Plane capillary flow of a viscous fluid with mutiply connected boundary in the Stokes approximation

A study is made of the process of relaxation to the equilibrium configuration of an isolated volume of a viscous incompressible Newtonian fluid under the influence of capillary forces. The fluid has the form of an infinite cylinder of arbitrary shape with a smooth compact and, in general, multiply connected boundary. In the course of relaxation, internal cavities collapse, and the cylinder acquires asymptotically a circular configuration. The quasisteady Stokes approximation [1] is used to describe the flow. First proposed by Frenkel' [2], this approximation has been used in the calculation of a dynamic boundary angle [3], the collapse of a circular cylinder [4], and the collapse of a hollow cylinder [5]. The analogy between the hydrodynamic equations in the Stokes approximation and the equations of elasticity theory made it possible [6] to describe the relaxation of a simply connected cylinder by a method close to the one employed by Muskheleshvili [7]. In the present paper, the approach of the author [8] based on work of Grinberg [9] and Vekua [10] is developed. It is shown that the true pressure distribution gives a minimum of the integral of the square of the pressure over the region for fixed integral of the pressure over the boundary. An explicit expression for the pressure is obtained in the form of the projection of a generalized function with support on the boundary onto the subspace of harmonic functions. The velocity field on the boundary of the region is calculated. An upper bound is found for the law of decrease of the perimeter of the region and for the time during which the number of connected components of the boundary remains unchanged.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 117–122, January–February, 1992.     

A boundary element method for steady viscous fluid flow using penalty function formulation

A new boundary element method is presented for steady incompressible flow at moderate and high Reynolds numbers. The whole domain is discretized into a number of eight-noded cells, for each of which the governing boundary integral equation is formulated exclusively in terms of velocities and tractions. The kernels used in this paper are the fundamental solutions of the linearized Navier–Stokes equations with artificial compressibility. Significant attention is given to the numerical evaluation of the integrals over quadratic boundary elements as well as over quadratic quadrilateral volume cells in order to ensure a high accuracy level at high Reynolds numbers. As an illustration, square driven cavity flows are considered for Reynolds numbers up to 1000. Numerical results demonstrate both the high convergence rate, even when using simple (direct) iterations, and the appropriate level of accuracy of the proposed method. Although the method yields a high level of accuracy in the primary vortex region, the secondary vortices are not properly resolved. © 1997 John Wiley & Sons, Ltd.     

Heat transfer and pressure drop for purely viscous non-Newtonian fluids in turbulent flow through rectangular passages

Experimental measurements of friction factor and heat transfer for the turbulent flow of purely viscous non-Newtonian fluids in a 21 rectangular channel are compared with results previously reported for the circular tube geometry. Comparisons are also made with available analytical and empirical predictions.It is found that the rectangular duct fully established friction factor measurements are within ± 5% of the Dodge-Metzner prediction if the Kozicki generalized Reynolds number is used. A modified form of the simpler explicit equation proposed by Yoo, [i.e.f=0.079n ^0.675(Re ^*)^–0.25], is found to yield predictions for both the rectangular duct and the circular tube geometries with approximately the same accuracy as the Dodge-Metzner equation.Fully developed Stanton numbers for the rectangular duct are in good agreement with the circular tube data over a range ofn from 0.37 to 0.88 for a given Prandtl number,Pr _a , when compared at a fixed value of the Reynolds number based on the apparent viscosity evaluated at the wall shear stress. In general, the experimental data are within ± 20% of Yoo's equation,St=0.0152Re _a ^–0.155 Pr _a ^–2/3 . A new equation is proposed to bring the prediction for circular pipes as well as rectangular channels into better agreement with generally accepted Newtonian heat transfer results.

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Wärmeübergang und Druckverlust für viskose nicht-Newtonsche Fluide in turbulenter Strömung durch rechteckige Kanäle

