Impulsive motion of a non-Newtonian fluid between two oscillating parallel plates

An exact solution for the flow of an incompressible viscoelastic fluid between two infinitely extended parallel plates, due to the harmonic oscillations of the upper plate and the impulsively started harmonic oscillations of the lower plate from rest, in the respective planes of the plates, has been obtained. The momentum transfer towards the central region and the skin friction of the lower plate are found to be greater for the viscoelastic fluid than that for viscous fluid. The effect of out-of-phase oscillations of the plates with different amplitudes on the flow characteristics has also been investigated.     

A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates

The fractional calculus approach in the constitutive relationship model of viscoelastic fluid is introduced. A generalized Maxwell model with the fractional calculus was considered. Exact solutions of some unsteady flows of a viscoelastic fluid between two parallel plates are obtained by using the theory of Laplace transform and Fourier transform for fractional calculus. The flows generated by impulsively started motions of one of the plates are examined. The flows generated by periodic oscillations of one of the plates are also studied.     

On unsteady unidirectional flows of a second grade fluid

Some properties of unsteady unidirectional flows of a fluid of second grade are considered for flows produced by the sudden application of a constant pressure gradient or by the impulsive motion of one or two boundaries. Exact analytical solutions for these flows are obtained and the results are compared with those of a Newtonian fluid. It is found that the stress at the initial time on the stationary boundary for flows generated by the impulsive motion of a boundary is infinite for a Newtonian fluid and is finite for a second grade fluid. Furthermore, it is shown that initially the stress on the stationary boundary, for flows started from rest by sudden application of a constant pressure gradient is zero for a Newtonian fluid and is not zero for a fluid of second grade. The required time to attain the asymptotic value of a second grade fluid is longer than that for a Newtonian fluid. It should be mentioned that the expressions for the flow properties, such as velocity, obtained by the Laplace transform method are exactly the same as the ones obtained for the Couette and Poiseuille flows and those which are constructed by the Fourier method. The solution of the governing equation for flows such as the flow over a plane wall and the Couette flow is in a series form which is slowly convergent for small values of time. To overcome the difficulty in the calculation of the value of the velocity for small values of time, a practical method is given. The other property of unsteady flows of a second grade fluid is that the no-slip boundary condition is sufficient for unsteady flows, but it is not sufficient for steady flows so that an additional condition is needed. In order to discuss the properties of unsteady unidirectional flows of a second grade fluid, some illustrative examples are given.     

Hall effects on the unsteady hydromagnetic oscillatory flow of a second-grade fluid

An exact solution of an oscillatory flow is constructed in a rotating fluid under the influence of an uniform transverse magnetic field. The fluid is considered as second-grade (non-Newtonian). The influence of Hall currents and material parameters of the second-grade fluid is investigated. The hydromagnetic flow is generated in the uniformly rotating fluid bounded between two rigid non-conducting parallel plates by small amplitude oscillations of the upper plate. The exact solutions of the steady and unsteady velocity fields are constructed. It is found that the steady solution depends on the Hall parameter but is independent of the material parameter of the fluid. The unsteady part of the solution depends upon both (Hall and material) parameters. Attention is focused upon the physical nature of the solution, and the structure of the various kinds of boundary layers is examined. Several results of physical interest have been deduced in limiting cases.     

Unsteady flows of a generalized second grade fluid with the fractional derivative model between two parallel plates

The fractional calculus approach in the constitutive relationship model of a generalized second grade fluid is introduced. Exact analytical solutions are obtained for a class of unsteady flows for the generalized second grade fluid with the fractional derivative model between two parallel plates by using the Laplace transform and Fourier transform for fractional calculus. The unsteady flows are generated by the impulsive motion or periodic oscillation of one of the plates. In addition, the solutions of the shear stresses at the plates are also determined. The project supported by the National Natural Science Foundation of China (10372007, 10002003) and CNPC Innovation Fund     

On the growth of unsteady hydromagnetic multiple boundary layers

Summary A study is made of the development of hydromagnetic multiple boundary layers in an electrically conducting rotating liquid bounded by an infinite insulating disk or by two parallel insulating disks in the presence of a uniform magnetic field. The velocity field of the unsteady boundary layer flow generated by non-torsional oscillations of the disk(s) is calculated for small and large values of time. The structures of the associated hydromagnetic boundary layers formed on the disk(s) are determined explicitly. The simultaneous effects of the external magnetic field and the Coriolis force with reference to weak and strong rotation have been explored. The characteristic features of the flow phenomena are examined in some detail. A comparison is made of the results of the present analysis with those of non-. rotating hydromagnetic flow and of hydrodynamic fluid flow. The physical interpretations of the mathematical results are given. The Laplace transform treatment together with the Heaviside expansion theorem has been employed to carry out the mathematical analysis.

