and fundamental conservation laws was presented firstly in [3]. Under the assumption that the gradients of potentials, physical quantities, and the intensity of different external sources of matter Q  , momentum i Q , tempetature T Q and salt S Q are small, the basic system describes the transfer of matter, the concentration of individual components, temperature, and momentum [2,3]…”

Modern Problems of Mathematical Physics and Their Applications

“…In [19], Chashechkin has tried to formulate a list of principles that substantiates the choice of axioms and methods for studying nature. He showed that the axiomatics of fluid flows are based on conservation laws in the frames of engineering mathematics and technical physics.…”