Showing papers by "Kumbakonam R. Rajagopal published in 2012"

On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

Abstract: We provide a thermodynamic basis for the development of models that are usually referred to as “phase-field models” for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive “phase-field models” both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631–651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier–Stokes–Fourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier–Stokes–Fourier fluid. As observed earlier in Heida and Malek (Int J Eng Sci 48(11):1313–1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn–Hilliard–Navier–Stokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn–Hilliard–Navier–Stokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).

Flow of fluids with pressure- and shear-dependent viscosity down an inclined plane

Abstract: In this paper we consider a fluid whose viscosity depends on both the mean normal stress and the shear rate flowing down an inclined plane. Such flows have relevance to geophysical flows. In order to make the problem amenable to analysis, we consider a generalization of the lubrication approximation for the flows of such fluids based on the development of the generalization of the Reynolds equation for such flows. This allows us to obtain analytical solutions to the problem of propagation of waves in a fluid flowing down an inclined plane. We find that the dependence of the viscosity on the pressure can increase the breaking time by an order of magnitude or more than that for the classical Newtonian fluid. In the viscous regime, we find both upslope and downslope travelling wave solutions, and these solutions are quantitatively and qualitatively different from the classical Newtonian solutions.

A numerical study of a plate with a hole for a new class of elastic bodies

Abstract: It has been shown recently that the class of elastic bodies is much larger than the classical Cauchy and Green elastic bodies, if by an elastic body one means a body incapable of dissipation (converting working into heat). In this paper, we study the boundary value problem of a hole in a finite nonlinear elastic plate that belongs to a subset of this class of the generalization of elastic bodies, subject to a uniaxial state of traction at the boundary (see Fig. 1). We consider several different specific models, including one that exhibits limiting strain. As the plate is finite, we have to solve the problem numerically, and we use the finite element method to solve the problem. In marked contrast to the results for the classical linearized elastic body, we find that the strains grow far slower than the stress.

On Kelvin-Voigt model and its generalizations

Abstract: We consider a generalization of the Kelvin-Voigt model where the elastic part of the Cauchy stress depends non-linearly on the linearized strain and the dissipative part of the Cauchy stress is a nonlinear function of the symmetric part of the velocity gradient. The assumption that the Cauchy stress depends non-linearly on the linearized strain can be justified if one starts with the assumption that the kinematical quantity, the left Cauchy-Green stretch tensor, is a nonlinear function of the Cauchy stress, and linearizes under the assumption that the displacement gradient is small. Long-time and large data existence, uniqueness and regularity properties of weak solution to such a generalized Kelvin-Voigt model are established for the non-homogeneous mixed boundary value problem. The main novelty with regard to the mathematical analysis consists in including nonlinear (non-quadratic) dissipation in the problem.

On the development and generalizations of Allen–Cahn and Stefan equations within a thermodynamic framework

Abstract: Starting from a simplified framework of the theory of interacting continua in which the mass balance equations are considered for each constituent but the balance of linear momentum and the balance of energy are considered for the mixture as a whole, we provide a thermodynamic basis for models that include the Allen–Cahn and Stefan equations as particular cases. We neglect the mass flux due to diffusion associated with the components of the mixture but permit the possibility of mass conversion of the phases. As a consequence of the analysis, we are able to show that the reaction (source) term in the mass balance equation leads to the Laplace operator that appears in the Allen–Cahn model and that this term is not related to a diffusive process. This study is complementary to that by Heida et al. (Zeitschrift fur Angewandte Mathematik und Physik (ZAMP) 63, 145–169, 2012), where we neglected mass conversion of the species but considered mass diffusion effects and derived the constitutive equations for diffusive mass flux (the framework suitable for capturing other interface phenomena such as capillarity and for generalizing the Cahn–Hilliard and Lowengrub–Truskinovsky models).

On implicit constitutive relations for materials with fading memory

Abstract: Starting with an implicit constitutive relation between the stress and relative deformation gradient histories to describe the response of a fluid, and using the assumption of fading memory, we show that both rate type and differential type fluid representations can be obtained, as approximations in retarded motions, that have the same form as the Maxwell, Oldroyd-B, Rivlin–Ericksen, and other popular fluid models. This result provides further evidence that the recently proposed implicit constitutive framework provides a very robust and general methodology to describe fluid response. Thus, fluids defined through an implicit constitutive relation between the stress and relative deformation gradient histories can be seen as an appropriate generalization of the notion of simple fluids and rate type fluids.

