Effect of random longitudinal vibrations on the Poiseuille flow of a complex liquid

In this work, the rectilinear Poiseuille flow of a complex liquid flowing in a vibrating pipe is analyzed. The pipe wall performs oscillations of small amplitude that can be adequately represented by a weakly stochastic process, for which a quasi-static perturbation solution scheme is suggested. The flow is analyzed using the Bautista–Manero–Puig constitutive equation, consisting on the upper-convected Maxwell equation coupled to a kinetic equation to account for the breakdown and reformation of the fluid structure. A drastic enhancement of the volumetric flow is predicted in the region where the fluid experiences pronounced shear-thinning. Finally, flow enhancement is predicted using experimental data reported elsewhere for wormlike micellar solutions of cetyl trimethyl ammonium tosilate.     

Using the perturbation method to solve the problem of separated incompressible flow past thin airfoils

A model for separated incompressible flow past thin airfoils in the neighborhood of the “shockless entrance” condition is constructed based on the averaging of the vortex shedding flow past the airfoil edges. By approximation of the vortex shedding by two vortex curves, determination of the average hydrodynamic parameters is reduced to a twofold solution of an integral singular equation equivalent to the equation describing steady-state nonseparated airfoil flow. In this case, the calculation time is two orders of magnitude smaller than the time required for the solution of the corresponding evolution problem. The results of a test calculation using the proposed method are in fair agreement with available results of calculations and experiments. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 49–63, May–June, 2006.     

A theoretical analysis of forced convection in a porous-saturated circular tube: Brinkman–Forchheimer model

Fully developed forced convection inside a circular tube filled with saturated porous medium and with uniform heat flux at the wall is investigated on the basis of a Brinkman–Forchheimer model. The matched asymptotic expansion method is applied at small Darcy numbers. For large Darcy numbers, the solution for the Brinkman–Forchheimer momentum equation is found in terms of an asymptotic expansion. Once the velocity distribution is determined, the energy equation is solved using the same asymptotic technique. The results for the two limiting cases of clear fluid and Darcy flow conditions show good agreement with those available in the literature.     

Linear problem of a hydrofoil moving under the free surface of a finite-depth fluid

A method of solving the plane linear problem of a steady-state irrotational flow about a body under the free surface of a heavy fluid of finite depth is developed. The boundary-value problem is formulated for a complex perturbed velocity and is reduced to a singular integral equation relative to the intensity of a vortex layer that models the hydrofoil. The kernel of the equation is the exact solution of the corresponding boundary-value problem for a vortex of unit intensity. The equation is solved by the discrete-vortex method. The effect of the parameters of the problem on the hydrodynamic characteristics of the elliptical hydrofoil and the shape of the free surface are estimated numerically. Omsk Division of the Sobolev Institute of Mathematics, Omsk 644099. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 85–90, November–December, 1998.     

The effect of thermal dispersion on free convection film condensation on a vertical plate with a thin porous layer

An analytical solution to the problem of condensation by natural convection over a thin porous substrate attached to a cooled impermeable surface has been conducted to determine the velocity and temperature profiles within the porous layer, the dimensionless thickness film and the local Nusselt number. In the porous region, the Darcy–Brinkman–Forchheimer (DBF) model describes the flow and the thermal dispersion is taken into account in the energy equation. The classical boundary layer equations without inertia and enthalpyterms are used in the condensate region. It is found that due to the thermal dispersion effect, the increasing of heat transfer is significant. The comparison of the DBF model and the Darcy–Brinkman (DB) one is carried out.     

Finite Element-Based Characterization of Pore-Scale Geometry and Its Impact on Fluid Flow

