Effects of slip on steady Bödewadt flow and heat transfer of an electrically conducting non-Newtonian fluid

The steady flow and heat transfer arising due to the rotation of a non-Newtonian fluid at a larger distance from a stationary disk is extended to the case where the disk surface admits partial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The momentum equation gives rise to a highly nonlinear boundary value problem. Numerical solution of the governing nonlinear equations are obtained over the entire range of the physical parameters. The effects of slip, non-Newtonian fluid characteristics and the magnetic interaction parameter on the momentum boundary layer and thermal boundary layer are discussed in detail and shown graphically. It is observed that slip has prominent effects on the velocity and temperature fields.     

Effects of partial slip,viscous dissipation and Joule heating on Von Kármán flow and heat transfer of an electrically conducting non-Newtonian fluid

The steady Von Kármán flow and heat transfer of an electrically conducting non-Newtonian fluid is extended to the case where the disk surface admits partial slip. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The momentum equations give rise to highly non-linear boundary value problem. Numerical solutions for the governing non-linear equations are obtained over the entire range of the physical parameters. The effects of slip, magnetic parameter and non-Newtonian fluid characteristics on the velocity and temperature fields are discussed in detail and shown graphically. Emphasis has been laid to study the effects of viscous dissipation and Joule heating on the thermal boundary layer. It is interesting to find that the non-Newtonian cross-viscous parameter has an opposite effect to that of the slip and the magnetic parameter on the velocity and the temperature fields.     

Problem of steady motion for a second-grade fluid in the hölder classes of functions

The paper studies a boundary-value problem (with the usual adherence boundary condition) for a stationary system of equations of motion of second-grade fluids in a bounded domain. This system is not elliptic and contains third-order derivatives of the velocity vector field. This leads to obvious difficulties in the analysis of the problem. It is known that the problem is reduced to the usual Stokes problem and to the transport equations or their analogs. We present a new easier method of such a reduction which allows us to prove the solvability of a stationary boundary-value problem for the equations of motion of second-grade fluids in the Hölder classes of functions in the case of small exterior forces. Bibliography: 6 titles.     

Sensitivity of the slip rate coefficient in fluid flow – poroelastic coupling conditions

To model flow-induced structural vibrations, an interface to couple fluid flow and poroelastic material in a finite element formulation has been developed. One parameter of this interface condition is the slip rate coefficient, resulting from the so-called Beavers-Joseph-Saffman condition. This condition states that the jump in tangential velocity at a fluid flow – porous interface is proportional to the shear stress. Up to now no a priori determination of this parameter exists, and the known parameter range has been deducted from measurements, i. e., in our case from the results of the pore-resolving simulations. When modeling realistic problems assuming incompressible fluids, vectorial flow velocity and scalar pressure interact with the poroelastic material. As the slip rate coefficient only influences the tangential contributions, its overall influence is not clear. In this work, the sensitivity of the slip rate coefficient regarding the interface's coupling conditions is evaluated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of the problem of the flow of a compressible viscous fluid past a bounded body in the hölder classes of functions

It is shown that the stationary exterior problem for the equations of motion of a viscous compressible fluid is uniquely solvable in weighted Hölder spaces if the exterior forces and the value of the velocity at infinity are sufficiently small. As a weight function, a power function (1+|x|)^?m, m>0, is taken. The proof, which is carried out by the method of decomposition, relies on the estimates of singular integrals and solutions of the transport equations in weighted spaces. Bibliography: 5 titles.     

Effects of the self- and cross-diffusion on positive steady states for a generalized predator–prey system

This paper deals with a generalized predator–prey system with cross-diffusion and homogeneous Neumann boundary condition, where the cross-diffusion is included in such a way that the prey runs away from the predator. We first give a priori estimate for positive steady states to the system. Then we obtain the non-existence result of non-constant positive steady states. Finally, we investigate the stability of constant equilibrium point and the existence of non-constant positive steady states. It is shown that the system admits a non-constant positive steady state provided that either of the self-diffusions is large or the cross-diffusion is small.     

Comment on “Effects of the slip boundary condition on non-Newtonian flows in a channel” [Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1377–1384]

Recently, Ellahi [1] discussed the slip effects on the flows of an Oldroyd 8-constant fluid using the homotopy analysis method. Crucial flaws in [1] are pointed out in this comment. The present paper provides an exact solution and a numerical solution by shooting method using Runge–Kutta algorithm of the flow problems considered in [1] with the correct nonlinear boundary conditions.     

Cattaneo–Christov heat flux model for three-dimensional flow of a viscoelastic fluid on an exponentially stretching surface

ABSTRACT

In this article, we explore the three-dimensional boundary-layer flow over an exponentially stretching surface in two parallel ways. Constitutive equations of a second-grade fluid are used. Instead of classical Fourier’s law, Cattaneo–Christov heat flux model is employed for the formulation of the energy equation. This model can predict the effects of thermal relaxation time on the boundary layer. The resulting partial differential equations are reduced into ordinary differential equations by similarity transformations. Homotopy Analysis Method (HAM) is employed to solve the non-linear problem. Physical impact of emerging parameters on the momentum and thermal boundary-layer thickness are studied.     

