Couple stress fluid flow with variable properties: A second law analysis

The present work examines the combined influence of variable thermal conductivity and viscosity on the irreversibility rate in couple stress fluid flow in between asymmetrically heated parallel plates. The dimensionless fluid equations are solved by using homotopy analysis method (HAM) and validated with Runge‐Kutta shooting method (RKSM). The convergent series solution is then used for the irreversibility analysis in the flow domain. The effects of thermal conductivity and viscosity variation parameters, couple stress parameter, Reynolds number, Grashof number, Hartmann number on the velocity profile, temperature distribution, entropy production, and heat irreversibility ratio are presented through graphs, and salient features of the solutions are discussed. The computations show that the entropy production rate decreases with increased magnetic field and thermal conductivity parameters, whereas it rises with increasing values of couple stress parameter, Brinkman number, viscosity variation parameter, and Grashof number. The study is relevant to lubrication theory.     

Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

In this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.     

On the embedding of variational inequalities

This work is devoted to the approximation of variational inequalities with pseudo-monotone operators. A variational inequality, considered in an arbitrary real Banach space, is first embedded into a reflexive Banach space by means of linear continuous mappings. Then a strongly convergent approximation procedure is designed by regularizing the embedded variational inequality. Some special cases have also been discussed.

     

A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow

The problem considered is that of determining the fluid velocity for linear hydrostatics Stokes flow of slow viscous fluids from measured velocity and fluid stress force on a part of the boundary of a bounded domain. A variational conjugate gradient iterative procedure is proposed based on solving a series of mixed well-posed boundary value problems for the Stokes operator and its adjoint. In order to stabilize the Cauchy problem, the iterations are ceased according to an optimal order discrepancy principle stopping criterion. Numerical results obtained using the boundary element method confirm that the procedure produces a convergent and stable numerical solution.     

The flow of a viscoelastic fluid in a wedge

Summary It is shown that the kinematics of the flow of a general viscoelastic fluid in a wedge, one plate of which is being stretched at a rate proportional to the distance from the wedge apex, is Newtonian in character. Existence proof is given when non-Newtonian effects are slight. Furthermore, the stress field is multivalued at the wedge apex and the pressure field is logarithmically singular there. The strength of this singularity increases with the Weissenberg number.     

Couette flow of a third-grade fluid with variable magnetic field

This investigation deals with the analytic solution for the time-dependent flow of an incompressible third-grade fluid which is under the influence of a magnetic field of variable strength. The fluid is in an annular region between two coaxial cylinders. The motion is induced due to an inner cylinder with arbitrary velocity. Group theoretic methods are employed to analyse the nonlinear problem and a solution for the velocity field is obtained analytically.     

On embedding the volume algorithm in a variable target value method

We employ the volume algorithm as a subgradient deflection strategy in a variable target value method for solving nondifferentiable optimization problems. Focusing on Lagrangian duals for LPs, we exhibit primal nonconvergence of the original method, establish convergence of the proposed algorithm in the dual space, and present related computational results.     

Complementary variational principles for poiseuille flow of an Oldroyd fluid

The complementary variational principles are given for the poiseuille flow of an Oldroyd fluid by taking the pressure gradient to be exponentially increasing with time and the bounds on the flux are obtained.     

Sliding and squeezing flow of a viscoelastic fluid in a wedge

The paper reports an exact three-dimensional similarity solution for the Oldroyd fluid B. The flow involved is generated by closing as well as sliding the boundaries of a two-dimensional wedge. It is found that the squeezing motion is independent of the sliding motion, but not vice-versa. The squeezing load is shown to be a decreasing function of the Weissenberg number, while the frictional coefficient is only weakly-dependent on the Weissenberg number.     

On the plane couette flow of a variable viscosity fluid

In the present paper the Plane Couette flow for three cases, (i) Walls at unequal temperatures, (ii) Walls at equal temperatures and (iii) One of the walls is insulated, has been studied by taking a non-linear relation between viscosity and temperature. Exact solutions of the conpled momentum and energy equations have been obtained. A modified law for the transfer of heat at the upper wall is found in the case of unequal wall temperatures.     

Vectorial variational principle with variable set-valued perturbation

We give a general vectorial Ekeland's variational principle, where the objective function is defined on an F-type topological space and taking values in a pre-ordered real linear space. Being quite different from the previous versions of vectorial Ekeland's variational principle, the perturbation in our version is no longer only dependent on a fixed positive vector or a fixed family of positive vectors. It contains a family of set-valued functions taking values in the positive cone and a family of subadditive functions of topology generating quasi-metrics. Hence, the direction of the perturbation in the new version is a family of variable subsets which are dependent on the ob jective function values. The general version includes and improves a number of known versions of vectorial Ekeland's variational principle. From the general Ekeland's principle, we deduce the corresponding versions of Caristi–Kirk's fixed point theorem and Takahashi's nonconvex minimization theorem. Finally, we prove that all the three theorems are equivalent to each other.     

