Viscoelastic Flow.Pdf

Journal of Petroleum Science and Engineering 76 (2011) 93–99

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Journal of Petroleum Science and Engineering

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Viscoelastic ﬂow and species transfer in a Darcian high-permeability channel

O. Anwar Bég a,⁎, O.D. Makinde b a Biomechanics, Biotechnology and Magnetohydrodynamics, Mechanical Engineering, Department of Engineering and Mathematics, Shefﬁeld Hallam University, Shefﬁeld, S1 1WB, England, UK b Institute for Advance Research in Mathematical Modelling and Computations, Cape Peninsula University of Technology, P. O. Box 1906, Bellville 7535, South Africa article info abstract

Article history: We study the two-dimensional steady, laminar flow of an incompressible, viscoelastic fluid with species Received 13 May 2010 diffusion in a parallel plate channel with porous walls containing a homogenous, isotropic porous medium Accepted 10 January 2011 with high permeability. The Darcy model is employed to simulate bulk drag effects on the flow due to the Available online 21 January 2011 porous matrix. The upper convected Maxwell model is implemented due to its accuracy in simulating highly elastic fluid flows at high Deborah numbers. The conservation equations are transformed into a pair of couple Keywords: nonlinear ordinary differential equations which are solved numerically using efficient 6th order Runge–Kutta viscoelastic flow mass transfer shooting quadrature in the computer algebra package system MAPLE. The effects of Darcy number (Da), upper convected Maxwell (UCM) fluid Deborah number (De), Schmidt number (Sc) and transpiration Reynolds number (ReT) on velocity and Deborah number species concentration distributions and also wall shear stress and concentration gradients are examined in MAPLE detail. The study finds applications in petroleum filtration dynamics, hydrocarbon fluid flow in geosystems, oil petroleum flows spill contamination in soils and also chemical engineering technologies. © 2011 Elsevier B.V. All rights reserved.

1. Introduction similarity transformation to model the UCM flow in an infinitely long cylinder whose surface has a velocity that increases in Viscoelastic flows are encountered in numerous areas of magnitude linearly with an axial coordinate showing that with an petrochemical, biomedical and environmental engineering includ- increasing elasticity of the fluid normal stress gradients in an elastic ing polypropylene coalescence sintering (Scribben et al., 2006), boundary layer near the accelerated surface aid in offseting dynamically-loaded journal bearings (Tichy, 1996), blood flow inertially-generated negative axial pressure gradients. Roberts and (Chien et al., 1975; Vlastos, 1998) and geological flows (Wouter et Walters (1992) obtained spectral numerical solutions for the three- al., 2005). A wide range of mathematical models have been dimensional flow of a UCM fluid in a journal bearing, operating developed to simulate the nonlinear stress–strain characteristics under static loading conditions, showing that a relaxation time of the of such fluids which exhibit both viscous and elastic properties. A order of 10−4 s is needed prior to viscoelasticity enhancing the load- detailed discussion of such models which include the upper bearing capacity. Maders et al. (1992) used a decoupled finite convected Maxwell model, the Walters-B model and the Reiner– element method to study the flow of a polymeric UCM fluid in a 2- Rivlin second-order model is provided in Zahorski (1982).Forhighly dimensional convergent geometry. Brown et al. (1993) have studied elastic fluids such as polymer melts, the upper convected Maxwell the linear stability of the planar Couette flow of a UCM fluid using a (UCM) model has proved to be very reliable. This viscoelastic flow mixed finite-element method obtaining stabile calculations for the model is a generalization of the Maxwell material model for the case values of the Deborah number in excess of 50. Khayat (1994) used a of large deformations and was derived by Oldroyd using an upper perturbation technique to analyze the two-dimensional incom- convected time derivative (Oldroyd, 1950). Many theoretical, pressible viscoelastic UCM flow between two parallel plates, with numerical and experimental studies have utilized this model. two straight free boundaries. The hyperbolic partial differential Horikawa (1987) presented finite difference solutions using a equations were solved using an implicit finite-difference procedure. perturbation method for the flow of a UCM fluid around an inclined Rahaman (1997) investigated the transient UCM viscoelastic flow in circular cylinder of finite length showing that the viscoelastic tends a rectangular duct. Avgousti and Renardy (1998) studied computa- to flow axially in the vicinity of the cylinder. Chiba et al. (1988) tionally the hydrodynamic stability of the eccentric Dean flow of a investigated analytically the effects of a wall transpiration of the UCM fluid. Xue et al. (1998) employed a 3-dimensional finite volume UCM fluid flow via a porous-walled tube. Larson (1988) used a numerical solver to model Lagrangian transient extensional flow of both a Phan-Thien Tanner (PTT) viscoelastic and a UCM viscoelastic fluid in a rectangular duct with a sudden contraction is carried out ⁎ Corresponding author. fi E-mail addresses: [email protected] (O.A. Bég), [email protected] using a three-dimensional (3-D) nite volume method (FVM). (O.D. Makinde). Further studies of the viscoelastic UCM flows were described by

