Computation of flow of viscoelastic fluids by parameter differentiation

A technique combining the features of parameter differentiation and finite differences is presented to compute the flow of viscoelastic fluids. Two flow problems are considered: (i) three-dimensional flow near a stagnation point and (ii) axisymmetric flow due to stretching of a sheet. Both flows are characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. The exact numerical solutions are obtained using the technique described in the paper. Also, the first-order perturbation solutions (in terms of the viscoelastic fluid parameter) are derived. A comparison of the results shows that the perturbation method is inadequate in predicting some of the vital characteristic features of the flows, which can possibly be revealed only by the exact numerical solution.     

Laminar flow through a channel with uniformly porous walls of different permeability

Summary The steady laminar flow of a viscous incompressible fluid through a two-dimensional channel, having fluid sucked or injected with different velocities through its uniformly porous parallel walls is considered. A solution for small suction Reynolds number has been given by the authors in a previous paper. The purpose of this paper is to present a solution valid for large Reynolds numbers for the cases of (i) suction at both walls, and (ii) suction at one wall and injection at the other. A technique of matching outer and inner expansions is used to obtain an asymptotic solution for both of these cases. Further a perturbation solution for the case of suction at one wall and injection at the other is obtained by choosing the difference between two wall velocities as the perturbation parameter. Both asymptotic and perturbation solutions are confirmed by exact numerical solutions. As expected, the resulting solutions show the presence of the usual suction boundary layers in both types of flow considered in this paper.     

Non-Darcy Mixed Convection in a Vertical Porous Channel with Boundary Conditions of Third Kind

Combined free and forced convection flow in a parallel plate vertical channel filled with porous matrix is analyzed in the fully developed region with boundary conditions of third kind. The flow is modeled using the Brinkman?CForchheimer-extended Darcy equations. The plates exchange heat with an external fluid. Both conditions of equal and different reference temperatures of the external fluid are considered. Governing equations are solved numerically by shooting technique that uses classical explicit Runge?CKutta scheme and Newton?CRaphson method as a correction scheme and analytically using perturbation series method for Darcy model. The velocity field, the temperature field and Nusselt numbers are obtained for governing parameters such as porous parameter, inertia term and perturbation parameter for equal and unequal Biot numbers and are displayed graphically. The dimensionless mean velocity and bulk temperature are also determined. It is found that the numerical solutions agree for small values of the perturbation parameter in the absence of the inertial forces.     

Numerical solution of the three‐dimensional fluid flow in a rotating heterogeneous porous channel

A numerical solution to the problem of the three‐dimensional fluid flow in a long rotating heterogeneous porous channel is presented. A co‐ordinate transformation technique is employed to obtain accurate solutions over a wide range of porous media Ekman number values and consequent boundary layer thicknesses. Comparisons with an approximate asymptotic solution (for large values of Ekman number) and with theoretical predictions on the validity of Taylor–Proudman theorem in porous media for small values of Ekman number show good qualitative agreement. An evaluation of the boundary layer thickness is presented and a power‐law correlation to Ekman number is shown to well‐represent the results for small values of Ekman number. The different three‐dimensional fluid flow regimes are presented graphically, demonstrating the distinct variation of the flow field over the wide range of Ekman numbers used. Copyright © 1999 John Wiley & Sons, Ltd.     

Oberbeck convection flow of a couple stress fluid through a vertical porous stratum

The flow and heat transfer characteristics of Oberbeck convection of a couple stress fluid in a vertical porous stratum is investigated. The perturbation method of solution is obtained in terms of buoyancy parameter N valid for small values of N. This limitation is relaxed through numerical solutions using the finite difference technique with an error of 0.1×10^-7. The effect of increase in the values of temperature difference between the plates, permeability parameter and couple stress parameter on velocity, temperature, mass flow rate, skin friction and rate of heat transfer are reported. A new achievement is explored to analyse the flow for strong, weak and comparable porosity with the couple stress parameter. It is noted that both the porous parameter and the couple stress parameter suppress the flow. Higher-temperature difference is required to achieve the mass flow rate equivalent to that of viscous flow.     

