Hydromagnetic rotating flow of a fourth‐order fluid past a porous plate

The rotating flow in the presence of a magnetic field is a problem belonging to hydromagnetics and deserves to be more widely studied than it has been to date. In the non‐linear regime the literature is scarce. We develop the governing equations for the unsteady hydromagnetic rotating flow of a fourth‐order fluid past a porous plate. The steady flow is governed by a boundary value problem in which the order of differential equations is more than the number of available boundary conditions. It is shown that by augmenting the boundary conditions based on asymptotic structures at infinity it is possible to obtain numerical solutions of the nonlinear hydromagnetic equations. Effects of uniform suction or blowing past the porous plate, exerted magnetic field and rotation on the flow phenomena, especially on the boundary layer structure near the plate, are numerically analysed and discussed. The flow behaviours of the Newtonian fluid and second‐, third‐ and fourth‐order non‐Newtonian fluids are compared for the special flow problem, respectively. Copyright © 2004 John Wiley & Sons, Ltd.     

A numerical simulation of unsteady blood flow through multi-irregular arterial stenoses

An unsteady mathematical model to study the characteristics of blood flowing through an arterial segment in the presence of a couple of stenoses with surface irregularities is developed. The flow is treated to be axisymmetric, with an outline of the stenoses obtained from a three dimensional casting of a mildly stenosed artery [1], so that the flow effectively becomes two-dimensional. The governing equations of motion accompanied by appropriate choice of boundary and initial conditions are solved numerically by MAC (Marker and Cell) method in cylindrical polar coordinate system in staggered grids and checked numerical stability with desired degree of accuracy. The pressure-Poisson equation has been solved by successive-over-relaxation (SOR) method and the pressure–velocity correction formulae have been derived. The flexibility of the arterial wall has also been accounted for in the present investigation. Further, in-depth study in the flow pattern reveals that the separation Reynolds number for the multi-irregular stenoses is lower than those for cosine-shaped stenoses and a long single irregular stenosis. The present results predict the excess pressure drop across the cosine stenoses than the irregular ones and show quite consistency with several existing results in the literature which substantiate sufficiently to validate the applicability of the model under consideration.     

Numerical simulation of laminar flow past a circular cylinder

The present paper focuses on the analysis of two- and three-dimensional flow past a circular cylinder in different laminar flow regimes. In this simulation, an implicit pressure-based finite volume method is used for time-accurate computation of incompressible flow using second order accurate convective flux discretisation schemes. The computation results are validated against measurement data for mean surface pressure, skin friction coefficients, the size and strength of the recirculating wake for the steady flow regime and also for the Strouhal frequency of vortex shedding and the mean and RMS amplitude of the fluctuating aerodynamic coefficients for the unsteady periodic flow regime. The complex three dimensional flow structure of the cylinder wake is also reasonably captured by the present prediction procedure.     

Numerical solution of laminar incompressible generalized Newtonian fluids flow

This paper deals with the numerical solution of laminar viscous incompressible flows for generalized Newtonian fluids in the branching channel. The generalized Newtonian fluids contain Newtonian fluids, shear thickening and shear thinning non-Newtonian fluids. The mathematical model is the generalized system of Navier-Stokes equations. The finite volume method combined with an artificial compressibility method is used for spatial discretization. For time discretization the explicit multistage Runge-Kutta numerical scheme is considered. Steady state solution is achieved for t → ∞ using steady boundary conditions and followed by steady residual behavior. For unsteady solution a dual-time stepping method is considered. Numerical results for flows in two dimensional and three dimensional branching channel are presented.     

Analytic solution of natural convection flow of a non‐newtonian fluid between two vertical flat plates by using decomposition method

An analysis has been performed to study the natural convection of a non‐Newtonian fluid between two infinite parallel vertical flat plates and the effects of the non‐Newtonian nature of fluid on the heat transfer are studied. The governing boundary layer and temperature equations for this problem are reduced to an ordinary form and are solved by Adomian decomposition method (ADM) and numerical method. Velocity and temperature profiles are shown graphically. The obtained results are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and non‐linear equations and so can be found widely applicable in engineering and science. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1384–1395, 2010     

The flow of a variable viscosity fluid between parallel plates with shear heating

A model for the flow of a fluid through a channel with parallel plates is investigated. The channel is narrow, so that the lubrication approximation may be applied. The channel walls are maintained at a constant temperature. Shear heating effects are included and the fluid viscosity decreases exponentially with temperature. When the flow is driven solely by shear stress or imposed velocity at the top, analytical progress is possible. When pressure gradient also drives the flow the problem is solved numerically.     