Diffusion of a fluid through an anisotropically chemically reacting thermoelastic body within the context of mixture theory

Abstract: A mixture theory approach is used to analyze the change in response characteristics of an anisotropic, non-linear viscoelastic fluid diffusing through a finitely deforming thermoelastic body of arbitrary symmetry wherein the fluid chemically reacts with the solid. It is assumed for simplicity that it is possible to characterize the reaction as having a single step and as one in which the reaction products remain within the body, thereby modifying the solid constituent. The effects of the reaction on the solid that render it anisotropic are quantified by means of a tensorial parameter that tracks the extent of the reaction in different directions. Both diffusion-dominated (diffusion of the reactants is far more rapid than the reaction) and reaction-dominated (the reaction is far more rapid than the diffusion of the reactants) processes are considered. Constitutive equations are derived from the requirement that the rate of entropy production be maximized, which makes it possible to characterize the behavio...

Role of pressure dependent viscosity in measurements with falling cylinder viscometer

Abstract: The falling cylinder viscometer is frequently used in measuring the dependence of the viscosity on the pressure. The viscosity is calculated using an indirect procedure, namely by appealing to the linear relation between the time taken for the fall and the viscosity. Under certain assumptions, the coefficient of proportionality can be derived analytically, and one gets the classical formula for the viscosity as a function of geometric parameters of the device, density of the fluid and the sinker, gravitational acceleration, and the distance and the time of the fall. Although the classical formula is valid only for fluids with constant viscosity, it is indiscriminately used even for fluids with pressure dependent viscosity. We investigate the role of variable viscosity, and we derive a heuristic correction to the classical formula for the case of fluids with pressure dependent viscosities. The systematic error introduced by the unwarranted application of the classical formula for fluids with pressure dependent viscosity is analysed, and it is shown it is measurable and it can in some cases significantly influence the experimental results.

On the inhomogeneous shearing of a new class of elastic bodies

Abstract: Recently, it has been shown that the class of bodies that can be called ‘elastic’ is far larger than classical Cauchy and Green elastic bodies. In order to assess the usefulness and viability of th...

Algorithms for synthesizing mechanical systems with maximal natural frequencies

Abstract: We consider a simpler version of an open problem in system realization theory, which has relevance to several important problems in biomedicine, altering the dynamic response of discrete and continuous systems, connectivity of Very Large Scale Integrated circuits, as well as the co-ordination of Unmanned vehicles. The fundamental question this article tries to answer is the following one: Given all the components of a system, how do we put these components together in order to obtain a desired response? In the simplest form, this basic question arises in mechanical systems where, the objective is to connect the masses with springs in a suitable way, and in the most general form, it arises in biomedicine where one is interested in engineering and achieving a desired output by either allowing certain new interactions or disallowing some interactions to take place between the proteins, nucleic acids and other cellular components. We formulate a simpler version of this problem in one dimension (i.e., all the masses and springs are arranged along a line), where the objective is to choose a set of springs to connect the masses so that the resulting “graph” structure is as stiff as possible. The system considered corresponds to an ungrounded structure and will always admit a rigid body mode; for that reason, the smallest natural frequency is zero and we use the smallest non-zero natural frequency as a metric for stiffness of the structure and we maximize this objective. Maximizing the smallest non-zero frequency increases all the natural frequencies thereby making the system stiffer. We develop an iterative primal-dual algorithm and a cutting plane algorithm to solve the problem and provide preliminary computational results on a network up to nine masses.

A model for the thermo-oxidative degradation of polyimides

Abstract: Polyimides, due to their superior mechanical behavior at high temperatures, are used in a variety of applications that include aerospace, automobile and electronic packaging industries, as matrices for composites, as adhesives etc. In this paper, we extend our previous model in S. Karra and K. Rajagopal (Mech. Mater. 43(1):54–61, 2011), to include thermo-oxidative degradation of these high temperature polyimides. Appropriate forms for the Helmholtz potential and the rate of dissipation are chosen to describe the degradation. The results for a specific boundary value problem, using our model, compares well with the experimental creep data for PMR-15 resin that is aged in air.

A note on novel generalizations of the Maxwell fluid model

Abstract: In this note we generalize the classical viscoelastic fluid model due to Maxwell to allow the relaxation time and the viscosity to depend on the stress. Such models are very useful in describing the response of geological and many other polymeric fluids. The procedure that is adopted to carry out the generalization allows one to have the springs and dashpots and their three dimensional generalizations, to be described by implicit constitutive theories thereby greatly enlarging the class of constitutive models that can be put into place to describe the response of complex viscoelastic fluids.

Revisiting total, matric, and osmotic suction in partially saturated geomaterials

Abstract: An accurate quantification of negative pore pressure (commonly referred to as ‘suction’) in the pore network is necessary for modeling the mechanical response of unsaturated geomaterials. Traditional definitions and formulations of total, matric, and osmotic suction suggest incorrect pore fluid pressures under certain conditions. In this paper, the notion of suction is revisited by deriving an expression for pore fluid pressure in a simple osmotic, capillary tube using the framework of mixture theory in conjunction with the fundamental laws of thermodynamics. Based on the derived expression for the tube, expressions are derived for total, matric, and osmotic suction for partially saturated geomaterials. Particular attention is given to osmotic suction since confusion regarding its mechanisms has apparently contributed to its misapplication in geomechanics. The new expressions derived herein adequately explain behavior that is incorrectly explained by the traditional formulations and unifies two approaches to modeling osmotic suction previously considered to be in contradiction.

