THE SINGULARITIES NEAR THE CORNER OF A VISCOELASTIC FLUID IN A 2 D CAVITY

In this article finite differences are used to study viscoelastic incompressible flow of a Criminale Erickson Filbey fluid in a square cavity flow domain. In this case, the nature of corner singularities is examined in which the fluid is contained and the flow generated by the motion of one or more walls. The governing equations are formulated in terms of stream function and vorticity equation and the corresponding radial parts are defined by a tourth-order non-linear differential equations for Stokes flow. In recent years that mathematical formulations of viscoelastic flows often remain very complex velocity and stress tield and then stress singularities are known to occur in several flows as in this article Therefore, singularity behaviour became a very imp0l1ant current issue in fluid dynamics However, this al1icle is set up with the aim of examining the corner singularities for cavity driven tlow in 2D for viscoelastic flow despite the Newtonian flow being weil known rhen we show that the viscoelastic fluid has different singularity behaviour than the viscous tluid near the corner with respect to the shear-rate The corner singularity problem of the non-Newtonian fllm in flow domain has been subject of experimental and numerical study for over 20 years [11 and is ,I valuable work in tluid dynamics These techniques were first used bv Dean and :VI<lntagnon [2] and later developments include the work of Moffat [3] lVloreover. DavIes [4] described the methods of how an investigation of the effects of elasticity on eddies could be made Because the local behaviour of flow variables in two-dimensional Stokes tlow has proved the existence of such eddies in wedges and non-re-entrant corners Many important studies were carried out but first study was done by Walters and Webster [5]. They reported that corner conditions had a greater effect to the flow in the case of non-Ne'wtonian tlows than corresponding Newtonian flows in their experiments. Some disadvantages can be expected for this problem tor nonNewtonian flows than their Newtonian counterparts in numerical studies due to numerical discretison errors occurs near the corner This can be supported by the nature of the governing equations which are being different type partial differential equations in each case For example, while the non-linear partial differential governing equations for non-Newtonian flows are of mixed elliptic-hyperbolic type for steady incompressible flows, the corresponding Newtonian flows are elliptic. Therefore, this problem gives undesirable results in the simulation of non-Newtonian flows, apart from a few works [6] in which special methods were used to define boundary singularity behaviour which can be existed near the corner, in velocity gradients, stress and pressure instead of using the numerical techniques [4]. In standard cavity flow, singularities exist near the top two corners because the shear-rate becomes increasingly large and therefore both shear-rate and vorticity become singular x Figure I. Diagram of the corner singularity To consider the standard cavity flow whose top plate moves with constant speed (I we assume that no-slip boundary conditions are applied. We next examine the nature of the corner singularities separately for both Newtonian and non-Newtonian flow 2.1. Newtonian Flow In steady two-dimensional VISCOUS incompressible flow the stream function and vOl1icity equations are ,,2 R })(;) v OJ = e-j){

Abstract-In this article finite differences are used to study viscoelastic incompressible flow of a Criminale Erickson Filbey fluid in a square cavity flow domain.In this case, the nature of corner singularities is examined in which the fluid is contained and the flow generated by the motion of one or more walls.The governing equations are formulated in terms of stream function and vorticity equation and the corresponding radial parts are defined by a tourth-order non-linear differential equations for Stokes flow.In recent years that mathematical formulations of viscoelastic flows often remain very complex velocity and stress tield and then stress singularities are known to occur in several flows as in this article Therefore, singularity behaviour became a very imp0l1ant current issue in fluid dynamics However, this al1icle is set up with the aim of examining the corner singularities for cavity driven tlow in 2D for viscoelastic flow despite the Newtonian flow being weil known rhen we show that the viscoelastic fluid has different singularity behaviour than the viscous tluid near the corner with respect to the shear-rate The corner singularity problem of the non-Newtonian fllm in flow domain has been subject of experimental and numerical study for over 20 years [11 and  Stokes tlow has proved the existence of such eddies in wedges and non-re-entrant corners Many important studies were carried out but first study was done by Walters and Webster [5].They reported that corner conditions had a greater effect to the flow in the case of non-Ne'wtonian tlows than corresponding Newtonian flows in their experiments.Some disadvantages can be expected for this problem tor non-Newtonian flows than their Newtonian counterparts in numerical studies due to numerical discretison errors occurs near the corner This can be supported by the nature of the governing equations which are being different type partial differential equations in each case For example, while the non-linear partial differential governing equations for non-Newtonian flows are of mixed elliptic-hyperbolic type for steady incompressible flows, the corresponding Newtonian flows are elliptic.Therefore, this problem gives undesirable results in the simulation of non-Newtonian flows, apart from a few works [6] in which special methods were used to define boundary singularity behaviour which can be existed near the corner, in velocity gradients, stress and pressure instead of using the numerical techniques [4].
In standard cavity flow, singularities exist near the top two corners because the shear-rate becomes increasingly large and therefore both shear-rate and vorticity become singular

r=-h(e) (7) r
where g(e) and h(e) incorporate!l(e) and its derivatives.As seen from ( 6) and ( 7), when r ~0 the vorticity and shear-rate become infinite and singularities exist at the corner points.
Similarly by using equation ( 4) in terms of polar co-ordinates and on using (5).we have the general form of In< (e) as where A, B, C and D are constants.In the special case m= I the solution takes the form (which is for the Stokesian flow equation) The boundaries may be rigid walls on which the velocity is defined, or surfaces on which the stress is defined.Therefore.
for standard cavity flow with one wall moving and the other stationary we have

Non-Newtonian Flow
To use Moffat's assumption, we need to see how inelastic non-Newtonian flow behaves near the corner In this case the flow equation is where M and L are defined •as before.Equation ( 13) is usually solved with equation (2).We consider the viscosity near the corner through the Cross-model for non-Newtonian viscous flow.Since we work with polar co-ordinates the velocity components are defined in terms of polar co-ordinates by

4]
We use a similar analysis as before and examine the steady viscoelastic fluid behaviour near the corner, for simplicity, whose model is denoted by CEF.It appears likely that the corner singularity of the viscoelastic fluids may create significant numerical prohlems We assume as previously the Moffat's assumption that the viscous force still dominate tIle inertial force as Re ~O.Under these circumstances, for the viscoelastic fluid

x
Figure I. Diagram of the corner singularity e = 0 and e = !!... represent moving wall and stationary wall respectively.The solution 2 of the Stokesian flow equation near the corner is \f'(r,e) = ( :.{/ ) (-Jr' sine+4ecose+2Jrsine) 11(8) = -(if + c ~~J elf 1'(8) = cf -s-' , 08 where I = II (8), s = sin 8 and c = cos8, The Cross-model viscosity, therefore, takes the form When we substitute the stream, vorticity and viscosity function, are defined in terms of polar co-ordinates, into (13) the equation of the non-Newtonian flow near the corner is I Here C(8) incorporates terms including I(8) since OJ has the form (u = -X(8), \7:w When r ~0, \7' OJ ~0 and \741.J! ~0, We can therefore say that the !low near the corner IS Stokesian for shear-dependent non-Newtonian viscous flow with these approximations2,3, The Corner Singularities of Viscoelastic FluidsThe flow of viscoelastic fluids in regions involving high stresses, it shows the different b~haviour from the viscous flow near the singularity It is a matter of fact that in non-Newtonian flows little is known about local behaviour near the corner apart from a few case[