Numerical analysis of the stability of nonisothermal couette flow

The stability of convective motion of a liquid between two rotating heated cylinders is investigated in the absence of external forces. The mathematical model for describing the convection is obtained from the general equations [1, 2] on the assumption that the density of the liquid, the thermal conductivity, the specific heat and the viscosity coefficients depend only on temperature, and that the work done by the pressure forces and the viscous dissipation are negligibly small. The thermal expansion coefficient of the liquid is not assumed to be small, which distinguishes the models in question from the classical Oberbeck-Boussinesq model [1, 3, 4]. Rostov-on-Don. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 70–76, September–October, 1988.     

Stability of nonisothermal couette flow

A study is made of the loss of stability of the stationary flow of a viscous fluid between two heated rotating cylinders. The linearized stability problem is studied in the Boussinesq approximation. The perturbations are assumed to be periodic in the time, and also in the axial and azimuthal directions. The neutral curves are calculated numerically. The ranges of variation of the parameters of the problem are found, and in them the most dangerous perturbations (in the class of spatially periodic perturbations) are those without rotational symmetry.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 167–170, January–February, 1980.I thank V. I. Yudovich for constant interest in the work.     

Calculation of auto-oscillations resulting from the loss of stability of a nonisothermal Couette flow

A study is made of the periodic (in time) auto-oscillatory regime (cycle) of azimuthal wave type which branches from a stationary nonisothermal Couette flow between rotating cylinders. Yudovich's method [1, 2] is used. It is shown that the cycle is unique (up to a shift along the axis z of the cylinders and rotation through any angle) and stable against spatially periodic three-dimensional perturbations. The results are given of a numerical calculation of the first two terms in the expansion of the cycle in Lyapunov-Schmidt series. The torque acting on the surface of the inner cylinder is calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 25–32, May–June, 1981.I thank V. I. Yudovich for constant interest in the work.     

New exact solution of the problem of rotationally symmetric Couette-Poiseuille flow

An exact solution is obtained for the problem of steady-state viscous incompressible flow under a pressure difference in the gap between coaxial cylinders for the case where the inner cylinder rotates at a constant angular velocity. The solution differs from the classical Couette-Poiseuille result by the presence of radial mass transfer, which provides for interaction between the poloidal and azimuthal circulations. The flow rate is found to depend linearly on the angular velocity of rotation of the inner cylinder. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 71–77, September–October, 2007.     

The stability of two stratified non-newtonian liquids in couette flow

Consideration is given to the stability of the interface between two Oldroyd liquids with shear-dependent viscosities, flowing in distinct layers while undergoing plane Couette flow. Results are presented as regions of stability in the plane determined by the logarithms of the viscosity and depth ratios. The work of previous authors for two Newtonian, power-law and constant-viscosity Oldroyd liquids is revealingly presented in a similar fashion. It is found that the dependence of the viscosities on shear-rate can drastically affect the regions of interfacial stability in a way over and above that due to just a change in the effective viscosity ratio. It is also found that for the Oldroyd liquids this viscosity variation affects the stability when it is present in the less-viscous layer.     

Two-phase couette flow

A study is made of plane laminar Couette flow, in which foreign particles are injected through the upper boundary. The effect of the particles on friction and heat transfer is analyzed on the basis of the equations of two-fluid theory. A two-phase boundary layer on a plate has been considered in [1, 2] with the effect of the particles on the gas flow field neglected. A solution has been obtained in [3] for a laminar boundary layer on a plate with allowance for the dynamic and thermal effects of the particles on the gas parameters. There are also solutions for the case of the impulsive motion of a plate in a two-phase medium [4–6], and local rotation of the particles is taken into account in [5, 6]. The simplest model accounting for the effect of the particles on friction and heat transfer for the general case, when the particles are not in equilibrium with the gas at the outer edge of the boundary layer, is Couette flow. This type of flow with particle injection and a fixed surface has been considered in [7] under the assumptions of constant gas viscosity and the simplest drag and heat-transfer law. A solution for an accelerated Couette flow without particle injection and with a wall has been obtained in [6]. In the present paper fairly general assumptions are used to obtain a numerical solution of the problem of two-phase Couette flow with particle injection, and simple formulas useful for estimating the effect of the particles on friction and heat transfer are also obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 42–46, May–June, 1976.     

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.     

