Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid

Summary. This paper deals with some steady unidirectional flows of an Oldroyd 8-constant magnetohydrodynamic (MHD) fluid in bounded domains. The fluid is electrically conducting in the presence of a uniform magnetic field. Three nonlinear flows are produced by the motion of a boundary or by sudden application of a constant pressure gradient or by the motion of a boundary and pressure gradient. The governing nonlinear differential equations are solved analytically using homotopy analysis method (HAM). Expressions for the velocity distribution are given. It is noted that for steady flow the solutions are strongly dependent on the non–Newtonian and magnetic parameters. The MHD solutions for a Newtonian fluid, as well as those corresponding to the Oldroyd 3 and 6-constant fluids, a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. Finally, a physical interpretation of the results is given with the help of several graphs.     

Wire coating analysis using MHD Oldroyd 8-constant fluid

The problem of wire coating by withdrawal from a bath of a magnetohydrodynamic Oldroyd 8-constant fluid is investigated. The fluid is electrically conducting in the presence of a uniform applied magnetic field. The obtained non-linear differential equation has been solved using homotopy analysis method. The solution is given in the form of a series. The convergence of the series is explicitly discussed. The effects of emerging non-Newtonian parameters and the Hartman number is seen. The results are presented graphically and discussed.     

Flow of an Oldroyd 8-constant fluid in a convergent channel

Summary The problem considered is the steady flow of an Oldroyd 8-constant fluid in a convergent channel. Using series expansions in terms of decreasing powers ofr given by Strauss [8] for the stream function and stress components, the governing equations of the problem are reduced to ordinary differential equations. The resulting differential equations have been solved by employing a numerical technique. It is shown that the streamline patterns are strongly dependent on the non-Newtonian parameters.     

Flow of an Oldroyd 6-constant fluid between intersecting planes, one of which is moving

Summary The problem dealing with the two-dimensional steady and slow flow of an Oldroyd 6-constant fluid between intersecting planes, one of which is fixed and the other moving, has been analysed. Using truncated series expansions given by Strauss [4] for the stream function and stress components, the governing equations of the problem are reduced to linear ordinary differential equations. These equations have been solved analytically subject to the relevant boundary conditions. The effects of the non-Newtonian parameters on the flow pattern are carefully delineated. There is, unlike the case of Newtonian fluid, a secondary flow near the corner.     

Exact solutions of non-Newtonian fluid flows with prescribed vorticity

Summary The equations of motion of a non-Newtonian second-grade fluid flow are highly nonlinear partial differential equations. For this reason, there exists only a limited number of exact solutions. Due to the complexity of the equations, inverse methods described by Nemenyi [1] have become attractive in the study of non-Newtonian fluids. In these methods, certain physical or geometrical properties of the flow field are assumed a priori.Lin and Tobak [2] studied steady plane viscous incompressible flows for a chosen vorticity function by decomposing the nonlinear fourth-order partial differential equation in the streamfunction. This excellent approach yielded two second-order linear partial differential equations in the streamfunction. Hui [3] used this approach to study unsteady plane viscous incompressible flows.During the past decade, there has been substantial interest in flows of viscoelastic liquids due to the occurrence of these flows in industrial processes. In this paper, we study the steady and unsteady incompressible viscous non-Newtonian second-grade fluid flows in which the vorticity is proportional to the streamfunction perturbed by a uniform stream. The solutions obtained are exact solutions and represent various non-parallel flows of second-grade fluids.The plan of this paper is as follows: In Sect. 2, the equations of motion of an unsteady plane incompressible second-grade fluid are given, and the vorticity function is assumed to be ² =A(–Ux–BUy ²). In Sect. 3, solutions for the steady flow are obtained. In Sect. 4, solutions for unsteady flow are obtained.     

Homotopy analysis of Couette and Poiseuille flows for fourth grade fluids

Summary The steady flow of a fluid, called a fourth grade fluid, between two parallel plates is considered. Depending upon the relative motion of the plates we analyze four types of flows: Couette flow, plug flow, Poiseuille flow and generalized Couette flow. In each case, the nonlinear differential equation describing the velocity field is solved using perturbation technique and homotopy analysis method. The pressure distribution is also found. It is observed that the homotopy analysis method is more efficient and flexible than the perturbation technique.     

Flow of a binary mixture of incompressible Newtonian fluids in a rectangular channel

The problems concerning some simple steady and unsteady flows of a mixture composed of two incompressible Newtonian fluids in an infinitely long channel of rectangular cross-section are examined. By means of finite Fourier sine transforms, the exact solutions of the field equations are obtained for the following four problems: (i) steady Couette flow in a rectangular channel, (ii) unsteady Couette flow in a rectangular channel, (iii) steady Poiseuille flow in a rectangular channel, (iv) unsteady Poiseuille flow in a rectangular channel.     