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Sommario Si studia lo sviluppo di strati limite multipli idromagnetici in un liquido elettroconduttore rotante limitato da un disco infinito isolante o da due dischi paralleli isolanti in presenza di un campo magnetico uniforme.Il campo di velocità del flusso transitorio di strato limite generato da oscillazioni non torsionali del disco o dei dischi è calcolato per valori piccoli e grandi del tempo. Si determinano esplicitamente le strutture degli strati limite idromagnetici associati sul disco o sui dischi.Si indagano gli effetti simultanei del campo magnetico esterno e della forza di Coriolis con riferimento a rotazione debole e forte. Si esaminano in dettaglio i fenomeni caratteristici del flusso. Si confrontano i risultati di questa analisi con quelli di flussi idromagnetici non rotanti e di flussi idrodinamici. Si danno le interpretazioni fisiche dei risultati matematici.Per eseguire l'analisi matematica si è impiegata la trasformata di Laplace unitamente al teorema d'espansione di Heaviside.

     

Computational study of unsteady turbulent flows around oscillating and ramping aerofoils

The aim of this work is to computationally investigate subsonic and transonic turbulent flows around oscillating and ramping aerofoils under dynamic‐stall conditions. The investigation is based on a high‐resolution Godunov‐type method and several turbulence closures. The Navier–Stokes and turbulence transport equations are solved in a strongly coupled fashion via an implicit‐unfactored scheme. We present results from several computations of flows around oscillating and ramping aerofoils at various conditions in order to (i) assess the accuracy of different turbulence models and (ii) contribute towards a better understanding of dynamic‐stall flows. The results show that the employed non‐linear eddy‐viscosity model generally improves the accuracy of the computations compared to linear models, but at low incidence angles the Spalart–Allmaras one‐equation model was found to provide adequate results. Further, the computations reveal strong similarities between laminar and high‐Reynolds number dynamic‐stall flows as well as between ramping and oscillating aerofoil cases. Investigation of the Mach number effects on dynamic‐stall reveals a delay of the stall angle within a range of Mach numbers. Investigation of the reduced frequency effects suggests the existence of an (almost) linear variation between pitch rate and stall angle, with higher slope at lower pitch rates. The pitch rate affects both the onset of dynamic‐stall as well as the evolution of the associated vortical structures. Copyright © 2003 John Wiley & Sons, Ltd.     

A note on unsteady unidirectional flows of a non-Newtonian fluid

Exact solutions are established for a class of unsteady unidirectional flows of an in compressible second grade fluid wherein inertial effects are not ignored. Amongst the several interesting flows which belong to this class are the flow due to a rigid plate oscillating in its own direction, the flow between two rigid boundaries one of which is suddenly started and the time-periodic Poiseuille flow due to an oscillating pressure gradient.     

On heat transfer effects of a viscous fluid squeezed and extruded between two parallel plates

The unsteady squeezing and extrusion of a viscous fluid between two parallel plates of constant temperature is examined. The dimensionless extrusion parameter,=U/V, is introduced to represent the effects of the extrusion on the squeezing velocities. The squeezing parameter=VH/, represents the effect of the inertial forces on heat and fluid flow characteristics. It is found that increasing the extrusion parameter will increase both the velocity and the heat transfer rates to the viscous fluid. Increasing the squeezing parameter had also decreased the fluid velocity and enhanced heat transfer rates. Increasing the viscous effects or the Eckert number E=U²/c_p (T_E – T_s) heated the fluid and consequently decreased the heat transfer rates. Different velocity profiles, temperature profiles, and Nusselt numbers against various dimensionless groups are drawn.     

Unsteady hydromagnetic Couette flow of dusty fluid with temperature dependent viscosity and thermal conductivity

In the present paper the unsteady Couette flow and heat transfer of a dusty conducting fluid between two parallel plates with temperature dependent viscosity and thermal conductivity are studied. A constant pressure gradient and an external uniform magnetic field are applied. The governing coupled momentum and energy equations are solved numerically using finite differences. The effect of the variable viscosity and thermal conductivity of the fluid and the uniform magnetic field on the velocity and temperature fields for both the fluid and dust particles is discussed.     

