Gyarmati's Principle Research Articles

Analysis of Heat and Mass Transfer on MHD Flow of a Nanofluid Past a Stretching Sheet

In this work, a variational technique is applied to MHD, radiative nanofluid flow over a non-isothermal stretching sheet with Brownian motion and thermophoresis effects by Gyarmati's principle. The heat and mass transfer effects have been investigated and analyzed by this technique. The flow fields inside the boundary layer are approximated as polynomial functions. Euler-Lagrange equations for the functional of variational principle are constructed. The non-linear boundary layer equations are simplified as simple polynomial equations in terms of momentum, thermal and concentration boundary layer thicknesses. The temperature, concentration profiles, local heat and mass transfer rates are analyzed and are compared with existing numerical results. The comparison shows remarkable accuracy.

On the relation between the Gyarmati, Prigogine and Hamilton principles

The paper shows that both the Hamilton and the Prigogine principle belong to a respective special form of Gyarmati's principle. The Prigogine and Hamilton principles are the limit cases of the Gyarmati principle, so that one can justify that the Gyarmatic principle should be valid not only for the irreversible process, but also for the nondissipative or mixed process. From the simplifled Gyarmati principle the classical expressions of the Hamilton principle can be generalized into a complete set of inequalities. The principle of minimum mechanical energy, the principle of minimum potential energy and the principle of minimum kinetic energy all belong to alternative forms of the Hamilton principle. The dissipative process should be a decelerated process, whereas the non-dissipative process should be an accelerated one. A possibility is suggested to solve the problem of the longitudinal profile of irreversible process by using an alternative expression of the Gyarmati principle.

On the extension of the Governing Principle of Dissipative Processes to nonlinear constitutive equations

On the basis of a special form of Gyarmati's principle (equivalent to that in the linear range) a new extension of the Governing Principle of Dissipative Processes and its local form is presented. It presumes neither the linearity of the constitutive equations, nor the existence of dissipation potentials or any other nonlinear generalisation of the Onsager Relations. The existence of a domain where this extremum problem possesses a unique solution is proven. The global (integral) form of the principle is also given. Some properties, requirements and interpretations of the principle are discussed.

Dissipative Model of the Universe

AbstractIn the present study a trial is carried out to generalize the non‐relativistic‐dynamic model of cosmology put forward by O. Heckmann. The generalization is performed under the framework of the Gyarmati principle of irreversible thermodynamics for anisotropic inhomogeneous and viscous case. The equation of motion of the dissipative Universe will be a Navier‐Stokes tensor equation leading to a Riccatian differential equation general in one dimension. The equations of the Heckmann model compatible with standard cosmology can be obtained from this by means of further simplifying assumption.

A nonlinear extension of the local form of Gyarmati's governing principle of dissipative processes

A proof of the validity of Gyarmati's Principle to non-Onsagerian constitutive equations is given. A generalisation of the Principle, independent of reciprocity relations, is suggested for nonlinear constitutive equations.

Gyarmati Principle and Open‐Channel Velocity Distribution

The relationship between the mixing‐length theory and the nonlinear theory is analyzed. The equation of motion for open‐channel flows can be obtained by the Gyarmati integral, which includes the linear and nonlinear terms of dissipation potential. The velocity distribution for laminar and turbulent flows in an open channel can be deduced by this integral. The relationship between the theory of minimum energy dissipation rate and the Gyarmati principle is analyzed. It shows that the constitutive structure of the mixing‐length model is valid in principle from the thermodynamic point of view. For the simple case of quasi‐uniform river flow, the Gyarmati integral of maximization should be reduced to the theory of minimum energy dissipation rate.

Thermodynamical derivation of equations of motion for multicomponent fluids

In this work the thermodynamics of multicomponent fluids is treated, based on the mechanics of superposed continua. Balances of diffusion’s momentum and energy are derived, and possible definitions and balances of internal energy are discussed. An entropy function is given, where the diffusion currents are the dynamic variables, thus the balances of the latter come from mechanical considerations. Constitutive equations of the components’ pressure tensor and friction, and of heat conduction are derived, applying the local form of the Gyarmati Principle. The equivalence of Newton’s Third Law and Onsager’s Reciprocity Relations is shown for diffusion. The equations of motion lead to a system of telegraphist’s equations, if viscosities are neglected. The results are compared with those of the “Wave Approach of Thermodynamics”, and with the Theory of Dynamic Variables.

Application of Gyarmati's variational principle to laminar stagnation flow problem

Gyarmati's principle for the evolution of irreversible processes has been applied to construct the variational solutions of laminar stagnation flow of incompressible Newtonian fluid in two dimensions. Variational solutions obtained in three different representations approximate closely the exact numerical result of longitudinal velocity component and the Gyarmati's principle in force representation yield the same result as obtained by Doty and Blick with the help of local potential method of Glansdorff and Prigogine.