We present a finite element (FEM) simulation method for pore geometry fluid flow. Within the pore space, we solve the single-phase Reynold’s lubrication equation—a simplified form of the incompressible Navier–Stokes equation yielding the velocity field in a two-step solution approach. (1) Laplace’s equation is solved with homogeneous boundary conditions and a right-hand source term, (2) pore pressure is computed, and the velocity field obtained for no slip conditions at the grain boundaries. From the computed velocity field, we estimate the effective permeability of porous media samples characterized by section micrographs or micro-CT scans. This two-step process is much simpler than solving the full Navier–Stokes equation and, therefore, provides the opportunity to study pore geometries with hundreds of thousands of pores in a computationally more cost effective manner than solving the full Navier–Stokes’ equation. Given the realistic laminar flow field, dispersion in the medium can also be estimated. Our numerical model is verified with an analytical solution and validated on two 2D micro-CT scans from samples, the permeabilities, and porosities of which were pre-determined in laboratory experiments. Comparisons were also made with published experimental, approximate, and exact permeability data. With the future aim to simulate multiphase flow within the pore space, we also compute the radii and derive capillary pressure from the Young–Laplace’s equation. This permits the determination of model parameters for the classical Brooks–Corey and van-Genuchten models, so that relative permeabilities can be estimated.     

Simple waves on a shear gas flow in a channel of constant cross section

The plane-parallel unsteady-state shear gas flow in a narrow channel of constant cross section is considered. The existence theorem of solutions in the form of simple waves of a set of equations of motion is proved for a class of isentropic flows with a monotone velocity profile over the channel depth. The exact solution described by incomplete beta-functions is found for a polytropic equation of state in a class of isentropic flows. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 36–43, January–February, 1999.     

Peristaltic flow of a Johnson-Segalman fluid through a deformable tube

To understand theoretically the flow properties of physiological fluids we have considered as a model the peristaltic motion of a Johnson–Segalman fluid in a tube with a sinusoidal wave traveling down its wall. The perturbation solution for the stream function is obtained for large wavelength and small Weissenberg number. The expressions for the axial velocity, pressure gradient, and pressure rise per wavelength are also constructed. The general solution of the governing nonlinear partial differential equation is given using a transformation method. The numerical solution is also obtained and is compared with the perturbation solution. Numerical results are demonstrated for various values of the physical parameters of interest.      

Inverse problem of wing aerodynamics in a supersonic flow

The inverse problem of wing aerodynamics—the determination of the lifting surface shape from a specified load—is solved within the framework of linear theory. Volterra's solution of the wave equation is used. Solutions are found in the class of bounded functions if certain conditions imposed on the governing parameters of the problem are satisfied. Solutions of inverse problems of supersonic flow are presented for an infinite-span wing, a triangular wing with completely subsonic edges, and a rectangular wing. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 86–91, May–June, 1998.     

Optimum Aerodynamic Design Using the Navier–Stokes Equations

This paper describes the formulation of optimization techniques based on control theory for aerodynamic shape design in viscous compressible flow, modeled by the Navier–Stokes equations. It extends previous work on optimization for inviscid flow. The theory is applied to a system defined by the partial differential equations of the flow, with the boundary shape acting as the control. The Fréchet derivative of the cost function is determined via the solution of an adjoint partial differential equation, and the boundary shape is then modified in a direction of descent. This process is repeated until an optimum solution is approached. Each design cycle requires the numerical solution of both the flow and the adjoint equations, leading to a computational cost roughly equal to the cost of two flow solutions. The cost is kept low by using multigrid techniques, in conjunction with preconditioning to accelerate the convergence of the solutions. The power of the method is illustrated by designs of wings and wing–body combinations for long range transport aircraft. Satisfactory designs are usually obtained with 20–40 design cycles. Received 5 February 1997 and accepted 30 May 1997     

Quasi-Neutral Limit for a Model of Viscous Plasma

We perform a rigorous analysis of the quasi-neutral limit for a model of viscous plasma represented by the Navier–Stokes–Poisson system of equations. It is shown that the limit problem is the Navier–Stokes system describing a barotropic fluid flow, with the pressure augmented by a component related to the nonlinearity in the original Poisson equation.     

Analysis of Momentum Transfer in a Lid-Driven Cavity Containing a Brinkman–Forchheimer Medium

The CE/SE (the space-time conservation element and solution element method) scheme with the second-order accuracy has been proposed. And the pretreatment method has been introduced to convert the parabolic equations to the hyperbolic equations, which are accurately solved by the CE/SE method. The lid-driven rectangular cavity containing a porous Brinkman–Forchheimer medium is studied in this numerical investigation. The Brinkman–Forchheimer equation is used such that both the inertial and viscous effects are incorporated. The governing equations are solved by the improved CE/SE approach. The characteristics of the flow are analyzed with emphasis on the influence of the Darcy number and the cavity depth. It is found that the porous medium effect decreases both the strength and the number of eddies, especially for deep cavities.     

