Stability of the inviscid plane couette flow in bubbly fluids

The aim of this article is to study the stability of shear flows in bubbly fluids. A mathematical model of bubbly fluids is presented. The stability of shear flows is studied by two methods: by using a spectral approach and by solving the initial-value problem. It is proved that the linear velocity profile is stable in the long wave approximation. Communicated by R. Grimshaw     

Stability of plane poiseuille and couette flow of a Maxwell fluid

The classical Orr-Sommerfeld analysis is extended to a Maxwell fluid in fully developed Poiseuille flow between two flat plates and Couette flow between two flat plates. For the Poiseuille flow problem eigenmodes that are anti-symmetric in position are considered to augment the literature results for the symmetric eigenmodes. A shooting method with a stiff integrator, orthonormalization, and Newton-Raphson iterations on the eigenvalue are used to find the eigenvalues. The most dangerous mode is the anti-symmetric one, and both symmetric and anti-symmetric modes are more dangerous when the wave number and the Weissenberg number are large. No unstable eigenvalues are found.     

Unsteady-state development of plane couette flow for viscoelastic fluids

Unsteady-state development of plane Couette flow for viscoelastic fluids is analyzed using a constitutive equation that can be obtained from molecular theory, in which the molecules are regarded as finitely extensible dumbbells. Typical features of the flow situation are as follows: (i) For a fluid with moderate elasticity, not only stress overshoot but also velocity overshoot are predicted. (ii) For suitable combinations of elasticity and gap width, and for some time intervals stress propagation and reflection phenomena are predicted. (iii) After a sufficient time has elapsed, the stress state behaves similarly to that corresponding to the start-up of a steady simple shear flow.     

THE APPLICATION OF MULTI－SCALE PERTURBATION METHOD TO THE STABILITY ANALYSIS OF PLANE COUETTE FLOW

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Stability of plane magnetohydrodynamic couette flow with asymmetrical velocity profile

The hydrodynamic stability of plane magnetohydrodynamic Couette flow with asymmetrical velocity profile formed by a transverse magnetic field is investigated within the framework of the linear theory. The complete spectrum of the small perturbations is studied for the characteristic Hartmann numbers. The perturbations are classified in accordance with their phase velocity at large wave numbers. It is established that the stability of the flow is controlled by only one type of perturbations. The critical parameters of the problem are determined. The instability in question recalls the instability of Hartmann flow against asymmetrical perturbations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 12–18, May–June, 1971.The author thanks M. A. Gol'dshtik for interest in the work and V. A. Sapozhnikov and V. N. Shtern for useful discussions.     

Criticality and transition for a steady reactive plane couette flow of a viscous fluid

Numerical integration is used to determine critical and transitional values of parameters for steady, reactive, viscous, one dimensional plane Couette flow of an incompressible, homogeneous fluid of third-grade with the lower plate at rest while the upper is in uniform motion. The solutions are found for the following cases: (i) Bimolecular (ii) Arrhenius and (iii) Sensitized temperature dependence. Specifically, it is shown that the parameter Λ controlling the non-Newtonian fluid does not affect the flow velocity in any sense while the influence on the viscous dissipation parameter Γ is examined. The results obtained are then compared with similar results in the literature.     

Nonisothermal flow of plasma in a plane MHD-channel

There is a large number of published papers (see the references given in the review [1]) in which various cases of the steady-state laminar flow of a conducting medium in a plane channel in the presence of a transverse magnetic field are considered. However, it has always been assumed that the transport coefficients were constants independent of the flow parameters such as the temperature. As a result, the dynamic and thermal problems were separable, and me temperature distribution had no effect on the dynamic flow parameters.In a low-temperature dense plasma, the conductivity is a very rapidly increasing function of temperature (it is approximately given by e^–A/T or T^10–13). It is clear that, in this case, it is necessary to take into account the fact that the transport coefficients are not constants, and the dynamic and thermal problems are not separable even for an incompressible fluid. We shall refer to such flows as nonisothermal, and contrast them with flows for which the transport coefficients are constants, and which we shall refer to as isothermal. for the sake of brevity.The importance of effects due to the nonisothermal nature of a flow was demonstrated in [2, 3], Hysteresis effects in friction and heat transfer, which were found in these papers, were discussed qualitatively in [4], Finally, the flow of a fluid with temperature-dependent conductivity in an MHD-channel was considered in [5] where it was. noted that the nonisothermal nature of the flow must be taken into account. In particular, the appearance of nonmonotonic velocity profiles with points of inflection was demonstrated [5], However, the latter paper included a number of conflicting assumptions. For example, when the propagation of heat was considered along the flow, it was assumed that the conductivity varied only across the channel. The temperature dependence of the conductivity used [5] was quite unrealistic, while the temperature dependence of viscosity and thermal conductivity was not taken into account at all.In the present paper we investigate the flow of plasma in a plane MHD-channel in the absence of a longitudinal flow of heat but with allowance for the temperature dependence of the transport coefficients. We shall use a more realistic form of the temperature dependence for the above parameters, and will take viscous energy dissipation into account.The authors are deeply indebted to M. V. Maslennikov and Yu. S. Sigov for valuable advice in connection with the numerical method of solution.     