Unsteady hydromagnetic channel flow of dusty fluid with temperature dependent viscosity and thermal conductivity

In the present paper the unsteady flow and heat transfer of a dusty conducting fluid between two parallel plates with temperature dependent viscosity and thermal conductivity are studied. A constant pressure gradient and an external uniform magnetic field is applied. The governing coupled momentum and energy equations are solved numerically using finite differences. The effect of the variable viscosity and thermal conductivity of the fluid and the uniform magnetic field on the velocity and temperature fields for both the fluid and dust particles is discussed.On leave from Department of Mathematics and Physics, Faculty of Engineering, El-Fayoum University, Egypt     

Temperature distribution in a Newtonian fluid injected between two semi-infinite plates

The analysis of the temperature distribution in time and place of a hot heat-conducting Newtonian fluid injected between two cooled parallel plates is presented. The 2-dimensional flow has a free flow front moving with constant velocity. The kernel of the fluid remains almost at the inlet temperature, but at the walls boundary layers occur with steeply descending temperature. The inner solutions inside these boundary layers are determined. To this end, the total region is divided into three distinct regions: the region G_I far behind the flow front, the flow front region G_II, and the intermediate region G_III between G_I and G_II. The asymptotics owing to each region are presented. The fundamental small parameter here is the thickness-to-length ratio of the 2-dimensional flow region. In most of the cases, similarity solutions are found. In the flow front region, for the formulation of the inner solution a Wiener–Hopf technique is used. Via matching procedures, the separate boundary layers are linked to each other to form one global boundary layer for the whole front region. All calculations in this paper are performed by analytical means, and all results are in analytical form. Comparison of our results with numerical solutions shows perfect agreement.     

A finite volume approach for unsteady viscoelastic fluid flows

A finite volume, time‐marching for solving time‐dependent viscoelastic flow in two space dimensions for Oldroyd‐B and Phan Thien–Tanner fluids, is presented. A non‐uniform staggered grid system is used. The conservation and constitutive equations are solved using the finite volume method with an upwind scheme for the viscoelastic stresses and an hybrid scheme for the velocities. To calculate the pressure field, the semi‐implicit method for the pressure linked equation revised method is used. The discretized equations are solved sequentially, using the tridiagonal matrix algorithm solver with under‐relaxation. In both, the full approximation storage multigrid algorithm is used to speed up the convergence rate. Simulations of viscoelastic flows in four‐to‐one abrupt plane contraction are carried out. We will study the behaviour at the entrance corner of the four‐to‐one planar abrupt contraction. Using this solver, we show convergence up to a Weissenberg number We of 20 for the Oldroyd‐B model. No limiting Weissenberg number is observed even though a Phan Thien–Tanner model is used. Several numerical results are presented. Smooth and stable solutions are obtained for high Weissenberg number. Copyright © 2002 John Wiley & Sons, Ltd.     

Application of local DFD method to simulate unsteady flows around an oscillating circular cylinder

In this paper, the recently proposed local domain‐free discretization (DFD) method is applied to simulate incompressible flows around an oscillating circular cylinder. It is found that it is very easy for the local DFD method to handle such moving boundary flow problems. This is because it does not need to move the mesh, which is indeed needed in traditional methods. Numerical experiments show that the present numerical results agree very well with the available data in the literature, and that the local DFD method is an effective tool for the computation of moving boundary flow problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Couple on a rotary oscillating sphere in a dusty gas

Rotary oscillations of a sphere about its vertical diameter in an infinite expanse of viscous, incompressible fluid with imbedded identical spherical particles are studied. The problem is solved by the method of separation of variables and explicit expressions for the velocity of fluid particles and the couple experienced by the sphere due to fluid stresses are obtained. The couple is expressed in terms of two parameters and graphs have been drawn to represent the variation in these parameters and some interesting conclusions are made.Nomenclature u ⁵ velocity of fluid particles - v ⁶ velocity of dust particles - P ⁵ pressure - ⁶ fluid density - t ⁹ time - m mass of a dust particle - N number density of dust particles - k ⁵ Stokes resistance coefficient - ³ viscosity - ⁶ kinematic viscosity - ⁷ particle relaxation time - ⁴ frequency of oscillation of the sphere - f ⁹ mass concentration of the dust particles     