Formulation of Gyarmati's principle for heat conduction equation

Gyarmati's principle is formulated in various pictures for the heat conduction phenomenon in solid. Since the heat current density and the internal energy function can be given in three different pictures for heat conduction phenomena, we get the nine forms of the principle from which the heat conduction equation can be derived. This formulation has been shown using the generalized Γ picture. In the subsequent section the principle is formulated in proper Γ picture from which three proper pictures namely Fourier, entropy and energy follow.

Application of Gyarmati principle to boundary layer flow

The paper deals with the development of a new variational method based on Gyarmati's principle for the solution of boundary layer flow along a flat plate. It is found that the variational solution differs by about 4% from exact result when a trial function of third degree is chosen for velocity profile. The results obtained with the help of the force representation of the principle are the same as obtained by the local potential method of Glansdorff and Prigogine. A polynomial of sixth degree with the help of universal form of the principle gives a result which differs by about 0.65% from the exact solution. The governing principle of dissipative processes, it seems, will prove itself quite satisfactory analytical method for the boundary layer flows.

On the phenomenological theory of heat conduction

In this paper the foundations of the phenomenological theory of heat conduction is generalized, then the principle of least dissipation of energy and the Gyarmati variational principle are treated. The second-order differential of Lagrangian of the Gyarmati principle, the general Γ picture and nonlinear cases are investigated. In the last part nonlinear constitutive theories are involved: the method of power series and the method of mollification. Generalizing the mollification method we demonstrate that results obtained by mollifications of different kinds are not equivalent, that is, this method is not unique.

Approximation Methods for the Solution of Heat Conduction Problems Using Gyarmati's Principle

AbstractAfter a brief description of GYARMATI's Governing Principle of Dissipative Processes, general approximation methods are developed for heat conduction phenomena on two scales. On the first scale the temperature and heat current density fields are approximated by two different sets of functions in such a way that the internal energy balance and the imposed boundary conditions be satisfied. On the second scale we introduce a new temperature field related to the heat current density through the constitutive equation. Since the dissipation potentials in GYARMATI's principle are connected with each other by LEGENDRE dual transformations we call “dual field methods” the approximation procedure based on the two temperature fields. It is shown that the equations of the GALERKIN method, the method of orthogonal projections, and the TREFFTZ method, when applied to the appropriate problems, are included in the approximation schemes. Due to their great importance a special paragraph is devoted to the treatment of eigenvalue problems. Finally, the approximation methods are illustrated by a simple example and the results are compared with the exact solution.

The treatment of non-isothermal multicomponent diffusion in electrostatic fields on the basis of gyarmati's variational principle

Fundamental equations of multi-component non-isothermal, diffusional systems are treated for an electric field. From the universal form of Gyarmati's principle the transport equations of these systems are derived. It is demonstrated that the principle remains valid also in the case of non-constant coefficients within the framework of the Gyarmati supplementary theorem. The general description of transport processes in the presence of an external electric field can be given in the form of a variational problem.

On the Non‐linear Generalization of the Gyarmati Principle and Theorem

AbstractThe validity of the Gyarmati theorem has been extended to non‐linear types of constitutive equations governing dissipative processes which have recently been suggested. It has been proved that by extending the validity of the theorem, the validity of Gyarmati's variational principle of thermodynamics is guaranted for non‐linear theories adequate to constitutive equations in question as well. The theory was applied for stationary temperature distribution in a rigid bar.

Deduction of the Quasi‐linear Transport Equations of Hydro‐thermodynamics from the Gyarmati Principle

AbstractFrom the universal form of Gyarmati's variational principle of thermodynamics the differential equations governing the internal energy and impulse transport of one component hydro‐thermodynamic systems are derived. In our particular case Gyarmati's “supplementary theorem” is confirmed, by which the validity of the universal form of Gyarmati's variational principle is guaranted also in non‐linear cases. Finally some problems of the Gyarmati principle and of non‐linear thermodynamics are discussed.

The Reformulation of the Gyarmati Principle in a Generalized “I” Picture

The Reformulation of the Gyarmati Principle in a Generalized “I” Picture

The Reformulation of the Gyarmati Principle in a Generalized “I” Picture

The Reformulation of the Gyarmati Principle in a Generalized “I” Picture

On the GYARMATI Principle and the Canonical Equations of Thermodynamics

AbstractOn the basis of the variational principle proposed by GYARMATI we determine the canonical equations of non‐equilibrium thermodynamics. It was demonstrated that the transport equations governing irreversible processes are similarly determined by GYARMATI's principle, further on the canonical equations equivalent to them, as in mechanics the wellknown Lagrangian equations of motion and their Hamiltonian forms are determined with the HAMILTON principle.