Linear stability of plane couette flow of an upper convected maxwell fluid

We investigate the linear stability of plane Couette flow of an upper convected Maxwell fluid using a spectral method to compute the eigenvalues. No instabilities are found. This is in agreement with the results of Ho and Denn [1] and Lee and Finlayson [2], but contradicts “proofs” of instability by Gorodtsov and Leonov [3] and Akbay and Frischmann [4,5]. The errors in those arguments are pointed out. We also find that the numerical discretization can generate artificial instabilities (see also [1,6]). The qualitative behavior of the eigenvalue spectrum as well as the artificial instabilities is discussed.     

A kinetic analysis of couette plasma flow in an electric field

Steady flow of a completely ionized plasma between parallel plates moving in their own plane in the presence of an electric field is examined. The distribution functions of the ions and electrons are derived from the kinetic Boltzmann equations supplemented by equations for the electric field. The solution is constructed by means of a variation of the method of moments; at the same time, it is assumed that momentum is transferred only by ions and heat is transferred by electrons. The analysis takes into account near collisions between particles for an arbitrary degree of rarefaction of the plasma. An example of calculation of the principal characteristics of the flow is given.The boundary problems connected with the presence of solid surfaces in the flow are of far less practical importance in plasma dynamics than in gasdynamics. This is because a plasma can exist only at very high temperatures which destroy the majority of materials to a greater or lesser extent. As a rule, in order to contain a plasma within definite bounds, strong magnetic fields are utilized, not solid walls. Nevertheless, consideration of problems with bounding surfaces is still of definite interest in the case of plasma. An approximate solution of one of the simplest problems of this kind is given below.     

Characteristics of a viscoplastic material in the couette flow

A model governing a steady flow of a viscoplastic material between coaxial cylinders is proposed. Nonlinear velocity sensitivity typical of superplastic materials is taken into account. An algorithm of calculating the characteristics of the material is developed. The algorithm is based on the experimental data on moments and angular velocities of the rotating coaxial cylinders. The stability of the algorithm to errors in the initial data is estimated.     

Critical Reynolds number of the couette flow of a vibrationally excited diatomic gas energy approach

An energy balance equation for plane-parallel flows of a vibrationally excited diatomic gas described by a two-temperature relaxation model is derived within the framework of the nonlinear energy theory of hydrodynamic stability. A variational problem of calculating critical Reynolds numbers Re_cr determining the lower boundary of the possible beginning of the laminar-turbulent transition is considered for this equation. Asymptotic estimates of Re_cr are obtained, which show the characteristic dependences of the critical Reynolds number on the Mach number, bulk viscosity, and relaxation time. A problem for arbitrary wave numbers is solved by the collocation method. In the realistic range of flow parameters for a diatomic gas, the minimum critical Reynolds numbers are reached on modes of streamwise disturbances and increase approximately by a factor of 2.5 as the flow parameters increase.     

Numerical simulation of plasma motion in a magnetic field. Two-dimensional case

A model of dynamics and heating of a plasma cloud in a magnetic field is considered in a two-temperature approximation. Based on a predictor-corrector-type implicit difference scheme, spreading of a plasma cloud in an external magnetic field is numerically simulated, and the influence of this field on spread dynamics is evaluated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 121–132, May–June, 2007.     

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).     

Numerical simulation of the transient shape of the red blood cell in microcapillary flow

The transient shape of a red blood cell (RBC) in a microcapillary flow is simulated under different initial conditions, including various axis orientations and centroid locations, using the LBM-DLM/FD method, which is derived from the lattice Boltzmann method and the distributed Lagrange multiplier/fictitious domain method. Although the terminal velocity is not sensitive to the initial configuration, the evolution of the velocity and the shape are determined by the initial conditions. The parachute and the slipper shape are the most probable shapes for a deformed RBC in the flow. An RBC with an initial axis orientation of 90 degrees exhibits a more complicated deformation. RBCs have a tendency to move to the centerline of a tube if an offset between the RBC centroid and the centerline exists. Our numerical results are validated by experiments, and some details beyond the experiment are provided.     

Mass transport for couette flow in a flat conduit

The mass transport is determined in a two-dimensional channel due to pure shear-Couette flow. Local concentration, mean concentration and flux at the walls of the conduit are determined analytically for arbitrary wall velocity ratios. The special case of a fixed lower wall and a moving upper wall of the channel has been numerically evaluated. In addition the case of linear homogeneous chemical reaction has been treated for shear flow of the liquid in the conduit.

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Stofftransport bei Couette-Strömung im Kanal

Zusammenfassung Es wird der Stofftransport in einem Kanal mit reiner Scher-Couette-Strömung analytisch bestimmt. Dabei wird die lokale Konzentration, die mittlere Konzentration, sowie der Konzentrationsflu\ über die WÄnde bei beliebigem GeschwindigkeitsverhÄltnis der WÄnde bestimmt. Die numerische Auswertung der analytischen Ergebnisse beschrÄnkte sich auf den Fall des Stofftransportes in einem Kanal, dessen untere Wand sich in Ruhe befindet, wÄhrend die obere Wand sich mit konstanter Geschwindigkeit bewegt. Es wird au\erdem die Konzentration bei linearer homogener chemischer Reaktion bei Scherströmung angegeben.

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Nomenclature c concentration - c _w wall concentration aty=0,h - c _i initial concentration atx = 0 - ¯ c (x) mean concentration - D diffusion coefficient - h width of the channel - k ^* homogeneous chemical reaction coefficient - u ₁ velocity of lower channel wally=0 - u ₂ velocity of upper channel wally=h - x axial coordinate - y coordinate perpendicular to channel (y ^*=y/h) - _n, _n Eigenvalues - J _±1/3 Besselfunction of ±1/3 order