solutions of polyethylene oxide flowing through beds of packed beads Notation i.e. porous media, showing that viscoelasticity was most prevalent at moderate flow rates. A reduced viscoelastic effect at higher flow rates C concentration of species diffusing in fluid was attributed to the dominance of extensional stresses in this C concentration at channel center (y=0). w regime. Deiber and Schowalter (1981) performed experiments on the C concentration at both plates H flow of dilute aqueous solutions of a polyacrylamide in a tube with (i.e. at y=H/2, y= −H/2) sinusoidal axial variations in diameter as a porous medium flow D species diffusivity model, showing that Lagrangian unsteadiness generates an increase in H channel width resistance to flow through the sinusoidal channel relative to that K permeability of the porous medium predicted for a purely viscous fluid. Two other excellent investigations u velocity in x-direction of the viscoelastic flow in Darcian porous media include the articles by v velocity in y-direction Durst et al. (1987) and the finite element study by Talwar and V/2 suction velocity at the plates Khomami (1992). Tan and Masuoka (2005) used a modified Darcy's x coordinate along channel center-line law for the Oldroyd-B viscoelastic fluid to study Stokes' first problem (parallel to plates) in a porous half space using a Fourier sine transform. They found that y coordinate normal to channel center-line the boundary layer thickness has a limited value and deviates from λ relaxation time of the UCM fluid the purely fluid case. Several studies have considered specifically the ν kinematic viscosity of the UCM fluid flow of Maxwell viscoelastic fluids in porous media. De Haro et al. (1996) used a volume averaging approach to study Maxwell flow in a rigid porous medium deriving the momentum equation with a time- Dimensionless parameters dependent permeability tensor. They simulated the viscoelasticity Da Darcy number effects by transforming the model to the frequency domain via a De Deborah number temporal Fourier transform and presented closed-form solutions for a F dimensionless stream function porous medium modeled as a bundle of capillary tubes. Other studies Re transpiration (suction) Reynolds number (N0) T include the papers by del Río et al. (1998) and Lopez et al. (2003). Sc Schmidt number Numerous technological applications exist wherein both viscoelastic X dimensionless coordinate along channel center-line flow and mass (species) diffusion take place, including the separation (parallel to plates) systems, polymer processing, haemodynamics, petroleum displace- Y dimensionless coordinate normal to channel ment in reservoirs etc. Flows may be both laminar or turbulent. Cho center-line and Hartnett (1981) employed the successive approximation tech- ϕ dimensionless concentration function nique to investigate the mass transfer entry length and maximum mass transfer reduction asymptote for the drag-reducing viscoelastic fluids obtaining a good correlation with the empirical mass transfer results for the predicted mass transfer rates and showing approxi- mately 56–75% reduction in the mass transfer rate compared to the Van Os and Gerritsma (2001); Ghosh and Sengupta (2002) with Newtonian values at the same Reynolds and Schmidt numbers. The magnetohydrodynamic effects and from a continuous stretching transient species transfer in a viscoleastic tube flow was studied by surface by Sadeghy et al. (2005). Evans (2005) discussed the steady Dalal and Mazumder (Dalal, 1998). Lo et al. (2003) studied the mass planar flow of the UCM fluid for the re-entrant corners with obtuse diffusion in curdlan viscoelastic gels. A detailed analysis has also been angles, obtaining a class of similarity solutions associated with the presented by Ramakrishnan (2004) of the non-Fickian species inviscid flow equations which arise from the dominance of the upper transfer in the polymeric viscoelastic flows. Herein we extend to convective stress derivative in theconstitutiveequations.Healso consider the non-reactive version of the study of Hayat and Abbas identified two classes of the boundary-layer structure, namely the (2007) but with the porous drag effects considered, to present Renardy single-layer structure and a new double-layer boundary extensive numerical solutions to the flow and dispersion of a species layer structure. Sadeghy et al. (Sadeghy, 2006)usedaChebyshev in a UCM viscoelastic-fluid saturated regime between the parallel pseudo-spectral collocation-point method to simulate the two- plates and wall transpiration. The case of the low Deborah number dimensional stagnation-point flow of the UCM viscoelastic fluid (De) i.e. weak elasticity, is considered. The shooting iteration indicating a thickening of the boundary layer and a drop in the wall technique together with the Runge–Kutta sixth-order integration skin friction coefficient with higher elasticity effects. Hayatetal. scheme is employed to solve the transformed system of the nonlinear (2006) used the homotopy analysis method to obtain series ordinary differential equations. Such a study we believe constitutes an solutions for the hydromagnetic boundary layer flow of a UCM important addition to the literature on the rheological flows in the fluid over a porous stretching sheet. petroleum geosystem applications. The above studies all omitted any consideration of porous media despite the frequent presence of such media in many petroleum and 2. Hydromechanics of the UCM viscoelastic fluid chemical engineering operations and systems. Filtration systems, petroleum geosystems, packed bed reactors and foodstuffs are several Zahorski (1982) has discussed extensively the mathematical examples of porous media in which viscoelastic flows may occur. aspects of the UCM viscoelastic model. This model is the most Generally the Darcy model is employed for low-velocity flows in elementary of the nonlinear viscoelastic models which accounts for porous media and relates the pressure drop in the porous medium to a frame invariance in the nonlinear flow regime. It amounts to a linear drag force. Wissler (1971) gave the first analytical explanation succinct amalgamation of the Newtonian law for viscous fluids and for the elongation stresses developed in the viscoelastic flow in a the derivatives of the Hooke's law for elastic solids and cannot Darcian porous medium, presenting a third-order perturbation simulate more complex effects which are reproduced in the more analysis and showing that viscoelastic (e.g. polymer and hydrocarbon elaborate viscoelastic formulations. Nevertheless in simple engi- derivatives) experience a reduced mobility in porous media. James neering flows, the UCM model is easily implemented and leads to and Mclaren (1975) discussed experiments relating to the measure- relatively fewer stability and convergence problems in computa- ments of the pressure drop and flow rate for dilute viscoelastic tion. The UCM model simulates purely elastic fluids with shear- O.A. Bég, O.D. Makinde / Journal of Petroleum Science and Engineering 76 (2011) 93–99 95

