Squeezing flows in the non-Newtonian fluids

Abstract

Newtonian fluids reflecting a linear between the stress and the rate of strain do not explain many materials occurred in diverse applications of industry. The constitutive equations of such liquids vary considerably in complexity. Different models have been discussed in context of these fluids. A special category of viscoelastic fluid attracted many researchers in the recent years. Exact/approximate solution can be expected in case of second grade fluid which is considered as a subclass of viscoelastic fluids. The rheologists have been able to provide a theoretical foundation in the form of a constitutive equation which can in principle, have any order. For applied mathematicians, modelers and computer scientists the challenge comes from a different quarter. The constitutive equations of even the simple viscoelastic fluids, namely, second grade fluids are such t hat the differential equations describing the motion have, in general, order higher t han those describing the motion of the Newtonian fluids but apparently there is no corresponding increase in t he number of boundary conditions. Applied mathematicians and computer scientists are thus forced with the so-called ill-posed boundary value problems which, 4 in theory, would have a family of infinitely many solutions for flows confined for finite domain. The task then becomes to select one of them under some plausible assumption. However for different flow configurations, the stated assumptions take different forms. Taking all such complexities in view, even various recent researchers are engaged in the advancement of flows of second grade fluids (see [1 - 10]). Further, geophysicists encounter MHD aspect through interaction of conducting fluids. The MHD concept is useful for the engineers in the design of heat exchangers, MHD pumps and fl ow meters, in space vehicle propulsion, control and reentry problems, in metallurgical process and polymer industry; in creating novel power generating systems and in controlling fusion process. The flows of second grade fluids in the presence of magnetic fi eld have been addressed by the very recent investigations [11 - 15]. No doubt, the model of second grade fluid describes t he normal stress effects but important features of shear t hinning/ t hi ckening cannot be explained through its implementation . In such situation, the model of third grade fluid is more appropriate. This fluid model even in steady fl ow predicts the shear thinning/ thickening properties. With this awareness, Feiz-Dizaj i et al.[16] examined t he flow of third grade fluid in the annulus. Some fundamental flows of third grade fluid in the presence of partial slip effects are discussed by Ellahi et al. [17]. Abelman et al. [18, 19] investigated such analysis in a rotating frame. Flow of third grade fluid over a stretching surface with heat transfer and partial slip is analyzed by Sahoo and Do [20]. They used the regular perturbation method for the solution analysis. The series solution to unsteady boundary layer equations in a special third grade fluid is presented by Abbasbandy and Hayat [21]. Convergent series solutions are developed by homotopy analysis method. Narain and Kara [22] made the analysis of conservation laws in third grade fluid . Keimanesh et al. [23] employed multi-step differential transform method for flow of t hird grade fluid between two parallel plates. Hayat et al. [24] constructed mathematical model for peristaltic flow of t hird grade fluid in a curved channel with heat and mass transfer. Ellahi and Afzal [25] studied the effect of variable viscosity on the flow of third grade fluid in a porous space. The convergent solutions of t he governing equations are obtained via homotopy analysis method. The second grade and third grade fluid models can not explain the relaxation and retardation times effects. Among the several models which have been employed to describe the rheological parameters exhibited by cert ain real fluids, the J effrey fluid has gained much support from the experimentalists and theoreticians.