Showing papers by "Kumbakonam R. Rajagopal published in 1980"

Thermodynamics and stability of fluids of third grade

Abstract: Today, even though the Clausius-Duhem inequality is widely considered to be of central importance in the subject of continuum thermomechanics, it is also believed to be a somewhat special interpretation of a more fundamental (second) law of thermodynamics. In this work, which is concerned with the relation between thermodynamics and stability for a class of non-Newtonian incompressible fluids of the differential type, we find it essential to introduce the additional thermodynamical restriction that the Helmholtz free energy density be at a minimum value when the fluid is locally at rest. As a background to our main considerations we begin by introducing the general class of Rivlin-Ericksen fluids of complexity n and obtain, for this class, a preliminary set of thermodynamical constitutive restrictions. We then give detailed attention to the special case of fluids of grade 3 and arrive at fundamental inequalities which restrict its (temperature dependent) material moduli. When the moduli are taken to be constant we find that these inequalities require that a body of such a fluid be stable in the sense that its total kinetic energy must tend to zero in time, no matter what its previous mechanical and thermal fields, provided it is both mechanically isolated and immersed in a thermally passive environment at constant temperature from some finite time onward. When the material constants of a fluid of grade 3 are such that the Clausius-Duhem inequality is satisfied but the free energy is not at a minimum in equilibrium, we show that for a broad class of reasonably posed problems the flows are necessarily asymptotically unbounded. Finally, we determine the stability character of non-trivial base flows for fluids of grade 3 with constant material moduli, and establish a uniqueness theorem for the initial-boundary value problem and a uniqueness theorem for problems involving sufficiently slow steady flows.

On the decay of vortices in a second grade fluid

Abstract: G. I. Taylor [ 1 ] showed that the flow representing a double array of vortices which has the same periodicity in both the x and y directions is a solution to the equations of motion in two dimensions of a linearly viscous fluid. It was shown in [2] that such a result is also true for 'second order' fluids if time scale which characterizes the memory of the fluid and the size of the vortices satisfy certain apriori restrictions. In this note we show that the results established by Taylor [1] for the linearly viscous fluid are uncondi t ional ly true, irrespective of the time scale which characterizes the fluid or the size of the vortices in the case of incompressible second grade fluids provided they are thermodynamically compatible. Also, in this analysis we investigate the relationship between the rate of decay of the vortices, and the periodicity of the vortices. It is found that if the periodicity is increased in the x or y directions, the vortices decay faster. It is also found, as is to be expected, that the vortices decay faster as the coefficient of viscosity # increases, while the decay is slower if the normal stress moduli oq is larger. The Cauchy stress T in an incompressible second grade fluid is assumed to be related to the fluid motion in the following manner [3 ]

On constitutive equations for branching of response with selectivity

Abstract: Materials, such as elastic-plastic, which exhibit distinct regimes of response are usually modeled by different constitutive equations in each regime. The present paper explores a method for the construction of a unified constitutive equation from these separate relations. The main idea is to write this unified equation in an implicit form which contains these separate solutions as non-unique solutions. The form is chosen in order to utilize the notions of branch points and branches. Different solutions, corresponding to constitutive equations for different regimes of response, are then regarded as bifurcations at branch points from the fundamental response. The choice of the appropriate branch at a branch point is governed by a selectivity condition which depends on the nature of the response under consideration. A detailed example is provided for elastic-plastic response, with and without the effect of strain rate dependence.

A boundary integral approach for determining the pressure error

Abstract: The rectilinear flow of a second-order fluid is considered between two infinitely wide and long parallel plates. The bottom plate is at rest and has a slot of depth ‘d’ and width ‘W’ while the top plate is flat and moves along the flow direction with constant speed. A boundary integral equation technique is developed to determine the pressure error due to such a flow. We find that our numerical results compare favorably with the exact solutions obtained byKearsley for a special case. Also our results lie within the analytical bounds established byRajagopal andHuilgol for the pressure error for such flows.