liquid from Eqs. (1) and (2), the appropriate conservation equations may be shown to take the form: Mass conservation:

∂u ∂v + =0 ð3Þ ∂x ∂y

Momentum conservation: "# Fig. 1. The physical model and a coordinate system. ∂u ∂u ∂2u ∂2u ∂2u ∂2u u u + v + λ u2 + v2 +2uv = ν −ν ð4Þ ∂x ∂y 1 ∂x2 ∂y2 ∂x∂y ∂y2 K independent viscosity. It is obtained by replacing the partial time derivative in the differential form of the linear Maxwell model with Species conservation (mass diffusion): the upper convected time derivative, viz: ∂C ∂C ∂2C u + v = D ð5Þ τ λ τˆ μ γ ð Þ 2 + 1 = 0 1 ∂x ∂y ∂y

∂τ where u, v are the x-and y-direction velocity components, λ is τˆ = + vd∇τ−∇ðÞv Td τ−τd∇v ð2Þ 1 ∂t Maxwell relaxation time, D is the species diffusivity, C is concentration of the diffusing species, K is the permeability of the porous medium. τ λ μ where is the extra stress tensor, 1 is the relaxation time, 0 is the The appropriate no-slip boundary conditions are identical to those in low-shear viscosity, γ is the rate-of-strain tensor, τˆ is the upper Hayat and Abbas (2007) and owing to symmetry take the form: convected time derivative of the stress tensor, t denotes time, v is the velocity vector, (d )T is the transpose of the tensor and ∇ν represents ∂u = v =0;C=C at y = 0 ð6Þ the fluid velocity gradient tensor. The convected derivative expresses ∂y w therateofchangeasafluid element moves and undergoes deformation. The first two terms in Eq. (2) designate the material or = = ð Þ u =0;v =V 2; C = CH at y = H 2 7 substantial derivative of the extra stress tensor, which is the time derivative following a material element and simulates the time where Cw is the species concentration at the channel center-line and changes at a particular element of the substance or material. The CH is the concentration at the upper plate. As such only a semi-region remaining terms in Eq. (2) correspond to deformation. As indicated by of the channel (upper half, 0≤y≤H) is considered. To facilitate Zahorski (1982), the presence of such terms which simulates the numerical solutions, we define the following transformations: convection, rotation and stretching of the fluid motion, guarantees that the principle of the frame invariance is satisfied i.e. the x y dFðÞ Y X = ; Y = ; u = −VX ; relationship between the stress tensor and deformation history of H H dY the UCM fluid is independent of the particular coordinate system employed for the description. The viscosity in the UCM fluid C−C v = VFðÞ Y ; ϕ = H : ð8Þ fl − represents the Boger uid and predicts a Newtonian elongation Cw CH viscosity which is three times the Newtonian shear viscosity. The UCM model may accurately simulate the first normal stress difference in whereallparametersaredefined in the nomenclature. The continuity the shear and strain hardening in the viscoelastic fluids; on the other Eq. (1) is satisfied identically with Eq. (8) and Eqs. (4) and (5) reduce to hand this model incorrectly predicts that the steady-state elonga- the following coupled, non-linear, third order, second degree ordinary tional viscosity is infinite at a finite elongation rate, a result in clear contradiction to the actual physical observation. Unlike other viscoelastic models which may accurately predict such phenomena (e.g. the Walters-B model) the UCM model does not generate the flow equations which are an order higher than the classical Navier–Stokes equations and the resulting differential model is easier to solve.

