Erdoğan S. Şuhubi

Isovector fields and similarity solutions for general balance equations

Abstract The symmetry groups associated with a system of balance equations of arbitrary order involving n independent and N dependent variables are discussed by employing exterior calculus. The infinitesimal generators of symmetry groups are components of isovectors fields of the closed ideal of certain exterior differential forms corresponding to an equivalent first order system of equations. Rather compact expressions which lead to the set of equations whose solutions determine isovector fields are derived. These expressions may be evaluated automatically by a computer for large systems. Similarity solutions are then obtainable as usual as group invariant solutions. An example is treated briefly to expose short cuts provided by the approach.

Similarity solutions of boundary layer equations for second order fluids

Abstract In a previous paper [ Int. J. Engng Sci . 30 , 523 (1992)] boundary layer equations for second order fluids are derived. The general isovector fields for these equations are found by using exterior calculus. From the four independent isovectors only one provides useful solutions which is a scaling transformation. Using this isovector, the ordinary differential equations leading to the similarity solutions are found. The numerical solution of the equations are presented and the profile corresponding to the solution is discussed. The shear stress of the similarity solution is also calculated.

Boundary layer theory for second order fluids

Abstract Two-dimensional equations of steady motion for second order fluids are expressed in a special coordinate system generated by the potential flow corresponding to an inviscid fluid. For the flow around an arbitrary object φ coordinates are the streamlines, ψ coordinates are the velocity potential lines. It is clear that the equations of motion so derived and boundary conditions become in a sense independent of the body shape immersed into the flow. Using the usual boundary layer assumptions the boundary layer equations are then deduced from the equations of motion by employing a technique of matched asymptotic expansion.

Equivalence groups for second order balance equations

The groups of equivalence transformations for a family of second order balance equations involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with different functions or parameters. Our approach to attack this problem is based on exterior calculus. The analysis is reduced to determine isovector fields of an ideal of the exterior algebra over an appropriate differentiable manifold dictated by the structure of the differential equations. The isovector fields induce point transformations, which are none other than the desired equivalence transformations, via their orbits which leave that particular ideal invariant. The general scheme is applied to a one-dimensional nonlinear wave equation and hyperelasticity. It is shown that symmetry transformations can be deduced directly from equivalence transformations.

A generalized theory of simple thermomechanical materials

Abstract We develop here a constitutive theory for the generalized simple thermomechanical materials by resorting to generalized thermodynamics in which the empirical temperature is replaced by the thermodynamic temperature which is itself a constitutive function. The restrictions imposed by the generalized Clausius-Duhem inequality on the constitutive relations are fully explored and the theory is applied to a special thermoviscoelastic material.

Structure of Weiss domains in ferroelectric crystals

Abstract The structure of Weiss domains in ferroelectric crystals in each of which the polarization vector is constant, is investigated through a new variational principle. The general field equations are obtained and it is shown that in the presence of external electric field the total electric field is also constant in Weiss domains but is usually different from that of the polarization field. Moreover, it is proved that domain walls can only be planar surfaces. Finally the case corresponding to pure polarization fields is also treated and an illustrative problem is considered.

Symmetry groups and similarity solutions for radial motions of compressible heterogeneous hyperelastic spheres and cylinders

Abstract The general formalism which was previously developed for partial differential equations of balance type is employed to determine the infinitesimal generators (or isovector fields in the language of exterior calculus) inducing the symmetry group associated with radial motions of a heterogeneous isotropic hyperelastic material. The most general group structure is found and the results obtained are applied to a heterogeneous Ko material in which the initial density and the modulus of the material vary according to certain power laws. The nonlinear ordinary differential equations satisfied by the similarity solution corresponding to the symmetry group admitted by the material are provided.

A class of similarity solutions for radial motions of compressible hyperelastic spheres and cylinders

A class of similarity solutions is obtained for radial motions of spherical and cylindrical bodies made of a certain type of compressible hyperelastic materials. The equations satisfied by the infinitesimal generators of the symmetry group of the unified governing first order field equations for spheres and cylinders are found. It is shown that these equations admit a special class of solutions which generate a five-parameter group of transformations. The form of the strain energy function Σ corresponding to the resulting symmetry group is evaluated. The similarity variable is determined and ordinary differential equations satisfied by similarity solutions are obtained. Numerical solutions are given for a Ko material which falls into the class of admissible materials.

Conservation laws for one-dimensional isentropic gas flows

Abstract The governing equations of one-dimensional isentropic gas flow are expressed in terms of a pair of exterior differential forms. By employing Cartans theory and the classical Frobenius theorem linear partial differential equations are obtained to determine conservation laws which are conjectured to be the key to detect completely integrable systems. By using a similarity technique explicit expressions are provided for polynomial type of conservation laws in terms of Gegenbauer and Chebyshev polynomials.

A local analysis of the Kolmogorov-Spiegel-Sivashinsky equation

Abstract Kolmogorov-Spiegel-Sivashinsky (KSS) equation arises in different two dimensional viscous flows. Kolmogorov flow and the large-scale turbulent solar convection can be modelled by this equation [ Los Alamos Sci. Spec. Iss . 218 (1987); Nucl. Phys. B (Proc. Suppl.) 2 , 453 (1987)]. In this article, we have studied the group invariant solutions to the KSS equation. Also, a local analysis of the dynamical system obtained by the group theoretical means are performed by employing normal form analysis. This approach allowed us to obtain in analytical form an approximate travelling wave solution possessing periodic, quasi-periodic and aperiodic features. Bifurcations pertaining to the latter have been identified.