Existence and uniqueness of strong–weak solutions for chemically reacting generalized second grade fluids in 2 space dimensions

The present paper is devoted to the analysis of a nonlinear system modeling unsteady flows of an incompressible non‐Newtonian fluid mixed with a reactant. We are interested on generalized second grade fluids, which are chemically reacting and whose viscosity depends both on the shear‐rate and the concentration. We prove existence and uniqueness of strong–weak solution for a flow filling in the plane and subject to space periodic boundary conditions. This result is established under the fulfillment of some assumptions on the viscosity stress tensor and the flux vector of the diffusion–convection equation reflecting the chemical reaction. Copyright © 2016 John Wiley & Sons, Ltd.     

Local strong solutions to a compressible non‐Newtonian fluid with density‐dependent viscosity

We consider the initial boundary problem for a compressible non‐Newtonian fluid with density‐dependent viscosity. The local existence of strong solution is established that is based on some compatibility condition. Moreover, it is also proved that the solutions are to blow up, and the maximum norm of velocity gradients controls the possible break down of the strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Heat transfer and Couette flow of a chemically reacting non‐linear fluid

In this paper we study the flow and heat transfer in a chemically reacting non‐linear fluid between two long horizontal parallel flat plates that are at different temperatures. The top plate is sheared, whereas the bottom plate is fixed. The fluid is modeled as a generalized power‐law fluid whose viscosity is also assumed to be a function of the concentration. The effects of radiation are neglected. The equations are made dimensionless and the boundary value problem is solved numerically; the velocity and temperature profiles are obtained for various dimensionless numbers. Published in 2009 by John Wiley & Sons, Ltd.     

Numerical solution of the generalized Stokes problem for the flow in a channel

In this article we introduce the separation of variables in the two-dimensional generalized Stokes problem. −νΔu + αu + inverted delta p = f, for the flow in a channel. Also for the first time, we discuss the implementation of the Incremental Unknowns Method with a data structure of Compressed Column Storage. Two examples of application of the Incremental Unknowns method for this problem are presented in which we compare the CPU times of three methods: Conjugate Gradient (CG), Incremental Unknowns (IU), and Uzawa Algorithm (Uzawa). © 1997 John Wiley & Sons, Inc.     

On the effective viscosity of a periodic suspension – analysis of primal and dual formulations for Newtonian and non‐Newtonian solvents

Computer simulations of the injection molding process of fiber‐reinforced plastics critically depend on the accuracy of the constitutive models. Of prime importance for the process simulation is the precise knowledge of the viscosity. Industrial applications generally feature both high shear rates and high fiber volume fractions. Thus, both the shear‐thinning behavior of the melt and the strong anisotropic effects induced by the fibers play a dominant role. Unfortunately, the viscosity cannot be determined experimentally in its full anisotropy, and analytical models cease to be accurate for the high fiber volume fractions in question. Computing the effective viscosity by a simplified homogenization approach serves as a possible remedy. This paper is devoted to the analysis of a cell problem determining the effective viscosity. We provide primal as well as dual formulations and prove corresponding existence and uniqueness theorems for Newtonian and Carreau fluids in suitable Sobolev spaces. In the Newtonian regime, the primal formulation leads to a saddle point problem, whereas a dual formulation can be obtained in terms of a coercive and symmetric bilinear form. This observation has deep implications for numerical formulations. As a by‐product, we obtain the invertibility of the effective viscosity, considered as a function, mapping the macroscopic shear rate to the macroscopic shear stress. Copyright © 2016 John Wiley & Sons, Ltd.     

Web‐spline‐based finite element approximation of some quasi‐newtonian flows: Existence‐uniqueness and error bound

This article deals with the web‐spline‐based finite element approximation of quasi‐Newtonian flows. First, we consider the scalar elliptic p‐Laplace problem. Then, we consider quasi‐Newtonian flows where viscosity obeys power law or Carreau law. We prove well‐posedness at the continuous as well as the discrete level. We give some error bounds for the solution of quasi‐Newtonian flow problem based on the web‐spline method. Finally, we provide the numerical results for the p‐Laplace problem. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 54–77, 2015     

Falling film flow of a viscoelastic fluid along a wall

In this paper, we study the heat transfer in the fully developed flow of a viscoelastic fluid, a slag layer, down a vertical wall. A new constitutive relation for the stress tensor of this fluid is proposed, where the viscosity depends on the volume fraction, temperature, and shear rate. For the heat flux vector, we assume the Fourier's law of conduction with a constant thermal conductivity. The model is also capable of exhibiting normal stress effects. The governing equations are non‐dimensionalized and numerically solved to study the effects of various dimensionless parameters on the velocity, temperature, and volume fraction. The effect of the exponent in the Reynolds viscosity model is also discussed. The different cases of shear‐thinning and shear‐thickening, cooling and heating, are compared and discussed. The results indicate that the viscous dissipation and radiation (at the free surface) cause the temperature to be higher inside the flow domain. Copyright © 2013 John Wiley & Sons, Ltd.     