3. Mathematical formulation and ﬂow model

The flow regime to be studied is illustrated below in Fig. 1, comprising a parallel plate channel with porous walls in an (x,y) coordinate system. The inter-plate region contains an incompressible, UCM viscoelastic fluid which saturates the rigid, isotropic, homogenous porous material intercalated between the plates which are a distance, H, apart. The x-axis is orientated along the channel center- line, parallel to both plates and the y-axis is normal to it. Following Hayat and Abbas (2007) the flow is assumed to be symmetric about both axes. The steady state flow and species diffusion occur in the channel with the fluid extraction or influx taking place at both plates, with velocity, V/2. For Vb0, an injection occurs at both plates and for

VN0, a suction. Implementing the shear-stress strain tensor for a UCM Fig. 2. dF/dY versus Y for De=0.5, Sc=0.2, ReT =5 with various Darcy numbers. 96 O.A. Bég, O.D. Makinde / Journal of Petroleum Science and Engineering 76 (2011) 93–99

Fig. 3. ϕ versus Y for De=0.5, Sc=0.2, ReT =5 for various Darcy numbers. Fig. 5. ϕ versus Y for De=0.5, Sc=2.0, ReT =5 with Darcy numbers.

differential equation for the momentum and the nonlinear, second version of the Newton–Raphson shooting method (Nachtsheim and order, first degree ordinary differential equation for the species: Swigert, 1965; Makinde, 2009a, 2009b) with Sc, Da, De, and ReT as Momentum: prescribed parameters. The computations were done by a program "# "# which uses a symbolic and computational language MAPLE (Betounes, 3 2 2 2 3 d F dF d F dF d F 2 d F 1 dF 2001). A step size of ΔY=0.001 was selected to be satisfactory for a +Re −F + De 2F −F + =0 − dY3 T dY dY2 dY dY2 dY3 Da dY convergence criterion of 10 7 in nearly all cases. ð9Þ 4. Results and discussion Species: In the present study we examine the influence of four key d2ϕ dϕ −Re ScF =0: ð10Þ parameters: Deborah number (De), Schmidt number (Sc), transpiration dY2 T dY Reynolds number (ReT) and Darcy number (Da) on dimensionless velocity ( dF) and concentration function (ϕ) distributions. The transformed boundary conditions (6), (7) are: dY Fig. 2 shows the velocity distribution with various Darcy numbers. At the center-line of the channel (Y=0) we observe that the velocity d2F F = =0;ϕ = 1 at Y = 0 ð11Þ is strongly accelerated with an increasing Darcy number i.e. with dY2 greater permeability. Note that only the upper half space,0≤Y≤0.5, of dF the channel is considered in the solution domain (the lower half space F =0:5; =0;ϕ = 0 at Y = 0:5 ð12Þ fl dY will be a re ection of this solution in the centerline Y=0). In the dimensionless momentum Eq. (9), the final term on the left hand side, ν N 1 dF fl where ReT =HV/ is the transpiration Reynolds number ( 0 for viz, Da dY, simulates the Darcian resistance to the ow. With the 2 suction and b0 for injection), De=λ1V /ν is the Deborah number, increasing values of Da the regime will contain progressively lesser Sc=ν/D is the Schmidt number and Da=K/H2 is the Darcy number. and lesser fibers to resist the flow — the Darcian force is inversely The system of the non-linear ordinary differential Eqs. (9),(10) with proportional to the Darcy number and physically this implies that for boundary conditions (11)–(12) has been solved numerically by using the sixth order Runge–Kutta integration scheme with a modified