Viscoelastic stresses in the stagnation flow of a dilute polymer solution

In this paper, we consider viscoelastic stresses T₁₁, T₁₂ and T₂₂ arising in the stagnation flow of a dilute polymer solution; in particular, we consider an upper convected Maxwell (UCM) fluid. We present exact solutions to the coupled partial differential equations describing the viscoelastic stresses and deduce the results for the stress T₂₂ of Becherer et al. [P. Becherer, A.N. Morozov, W. van Saarloos, Scaling of singular structures in extensional flow of dilute polymer solutions, J. Non-Newtonian Fluid Mech. 153 (2008) 183–190]. As we considered the viscoelastic stresses over two spatial variables, we are able to study the effect of variable boundary data at the inflow. As such, our results are applicable to a wider range of fluid flow problems.     

A singular perturbation problem of laminar flow in a uniformly porous channel with large injection and with an applied transverse magnetic field

Summary The problem of the steady flow of an electrically conducting viscous fluid through porous walls of a channel in the presence of an applied transverse magnetic field is considered. A solution for the case of small M ²/R (where M = Hartmann number, R = suction Reynolds number) with large blowing at the walls has been given by Terrill and Shrestha [3]. Their solution, on differentiating three times, is found to become infinite at the centre of the channel. Physically this means that there must be a viscous layer at the centre of the channel and Terrill and Shrestha are neglecting the shear layer. In this paper the solution given by Terrill and Shrestha is extended by obtaining an extra term of the series of expansion and the method of inner and outer expansion is used to obtain the complete solution which includes the viscous layer. The resulting series solutions are confirmed by numerical results.     

Analytical Solution for Forced Convection in a Semi-Circular Channel Filled with a Porous Medium

Fully developed laminar forced convection inside a semi-circular channel filled with a Brinkman-Darcy porous medium is studied. Analytical solutions for flow and constant flux heat transfer are found using a mixture of Cartesian and cylindrical coordinates. The problem depends on a parameter s, which is proportional to the inverse square of the Darcy number. Velocity boundary layers exist when s is large. Both friction factor-Reynolds number product and Nusselt number are determined. Closed form expressions for the clear fluid () limit are found. Rare analytical solutions not only describe fundamental channel flows, but also serve as a check for more complicated numerical solutions.     

Asymptotic solutions of the flow of a Johnson-Segalman fluid through a slowly varying pipe using double perturbation strategy

A double perturbation strategy is presented to solve the asymptotic solutions of a Johnson-Segalman (J-S) fluid through a slowly varying pipe. First, a small parameter of the slowly varying angle is taken as the small perturbation parameter, and then the second-order asymptotic solution of the flow of a Newtonian fluid through a slowly varying pipe is obtained in the first perturbation strategy. Second, the viscoelastic parameter is selected as the small perturbation parameter in the second perturbation strategy to solve the asymptotic solution of the flow of a J-S fluid through a slowly varying pipe. Finally, the parameter effects, including the axial distance, the slowly varying angle, and the Reynolds number, on the velocity distributions are analyzed. The results show that the increases in both the axial distance and the slowly varying angle make the axial velocity slow down. However, the radial velocity increases with the slowly varying angle, and decreases with the axial distance. There are two special positions in the distribution curves of the axial velocity and the radial velocity with different Reynolds numbers, and there are different trends on both sides of the special positions. The double perturbation strategy is applicable to such problems with the flow of a non-Newtonian fluid through a slowly varying pipe.     

Peristaltic motion of third grade fluid in curved channel

＂ Analysis is performed to study the slip effects on the peristaltic flow of non-Newtonian fluid in a curved channel with wall properties. The resulting nonlinear partial differential equations are transformed to a single ordinary differential equation in a stream function by using the assumptions of long wavelength and low Reynolds number. This differential equation is solved numerically by employing the built-in routine for solving nonlinear boundary value problems （BVPs） through the software Mathematica. In addition, the analytic solutions for small Deborah number are computed with a regular perturbation technique. It is noticed that the symmetry of bolus is destroyed in a curved channel. An intensification in the slip effect results in a larger magnitude of axial velocity. Further, the size and circulation of the trapped boluses increase with an increase in the slip parameter. Different from the case of planar channel, the axial velocity profiles are tilted towards the lower part of the channel. A comparative study between analytic and numerical solutions shows excellent agreement.     