Numerical Solutions for Unsteady Flows of a Magnetohydrodynamic Jeffrey Fluid Between Parallel Plates Through a Porous Medium

In the present article, the numerical solutions for three fundamental unsteady flows (namely Couette, Poiseuille, and generalized Couette flows) of an incompressible magnetohydrodynamic Jeffrey fluid between two parallel plates through a porous medium are presented using differential quadrature method. The equations governing the flow of Jeffrey fluid are modeled in Cartesian coordinate system. The resulting non-dimensional differential equations are approximated by using a new scheme that is trigonometric B-spline differential quadrature method. The scheme is based on the differential quadrature method in which the weighting coefficients are obtained by using trigonometric B-splines as a set of basis functions. This scheme reduces the equation into the system of first-order ordinary differential equation which is solved by adopting strong stability-preserving time-stepping Runge–Kutta scheme. The effects of the sundry parameters of interest on the velocity profiles are studied and the results are presented through graphs. It is observed that, the velocity increases from the horizontal channel to vertical channel. The velocity is a decreasing function of magnetic parameter. With an increase in time, the velocity increases.     

A boundary-element analysis for transient viscoelastic blade coating flow

The one-dimensional plane Couette flow is first determined for a large class of Oldroyd fluids with added viscosity, which typically represent polymer solutions composed of a Newtonian solvent and a polymeric solute. Next, the determined channel velocity profile is used as the boundary condition at the channel exit for the blade coating flow. The free-surface evolution of the flow at the channel exit is simulated using the boundary element method. It is argued that the free surface flow can be assumed to be Newtonian. For the channel flow, the problem is reduced to a non-linear dynamical system using the Galerkin projection method. Stability analysis indicates that the velocity profile at the inlet may be linear or non-linear depending on the range of the Weissenberg number.     

Application of the Lambert W function to steady shearing flows of the Papanastasiou model

Using the Lambert W function, the constitutive relation of the Papanastasiou model is inverted so that the second invariant of the first Rivlin-Ericksen tensor can be expressed as a function of the second invariant of the extra stress tensor. In steady shearing flows, this results in the magnitude of the shear rate becoming a function of the magnitude of the shear stress. Since the distribution of the latter is known explicitly in channel, Poiseuille and Couette flows, one can investigate the nature of analytical solutions in these flows. It is shown that explicit answers are found for channel and Poiseuille flows only, with the Couette flow requiring a numerical solution in general. From the channel flow results, it is obvious that there is a great amount of congruence between the predictions of the Papanastasiou model and the Bingham fluid. In turn, this lends further confidence to the application of the Papanastasiou model to study the flows of Bingham fluids.     

STEADY VISCOELASTIC FLOW PAST A SPHERE USING SPECTRAL ELEMENTS

The steady flow of a viscoelastic fluid past a sphere in a cylindrical tube is considered. A spectral element method is used to solve the system of coupled non-linear partial differential equations governing the flow. The spectral element method combines the flexibility of the traditional finite element method with the accuracy of spectral methods. A time-splitting algorithm is used to determine the solution to the steady problem. Results are presented for the Oldroyd B model. These show excellent agreement with the literature. The results converge with mesh refinement. A limiting Deborah number of approximately 0⋅6 is found, irrespective of the spatial resolution.     

SPH Simulation of Low Reynolds Number Planar Shear Flow and Heat Convection

The present study reports on a set of computer programmable SPH formulations, which are used to simulate transient planar shear flows, and in particular Poiseuille flow and Couette flow with different types of body forces. The flows examined have Reynolds numbers within the range 0.05∼50. SPH results agree well with analytical solutions for those situations amenable to an analytical treatment, with the largest deviation being less than 2.0 %. The accuracy of a SPH formulation for heat convection, with particular emphasis in the viscous heat dissipation, is also tested via a steady convective heat transfer case for a combined Poiseuille and Couette flow.     

Exact solutions for Couette and Poiseuille flows for fourth grade fluids

Summary Exact solutions for four types of flows between two parallel plates are presented, viz. Couette flow, plug flow, Poiseuille flow and generalized Couette flow. The nonlinear second-order ordinary differential equation for the velocity field is solved exactly in each case. These solutions are compared to those found by perturbation and homotopy analysis methods by Siddiqui et al. [1].     