Fig. 4. dF/dY versus Y for De=0.5, Sc=2.0, ReT =5 with Darcy numbers. Fig. 6. dF/dY versus Y for Da=0.1, Sc=0.2, ReT =5 with Deborah numbers. O.A. Bég, O.D. Makinde / Journal of Petroleum Science and Engineering 76 (2011) 93–99 97

Fig. 7. ϕ versus Y for Da=0.1, Sc=0.2, ReT =5 with Deborah numbers. Fig. 9. ϕ versus Y for Da=0.1, Sc=0.2, De=0.5 with Reynolds numbers (ReT). a greater permeability (higher Da values), the Darcian drag will Fig. 5 illustrates the species variation with different Darcy numbers decrease. This explains the strong acceleration in the flow. As we near across the upper channel half space, however for Sc=2. In this case the upper channel wall however, a transition in response is observed the viscous diffusion rate is twice the molecular (species) diffusion i.e. the velocity is found to decrease with increasing permeability. The rate. A noticeable decrease in the species diffusion is observed with a wall effect has a major influence on this behavior. Fig. 2 corresponds decrease in the Darcy number i.e. for the lowest permeability case to De=0.5, Sc=0.2, and ReT =5. For a low Deborah number (De) the (Da=0.05), the lowest values of the species concentration are liquid is more fluid and elastic effects are dominated by viscous computed — this trend is consistent across the entire channel half effects. For higher Deborah numbers (considered later) the opposite is space. Clearly therefore with ScN1, the species diffusion field will be apparent i.e. elasticity effects will dominate viscous effects. affected (if only slightly) by the permeability of the regime. It is Fig. 3 presents the species (concentration) distribution in the expected that with even higher values the influence will be upper channel half-space with a variation in the Darcy number. Very accentuated. little influence is observed. Although in the species diffusion Eq. (10), Fig. 6 shows the velocity response with a distance across the the velocity field is coupled with the species diffusion field via the channel (again for the upper half space) to a change in the viscoelastic − dφ material parameter i.e. Deborah number (De=λ V2/ν). The visco- term, ReT ScF dY, since there are no buoyancy forces present (no 1 species buoyancy force term arises in the momentum equation), it is elastichi effect in the momentum equation is simulated via the terms, fi fl dF d2F − 2 d3F expected that permeability will not exert a signi cant in uence on the De 2F dY dY2 F dY3 , which are complex, nonlinear and in multiple diffusion of the species, especially for low Schmidt numbers i.e. for order. The Deborah number (De) represents the ratio of a relaxation Scb1, for which viscous diffusion is dominated by molecular (species) time (characterizing the intrinsic fluidity of a material), and the diffusion. characteristic time scale of an experiment (or a computer simulation) Fig. 4 again presents the velocity distribution with various Darcy probing the response of the material. The smaller the Deborah numbers, however with a much higher Schmidt number, Sc=2. A number, the more fluid the material appears. Higher De values imply a negligible difference is observed compared with the case for Sc=0.2 strongly elastic behavior. The Newtonian fluids possess no relaxation (Fig. 2). Therefore the molecular diffusivity of the species exerts no time i.e. De=0. For De=1 the relaxation time and characteristic time tangible influence on the velocity response for different constitutions scale are equal. The effect of De on the flow is therefore also complex. of the filtration material i.e. different permeabilities. At the channel center, velocities are maximized for De=0 i.e. the

Fig. 8. dF/dY versus Y for Da=0.1, Sc=0.2, De=0.5 with Reynolds numbers (ReT). Fig. 10. dF/dY versus Y for Da=0.1, ReT =5, De=0.5 with Schmidt numbers. 98 O.A. Bég, O.D. Makinde / Journal of Petroleum Science and Engineering 76 (2011) 93–99

Fig. 11. ϕ versus Y for Da=0.1, ReT =5, De=0.5 with Schmidt numbers. Fig. 13. ϕ versus Y for Da=0.1, ReT =5, De=0.05 with Schmidt numbers.