On elastic effects in unsteady pipe flows

Summary Theoretical consideration is given to two unsteady pipe flows. In the first, a combined steady and oscillatory shear flow is generated by a pulsatile pressure gradient in a stationary pipe. In the second, the pressure gradient is constant, but the pipe wall executes axial vibrations.The theoretical analysis is carried out for an upper convected Maxwell model characterized by a viscosity function and one relaxation time. A conventional perturbation method of solution is concluded to be inadequate to describe some of the interesting experimental observations and most of the work employs a finite-difference formulation based onTownsend's work. In the vibrating-pipe analysis, it is concluded that the flow must be considered to be dominated by the axial movement of the pipe if the interesting experiments ofManero andMena are to be properly interpreted.We have still not reached the stage where existing experimental results on pulsatile flow and the vibratingpipe situation can bequantitatively predicted.

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Zusammenfassung Es werden zwei Typen instationärer Rohrströmungen theoretisch betrachtet. Bei dem ersten wird eine kombinierte stationäre und oszillierende Scherströmung dadurch erzeugt, daß bei ruhendem Rohr der Strömung ein pulsierender Druckgradient aufgeprägt wird. Bei dem zweiten ist der Druckgradient konstant, dagegen führt das Rohr axiale Schwingungen aus.Die theoretische Analyse wird für das Modell einer verallgemeinerten Maxwell-Oldroyd-FlüssigkeitB (d. h. mit kontravarianter konvektiver Zeitableitung) durchgeführt, welches durch eine Viskositätsfunktion, aber nur eine einzige Relaxationszeit gekennzeichnet ist. Es wird gezeigt, daß die üblichen Störungsmethoden nicht geeignet sind, einige der interessantesten experimentellen Beobachtungen zu beschreiben. Daher wird in den meisten Fällen eine auf den Arbeiten vonTownsend basierenden Finite-Differenzen-Methode angewendet. Bei der Analyse des vibrierenden Rohrs wird geschlossen, daß die Strömung maßgeblich durch die Axialbewegung des Rohrs bestimmt sein muß, wenn die interessanten Experimente vonManero undMena angemessen interpretiert werden sollen.Zur Zeit ist noch nicht eine solche Stufe des theoretischen Verständnisses erreicht, von wo aus einequantitative Voraussage der bekannten experimentellen Ergebnisse bei pulsierenden Strömungen bzw. einem vibrierenden Rohr geleistet werden könnte.

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With 14 figures     

Performance of hydromagnetic squeeze films between conducting porous rough conical plates

An endeavor has been made to discuss the behavior of hydromagnetic squeeze film between two conducting rough porous conical plates. The plates are considered to be electrically conducting and the clearance space between them is filled by an electrically conducting lubricant. A transverse magnetic field is applied between the plates. Efforts have been made to solve the concerned Reynolds’ equation with the associated boundary conditions to get the pressure distribution. This in turn, is used to obtain the expression for load carrying capacity leading to the calculation of the response time. The results are presented graphically as well as in tabular form. It is suggested by the results that the bearing system records an enhanced performance as compared to that of a bearing system working with a conventional lubricant. It is noticed that the pressure, load carrying capacity and the response time increase steadily with increasing values of the magnetization parameter. In general, the bearing suffers owing to transverse surface roughness. However, the negatively skewed roughness tends to better the performance of the bearing system marginally. This performance gets further improved especially, when the negative variance is involved. It is observed that the semi-vertical angle increases the load carrying capacity. Besides, the conductivity also increases the load carrying capacity significantly. In addition, it is revealed that the negative effect induced by the porosity can be neutralized to a nominal extent by the positive effect of the magnetization parameter in the case of negatively skewed roughness in the presence of negative variance. Thus, this study provides ample scopes for improving the performance of the bearing system considerably by choosing a suitable combination of magnetization parameter, semi-vertical angle and the conductivities of the plates.     

Exact solutions for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit

The exact solutions are obtained for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit. The fractional calculus in the constitutive relationship of a non-Newtonian fluid is introduced. We construct the solutions by means of Fourier transform and the discrete Laplace transform of the sequential derivatives and the double finite Fourier transform. The solutions for Newtonian fluid between two infinite parallel plates appear as limiting cases of our solutions.