Analytical solutions for non-Newtonian fluid flows in pipe-like domains

This paper deals with some unsteady unidirectional transient flows of an Oldroyd-B fluid in unbounded domains which geometrically are axisymmetric pipe-like. An expansion theorem of Steklov is used to obtain exact solutions for flows satisfying no-slip boundary conditions. The well known solutions for a Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid and a second grade one, appear as limiting cases of our solutions. The steady state solutions are also obtained for t→∞.     

Unsteady flows of Oldroyd-B fluids in a channel of rectangular cross-section

The aim of this note is to present the exact solutions corresponding to two types of unsteady flows of an Oldroyd-B fluid in a channel of rectangular cross-section. The solutions that have been obtained satisfy both the associate partial differential equations and all imposed initial and boundary conditions. For λ_r or λ→0 they tend toward similar solutions for a Maxwell or second-grade fluid. If both λ_r and λ→0, the solutions for Navier-Stokes fluids are recovered.     

Radiative mixed convection flow of an Oldroyd-B fluid over an inclined stretching surface

A mixed convection flow of an Oldroyd-B fluid in the presence of thermal radiation is investigated. The flow is induced by an inclined stretching surface. The boundary layer equations of the Oldroyd-B fluid in the presence of heat transfer are used. Appropriate transformations reduce partial differential equations to ordinary differential equations. A computational analysis is performed for convergent series solutions. The values of the local Nusselt number are numerically analyzed. The effects of various parameters on velocity and temperature are discussed.     

Application of differential constraint method to exact solution of second-grade fluid

A differential constraint method is used to obtain analytical solutions of a second-grade fluid flow. By using the first-order differential constraint condition, exact solutions of Poiseuille flows, jet flows and Couette flows subjected to suction or blowing forces, and planar elongational flows are derived. In addition, two new classes of exact solutions for a second-grade fluid flow are found. The obtained exact solutions show that the non-Newtonian second-grade flow behavior depends not only on the material viscosity but also on the material elasticity. Finally, some boundary value problems are discussed.     

Nonlinear three-dimensional stretched flow of an Oldroyd-B fluid with convective condition,thermal radiation,and mixed convection

The effect of non-linear convection in a laminar three-dimensional Oldroyd-B fluid flow is addressed. The heat transfer phenomenon is explored by considering the non-linear thermal radiation and heat generation/absorption. The boundary layer assumptions are taken into account to govern the mathematical model of the flow analysis. Some suitable similarity variables are introduced to transform the partial differential equations into ordinary differential systems. The Runge-Kutta-Fehlberg fourth-and fifth-order techniques with the shooting method are used to obtain the solutions of the dimensionless velocities and temperature. The effects of various physical parameters on the fluid velocities and temperature are plotted and examined. A comparison with the exact and homotopy perturbation solutions is made for the viscous fluid case, and an excellent match is noted. The numerical values of the wall shear stresses and the heat transfer rate at the wall are tabulated and investigated. The enhancement in the values of the Deborah number shows a reverse behavior on the liquid velocities. The results show that the temperature and the thermal boundary layer are reduced when the nonlinear convection parameter increases. The values of the Nusselt number are higher in the non-linear radiation situation than those in the linear radiation situation.     

Exact solutions for generalized Burgers’ fluid in an annular pipe

This paper deals with some unsteady unidirectional transient flows of generalized Burgers’ fluid in an annular pipe. Exact solutions of some unsteady flows of generalized Burgers’ fluid in an annular pipe are obtained by using Hankel transform and Laplace transform. The following two problems have been studied: (1) Poiseuille flow due to a constant pressure gradient; (2) axial Couette flow in a annulus. The well known solutions for Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid, a second grade fluid and an Oldroyd-B fluid appear as limiting cases of our solutions.     

Coaxial-disk flow of an Oldroyd-B fluid: exact solution and stability

This paper reports an exact solution for the coaxial disk flow of an Oldroyd-B fluid. The flow is approximately generated by the parallel-plate viscometer. Asymptotic and numerical solutions are reported showing that there is a critical Weissenberg number based on the angular velocity and the Maxwellian relaxation time, above which the flow is unstable. A linearized stability analysis for the basic inertialess flow confirms this numerical instability and yields the critical Weissenberg number.     