The stability of two stratified ”power-law“ liquids in couette flow

Consideration is given to the stability of the flow of two ”power-law“ liquids between two infinite parallel planes when one of the planes moves with constant velocity in its own plane. It is found that the ratios of the power-law parameters for each layer have a dramatic effect and can be chosen to destabilize the flow.     

The stability of nonisothermal plasma flow in a flat MHD channel

Papers [1, 2] were devoted to questions of the stability of the laminar flow of a conducting fluid in a transverse magnetic field with Hartmann flow. It was assumed in these papers, however, that the transport coefficients are quantities independent of the flow characteristics; in particular, the temperature and the effect of energy dissipation were not taken into account. When these factors are allowed for it turns out that even for relatively small subsonic velocities, when the medium may be regarded as incompressible, the temperature distribution exerts a considerable influence on the dynamic flow characteristics. Papers [3,4] deal with this type of flow in an MHD channel which will be called nonisothermal in what follows. It has been shown that under specific conditions the velocity profiles are grossly deformed, and non-monotonic profiles with inflection points may even appear.However, the influence of nonisothermal flow on stability is not confined to an alteration of the stability criteria as a result of the change in the velocity profile. When energy dissipation and the fact that the transport coefficients are not constant are taken into account new dissipative instability branches appear, as, for example, the overheat instability [5, 8], This article considers the problem of the hydrodynamic stability of a nonisothermal plasma flow in constant crossed electric and magnetic fields in a flat channel with dielectric walls. The system of equations derived in this paper for the perturbations does, of course, take into account all the instability mechanisms mentioned above, but is difficult to solve. The general system of equations may be investigated in two limiting cases corresponding to the overheat and hydrodynamic instabilities.The author is most grateful to V. Kalitenko for writing the computer programs and to S. Filippov for advice and discussions.     

Secondary regimes in couette flow

We consider the asymptotic solutions of secondary steady flows in a fluid contained between cylinders rotating in the same direction for large Reynolds numbers.The existence of secondary axisymmetric steady flows in a fluid contained between cylinders rotating in the same direction was shown in [1, 2]. In the following we present the asymptotic behavior of such solutions for the case of large Reynolds numbers. The construction follows the scheme suggested in [3].     

Asymptotic investigation of two-phase couette flow

We consider plane and cylindrical Couette flow for a two-phase medium. The motion of the medium is described by the equations obtained in [1]. Collisions between the particles are disregarded, and their motion, in addition to the inertial forces, is determined by the pressure gradient of the carrying phase and the forces of viscous interaction between the carrying phase and the particles. We obtain simple asymptotic solutions of the indicated problems for small and large values of the dimensionless determining parameters. In a number of cases the solution has the nature of a boundary layer on solid walls.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 67–73, July–August, 1978.     

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

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Oscillatory flow and mass transfer within asymmetric and symmetric channels with sinusoidal wavy walls

Mass transfer for oscillatory flow was studied experimentally in channels with two different geometries, i.e., a periodically converging-diverging channel and a serpentine channel, both having sinusoidal wavy walls. The experiments were carried out under the following conditions: 10<Re<500 and 0.008<St<0.05. The channel geometries were found to have an important effect on the flow patterns and the mass transfer rates. At low Strouhal numbers of less than 0.023, the mass transfer rates for both channels were almost identical, regardless of different flow patterns and wall shear stresses. At high Strouhal numbers, however, the serpentine channel had a smaller mass transfer rate than the converging-diverging channel. The mass transfer characteristics were explained in terms of the vortex dynamics, wall shear stresses and fluid mixing based on numerical analysis and flow visualizations. The serpentine channel yields a better mass transfer and pumping power performance than the converging and diverging channel at low Strouhal numbers.

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Schwingungsströmung und Stofftransport innerhalb asymmetrischer und symmetrischer Kanäle mit sinusförmig gewellten Wandungen