Zusammenfassung Es werden Messungen des Reibungsfaktors und des Wärmeübergangs bei turbulenter Strömung viskoser nicht-Newtonscher Fluide in einem rechteckigen Kanal mit dem Seitenverhältnis 21 verglichen mit früheren Ergebnissen, die an runden Rohren gewonnen wurden. Weiterhin werden Vergleiche mit aus der Literatur verfügbaren analytischen und empirischen Beziehungen gemacht.Es zeigte sich, daß die Messungen des Reibungsfaktors im rechteckigen Kanal bei vollausgebildeter Strömung auf ± 5% mit der Vorhersage von Dodge-Metzner übereinstimmen, wenn die von Kozicki verallgemeinerte Reynolds-Zahl verwendet wird. Eine modifizierte Form der einfachen von Yoo vorgeschlagenen einfachen Gleichung in explizierter Form (f=0,079n ^0,675(Re ^*)^–0,25) bewies, daß sie sowohl für den rechteckigen Kanal als auch das runde Rohr die Werte mit fast der gleichen Genauigkeit wie die Methode von Dodge-Metzner vorhersagen kann.Die Stanton-Zahlen für den rechteckigen Kanal bei vollausgebildeter Strömung sind in guter Übereinstimmung mit den Werten für das runde Rohr in einem Bereich vonn= 0,37 – 0,88 für eine gegebene Prandtl-Zahl, wenn man den Vergleich bei einem vorgegebenen Wert der Reynolds-Zahl anstellt, die auf die scheinbare Viskosität — abgeleitet aus der Wandschubspannungbezogen ist. Generell läßt sich sagen, daß die Werte auf ± 20% mit der Gleichung von Yoo (St=0,0152Re _a ^–0,155 )Pr _a ^–2/3 ) übereinstimmen. Es wird eine neue Gleichung vorgeschlagen, welche sowohl die Werte für runde Rohre als auch die für rechteckige Kanäle in bessere Übereinstimmung bringt mit den in der Literatur üblichen Ergebnissen für den Wärmeübergang an Newtonsche Fluide.

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Nomenclature a constant in Eq. (8) - A area of cross-section of channel [m²] - b constant in Eq. (8) - c _p specific heat of test fluid [J kg^–1 K^–1] - d capillary tube diameter [m] - D _h hydraulic diameter, 4A/P[m] - f Fanning friction factor, _w/(g9 V²/2) - h axially local (spanwise averaged) heat transfer coefficient,q _w /(T_wi-T_b) [Wm^–2 K^–1] - k _f thermal conductivity of test fluid [Wm^–1K^–1] - K consistency index of power law fluid - n power law index - Nu fully established, local (spanwise averaged) Nusselt numberh D _h /k _f - P perimeter of channel [m] - Pr _a Prandtl number based on apparent viscosjity, c _p /k _f - Pr ^* defined as (Re _a Pr _a )/Re ^* - q _w wall heat flux [Wm^–2] - Re _a Reynolds number based on apparent viscosity, VD _h/ - Re Metzner's generalized Reynolds number in Eq. (2) - Re ^* Reynolds number defined in Eq. (8) - St Stanton number,h/( V c_p) - T _b local bulk temperature of the fluid [K] - T _wi local inside wall temperature [K] - T _wo local outside wall temperature [K] - V bulk flow velocity [m s^–1] - x distance from the inlet of channel along flow direction [m] Greek symbols shear rate [s^–1] - apparent viscosity [Pa s] - density of test fluid [kg m^–3] - shear stress [Pa] - _w shear stress at the wall [Pa] Dedicated to Prof. Dr.-Ing. U. Grigull's 75th birthday     

Two-dimensional stokes flow of a viscous fluid with a free boundary under the effect of capillary forces

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 5, pp. 47–51, September–October, 1992.     

Cylinder in viscous incompressible fluid flow

We present a technique for calculating the temperature field in the vicinity of a cylinder in a viscous incompressible fluid flow under given conditions for the heat flux or the cylinder surface temperature. The Navier-Stokes equations and the energy equation for the steady heat transfer regime form the basis of the calculations. The numerical calculations are made for three flow regimes about the cylinder, corresponding to Reynolds numbers of 20, 40, and 80. The pressure distribution, voracity, and temperature distributions along the cylinder surface are found.It is known that for a Reynolds number R>1 the calculation of cylinder drag within the framework of the solution of the Oseen and Stokes equations yields a significant deviation from the experimental data. In 1933 Thom first solved this problem [1] on the basis of the Navier-Stokes equations. Subsequently several investigators [2, 3] studied the problem of viscous incompressible fluid flow past a cylinder.It has been established that a stable solution of the Navier-Stokes equations exists for R40 and that in this case the calculation results are in good agreement with the experimental data. According to [2], a stable solution also exists for R=44. The possibility of obtaining a steady solution for R>44 is suggested.Analysis of the results of [2] permits suggesting that the questions of constructing a difference scheme with a given order of approximation of the basic differential relations which will permit obtaining the sought solution over the entire range of variation of the problem parameters of interest are still worthy of attention.Calculation of the velocity field in the vicinity of a cylinder also makes possible the calculation of the cylinder temperature regime for given conditions for the heat flux or the temperature on its surface. However, we are familiar only with experience in the analytic solution of several questions of cylinder heat transfer with the surrounding fluid for large R within the framework of boundary layer theory [4].     

