R.R. Huilgol

Anomalous stability behavior of a properly invariant constitutive equation which generalises fractional derivative models

Viscoelastic materials like amorphous polymers or organic glasses show a complex relaxation behavior in the softening dispersion region, i.e. from glass transition to the α relaxation zone. It is known that a uni-dimensional Maxwell model, modified within the conceptual framework of fractional calculus, has been found to predict experimental data in this range of temperatures. After developing a fully objective constitutive relation for an incompressible fluid, it is shown here that the fractional derivative Maxwell model results from the linearization of this objective equation about the state of rest, when some assumptions about the memory kernels are made. Next, it is demonstrated that the three dimensional, linearized version of the frame indifferent equation exhibits anomalous stability characteristics, namely that the rest state is neither stable nor unstable under exponential disturbances. Also, the material cannot support purely harmonic excitations either. Consequently, it appears that fractional derivative constitutive equations may be used to study a very limited category of flows in rheology rather than the whole spectrum.

On the determination of the plug flow region in Bingham fluids through the application of variational inequalities

Abstract Numerical methods developed through the theory of variational inequalities are used to determine the plug flow regions in two classes of pipe flow problems in Binv first set, including a test problem in a concentric annulus, deals with flows in eccentric annuli in general, and the second one examines the flow in an L-shaped region.

Variational inequalities in the flows of yield stress fluids including inertia: Theory and applications

Using the constitutive relations for a yield stress fluid in which the viscosity is shear rate dependent while the yield stress is a constant, a systematic procedure to include the local and convected acceleration components into a variational inequality applicable to each and every flow of such a yield stress fluid is developed. As examples of motions for which variational inequalities can be derived quite easily, several flows including some with free surfaces are exhibited. As an application of the variational inequality, a new nondimensional number, called the halting number, which determines whether a fluid in a pipe flow will come to rest in a finite amount of time is found, and is shown to be related to the minimum pressure drop per unit length necessary for a pipe flow to exist. As additional applications of the variational inequality, the rate at which an unsteady pipe flow approaches the steady state, and an upper bound for the steady flow rate in a pipe are investigated, both for Bingham fluids and more general viscoplastic fluids, with the halting number playing a part here as well.

Finite stopping time problems and rheometry of Bingham fluids

Abstract It is shown that steady channel, simple shear and Couette flows of a Bingham fluid come to rest in a finite amount of time, if either the applied pressure falls below a critical value, or the moving boundaries are brought to rest. An explicit formula for a bound on the finite stopping time in each case is derived. This bound depends on the density, the viscosity, the yield stress, a new geometric constant, and the least eigenvalue of the second order linear differential operator for the interval under consideration.

Variational principle and variational inequality for a yield stress fluid in the presence of slip

Abstract Minimum and maximum principles, which have been derived for the creeping flows of a yield stress fluid, are extended to flows when wall slip is present. In addition, the minimum principle is shown to lead to a variational inequality when wall slip is absent, and the latter is generalised to include inertial effects in specific cases, including flows in pipes and the flow past a body at rest due to a uniform flow at infinity. Moreover, the variational inequality is extended to deal with problems where wall slip may be present. Finally, the squeezing flow between two co-axial and parallel disks is re-examined as an application of the variational principles obtained here.

Recent advances in the continuum mechanics of viscoelastic liquids

Abstract The kinematics of viscoelastic fluid flows, the development of constitutive relations and their use in viscometric and nonviscometric flows is given. Experimental data in viscometric flows, extensional flows and the eccentric rotating disk motion along with oscillatory shear flows are presented and compared with theoretical predictions. The flow classification scheme for the selection of the appropriate constitutive equation, the perturbation schemes applicable to fixed and variable domains are described. These are applied to review the literature on particle motions, lubrication problems and rotating flows. Stability of the flows is discussed along with some recent work on existence, uniqueness and the use of dynamical system and hyperbolic theory in connection with propagating singular surfaces.

Non-axisymmetric flow of a viscoelastic fluid between rotating disks

Abstract It has been known for many years that two functions of the axial co-ordinate determine the axi-symmetric velocity field in the rotating disk problem in an incompressible Newtonian fluid. Recently, in the same configuration, the existence of a non-axisymmetric velocity field, determined by four functions of the axial co-ordinate, has been demonstrated. Because two functions of the axial co-ordinate determine the axi-symmetric velocity field in the above configuration for a class of viscoelastic fluids as well, it could be conjectured that four functions solve the non-axisymmetric problem for the same class of viscoelastic fluids. Here, this conjecture is shown to be true.

Propagation of a vortex sheet in viscoelastic liquids—the Rayleigh problem

Abstract It is shown by an example that unsteady shearing motions can give rise to vortex sheets in a special class of viscoelastic liquids only. Further, the congruence between the quantitative predictions of an initial—boundary value problem, namely the Reyleigh problem, and the theory of propagation of singular surfaces is demonstrated.

Variational principles and variational inequalities for the unsteady flows of a yield stress fluid

A minimum principle, which has been derived for the steady, creeping flows of a yield stress fluid with shear-dependent viscosity, is extended to flows when the yield stress is also shear dependent, and the flow may be unsteady. As an application of the minimum principle, the unsteady squeezing flow between two co-axial and parallel disks is examined. Next, the variational principle is extended to a variational inequality, and situations where inertia may be incorporated into the latter are discussed. Using this, the specific forms of the variational inequalities are derived for five flows: unsteady pipe flows, flow past a solid at rest, the reservoir problem, the cavity driven flow, and, finally, for a class of problems with free surfaces. Further, the variational principle and the inequality are extended to deal with those problems where wall slip may be present. In a manner similar to the way the minimum principle has been extended, a maximum principle for the stress in the above class of yield stress fluids is established, and is easily reworded to include the case of wall slip as well. In addition, this principle is converted to a variational inequality for the stress. Finally, it is shown that the mimimum velocity functional and the maximum stress functional are identical when the velocity and stress fields satisfy the equations of motion and the relevant boundary conditions.

Nearly extensional flows

Abstract After defining a nearly extensional flow, the stresses which arise in an incompressible simple fluid undergoing such a motion are determined. Since the linear functionals which arise in this perturbation are similar to those in the infinitesimal viscoelasticity theory of a transversely isotropic solid with rotational and reflectional symmetries, the non-zero linear functionals and interralations between them are determined quite easily. It is then shown that self-consistency demands that certain relations exist between the extensional modulus and these linear functionals. As an application, the speed of propagation of an acceleration wave in a fluid undergoing an extensional flow is considered. Finally, the nearly extensional flow theory is cast in terms of small displacements superposed on the extensional flow. In this form, it may be useful in the study of melt spinning.