Newtonian case. With increasing elastic effects i.e. as De rises from 0 However as the suction Reynolds number rises, a distinct increase in through 0.5, 1, 2 to 3, there is a marked decrease in the velocity at the the concentration values across the channel half space is observed — channel center and for some distance towards the upper plate. Clearly peak values of ϕ correspond to the maximum ReT value of 10. The therefore in the central region of the channel the flow is retarded with upper channel wall suction therefore has a positive influence on the an increasing elasticity i.e. higher Deborah numbers. As we progress diffusion of species in the upper semi-channel region. towards the channel wall however a transition arises — increasing the Figs. 10–13 show the velocity and species distributions for various elasticity (higher Deborah numbers) serves to accelerate the flow Schmidt numbers. Figs. 10 and 11 correspond to an intermediate nearer to the upper channel wall. In this region the Newtonian fluid value of the Deborah number (De=0.5) whereas Figs. 12 and 13 are velocity is minimized. associated with a very low De value (De=0.05). Negligible difference Fig. 7 presents the species distribution across the upper channel is observed between the respective plots for velocity and species half space with various De values. The change in ϕ is insignificant with concentration function for De=0.5 and De=0.05, indicating that, as a large increase in the De values. This stands to reason since the discussed earlier, viscoelasticity will not influence the diffusion of viscoelasticity of the liquid is not expected to influence the diffusion species. The inspection of Fig. 10 shows that with an increase in the field. No viscoelastic terms arise in the species diffusion Eq. (10), and Schmidt number, Sc, exerts no major influence on the velocity field. the coupling with the momentum Eq. (9) is very weak. Fig. 11 reveals that with an increase in Sc there is a very strong rise in Fig. 8 shows the effect of the transpiration Reynolds number (Re) the concentration values (ϕ) across the upper semi-channel region. on the velocity distribution. ReT =HV/ν is always positive since we Appropriate values of the Schmidt number for the species diffusing study the suction case only i.e. the material removal from the channel are Sc=0.2 (Hydrogen gas), Sc=0.3 (Helium gas), Sc=1 (denser wall. At the channel center (Y=0) we observe that as ReT increases gasses) and Sc=2 (hydrocarbon derivatives). For Sc=1, the viscous there is a corresponding decrease in the velocity (dF/dY). As we and molecular diffusion rates are equivalent. For Sc b1 species approach the upper wall (Y=0.5), the velocity is however enhanced diffusivity exceeds viscosity. For ScN1 viscosity exceeds molecular considerably with the rising suction Reynolds number. diffusivity. Clearly hydrocarbons will diffuse better than the low Fig. 9 illustrates the species function distribution, ϕ, with a molecular weight gasses in the porous regime. distance across the upper channel half space, and with various suction Reynolds numbers. For all ReT values there is a continuous 5. Conclusions decrease in ϕ from the channel center line to the upper channel wall. This study has analyzed theoretically the steady, incompressible fully developed flow of a UCM viscoelastic fluid with species diffusion in a Darcian porous medium channel. The transformed similarity of the ordinary differential equations has been solved numerically using the Runge–Kutta integration scheme with a modified version of the Newton–Raphson shooting method in the MAPLE software. Physically realistic data values are also described briefly which have been used in the computations. The computations have shown that in the central region of the channel the flow is accelerated with an increase in the Darcy number (i.e. increase in permeability) but decelerated with an increase in the Deborah number (owing to an increasing elasticity). With the increasing suction of the Reynolds number the flow is decelerated near the channel center whereas the species concentration is continuously increased throughout the upper semi-channel region. No significant influence of the Darcy number or the Deborah number is observed on the species diffusion values. An increase in the Schmidt number is observed to enhance the species concentration values but has no marked effect on the velocity field. The current study has potential applications in simulating polymeric processing, chemical engineering and also environmental contamination in the

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