Transient free‐surface flow of a viscoelastic fluid in a narrow channel

The interplay between inertia and elasticity is examined for transient free‐surface flow inside a narrow channel. The lubrication theory is extended for the flow of viscoelastic fluids of the Oldroyd‐B type (consisting of a Newtonian solvent and a polymeric solute). While the general formulation accounts for non‐linearities stemming from inertia effects in the momentum conservation equation, and the upper‐convected terms in the constitutive equation, only the front movement contributes to non‐linear coupling for a flow inside a straight channel. In this case, it is possible to implement a spectral representation in the depthwise direction for the velocity and stress. The evolution of the flow field is obtained locally, but the front movement is captured only in the mean sense. The influence of inertia, elasticity and viscosity ratio is examined for pressure‐induced flow. The front appears to progress monotonically with time. However, the velocity and stress exhibit typically a strong overshoot upon inception, accompanied by a plug‐flow behaviour in the channel core. The flow intensity eventually diminishes with time, tending asymptotically to Poiseuille conditions. For highly elastic liquids the front movement becomes oscillatory, experiencing strong deceleration periodically. A multiple‐scale solution is obtained for fluids with no inertia and small elasticity. Comparison with the exact (numerical) solution indicates a wide range of validity for the analytical result. Copyright © 2004 John Wiley & Sons, Ltd.     

Influence of inertia,topography and gravity on transient axisymmetric thin‐film flow

This study examines theoretically the development of early transients for axisymmetric flow of a thin film over a stationary cylindrical substrate of arbitrary shape. The fluid is assumed to emerge from an annular tube as it is driven by a pressure gradient maintained inside the annulus, and/or by gravity in the axial direction. The interplay between inertia, annulus aspect ratio, substrate topography and gravity is particularly emphasized. Initial conditions are found to have a drastic effect on the ensuing flow. The flow is governed by the thin‐film equations of the ‘boundary‐layer’ type, which are solved by expanding the flow field in terms of orthonormal modes in the radial direction. The formulation is validated upon comparison with the similarity solution of Watson (J. Fluid Mech 1964; 20 :481) leading to an excellent agreement when only 2–3 modes are included. The wave and flow structure are examined for high and low inertia. It is found that low‐inertia fluids tend to accumulate near the annulus exit, exhibiting a standing wave that grows with time. This behaviour clearly illustrates the difficulty faced with coating high‐viscosity fluids. The annulus aspect is found to be influential only when inertia is significant; there is less flow resistance for a film over a cylinder of smaller diameter. For high inertia, the free surface evolves similarly to two‐dimensional flow. The substrate topography is found to have a significant effect on transient behaviour, but this effect depends strongly on inertia. It is observed that the flow of a high‐inertia fluid over a step‐down exhibits the formation of a secondary wave that moves upstream of the primary wave. Gravity is found to help the film (coating) flow by halting or prohibiting the wave growth. The initial film profile and velocity distribution dictate whether the fluid will flow downstream or accumulate near the annulus exit. Copyright © 2004 John Wiley & Sons, Ltd.     

Three‐dimensional boundary element analysis of drop deformation in confined flow for Newtonian and viscoelastic systems

An adaptive (Lagrangian) boundary element approach is proposed for the general three‐dimensional drop deformation in confined flow. The adaptive method is stable as it includes remeshing capabilities of the deforming interface between drop and suspending fluid, and thus can handle large deformations. Both drop and surrounding fluid are viscous incompressible and can be Newtonian or viscoelastic. A boundary‐only formulation is implemented for fluids obeying the linear Jeffrey's constitutive equation. Similarly to the formulation for two‐dimensional Newtonian fluids (Khayat RE, Luciani A, Utracki LA. Boundary element analysis of planar drop deformation in confined flow. Part I. Newtonian fluids. Engineering Analysis of Boundary Elements 1997; 19 : 279), the method requires the solution of two simultaneous integral equations on the interface between the two fluids and the confining solid boundary. Although the problem is formulated for any confining geometry, the method is illustrated for a deforming drop as it is driven by the ambient flow inside a cylindrical tube. The accuracy of the method is assessed by comparison with the analytical solution for two‐phase radial spherical flow, leading to good agreement. The influence of mesh refinement is examined for a drop in simple shear flow. Copyright © 2000 John Wiley & Sons, Ltd.     