Fourier–Bessel theory on flow acoustics in inviscid shear pipeline fluid flow

Flow acoustics in pipeline is of considerable interest in both industrial application and scientific research. While well-known analytical solutions exist for stationary and uniform mean flow, only numerical solutions exist for shear mean flow. Based on potential theory, a general mathematical formulation of flow acoustics in inviscid fluid with shear mean flow is deduced, resulting in a set of two second-order differential equations. According to Fourier–Bessel theory which is orthogonal and complete in Lebesgue Space, a solution is proposed to transform the differential equations to linear homogeneous algebraic equations. Consequently, the axial wave number is numerically calculated due to the existence condition of non-trivial solution to homogeneous linear algebraic equations, leading to the vanishment of the corresponding determinant. Based on the proposed method, wave propagation in laminar and turbulent flow is numerically analyzed.     

On flow of a Navier–Stokes fluid in curved pipes. Part I: Steady flow

We consider the steady, fully developed motion of a Navier–Stokes fluid in a curved pipe of cross-section $D$ under a given axial pressure gradient $G$. We show that, if $G$ is constant, this problem has a smooth steady solution, for arbitrary values of the Dean’s number $\kappa$, for $D$ of arbitrary shape and for any curvature ratio $\delta$ of the pipe. This solution is also unique for $\kappa$ sufficiently small. Moreover, we prove that the solution is unidirectional (no secondary motion) if and only if $\kappa =0$. Finally, we show the same properties for the approximations to the Navier–Stokes equations called “Dean’s equations” and provide a rigorous way in which solutions to the full Navier–Stokes equations approach those to this approximation in the limit of $\delta \to 0$.     

Energy criterion of nonlinear stability of a flow of the Rivlin–Ericksen fluid in porous medium

Nonlinear stability of the motionless state of the thermosolutal Rivlin–Ericksen fluid in porous medium for the case of stress-free boundaries is studied by generalized energy method. By means of introducing an energy functional we will prove a sufficient condition for unconditional nonlinear stability of the motionless state. The nonlinearly stabilizing effect of concentration on the system is proved. Furthermore, for certain range of system parameters our sufficient condition for stability is also necessary.     

A Remark on a Theorem of Szegö

Let be a polynomial with complex coefficients and define, for , where ||P|| is the euclidean norm of the polynomial P. By a theorem of Szegö where is the Mahler measure of F. Recently, J. Dégot proved an effective version of this result. In this paper we sharpen Dégot's result, under the additional hypotheses that F is a square-free polynomial with integer coefficients and without reciprocal factors.     

Steady flow of a viscous incompressible fluid in an unbounded «funnel-shaped» domain

In this paper we consider a domain which is tube-like at one exit to infinity and the halfspace at the other side. We prove existence of steady motions for the Navier-Stokes problem and for the case in which the fluid is moving through a porous medium at rest filling . In both cases the proof holds for arbitrary fluxes. We describe the asymptotic behaviour of the solutions in the halfspace for both problems.     

Cavity flow simulation of Carreau–Yasuda non-Newtonian fluids using PIM meshfree method

Lid-driven cavity flow of a purely-viscous non-Newtonian fluid obeying Carreau–Yasuda rheological model is studied numerically using the PIM meshfree method combined with the Characteristic-Based Split-A algorithm. Results are reported for the velocity and pressure profiles at Reynolds numbers as high as 1000 for a non-Newtonian fluid obeying Carreau–Yasuda rheological model. For Newtonian fluids, results obtained from our PIM–CBS-A method show good agreement with benchmark results published in the literature and obtained using finite difference and/or Finite Element Methods. Our numerical results are also consistent with recent published results obtained using another meshfree method called LSM with the advantage that PIM needs less “points” to achieve the same degree of accuracy. Results obtained for the Carreau–Yasuda model reveals the strong effect of the shear-thinning behavior of a fluid on its flow kinematics within the cavity.     

Quantitative properties on the steady states to a Schrödinger–Poisson–Slater System

In this paper, we obtain the existence, uniqueness and asymptotic behavior of steady states to a class of Schrödinger-Poisson-Slater System.     

Unsteady MHD two-phase Couette flow of fluid–particle suspension

The problem of two-phase unsteady MHD Couette flow between two parallel infinite plates has been studied taking the viscosity effect of the two phases into consideration. Unified closed form expressions are obtained for the velocities and the skin frictions for both cases of the applied magnetic field being fixed to either the fluid or the moving plate. The novelty of this study is that we have obtained the solution of the unsteady flow using the Laplace transform technique, D’Alemberts method and the Riemann-sum approximation method. The solution obtained is validated by assenting comparisons with the closed form solutions obtained for the steady states which have been derived separately and also by the implicit finite difference method. Graphical result for the velocity of both phases based on the semi-analytical solutions are presented and discussed. A parametric study of some of the physical parameters involved in the problem is conducted. The skin friction for both the fluid and the particle phases decreases with time on both plates until a steady state is reached, it is also observed to decrease with increase in the particle viscosity on the moving plate while an opposite behaviour has been noticed on the stationary plate.