Unsteady jet flow of a viscous fluid

A solution is obtained, within the framework of the boundary layer theory, to the problem of the unsteady flow created by a two-dimensional jet source for a given momentum flux variation with time. In particular, aperiodic and periodic momentum variations are examined, and a qualitative analog with turbulent flow is noted for the latter.     

MOTION AND DEFORMATION OF VISCOUS DROP IN STOKES FLOW NEAR RIGID WALL

A boundary integral method was developed for simulating the motion and deformation of a viscous drop in an axisymmetric ambient Stokes flow near a rigid wall and for direct calculating the stress on the wall. Numerical experiments by the method were performed for different initial stand-off distances of the drop to the wall, viscosity ratios, combined surface tension and buoyancy parameters and ambient flow parameters. Numerical results show that due to the action of ambient flow and buoyancy the drop is compressed and stretched respectively in axial and radial directions when time goes. When the ambient flow action is weaker than that of the buoyancy the drop raises and bends upward and the stress on the wall induced by drop motion decreases when time advances. When the ambient flow action is stronger than that of the buoyancy the drop descends and becomes flatter and flatter as time goes. In this case when the initial stand-off distance is large the stress on the wall increases as the drop evolutes but when the stand-off distance is small the stress on the wall decreases as a result of combined effects of ambient flow, buoyancy and the stronger wall action to the flow. The action of the stress on the wall induced by drop motion is restricted in an area near the symmetric axis, which increases when the initial stand-off distance increases. When the initial stand-off distance increases the stress induced by drop motion decreases substantially. The surface tension effects resist the deformation and smooth the profile of the drop surfaces. The drop viscosity will reduce the deformation and migration of the drop.     

Gravity-induced spreading of a drop of a viscous fluid over a surface

The unsteady-state nonlinear problem of spreading of a drop of a viscous fluid on the horizontal surface of a solid under the action of gravity and capillary forces is considered for small Reynolds numbers. The method of asymptotic matching is applied to solve the axisymmetrical problem of spreading when the gravity exerts a significant effect on the dynamics of the drop. The flow structure in the drop is determined at large times in the neighborhood of a self-similar solution. The ranges of applicability of the quasiequilibrium model of drop spreading with a dynamic edge angle and a self-similar solution are found. It is shown that the transition from one flow model to another occurs at very large Bond numbers. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences, Tyumen’ 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 59–67, May–June, 1999.     

Energy-optimal time-dependent regimes of viscous incompressible fluid flow

The hydrodynamic equations of a viscous incompressible fluid are modified for axisymmetric flows in a pipe of time-varying radius. A new exact time-dependent solution of these equations which generalizes the well-known classic steady-state Hagen–Poiseuille solution for flow in a pipe of constant radius (independent of time) is obtained. It is shown that the law of time variation in the pipe radius can be determined from the condition of the minimum work done to pump a given fluid volume through such a pipe during the radius variation cycle period. A generalization of the optimal branching pipeline in which, instead of the Poiseuille law, its modification based on the use of the exact solution corresponding to the time-dependent M-shaped regime is employed is suggested. It is shown that the hydraulic resistance can be reduced over a certain range of the parameters of the time-dependent flow regime as compared with the steady-state pipe flow regime. The conclusion obtained can be used for the development of the hydrodynamic basis for simulating the optimal hydrodynamic blood flow regime.     

One-dimensional flow of a fluid through a porous skeleton with consideration of the Darcy and frontal pressure interaction forces

The problem of one-dimensional steady flow of a compressible fluid through a solid porous skeleton made of an incompressible material is formulated and numerically solved with consideration of the Darcy and frontal pressure interaction forces.     

One-dimensional simulation of viscous fluid displacement from a porous medium with consideration of lateral inflow

A one-dimensional model of fluid displacement in a porous medium is discussed with consideration of lateral inflow. The time period required for the complete displacement of an initially injected fluid from a region is studied. Some numerical results obtained for two formulations of the problem are given; these results are in good agreement with the estimates considered in this paper. The problem under study is of interest in practice for enhancing the oil recovery from oil fields.     