A boundary‐only approach to the deformation of a shear‐thinning drop in extensional Newtonian flow

The influence of shear thinning on drop deformation is examined through a numerical simulation. A two‐dimensional formulation within the scope of the boundary element method (BEM) is proposed for a drop driven by the ambient flow inside a channel of a general shape, with emphasis on a convergent–divergent channel. The drop is assumed to be shear thinning, obeying the Carreau–Bird model and the suspending fluid is Newtonian. The viscosity of the drop at any time is estimated on the basis of a rate‐of‐strain averaged over the region occupied by the drop. The viscosity thus changes from one time step to the next, and it is strongly influenced by drop deformation. It is found that small drops, flowing on the axis, elongate in the convergent part of the channel, then regain their spherical form in the divergent part; thus confirming experimental observations. Newtonian drops placed off‐axis are found to rotate during the flow with the period related to the initial extension, i.e. to the drop aspect ratio. This rotation is strongly prohibited by shear thinning. The formulation is validated by monitoring the local change of viscosity along the interface between the drop and the suspending fluid. It is found that the viscosity averaged over the drop compares, generally to within a few per cent, with the exact viscosity along the interface.     

A boundary element analysis of multiply connected three‐dimensional cavity mixing flow of polymer solutions

The emergence of non‐linear dynamics in cavity mixing is examined using the boundary element method (BEM). The method is implemented for the simulation of three‐dimensional transient creeping flow of Newtonian or linear viscoelastic fluids of the Jeffreys type. A boundary only formulation in the time domain is proposed for viscoelastic flow. Special emphasis is placed on cavity flow involving multiply connected moving domains. The BEM becomes particularly suited for this case, when part of the boundary (stirrer or rotor) is moving, and the remaining outer part (cavity) is at rest. In contrast to conventional volume methods, the BEM is shown to be much easier to implement since the kinematics of the elements bounding the fluid is known (imposed). It is found that, for a simple cavity flow induced by a rotating vane at constant angular velocity, the tractions at the vane tip and cavity face exhibit non‐linear periodic dynamical behaviour with time for fluids obeying linear constitutive equations. Copyright © 1999 John Wiley & Sons, Ltd.     

Hydromagnetic flow through uniform channel bounded by porous media

The combined effects of the magnetic field, permeable walls, Darcy velocity, and slip parameter on the steady flow of a fluid in a channel of uniform width are studied. The fluid flowing in the channel is assumed to be homogeneous, incompressible,and Newtonian. Analytical solutions are constructed for the governing equations using Beavers-Joseph slip boundary conditions. Effects of the magnetic field, permeability,Darcy velocity, and slip parameter on the axial velocity, slip velocity, and shear stress are discussed in detail. It is shown that the Hartmann number, Darcy velocity, porous parameter, and slip parameter play a vital role in altering the flow and in turn the shear stress.     

Three‐dimensional transient free‐surface flow of viscous fluids inside cavities of arbitrary shape

The three‐dimensional transient free‐surface flow inside cavities of arbitrary shape is examined in this study. An adaptive (Lagrangian) boundary‐element approach is proposed for the general three‐dimensional simulation of confined free‐surface flow of viscous incompressible fluids. The method is stable as it includes remeshing capabilities of the deforming free‐surface, and thus can handle large deformations. A simple algorithm is developed for mesh refinement of the deforming free‐surface mesh. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. The method is used to determine the flow field and free‐surface evolution inside cubic, rectangular and cylindrical containers. These problems illustrate the transient nature of the flow during the mixing process. Surface tension effects are also explored. Copyright © 2003 John Wiley & Sons, Ltd.     