Application of differential quadrature method to solve entry flow of viscoelastic second‐order fluid

The entry flow of viscoelastic second‐order fluid between two parallel plates is discussed. The governing equations of vorticity and the streamfunction are expanded with respect to a small parameter that characterizes the elasticity of the fluid by means of the standard perturbation method. By using the differential quadrature method with only a few grid points, high‐accurate numerical solutions are obtained. The numerical results show a lot of the features of a viscoelastic second‐order fluid. Copyright © 1999 John Wiley & Sons, Ltd.     

Extended homotopy perturbation method and the axisymmetric flow past a porous stretching sheet

The extended homotopy perturbation method, which is an extension of the celebrated homotopy perturbation method (HPM), is applied to obtain a solution to the problem of the steady, laminar, axisymmetric flow of a viscous, incompressible fluid past a porous stretching sheet. The solution so obtained is totally analytical and is expressible in terms of the cross‐flow velocity of the fluid past the stretching sheet. Its hallmark is that it does not depend upon computation of any auxiliary parameter for enlarging the convergence region of the solution. Rather, it calculates the solution automatically adjusting the scaling factor of the independent similarity variable normal to the sheet. The results obtained by the extended HPM are in excellent agreement with the exact numerical solution. Also, an asymptotic solution valid for large suction parameter is developed, which matches well with the exact solution even for moderate values of the suction parameter. Copyright © 2011 John Wiley & Sons, Ltd.     

Flow and Heat Transfer of Viscous Fluids Saturated in Porous Media over a Permeable Non-isothermal Stretching Sheet

The present paper deals with the flow and heat transfer of a viscous fluid saturated in a porous medium past a permeable and non-isothermal stretching sheet with internal heat generation or absorption and radiation. Closed-form solutions to steady, two dimensional momentum equations with neglecting quadratic inertia terms and heat transfer equation are found using a similarity transformation. Asymptotic expressions of the temperature functions are also presented valid for both very large and very small modified Prandtl numbers. Attention is focused on the effects of porous parameter K, suction parameter R, radiation parameter Nr, viscosity ratio Λ, internal heat parameter α and Prandtl number P to the characteristics of flow and heat transfer.     

Unsteady three‐dimensional boundary layer flow due to a stretching surface in a micropolar fluid

In this paper, the unsteady three‐dimensional boundary layer flow due to a stretching surface in a viscous and incompressible micropolar fluid is considered. The partial differential equations governing the unsteady laminar boundary layer flow are solved numerically using an implicit finite‐difference scheme. The numerical solutions are obtained which are uniformly valid for all dimensionless time from initial unsteady‐state flow to final steady‐state flow in the whole spatial region. The equations for the initial unsteady‐state flow are also solved analytically. It is found that there is a smooth transition from the small‐time solution to the large‐time solution. The features of the flow for different values of the governing parameters are analyzed and discussed. The solutions of interest for the skin friction coefficient with various values of the stretching parameter c and material parameter K are presented. Copyright © 2011 John Wiley & Sons, Ltd.     

Existence of a Weak Solution to a Nonlinear Fluid–Structure Interaction Problem Modeling the Flow of an Incompressible, Viscous Fluid in a Cylinder with Deformable Walls

We study a nonlinear, unsteady, moving boundary, fluid–structure interaction (FSI) problem arising in modeling blood flow through elastic and viscoelastic arteries. The fluid flow, which is driven by the time-dependent pressure data, is governed by two-dimensional incompressible Navier–Stokes equations, while the elastodynamics of the cylindrical wall is modeled by the one-dimensional cylindrical Koiter shell model. Two cases are considered: the linearly viscoelastic and the linearly elastic Koiter shell. The fluid and structure are fully coupled (two-way coupling) via the kinematic and dynamic lateral boundary conditions describing continuity of velocity (the no-slip condition), and the balance of contact forces at the fluid–structure interface. We prove the existence of weak solutions to the two FSI problems (the viscoelastic and the elastic case) as long as the cylinder radius is greater than zero. The proof is based on a novel semi-discrete, operator splitting numerical scheme, known as the kinematically coupled scheme, introduced in Guidoboni et al. (J Comput Phys 228(18):6916–6937, 2009) to numerically solve the underlying FSI problems. The backbone of the kinematically coupled scheme is the well-known Marchuk–Yanenko scheme, also known as the Lie splitting scheme. We effectively prove convergence of that numerical scheme to a solution of the corresponding FSI problem.     