Application of variational iteration method and homotopy perturbation method to the modified Kawahara equation

In this paper, we present the variational iteration method and homotopy perturbation method to solve the modified Kawahara equations. Both methods provide remarkable accuracy for the approximate solutions when compared to the exact solutions. Numerical results demonstrate that the methods provide efficient approaches to solving the modified Kawahara equation.     

MHD boundary-layer flow of a micropolar fluid past a wedge with constant wall heat flux

The steady laminar magnetohydrodynamic (MHD) boundary-layer flow past a wedge with constant surface heat flux immersed in an incompressible micropolar fluid in the presence of a variable magnetic field is investigated in this paper. The governing partial differential equations are transformed into a system of ordinary differential equations using similarity variables, and then they are solved numerically by means of an implicit finite-difference scheme known as the Keller-box method. Numerical results show that micropolar fluids display drag reduction and consequently reduce the heat transfer rate at the surface, compared to the Newtonian fluids. The opposite trends are observed for the effects of the magnetic field on the fluid flow and heat transfer characteristics.     

Falkner-Skan boundary layer flow of a power-law fluid past a stretching wedge

The steady two-dimensional laminar boundary layer flow of a power-law fluid past a permeable stretching wedge beneath a variable free stream is studied in this paper. Using appropriate similarity variables, the governing equations are reduced to a single third order highly nonlinear ordinary differential equation in the dimensionless stream function, which is solved numerically using the Runge-Kutta scheme coupled with a conventional shooting procedure. The flow is governed by the wedge velocity parameter λ, the transpiration parameter f₀, the fluid power-law index n, and the computed wall shear stress is f″(0). It is found that dual solutions exist for each value of f₀, m and n considered in λ − f″(0) parameter space. A stability analysis for this self-similar flow reveals that for each value of f₀, m and n, lower solution branches are unstable while upper solution branches are stable. Very good agreements are found between the results of the present paper and that of Weidman et al. [28] for n = 1 (Newtonian fluid) and m = 0 (Blasius problem [31]).     

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.     

Application of variational inequalities to the moving-boundary problem in a fluid model for biofilm growth

We consider a moving-boundary problem associated with the fluid model for biofilm growth proposed by J. Dockery and I. Klapper, Finger formation in biofilm layers, SIAM J. Appl. Math. 62 (3) (2001) 853–869. Notions of classical, weak, and variational solutions for this problem are introduced. Classical solutions with radial symmetry are constructed, and estimates for their growth given. Using a weighted Baiocchi transform, the problem is reformulated as a family of variational inequalities, allowing us to show that, for any initial biofilm configuration at time t=0$t=0$ (any bounded open set), there exists a unique weak solution defined for all t≥0$t\ge 0$.     

Fluctuating flow of a Maxwell fluid past a porous plate with variable suction

The problem dealing with the two-dimensional flow of an incompressible viscoelastic Maxwell fluid past an infinite porous plate is investigated. It is assumed that the suction velocity is normal to the plate and oscillates about a mean value. The external free-stream velocity varies periodically in time. The resulting differential equation subject to the relevant boundary and initial conditions is numerically solved by means of a numerical technique, in which a coordinate transformation is employed to transform the semi-infinite physical space to a bounded computational domain. The effects of various values of the emerging parameters, e.g. the elasticity parameter, the oscillation amplitude and frequency of the external flow and the suction velocity, on the time series of velocity, especially on the boundary-layer structure near the plate, are discussed. The nature of the shear stress engendered due to the flow is also investigated.     

A variational approach to mean curvature flow with Dirichlet conditions

We study a variational approach, called Generalized Minimizing Movements (GMM), to evolution of hypersurfaces by mean curvature in the case of a Dirichlet boundary datum. We prove an existence theorem of a GMM when on the initial solid are made suitable geometric hypotheses.

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Sunto In questo lavoro studiamo un approccio variazionale, detto Movimenti Minimizzanti Generalizzati (GMM), all’evoluzione di ipersuperfici per curvatura media nel caso di dato al bordo di Dirichlet. Dimostriamo un teorema di esistenza per un GMM quando vengono assunte ipotesi opportune sul solido iniziale.