Stagnation flows for the Oldroyd-B fluid

Exact solutions to the plane and axi-symmetric stagnation flows of an Oldroyd-B fluid are reported. It is found that a steady flow is possible if the Weissenberg numberWi, defined by the product of the Maxwellian relaxation time and the shear rate at infinity, satisfies – 1/2 <Wi < 1/m, wherem = 1 in an axisym-metric flow andm = 2 in a plane flow. Furthermore, the fluid elasticity always decreases the boundary-layer thickness. An Oldroyd-B fluid with the parameters matched those of a typical Boger fluid behaves essentially like a Newtonian fluid in a stagnation flow.     

On the consequence of discretization errors in the numerical calculation of viscoelastic flow

Having recalled some theorems pertaining to strong solutions of the viscoelastic flow problem, we establish a parallel between these theorems and observations made in the numerical calculation of such flows by means of a mixed method. It is shownt that numerical errors in the evaluation of the extra-stress tensor have dramatic consequences upon the other field variables for the flow of a Maxwell fluid, and that the damage is limited with an Oldroyd-B fluid.     

Exact solutions in generalized Oldroyd-B fluid

This investigation deals with the influence of slip condition on the magnetohydrodynamic (MHD) and rotating flow of a generalized Oldroyd-B (G.Oldroyd-B)fluid occupying a porous space.Fractional calcul...     

Axial Couette flow of an Oldroyd-B fluid in an annulus

This paper establishes the velocity field and the adequate shear stress corresponding to the motion of an Oldroyd-B fluid between two infinite coaxial circular cylinders by means of finite Hankel transforms. The flow of the fluid is produced by the inner cylinder which applies a time-dependent longitudinal shear stress to the fluid. The exact analytical solutions, presented in series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. The general solutions can be easily specialized to give similar solutions for Maxwell, second grade and Newtonian fluids performing the same motion. Finally, some characteristics of the motion as well as the influence of the material parameters on the behavior of the fluid motion are graphically illustrated.     

Transient viscoelastic helical flows in pipes of circular and annular cross-section

Start-up helical flows for Oldroyd-B and upper-convected Maxwell fluids are studied in straight pipes of circular and annular cross-section. The differential form of the constitutive equation leads to partial differential equations which are second-order in space and time. Apart from the condition that the fluid is initially at rest another initial condition is required to complete the solution process. By comparing results derived from the integral form of the constitutive equation we show that an appropriate initial condition may be found. Numerical results for start-up rotational flow in pipes of annular cross-section are presented.     

纤维悬浮液搅拌流动的数值模拟

由于缺乏适当的本构方程，对纤维悬浮液流动的研究一直局限于纤维的牛顿流体悬浮液。本文采用ＭＵＣＭ模型对作者最近提出的纤维Ｏｌｄｒｏｙｄ－Ｂ流体悬浮液的本构方程作了改进，并对锚式桨搅拌槽的二维Ｏｌｄｒｏｙｄ－Ｂ流体和牛顿流体纤维悬浮液搅拌流动作了数值模拟。模拟的结果表明，本文所用的模型和方法能有效地抑制过大局部应力的影响并合理地处理流体的记忆效应。     

Stokes’ first problem for some non-Newtonian fluids: Results and mistakes

The well-known problem of unidirectional plane flow of a fluid in a half-space due to the impulsive motion of the plate it rests upon is discussed in the context of the second-grade and the Oldroyd-B non-Newtonian fluids. The governing equations are derived from the conservation laws of mass and momentum and three correct known representations of their exact solutions given. Common mistakes made in the literature are identified. Simple numerical schemes that corroborate the analytical solutions are constructed.     

Three dimensional finite element analysis of the flow of polymer melts

The finite element simulation of a selection of two- and three-dimensional flow problems is presented, based upon the use of four different constitutive models for polymer melts (Oldroyd-B, Rolie-Poly, Pom-Pom and XPP). The mathematical and computational models are first introduced, before their application to a range of visco-elastic flows is described. Results demonstrate that the finite element models used here are able to re-produce predictions made by other published numerical simulations and, significantly, by carefully conducted physical experiments using a commercial-grade polystyrene melt in a three-dimensional contraction geometry. The paper also presents a systematic comparison and evaluation of the differences between two- and three-dimensional simulations of two different flow regimes: flow of an Oldroyd-B fluid around a cylinder and flow of a Rolie-Poly fluid into the contraction geometry. This comparison allows new observations to be made concerning the relatively poor quality of two-dimensional simulations for flows in even quite deep channels.     

