On a methodology to determine Navier's slip-parameter in Navier-Stokes fluid flows at a solid boundary

While the assumption of the ``no-slip" condition at a solid boundary is unquestioningly applied to study the flow characteristics of the Navier-Stokes fluid, there was considerable debate amongst the early pioneers of fluid mechanics, Du Buat, Girard, Navier, Coulomb, Poisson, Prony, Stokes and others, as to the proper condition that pertains at a solid boundary due to a fluid, such as water flowing adjacent to the same. Contemporary usage of the ``no-slip" boundary condition notwithstanding, M\'alek and Rajagopal outlined a methodology to test the validity of the assumption. In this study, we continue the investigation further by providing a scheme for determining the slip-parameter that determines the extent of slip, if one presumes that Navier-slip obtains at the boundary. We find that depending on whether the volumetric flow rate is greater or less than the volumetric flow rate corresponding to the no-slip case, different scenarios present themselves regarding what transpires at the boundary

A note on the drag for fluids of grade three

An expression is obtained for the traction vector t on a solid surface which is adjacent to an incompressible fluid of grade three which is compatible with thermodynamics. It is found that unlike fluids of grade two wherein there is no additional drag due to the non-Newtonian nature of the fluid for bodies with certain geometric symmetries (e.g. sphere), fluids of grade three provide an additional drag which is of the same sign as that provided by the viscous terms, provided certain symmetry conditions are met by the velocity field.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/23724/1/0000696.pd

Viscometric flows of third grade fluids

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/23349/1/0000292.pd

On the decay of vortices in a second grade fluid

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43225/1/11012_2005_Article_BF02128929.pd

On a class of exact solutions to the equations of motion of a second grade fluid

A class of exact solutions to the equations of motion of a second grade fluid is exhibited wherein the non-linearities which occur in the equations of motion are self-cancelling though individually nonvanishing. These flows are those in which the vorticity and the Laplacian of the vorticity remain constant along stream lines.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24577/1/0000860.pd

Shear induced redistribution of fluid within a uniformly swollen nonlinear elastic cylinder

The theory of mixtures is applied to the determination of equilibrium states of a solid-fluid mixture which is isolated from contact with a fluid bath. In the particular problem considered, a hollow rubber cylinder first undergoes unconstrained swelling in a fluid bath. The solid is homogeneously deformed to a larger cylindrical shape, the fluid is uniformly dispersed and the mixture is in a saturated equilibrium state. The mixture is then bonded to rigid impermeable membranes at its inner and outer surfaces. Rigid impermeable flat plates restrain motion at its ends. While the swollen length is held fixed, relative rotation of the membranes induces shear distortion in the rubber-fluid mixture. The resulting normal stresses cause a change in the mixture from its initial equilibrium state in which the system is homogeneously swollen but unsheared to a new equilibrium state in which there is radial variation of both the solid deformation and fluid density. A numerical example, using properties for a particular rubber-fluid mixture, shows that the volume of the mixture and the fluid density decrease near the inner wall of the cylinder and increase near the outer wall.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29779/1/0000118.pd

a novel approach to the description of constitutive relations

Recent advances in the development of implicit constitutive relations to describe the response of both solids and fluids have greatly increased the repertoire of the modeler in his ability to describe natural phenomena more faithfully than hitherto possible. It would not be an exaggeration to claim that such constitutive relations have the potential to lead to breakthroughs in mechanics as they provide very promising novel means to study two of the most important and ill-understood problems in mechanics, that of fracturing of solids, and of turbulence in fluids, in addition to providing a means to describe a plethora of phenomena that have eluded explanation in biomechanics, response of colloids and mixtures, etc. In this article we describe these recent developments within the context of both fluid and solid mechanics. A Novel approach to the description of constitutive relation

A constitutive equation for non-linear electro-active solids

Electro-active solids are solids that are either infused with electrorheological fluids or embedded with electrically conducting particles, the body as a whole however conducting negligible current. In this paper, we provide a mathematical framework, within the context of continuum mechanics, for the study of electro-active solids. The theory assumes that the body can be considered as a continuum, in the sense of homogenization, which is isotropic, incompressible, elastic and is capable of responding to an electric field. Appealing to standard techniques in continuum mechanics, we obtain a constitutive relation for the stresses in terms of the deformation and electric field. This is used in a study of triaxial extension, simple shear and anisotropy induced by the electric field.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41704/1/707_2005_Article_BF01305753.pd

Logarithmic convexity for third order in time partial differential equations

In this short note, we want to describe the logarithmic convexity argument for third-order in time partial differential equations. As a consequence, we first prove a uniqueness result whenever certain conditions on the parameters are satisfied. Later, we show the instability of the solutions if the initial energy is less or equal than zeroPeer ReviewedPostprint (author's final draft