Flow simulation on moving boundary‐fitted grids and application to fluid–structure interaction problems

We present a method for the parallel numerical simulation of transient three‐dimensional fluid–structure interaction problems. Here, we consider the interaction of incompressible flow in the fluid domain and linear elastic deformation in the solid domain. The coupled problem is tackled by an approach based on the classical alternating Schwarz method with non‐overlapping subdomains, the subproblems are solved alternatingly and the coupling conditions are realized via the exchange of boundary conditions. The elasticity problem is solved by a standard linear finite element method. A main issue is that the flow solver has to be able to handle time‐dependent domains. To this end, we present a technique to solve the incompressible Navier–Stokes equation in three‐dimensional domains with moving boundaries. This numerical method is a generalization of a finite volume discretization using curvilinear coordinates to time‐dependent coordinate transformations. It corresponds to a discretization of the arbitrary Lagrangian–Eulerian formulation of the Navier–Stokes equations. Here the grid velocity is treated in such a way that the so‐called Geometric Conservation Law is implicitly satisfied. Altogether, our approach results in a scheme which is an extension of the well‐known MAC‐method to a staggered mesh in moving boundary‐fitted coordinates which uses grid‐dependent velocity components as the primary variables. To validate our method, we present some numerical results which show that second‐order convergence in space is obtained on moving grids. Finally, we give the results of a fully coupled fluid–structure interaction problem. It turns out that already a simple explicit coupling with one iteration of the Schwarz method, i.e. one solution of the fluid problem and one solution of the elasticity problem per time step, yields a convergent, simple, yet efficient overall method for fluid–structure interaction problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Hydromagnetic thin film flow of Casson fluid in non-Darcy porous medium with Joule dissipation and Navier's partial slip

In this paper, the effects of viscous and Ohmic heating and heat generation/absorption on magnetohydrodynamic flow of an electrically conducting Casson thin film fluid over an unsteady horizontal stretching sheet in a non-Darcy porous medium are investigated. The fluid is assumed to slip along the boundary of the sheet. Similarity transformation is used to translate the governing partial differential equations into ordinary differential equations. A shooting technique in conjunction with the 4 th order Runge-Kutta method is used to solve the transformed equations. Computations are carried out for velocity and temperature of the fluid thin film along with local skin friction coefficient and local Nusselt number for a range of values of pertinent flow parameters. It is observed that the Casson parameter has the ability to enhance free surface velocity and film thickness, whereas the Forchheimer parameter, which is responsible for the inertial drag has an adverse effect on the fluid velocity inside the film. The velocity slip along the boundary tends to decrease the fluid velocity. This investigation has various applications in engineering and in practical problems such as very large scale integration(VLSI) of electronic chips and film coating.     

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

A finite difference scheme using a modified marker‐and‐cell (MAC) method is applied to investigate the characteristics of non‐linear wave motions and their interactions with a stationary three‐dimensional body inside a numerical wave tank (NWT). The Navier–Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co‐ordinate system. The viscous stresses and surface tension are neglected in the dynamic free‐surface condition, and the fully non‐linear kinematic free‐surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS–MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential‐based fully non‐linear NWT, and the second‐order diffraction computation. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical simulation of transient planar flow of a viscoelastic material with two moving free surfaces

In this study, we examine the numerical simulation of transient viscoelastic flows with two moving free surfaces. A modified Galerkin finite element method is implemented to the two-dimensional non-steady motion of the fluid of the Oldroyd-B type. The fluid is initially placed between two parallel plates and bounded by two straight free boundaries. In this Lagrangian finite element method, the spatial mesh deforms in time along with the moving free boundaries. The unknown shape of the free surfaces is determined with the flow field u, v, τ, p by the deformable finite element method, combined with a predictor-corrector scheme in an uncoupled fashion. The moving free surfaces and fluid motion of both Newtonian and non-Newtonian flows are investigated. The results include the influence of surface tension, fluid inertia and elasticity.     

An adaptive boundary element approach to transient free surface flow as applied to injection molding

An adaptive (Lagrangian) boundary element approach is proposed for the general two‐dimensional simulation of confined moving‐boundary flow of viscous incompressible fluids. Only the quasi‐steady creeping (Stokes) flow of a Newtonian fluid is examined. The method is stable as it includes remeshing capabilities of the deforming moving boundary, and thus it can handle large deformations. An algorithm is developed for mesh refinement of the deforming moving‐boundary mesh. Several flow problems are presented to illustrate the utility of the approach, with particular emphasis on cavity filling and viscous fingering, as applied to conventional and gas‐assisted injection molding. The accuracy of the method is assessed through the problem of jet flow and the transient fountain flow between two flat plates. Copyright © 2000 John Wiley & Sons, Ltd.     