Effect of the aspect ratio on the flow characteristics of magnetohydrodynamic (MHD) third grade fluid flow through a rectangular channel

The magnetohydrodynamic (MHD) flow of a third grade fluid through a rectangular channel, considering the effect of aspect ratio, has been investigated. The flow considered is steady, laminar, incompressible and hydro-dynamically fully developed. The equation, describing the flow, is a highly non-linear partial differential equation (PDE) with remote possibility of having an exact solution and even numerical solution also is very difficult to obtain. A combination of the homotopy perturbation method (HPM) and integral method (IM) has been employed to solve the non-linear PDE which is scarce in open literature. The results of the present study are compared with the results obtained by the least square method (LSM) of the MHD third grade fluid flow through a rectangular channel, without the effect of aspect ratio and are found to be in close agreement. The results indicate that the flow field is significantly affected by the aspect ratio which should be considered for practical applications. In all the available literatures of the third grade fluid flow, the aspect ratio effect is neglected and this simplifying assumption reduces the highly complicated non-linear PDE to a non-linear ordinary differential equation (ODE). The novelty of the subject work lies in the inclusion of the effects of aspect ratio in the governing equation describing the flow of a third grade fluid through a channel and solving this by a combined analytical method (HPM and IM). Further, the effects of the Hartmann number and non-Newtonian third grade fluid parameter on the flow filed are discussed.     

Multi-domain dual reciprocity for the solution of inelastic non-Newtonian flow problems at low Reynolds number

An efficient boundary element solution of the motion of inelastic non-Newtonian fluids at low Reynolds number is presented in this paper. For the numerical solution all the domain integrals of the boundary element formulation have been transformed into equivalent boundary integrals by means of the dual reciprocity method (DRM). To achieve an accurate approximation of the non-linear and non-Newtonian terms two major improvements have been made to the DRM, namely the use of augmented thin plate splines as interpolation functions, and the partition of the entire domain into smaller subregions or domain decomposition. In each subregion or domain element the DRM was applied together with some additional equations that ensure continuity on the interfaces between adjacent subdomains. After applying the boundary conditions the final systems of equations will be sparse and the approximation of the nonlinear terms will be more localised than in the traditional DRM. This new method known as multidomain dual reciprocity (MD-DRM) has been used to solve several non-Newtonian problems including the pressure driven flow of a power law fluid, the Couette flow and two simulations of industrial polymer mixers. Received 7 February 2001     

On some similarity solutions to shear flows of non-Newtonian power law fluids

Summary The nonlinear rheological effects of non-Newtonian power law fluids on some shear flows are addressed. Exact similarity solutions to Stokes' first problem for unsteady flow generated by the vertical motion of a slender cylinder in an unbounded fluid are presented. The nonlinear effects on the velocity and shear stress distributions are shown and discussed. These reveal the existence of traveling wave characteristics for a shear thickening fluid, which determine a moving shear front for which the shear disturbances propagate with a finite velocity.     

Unsteady boundary layer flow of power-law fluid on stretching sheet surface

Two-dimensional unsteady boundary layer equations of non-Newtonian fluids are treated. Flow of a thin fluid film of a power-law caused by stretching of surface is investigated by using a similarity transformation. By using this transformation, we reduce the unsteady boundary layer equations to a non-linear ordinary differential equation system. Numerical solutions of outcoming nonlinear differential equations are found by using a combination a Runge–Kutta algorithm and shooting technique. Boundary layer thickness is explored numerically for different values of power-law index.     

Electrohydrodynamic flow between parallel dielectric channels

Steady, accelerated, and pulsating electro-dynamic flows in a plane dielectric channel are considered, along with Couette flow. It is shown that for these types of electrohydrodynamic flows the effect is concentrated in a thin layer near the walls, which can considerably change the friction stress on the walls. Some exact solutions of the energy equation are obtained.     

Hall effects on the unsteady hydromagnetic flows of an Oldroyd-B fluid

The influence of Hall currents and rotation on the oscillatory flows of an infinite plate is investigated. Exact solutions for the two problems are obtained.The fluid considered is a homogeneous Oldroyd-B. During the mathematical analysis it is found that governing differential equation for steady flow in an Oldroyd-B fluid is identical to that of viscous fluid. Further, it is observed that in absence of the strength of transverse magnetic field (B₀) the solution in resonance case does not satisfy the boundary condition at infinity. Physical significance of mathematical results is also discussed.     

Flow of electro-rheological materials

Summary An electro-rheological fluid is a material in which a particulate solid is suspended in an electrically non-conducting fluid such as oil. On the application of an electric field, the viscosity and other material properties undergo dramatic and significant changes. In this paper, the particulate imbedded fluid is considered as a homogeneous continuum. It is assumed that the Cauchy stress depends on the velocity gradient and the electric field vector. A representation for the constitutive equation is developed using standard methods of continuum mechanics. The stress components are calculated for a shear flow in which the electric field vector, is normal to the velocity vector. The model predicts (i) a viscosity which depends on the shear rate and electric field and (ii) normal stresses due to the interaction between the shear flow and the electric field. These expressions are used to study several fundamental shear flows: the flow between parallel plates, Couette flow, and flow in an eccentric rotating disc device. Detailed solutions are presented when the shear response is that of a Bingham fluid whose yield stress and viscosity depends on the electric field.