Nonlinear viscoelastic response of asphalt binders in transient tests

Abstract: This paper presents the results of a set of transient experiments conducted on asphalt binders that serve to highlight the nonlinearity of the viscoelastic response of asphalt binders. Two grades of binders were used in this study. A transient test that combines both creep and stress relaxation tests was conducted using a dynamic shear rheometer at several loading conditions and at different temperatures. The acquired experimental data was satisfactorily fit using a nonlinear six-parameter viscoelastic model.

Extension, inflation and circumferential shearing of an annular cylinder for a class of compressible elastic bodies:

Abstract: This paper studies the classical problem of extension, inflation, and circumferential shearing of an annular cylinder for a new class of compressible elastic bodies wherein the left Cauchy–Green deformation tensor is given as a function of the Cauchy stress tensor. We use a semi-inverse method to study the problem by assuming forms for both the deformation field and the stress field. Focusing our attention on to three specific constitutive relations and two geometries corresponding to thick and thin annular cylinders, we study the qualitative features of the governing differential equations. The models are chosen so that they exhibit qualitatively different response features, one of them displaying a limiting stretch. The classical assumption that the hoop and axial stresses are nearly constant through the thickness of the thin annular cylinder subjected to inflation holds for this class of elastic bodies too. However, for thick-walled annular cylinders subjected to inflation at constant length and for a ...

Mechanical oscillators described by a system of differential-algebraic equations

Abstract: The classical framework for studying the equations governing the motion of lumped parameter systems presumes one can provide expressions for the forces in terms of kinematical quantities for the individual constituents. This is not possible for a very large class of problems where one can only provide implicit relations between the forces and the kinematical quantities. In certain special cases, one can provide non-invertible expressions for a kinematical quantity in terms of the force, which then reduces the problem to a system of differential-algebraic equations.

Flow of an electrorheological fluid between eccentric rotating cylinders

Abstract: Electrorheological fluids have numerous potential applications in vibration dampers, brakes, valves, clutches, exercise equipment, etc. The flows in such applications are complex three-dimensional flows. Most models that have been developed to describe the flows of electrorheological fluids are one-dimensional models. Here, we discuss the behavior of two fully three-dimensional models for electrorheological fluids. The models are such that they reduce, in the case of simple shear flows with the intensity of the electric field perpendicular to the streamlines, to the same constitutive relation, but they would not be identical in more complicated three-dimensional settings. In order to show the difference between the two models, we study the flow of these fluids between eccentrically placed rotating cylinders kept at different potentials, in the setting that corresponds to technologically relevant problem of flow of electrorheological fluid in journal bearing. Even though the two models have quite a different constitutive structure, due to the assumed forms for the velocity and pressure fields, the models lead to the same velocity field but to different pressure fields. This finding illustrates the need for considering the flows of fluids described by three-dimensional constitutive models in complex geometries, and not restricting ourselves to flows of fluids described by one-dimensional models or simple shear flows of fluids characterized by three-dimensional models.

Synthesizing robust communication networks for UAVs

Abstract: This article addresses the problem of synthesizing communication networks with maximum algebraic connectivity in the presence of constraints which limit the total number of communication links present in the network. This problem arises in Unmanned Aerial Vehicle (UAV) monitoring applications where some UAVs have to be deployed to relay and transmit time sensitive information between all the vehicles and the control stations. This network synthesis problem is a difficult optimization problem because of its non-linear objective coupled with the possibility that the number of feasible solutions increases rapidly with the size of the graph. The network synthesis problem is formulated as a mixed-integer, semi-definite program, and an algorithm to find the optimal solution is developed based on cutting plane and bisection methods. Some computational results are also presented to corroborate the performance of the proposed algorithm. Two other heuristics are presented along with numerical results corroborating their performance. Since these heuristics are based on evaluating the spectrum of the graph, they can be applied to large networks.

Further mathematical results concerning Burgers fluids and their generalizations

Abstract: In this paper, we extend the earlier work by Quintanilla and Rajagopal (Math Methods Appl Sci 29: 2133–2147, 2006) and establish qualitative new results for a proper generalization of Burgers’ original work that stems form a general thermodynamic framework. Such fluids have been used to describe the behavior of several geological materials such as asphalt and the earth’s mantle as well as polymeric fluids. We study questions concerning stability, uniqueness and continuous dependence on initial data for the solutions of the flows of these fluids. We show that if certain conditions are not satisfied by the material moduli, the solutions could be unstable. The spatial behavior of the solutions is also analyzed.