A finite element analysis of laminar unsteady flow of viscoelastic fluids through channels with non-uniform cross-sections

The effects of non-Newtonian behaviour of a fluid and unsteadiness on flow in a channel with non-uniform cross-section have been investigated. The rheological behaviour of the fluid is assumed to be described by the constitutive equation of a viscoelastic fluid obeying the Oldroyd-B model. The finite element method is used to analyse the flow. The novel features of the present method are the adoption of the velocity correction technique for the momentum equations and of the two-step explicit scheme for the extra stress equations. This approach makes the computational scheme simple in algorithmic structure, which therefore implies that the present technique is capable of handling large-scale problems. The scheme is completed by the introduction of balancing tensor diffusivity (wherever necessary) in the momentum equations. It is important to mention that the proper boundary condition for pressure (at the outlet) has been developed to solve the pressure Poisson equation, and then the results for velocity, pressure and extra stress fields have been computed for different values of the Weissenberg number, viscosity due to elasticity, etc. Finally, it is pertinent to point out that the present numerical scheme, along with the proper boundary condition for pressure developed here, demonstrates its versatility and suitability for analysing the unsteady flow of viscoelastic fluid through a channel with non-uniform cross-section.     

An adaptive Lagrangian boundary element approach for three‐dimensional transient free‐surface Stokes flow as applied to extrusion,thermoforming, and rheometry

An adaptive (Lagrangian) boundary element approach is proposed for the general three‐dimensional simulation of confined free‐surface Stokes flow. The method is stable as it includes remeshing capabilities of the deforming free surface and thus can handle large deformations. A simple algorithm is developed for mesh refinement of the deforming free‐surface mesh. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several flow problems are presented to illustrate the utility of the approach, particularly as encountered in polymer processing and rheology. These problems illustrate the transient nature of the flow during the processes of extrusion and thermoforming, the elongation of a fluid sample in an extensional rheometer, and the coating of a sphere. Surface tension effects are also explored. Copyright © 2001 John Wiley & Sons, Ltd.     

MHD free-convective flow of micropolar and Newtonian fluids through porous medium in a vertical channel

We have studied the fully-developed free-convective flow of an electrically conducting fluid in a vertical channel occupied by porous medium under the influence of transverse magnetic field. The internal prefecture of the channel is divided into two regions; one region filled with micropolar fluid and the other region with a Newtonian fluid or both the regions filled by Newtonian fluids. Analytical solutions of the governing equations of fluid flow are found to be in excellent agreement with analytical prediction. Analytical results for the details of the velocity, micro-rotation velocity and temperature fields are shown through graphs for various values of physical parameters. It is noticed that Newtonian fluids prop up the linear velocity of the fluid in contrast to micropolar fluid. Also the skin friction coefficient at both the walls is derived and its numerical values are offered through tables.     

Electrokinetic Mixing and Displacement of Charged Droplets in Hydrogels

Mixing in droplets is an essential task in a variety of microfluidic systems. Inspired by electrokinetic mixing, electric field-induced hydrodynamic flow inside a charged droplet embedded in an unbounded polyelectrolyte hydrogel is investigated theoretically. In this study, the polyelectrolyte hydrogel is modeled as a soft, and electrically charged porous solid saturated with a salted Newtonian fluid, and the droplet is considered an incompressible Newtonian fluid. The droplet-hydrogel interface is modeled as a surface, which is located at the plane of shear, with the electrostatic potential \(\zeta \) . The fluid inside the droplet attains a finite velocity owing to hydrodynamic coupling with the electroosmotic flow arising from the droplet and polymer charge. The fluid velocity inside the droplet is linearly proportional to the electroosmotic flow velocity in the charged gel and the electroosmotic flow velocity beyond the electrical double layer of a charged interface. It is found that the polymer boundary condition at the droplet surface and the viscosities of the fluids inside and outside the droplet significantly modulate the interior fluid flow. The ionic strength and the permeability of the polymer network impact the flow differently depending on whether the flow arises from the droplet or polymer charge. Finally, the displacement of a charged droplet embedded in a gel under the influence of an external electric field is undertaken. This work is motivated by experimental attempts, which can register sub-nanometer-scale inclusion displacements in hydrogels, to advance electrical microrheology as a diagnostic tool for probing inclusion-hydrogel interfaces. In the absence of polymer charge, a close connection is found between the electrical response of a charged droplet when it is immobilized in an uncharged incompressible gel and when it is dispersed in a Newtonian electrolyte.     