Zusammenfassung Der Stofftransport bei Schwingungsströmungen in Kanälen mit zwei verschiedenen Geometrien experimentell untersucht, d.h. in einem periodisch konvergierenden und divergierenden Kanal und einem schlangenförmig gewundenen Serpentinenkanal, wobei die Kanäle jeweils sinus förmig gewellte Wandungen aufwiesen. Die Versuche wurden unter folgenden Bedingungen, ausgeführt; 10<Re<500 und 0.008<St<0.05. Es zeigte sich, daß die Kanalgeometrien einen erheblichen Einfluß auf die Strömungsmuster und Stofftransportraten haben. Bei niedrigen Strouhal-Zahlen unter 0.023 waren die Stofftransportraten beider Kanäle praktisch identisch, und zwar unabhängig von unterschiedlichen Strömungsmustern und Wand-Scherspannungen. Bei hohen Strouhal-Zahlen dagegen zeigte der Serpentinenkanal eine geringere Stofftransportrate als der Konvergenz-Divergenz-Kanal. Diese Stofftransport-Charakteristik wurde, basierend auf numerischer Analyse und Sichtbarmachung der Strömung, erklärt in Form von Vortexdynamic, Wand-Scherspannungen und Flüssigkeitsmischung. Bei niedrigen Strouhal-Zahlen erbringt der Serpentinenkanal ein besseres Stofftransport- und Pumpleistungsverhalten als der Konvergenz-Divergenz-Kanal.

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Nomenclature A area of mass transfer surface - a wave amplitude of wavy wall - c _b concentration of the ferricyanide ion - D diameter of piston - D molecular diffusivity of the ferricyanide ion - F Faraday constant - f frequency of oscillation - H _min minimum spacing between wavy walls - Re Reynolds number, Eq. (4) - s length of stroke of piston - Sc Schmidt number - Sh Sherwood number, Eq. (5) - St Strouhal number, Eq. (3) - T period of oscillation - U _p peak velocity based onH _min - W width of wavy wall Greek symbols wavelength of wavy wall - kinematic viscosity - _w wall vorticity - _max vortex strength - angle of misalignment of the two channel walls This work was supported in part by a Grant-in-Aid for Science Research (No. 63750889 and No. 03302031) from the ministry of Education, Science and Culture of Japan. The author acknowledge with thanks the assistance of graduate student Shigeki Matsune in the computations.     

Natural convection in unsteady MHD couette flow

 The combined effect of natural convection and uniform transverse magnetic field on the couette flow of an electrically conducting fluid between two parallel plates for impulsive motion of one of the plates in discussed. Under the assumption of negligible induced magnetic field and applied magnetic field being fixed relative to the fluid or plate, the governing equations have been solved exactly, and the expressions for velocity and temperature field have been presented for two different cases. A comparative study is made between the velocity field for magnetic field fixed with respect to plate and magnetic field fixed with respect to fluid. Received on 12 July 1999     

Oscillatory flow in microchannels

Commonly used, lumped-parameter expressions for the impedance of an incompressible viscous fluid subjected to harmonic oscillations in a channel were compared with exact expressions, based on solutions of the Navier-Stokes equations for slots and channels of circular and rectangular cross-section, and were found to differ by as much as 30% in amplitude. These differences resulted in predicted discrepancies by as much as 400% in frequency response amplitude for simple second-order systems based on size scales and frequencies encountered in microfluidic devices. These predictions were verified experimentally for rectangular microchannels and indicate that underdamped fluidic systems operating near the corner frequency of any included flow channel should be modeled with exact expressions for impedance to avoid potentially large errors in predicted behavior.List of symbols A Channel cross-sectional area (m²) - A_c Membrane area (m²) - a Rectangular duct and slot half-width or radius (m) - b Rectangular duct half-depth and slot depth (m) - C Capacitance (m³/Pa) - C - D_h Channel hydraulic diameter (m) - E Voltage (V) - f Darcy friction factor - F Force (N) - I Channel inertance (Pa s²/m³) - i - Imaginary part of a complex number - J_k Bessel function of the first kind of order k - System transfer function - K Sum of minor loss factors - k Membrane stiffness (N/m) - L Channel length (m) - n Outward unit normal vector - P Fluid pressure (Pa) - p_n - Q Volumetric flow rate (m³/s) - R Channel resistance (Pa s/m³) - Real part of a complex number - Re Reynolds number, - V Velocity (m/s) - _V Volume (m³) - w Axial component of velocity (m/s) - Harmonic amplitude of membrane centerline displacement - Fluid impedance (kg/m⁴ s) - Duct aspect ratio, b/a - ² Nondimensional frequency parameter, - Nondimensional corner frequency, - Membrane shape factor - C/C - µ Fluid dynamic viscosity (Pa s) - Fluid kinematic viscosity (m²/s) - Mass density (kg/m³) - Radian frequency - _c R_s/I_s cutoff or corner frequency - _n Undamped natural frequency - Channel shape parameter in Eqs. 29 and 30 - Damping ratio - ( )_e Exact property - ( )_s Simplified property - (^–) Spatial average - Complex quantity