Development of A CFD 3D Model To Determine The Effect of The Mixing Quality on The CNG-Diesel Engine Performance

Abstract: In a CNG-DIESEL DUAL FEED ENGINE, the CNG-AIR MIXER plays a vital role. The CNG-AIR MIXER is an important part of the fuel system of CNGDIESEL DUAL FEED ENGINE. The basic operation of CNG-AIR MIXER mainly depends on restriction barrel known as venturi. When air flows through venturi, its speed increases and pressure decreases. CFD approach is applied to investigate the flow behavior of methane and air in a CNG-AIR MIXER to be used for CNG-DIESEL DUAL FEED ENGINE.CFD analysis is done on 8 hole cng-air mixer and results of simulations showed that 8-hole cng-air mixer gives superior performance. Also mixing quality at outlet in terms of spread parameter of CNG-MASS fraction will be calculated. INTRODUCTION Now a day’s many efforts are being done to reduce the emissions from conventional diesel engine. One of the techniques that can be used to reduce emission without too many modifications to conventional diesel engine is “CNGInternational Journal of Engineering Research & Technology (IJERT) Vol. 1 Issue 5, July 2012 ISSN: 2278-0181

Flow and Heat Transfer over an Exponential Stretching Sheet under the Effects of a Temperature Gradient Dependent Heat Sink and Thermal Radiation.

Abstract: The problem of MHD flow and heat transfer in a Newtonian viscous incompressible fluid over a stretching sheet with temperature gradient dependent heat sink/source and radiation is investigated. The governing partial differential equations are converted into ordinary differential equations by similarity transformation technique . The effects of viscous dissipation, work due to deformation, thermal radiation are considered in the energy equation and the variations of dimensionless surface temperature as well as the heat transfer characteristics with various values of non-dimensional parameters like Prandtl number, suction parameter , radiation parameter, temperature gradient dependent heat sink parameter are graphed and tabulated. The heating process of the type i ) the sheet with prescribed surface temperature (PST case) is studied.

Algorithms for Finding Diameter-constrained Graphs with Maximum Algebraic Connectivity

Abstract: This article addresses a problem of synthesizing robust networks in the presence of constraints which limit the maximum number of links that connect any two nodes in the network. This problem arises in surveillance and monitoring applications where wireless sensor networks have to be deployed to collect and exchange time sensitive information among the vehicles. This network synthesis problem is formulated as a mixed-integer, semi-definite program, and an algorithm for finding the optimal solution is developed based on cutting plane and bisection methods. Computational results are presented to corroborate the performance of the proposed algorithm.

Degradation and healing in a generalized neo-Hookean solid due to infusion of a fluid

Abstract: The mechanical response and load bearing capacity of high performance polymer composites changes due to degradation or healing associated with diffusion of a fluid, temperature, oxidation or the extent of the deformation. Hence, there is a need to study the response of bodies under such degradation/healing mechanisms. In this paper, we study the effect of degradation and healing due to the diffusion of a fluid on the response of a solid which prior to the diffusion can be described by the generalized neo-Hookean model. We show that a generalized neo-Hookean solid—which behaves like an elastic body (i.e., it does not produce entropy) within a purely mechanical context—creeps and stress relaxes due to degradation/healing when infused with a fluid and behaves like a body whose material properties are time dependent. We specifically investigate the torsion of a degrading/healing generalized neo-Hookean circular cylindrical annulus infused with a fluid. The equations of equilibrium for a generalized neo-Hookean solid are solved together with the convection–diffusion equation for the fluid concentration. Different boundary conditions for the fluid concentration are also considered. We also solve the problem for the case when the diffusivity of the fluid depends on the deformation of the generalized neo-Hookean solid.

Thermal stresses investigation of a gas turbine blade

Abstract: The analysis of structural and thermal stress values that are produced while the turbine is operating are the key factors of study while designing the next generation gas turbines. The present study examines structural, thermal, modal analysis of the first stage rotor blade of a two stage gas turbine. The design features of the turbine segment of the gas turbine have been taken from the preliminary design of a power turbine for maximization of an existing turbojet engine with optimized dump gap of the combustion chamber, since the allowable temperature on the turbine blade dependents on the hot gas temperatures from the combustion chamber. In the present paper simplified 3-D Finite Element models are developed with governing boundary conditions and solved using the commercial FEA software ANSYS. As the temperature has a significant effect on the overall stress on the rotor blades, a detail study on mechanical and thermal stresses are estimated and evaluated with the experimental values.

Controlling the deformation in elastic and viscoelastic structures due to temperature and moisture changes using piezoelectric actuators

Abstract: We study the placement of surface-bonded piezoelectric actuators to control or minimize the deformation due to changes in temperature and moisture content in linear elastic and linear viscoelastic ...