Properties of the nonequilibrium high-velocity “Tails” of the molecular distribution function in plane couette flow of a compressible gas

The results of an analytic and numerical investigation of the properties of the high-velocity “tails” of the distribution function are given for the solution of the BGK model of the kinetic Boltzmann equation for plane Couette flow of a compressible gas. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 183–190, July–August, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-00573; grant in support of leading science schools No. 96-15-9603).     

Slow motion of a granular layer on an inclined plane

The shape of the free surface of a layer of granular material moving on an inclined plane is studied on the basis of a model of a non-Newtonian fluid with a nonlinear relation between the stress tensor and the shear rate of the flow. For small but finite elevations of the free surface, the governing equations are reduced to a quasilinear Burgers equation. Results of a numerical solution are presented for the case of arbitrary elevations. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 117–120, March–April, 1998.     

Asymptotic Behavior of Solutions to the Compressible Navier–Stokes Equation Around a Parallel Flow

The initial boundary value problem for the compressible Navier–Stokes equation is considered in an infinite layer of . It is proved that if the Reynolds and Mach numbers are sufficiently small, then strong solutions to the compressible Navier–Stokes equation around parallel flows exist globally in time for sufficiently small initial perturbations. The large time behavior of the solution is described by a solution of a one-dimensional viscous Burgers equation. The proof is given by a combination of spectral analysis of the linearized operator and a variant of the Matsumura–Nishida energy method.     

Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 165–175, May–June, 2006.     

Three-dimensional motion of a viscous incompressible fluid in a narrow tube

An approximate solution of the problem of unsteady motion of a viscous incompressible fluid in a long narrow deformable tube at low Reynolds numbers is obtained. Pressure oscillations and tube deformation are shown to be related by an integrodifferential equation. The solution obtained extends the Poiseuille solution in elliptic tubes to the case of comparatively arbitrary small deformations in terms of the tube length and angle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 28–32, July–August, 2009.     

Wave regimes on a nonisothermal film of a viscous liquid flowing down a vertical plane

A thin film flow of a viscous liquid flowing down a vertical wall in the field of the gravity force is studied. The values of temperatures on the solid wall and on the free surface are constant. The viscosity and thermal diffusivity are functions of temperature. An equation that describes the evolution of surface disturbances is derived for small flow rates in the long-wave approximation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 89–97, March–April, 2008.     

A dual-porosity model for gas reservoir flow incorporating adsorption behaviour—Part II. Numerical algorithm and example applications

In the present study, an algorithm is presented for the dual-porosity model formulated in Part I of this series. The resultant flow equation with the dual-porosity formulation is of an integro-(partial) differential equation involving differential terms for the Darcy flow in large fractures and integrals in time for diffusion within matrix blocks. The algorithm developed here to solve this equation involves a step-by-step finite difference procedure combined with a quadrature scheme. The quadrature scheme, used for the integral terms, is based on the trapezoidal method which is of second-order precision. This order of accuracy is consistent with the first- and second-order finite difference approximations used here to solve the differential terms in the derived flow equation. In an approach consistent with many petroleum reservoir and groundwater numerical flow models, the example formulation presented uses a first-order implicit algorithm. A two-dimensional example is also demonstrated, with the proposed model and numerical scheme being directly incorporated into the commercial gas reservoir simulator SIMED II that is based on a fully implicit finite difference approach. The solution procedure is applied to several problems to demonstrate its performance. Results from the derived dual-porosity formulation are also compared to the classic Warren–Root model. Whilst some of this work confirmed previous findings regarding Warren–Root inaccuracies at early times, it was also found that inaccuracy can re-enter the Warren–Root results whenever there are changes in boundary conditions leading to transient variation within the domain.     

Waves on a viscous-fluid film flowing down a vibrating vertical plate

A thin film of a viscous fluid flowing down a vertical plane in a gravitational field is considered. The plane executes harmonic oscillations in the direction normal to itself. An equation that describes the evolution of surface disturbances at small fluid flow rates is obtained. Some solutions of this equation are found. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 90–98, July–August, 1999.