Variational approach to unsteady flow and heat transfer in a channel

The transient thermal response of a Newtonian fluid to several assumed forms of pressure pulse in a channel is analyzed by the semi-direct variational method of Kantorovich. The temperature distribution in unsteady laminar flow in a channel is investigated when the pressure gradient is an arbitrary function of time. Results are obtained for the linearly varying as well as the suddenly applied pressure gradient. Received on 1 July 1997     

A transient natural convection of micropolar fluids over a vertical cylinder

A transient free convective boundary layer flow of micropolar fluids past a semi-infinite cylinder is analysed in the present study. The transformed dimensionless governing equations for the flow, microrotation and heat transfer are solved by using the implicit scheme. For the validation of the current numerical method heat transfer results for a Newtonian fluid case where the vortex viscosity is zero are compared with those available in the existing literature, and an excellent agreement is obtained. The obtained results concerning velocity, microrotation and temperature across the boundary layer are illustrated graphically for different values of various parameters and the dependence of the flow and temperature fields on these parameters is discussed. An increase in the vortex viscosity tends to increase the magnitude of microrotation and thus decreases the peak velocity of fluid flow. An increase in the vortex viscosity in micropolar fluids is shown to decrease the heat transfer rate.     

Velocity field of convergent flow into a rectangular slit for a polymeric fluid

Three-dimensional velocity distributions in the entry region of a rectangular slit contraction were investigated using a dual-beam laser Doppler velocimeter. The flow of a silicone oil (a Newtonian fluid) and a solution of silicone rubber in the same silicone oil (a viscoelastic fluid) was studied at low Reynolds numbers (Re < 0.5). In contrast to the usual velocity distribution of a Newtonian fluid, the viscoelastic fluid showed the following characteristic features: (1) a pronounced axial velocity overshoot immediately after the slit entrance and a maximum before the slit exit; (2) appearance of an axial flow deceleration region just before the sharp acceleration near the slit entrance. Even more remarkably, a saddle form of velocity profile was found in the entrance region. This flow pattern is completely different from that found for Newtonian fluids and has not yet been explained using existing rheological analysis.Parts of this paper were presented at the IX. Intern. Congress on Rheology at Acapulco (Mexico), October 8–13, 1984     

A 2D boundary element method for simulating the deformation of axisymmetric compound non‐Newtonian drops

The boundary integral formulation of the solution to the Stokes equations is used to describe the deformation of small compound non‐Newtonian axisymmetric drops suspended in a Newtonian fluid that is subjected to an axisymmetric flow field. The non‐Newtonian stress is treated as a source term in the Stokes equations, which yields an extra integral over the domains containing non‐Newtonian material. By transforming the integral representation for the velocity to cylindrical co‐ordinates and performing the integration over the azimuthal direction analytically, the dimension of the problem can be reduced from three to two. A boundary element method for the remaining two‐dimensional problem aimed at the simulation of the deformation of such axisymmetric compound non‐Newtonian drops is developed. Apart from a numerical validation of the method, simulation results for a drop consisting of an Oldroyd‐B fluid and a viscoelastic material are presented. Moreover, the method is extended to compound drops that are composed of a viscous inner core encapsulated by a viscoelastic material. The simulation results for these drops are verified against theoretical results from literature. Moreover, it is shown that the method can be used to identify the dominant break‐up mechanism of compound drops in relation to the specific non‐Newtonian character of the membrane. Copyright © 1999 John Wiley & Sons, Ltd.