Cavitation flow calculations for a viscous and capillary fluid

Developed cavitation calculations, where the cavity forms a void directly adjoining and stationary relative to the body, have been carried out almost exclusively within the framework of ideal fluid mechanics [1, 2]. Experiments (for example, [2, 3]), however, indicate that viscosity and capillarity have an undoubted influence on cavitation flows. In the case of developed cavities behind nonlifting bodies this effect has been taken into account [4] in terms of the dependence of the arc abscissa of the beginning of the cavity on the Weber and Reynolds numbers We and Re for a given value of the cavitation number. In calculating a partial cavity (of a length not exceeding that of the body in the flow) it is necessary to take into account the development of the boundary layer on the cavity and the presence of viscous separation zones not only in front of but also behind it. In this paper a method of calculating partial cavitation satisfying these requirements is proposed, and problems relating to the justification of the method are discussed. The cavitation calculations presented employ the flow model described in [5], which takes into account the presence of the boundary layer on the body and the cavity, together with the viscous separation zones. The calculation method is a development of that described in [6] and makes important use of an idea derived from [2, 7]. In this connection, the fact that the characteristics of the boundary layer in cavitation flow past bodies have not been sufficiently studied has made it necessary to resort to a numerical experiment to close the semiempirical relations used in the calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 45–51, November–December, 1985.     

Magnetoaerodynamic boundary layer flow with pressure gradient and separation

This study considers the large interaction parameter magnetoaerodynamic boundary layer associated with the free stream flow of a conducting fluid over an infinitely long circular insulator cylinder with the applied magnetic field normal to the distant free stream flow. The investigation is conducted in two parts; a theoretical solution of the associated boundary layer equations and a qualitative experimental investigation to allow visualization of flow separation caused by the magnetic field. The general integral formulation of Galerkin-Kantorovich-Dorodnitsyn is used to determine the boundary layer thickness, momentum thickness, displacement thickness, approximate separation point, and velocity profiles.     

Boundary condition formulation for viscous fluid flow problems

In the solution of the Navier-Stokes equations by difference methods in infinite regions, the question arises as to the nature of the approximate boundary conditions at those portions of the computational region boundary where these conditions are not determined directly by the formulation of the basic problem. In certain cases of practical importance, these boundary conditions may be obtained by coupling the N-S equations with equations which are similar to the boundary-layer equations.In the present paper, we propose boundary conditions for the case of viscous incompressible fluid flow. Their application is illustrated for the problem of flow past the leading edge of a semi-infinite flat plate.The author wishes to thank I. Yu. Brailovskaya and L. A. Chudov for helpful suggestions in the course of this investigation.     

Motion and shape of an axisymmetric viscoplastic drop slowly falling through a viscous fluid

Slow sedimentation of a deformable drop of Bingham fluid in an unbounded Newtonian medium is studied using a variation of the integral equation method (Toose et al., J Eng Math 30:131–150, 1996, Int J Numer Methods Fluids 30:653–674, 1999). The Green function for the Stokes equation is used, and the non-Newtonian stress is treated as a source term. The computations are performed for a range of physical parameters of the system. It is demonstrated that initially deformed drop similar to Newtonian ones breaks up for high capillary number, Ca, and stabilizes to steady shapes at low Ca. Estimations of critical capillary number for specific initial deformations demonstrated its growth (increase in the stability of the drop) with the yield stress magnitude both for prolate and oblate initial shapes. Prolate initial shapes become more stable with the increase of the plastic viscosity. In contrast to this, for low yield stress, oblate shapes are destabilized with the growth of the plastic viscosity. This effect is similar to the effect of the viscosity of a Newtonian drop on its stability. However, at higher yield stress, the effect of plastic viscosity is reversed.     

Effects of outflow boundary condition on convective heat transfer with strong recirculating flow

Three practices of treating outflow boundary condition were adopted in computations for convective heat transfer of a two-dimensional jet impinging in a rectangular cavity. The three practices were local mass conservation method, local one-way method and fully developed assumption. The numerical solutions of the three methods were compared with test data obtained via, naphthalene sublimation technique. It was found that the fully developed assumption was inappropriate, and the local one-way method could provide reasonably good results for the cavity bottom, while for the lateral wall the results with this method qualitatively differed from the test data. The solution with the local mass conservation method was the best. It thus suggested that for a problem expected with a strong recirculating flow at the exit of the computation domain, the local mass conservation method be adopted to treat the outflow boundary condition.