Dynamic response of pulsatile flow of blood in a stenosed tapered artery

The aim of this paper is to throw some light on the rheological study of pulsatile blood flow in a stenosed tapered arterial segment. Arterial wall is considered to be rigid and flexible separately for improving the similarity to the in vivo situation. The streaming blood is considered to be Newtonian. The governing nonlinear equations of motion are sought using the well‐known stream function‐vorticity method and are solved numerically by finite difference technique. Important rheological parameters, such as axial velocity component, wall shear stress, and flow separation region are estimated in the neighborhood of the stenosis. Effects of stenosis height, vessel tapering, and wall flexibility on the blood flow are investigated properly and are explained in detail through their graphical representations.     

Effects of variable viscosity on pulsatile flow of blood in a tapered stenotic flexible artery

The objective of the present study is to investigate the effects of variable viscosity on incompressible laminar pulsatile flow of blood through an overlapping doubly constricted tapered artery. To mimic the realistic situation, wall of the artery is taken to be flexible, and physiologically relevant pulsatile flow is introduced. The governing equations of blood flow are made dimensionless. A coordinate transformation is used to make the overlapping doubly constricted wall geometry of tube to a straight tube. Taking advantage of the Stream function–Vorticity formulation, the system of partial differential equations is then solved numerically by finite difference approximations. Effects of Reynolds number, Strouhal number, degree of contraction, tapering angle, and viscosity parameters are presented graphically and analyzed. The results show that formation of stenosis and tapering disturb the flow field significantly, and degree of stenosis is more important in influencing blood flow compared with tapering.     

Flow of a non‐linear (density‐gradient‐dependent) viscous fluid with heat generation,viscous dissipation and radiation

In this paper, we study the flow of a compressible (density‐gradient‐dependent) non‐linear fluid down an inclined plane, subject to radiation boundary condition. The convective heat transfer is also considered where a source term, similar to the Arrhenius type reaction, is included. The non‐dimensional forms of the equations are solved numerically and the competing effects of conduction, dissipation, heat generation and radiation are discussed. Copyright © 2008 John Wiley & Sons, Ltd.     

Indirect boundary integral formulation for the solution of shear‐thinning flow inside single rotor mixers

An efficient indirect boundary integral formulation for the evaluation of inelastic non‐Newtonian shear‐thinning flows at low Reynolds number is presented in this article. The formulation is based on the solution of a homogeneous Stokes flow field and the use of a particular solution for the nonlinear non‐Newtonian terms that yields the complete solution to the problem. Matrix multiplications are reduced in comparison to other means of handling nonlinear terms in boundary integral formulations such as the dual reciprocity method. The iterative solution of the nonlinear system of equations has been performed with a modified Newton‐Raphson method obtaining accurate results for values of the power law index as low as 0.4 without domain partitioning. Geometries such as Couette flow and a typical industrial polymer mixer have been analyzed with the proposed method obtaining good results with a reduction in computational cost compared with other equivalent formulations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:1610–1627, 2011     

Numerical simulation of a flexible fiber deformation in a viscous flow by the immersed boundary-lattice Boltzmann method

In this paper, deformation of a mass-less elastic fiber with a fixed end, immersed in a two-dimensional viscous channel flow, is simulated numerically. The lattice-Boltzmann method (LBM) is used to solve the Newtonian flow field and the immersed-boundary method (IBM) is employed to simulate the deformation of the flexible fiber interacting with the flow. The results of this unsteady simulation including fiber deformation, fluid velocity field, and variations of the fiber length are depicted in different time-steps through the simulation time. Similar trends are observed in plots representing length change of fibers with different values of stretching constant. Also, the numerical solution reaches a steady state equivalent to the fluid channel flow over a flat plate.     

Unconditional non‐linear stability for a fluid of third grade

The thermal convection in a layer of a third grade fluid is investigated, with viscosity being a general function of temperature. We develop a non‐linear stability analysis and prove that unconditional non‐linear stability criterion is achieved using a natural energy approach. This shows that, in some sense, the equations for a fluid of third grade are preferable to those for a fluid of second grade or a dipolar fluid. Copyright © 2004 John Wiley & Sons, Ltd.     

On a new quasi‐linearization method for systems of nonlinear boundary value problems

We propose a new quasi‐linearization technique for solving systems of nonlinear equations. The method finds recursive formulae for higher order deformation equations which are then solved using the Chebyshev spectral collocation method. The implementation of the method is demonstrated by solving the coupled nonlinear equations that govern the injection of a non‐Newtonian fluid through the sides of a vertical channel. The equations are also solved numerically and comparison made with the results in the literature. The linearization method is found to be computationally efficient and accurate with a rapidly convergent series solution. Copyright © 2011 John Wiley & Sons, Ltd.