Laminar axisymmetric flow of a weakly compressible viscoelastic fluid

The combined effects of weak compressibility and viscoelasticity in steady, isothermal, laminar axisymmetric Poiseuille flow are investigated. Viscoelasticity is taken into account by employing the Oldroyd-B constitutive model. The fluid is assumed to be weakly compressible with a density that varies linearly with pressure. The flow problem is solved using a regular perturbation scheme in terms of the dimensionless isothermal compressibility parameter. The sequence of partial differential equations resulting from the perturbation procedure is solved analytically up to second order. The two-dimensional solution reveals the effects of compressibility and the other dimensionless numbers and parameters in the flow. Expressions for the average pressure drop, the volumetric flow rate, the total axial stress, as well as for the skin friction factor are also derived and discussed. The validity of other techniques used to obtain approximate solutions of weakly compressible flows is also discussed in conjunction with the present results.     

Numerical simulation of viscoelastic flows with Oldroyd-B constitutive equations and novel algebraic stress models

The purpose of the present study is to compare numerical simulations of viscoelastic flows using the differential Oldroyd-B constitutive equations and two newly devised simplified algebraic explicit stress models (AES-models). The flows of a viscoelastic fluid in a 180° bent planar channel and in a 4:1 planar contraction are considered to illustrate and support the underlying theory. The flow in the bent channel is used to illustrate the frame-invariant property of the new models in a pure shear flow exhibiting strong streamline curvature. The flow in the 4:1 contraction serves as a benchmark test in a situation where strong elongation occurs. For both geometries, it is found that the predictions of the new AES-models are in good agreement with Oldroyd-B up to Deborah numbers of order 0.5, with a significant reduction in computational effort.     

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.     

Rotational flow of Oldroyd-B nanofluid subject to Cattaneo-Christov double diffusion theory

A nanofluid is composed of a base fluid component and nanoparticles, in which the nanoparticles are dispersed in the base fluid. The addition of nanoparticles into a base fluid can remarkably improve the thermal conductivity of the nanofluid, and such an increment of thermal conductivity can play an important role in improving the heat transfer rate of the base fluid. Further, the dynamics of non-Newtonian fluids along with nanoparticles is quite interesting with numerous industrial applications. The present predominately predictive modeling studies the flow of the viscoelastic Oldroyd-B fluid over a rotating disk in the presence of nanoparticles. A progressive amendment in the heat and concentration equations is made by exploiting the Cattaneo-Christov heat and mass flux expressions. The characteristic of the Lorentz force due to the magnetic field applied normal to the disk is studied. The Buongiorno model together with the Cattaneo-Christov theory is implemented in the Oldroyd-B nanofluid flow to investigate the heat and mass transport mechanism. This theory predicts the characteristics of the fluid thermal and solutal relaxation time on the boundary layer flow. The von K′arm′an similarity functions are utilized to convert the partial differential equations(PDEs) into ordinary differential equations(ODEs). A homotopic approach for obtaining the analytical solutions to the governing nonlinear problem is carried out. The graphical results are obtained for the velocity field, temperature, and concentration distributions. Comparisons are made for a limiting case between the numerical and analytical solutions, and the results are found in good agreement. The results reveal that the thermal and solutal relaxation time parameters diminish the temperature and concentration distributions, respectively. The axial flow decreases in the downward direction for higher values of the retardation time parameter. The impact of the thermophoresis parameter boosts the temperature distribution.     

Boundary layer flow of Oldroyd-B fluid by exponentially stretching sheet

The present paper investigates the steady flow of an Oldroyd-B fluid. The fluid flow is induced by an exponentially stretched surface. Suitable transformations reduce a system of nonlinear partial differential equations to a system of ordinary differential equations. Convergence of series solution is discussed explicitly by a homotopy analysis method (HAM). Velocity, temperature and heat transfer rates are examined for different involved parameters through graphs. It is revealed that for a larger retardation time constant, the velocity is enhanced and the temperature is lowered. It is noted that relaxation time constant and the Prandtl number enhance the heat transfer rate.