A regular boundary element method for fluid flow

The Boundary Element Method is now well established as a valid numerical technique for the solution of field problems, equal to the Finite Element Method in generality and surpassing it in computational efficiency in some cases.¹ In this paper is presented a 'Regular Boundary Element Method' as applied to inviscid laminar fluid flow problems. It involves the formation of a system of regular integral equations obtained by moving the singularity outside the domain of the given problem. It is also shown that non-conforming elements may be used whereby freedoms are not defined at the geometric nodes under the boundary element discretization. A linear element is developed here; higher order variants could easily be defined. Satisfactory numerical results have been obtained using the proposed regular method with both conventional (continuous across the boundary) and non-conforming boundary elements for two-dimensional inviscid laminar fluid flow problems having regular and singular solutions.     

Transient free convection flow between long vertical parallel plates with ramped wall temperature at one boundary in the presence of thermal radiation and constant mass diffusion

The unsteady laminar free convection flow between two long vertical parallel plates with ramped wall temperature at one boundary has been investigated in the presence of thermal radiation and chemical species concentration. The exact solutions of the momentum, energy and concentration equations have been obtained using the Laplace transform technique. The velocity and temperature profiles, skin-friction and Nusselt number variations are shown graphically and the numerical values of the volume flow rate, the total heat rate and species rate added to the fluid are presented in a table. The influence of different system parameters such as the radiation parameter (R), buoyancy ratio parameter (N), Schmidt number (Sc) and time (t) has been analyzed carefully. A critical analysis of the coupled heat and mass transfer phenomena is provided. The free convective flow due to ramped wall temperature has also been compared with the baseline case of flow due to constant wall temperature.     

MHD free convection flow of visco-elastic fluid past an infinite vertical porous plate

 This work provides a comprehensive theoretical analysis of a two-dimensional unsteady free convection flow of an incompressible, visco-elastic fluid past an infinite vertical porous plate. Solutions for the zero order perturbation velocity profile, the first order perturbation velocity and temperature profiles in closed form are obtained with the help of Laplace transform technique. The numerical solutions are carried out for the Prandtl number 0.1, 0.72, 1.0, 1.5 and 2.0 which are appropriate for different types of liquid metals and for different values of magnetic field parameter, M. Received on 1 September 1999     

Rotating electroosmotic flow of an Eyring fluid

A perturbation analysis is presented in this paper for the electroosmotic(EO) flow of an Eyring fluid through a wide rectangular microchannel that rotates about an axis perpendicular to its own. Mildly shear-thinning rheology is assumed such that at the leading order the problem reduces to that of Newtonian EO flow in a rotating channel, while the shear thinning effect shows up in a higher-order problem.Using the relaxation time as the small ordering parameter,analytical solutions are deduced for the leading-as well as first-order problems in terms of the dimensionless Debye and rotation parameters. The velocity profiles of the Ekman–electric double layer(EDL) layer, which is the boundary layer that arises when the Ekman layer and the EDL are comparably thin, are also deduced for an Eyring fluid. It is shown that the present perturbation model can yield results that are close to the exact solutions even when the ordering parameter is as large as order unity. By this order of the relaxation time parameter, the enhancing effect on the rotating EO flow due to shear-thinning Eyring rheology can be significant.     

Numerical simulation of vortical ideal fluid flow through curved channel

A numerical algorithm to study the boundary‐value problem in which the governing equations are the steady Euler equations and the vorticity is given on the inflow parts of the domain boundary is developed. The Euler equations are implemented in terms of the stream function and vorticity. An irregular physical domain is transformed into a rectangle in the computational domain and the Euler equations are rewritten with respect to a curvilinear co‐ordinate system. The convergence of the finite‐difference equations to the exact solution is shown experimentally for the test problems by comparing the computational results with the exact solutions on the sequence of grids. To find the pressure from the known vorticity and stream function, the Euler equations are utilized in the Gromeka–Lamb form. The numerical algorithm is illustrated with several examples of steady flow through a two‐dimensional channel with curved walls. The analysis of calculations shows strong dependence of the pressure field on the vorticity given at the inflow parts of the boundary. Plots of the flow structure and isobars, for different geometries of channel and for different values of vorticity on entrance, are also presented. Copyright © 2003 John Wiley & Sons, Ltd.