Sensitivity of plane Poiseuille flow to vibration of the channel walls

An accurate quantitative investigation of the disturbances induced in a Poiseuille flow by vibration of the walls is made on the basis of the Fourier transformation method.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 12–19, July–August, 1992.     

Relative periodic orbits in plane Poiseuille flow

A branch of relative periodic orbits is found in plane Poiseuille flow in a periodic domain at Reynolds numbers ranging from Re=3000$\mathit{Re}=3000$ to Re=5000$\mathit{Re}=5000$. These solutions consist in sinuous quasi-streamwise streaks periodically forced by quasi-streamwise vortices in a self-sustained process. The streaks and the vortices are located in the bulk of the flow. Only the amplitude, but not the shape, of the averaged velocity components does change as the Reynolds number is increased from 3000 to 5000. We conjecture that these solutions could therefore be related to large- and very large-scale structures observed in the bulk of fully developed turbulent channel flows.     

Stability of plane Poiseuille flow of a slightly viscoelastic fluid

Consideration is given to the stability of planePoiseuille flow of a slightly viscoelastic fluid which has a constant viscosity and normal stress differences varying nearly with the shear rate. It is shown that the presence of elasticity lowers the criticalReynolds number at which instability occurs.     

The stability of the plane Poiseuille flow of a non-Newtonian fluid

Rheologica Acta - This paper considers the flow and the stability of the flow of a fluid whose viscosity depends on the shear in the form $$\nu = {\nu _0}\left\{ {r - s{{\left( {\frac{{d\bar...     

Stability of plane Poiseuille flow of slightly viscoelastic fluids

Summary The title subject has been examined by the author in a series of papers (Cousins, 1970, 1972a, b), and the assumptions and principal results of those papers are discussed here. The work is motivated by the phenomenon evinced in fluid flow situations, of turbulent drag reduction by certain polymer additives. From a survey of experimental work it is clear that molecular elongation plays an important role in reducing drag by suppressing transverse motions. This effect may be interpreted as a normal stress effect in a continuum theory. A second-order fluid, which is a simple model exhibiting such a property, is used in a linear analysis of disturbances to planePoiseuille flow. Unlike theNewtonin case Squire's theorem is not valid (Lockett, 1969a) and a three-dimensional analysis is required. The viscoelastic terms are in general destabilising. Under certain conditions the first growing disturbance will propagate at an angle to the basic flow, giving a longitudinal vortex structure close to the channel boundaries not present at the onset of instability in aNewtonian fluid. The analysis is extended to finite-amplitude disturbances by introducing a time-dependent amplitude, but calculations are here confined to the simpler two-dimensional case. Disturbances which would decay under linear theory may in fact grow provided the initial amplitude is sufficiently large. A threshold amplitude for instability is found as a function ofReynolds number. The viscoelastic terms are again found to be destabilising. Finally, a further viscoelastic property, that of stress relaxation, is introduced through an integral representation of the stress. A linear analysis is developed and stress relaxation is also shown to be a destabilising influence.With 6 figures     

Dynamics of an elastic capsule in moderate Reynolds number Poiseuille flow

The dynamic motions and lateral equilibrium positions of a two-dimensional elastic capsule in a Poiseuille flow were explored at moderate Reynolds number (10 ⩽ Re ⩽ 100) as a function of the initial lateral position (y₀), Re, aspect ratio (ɛ), size ratio ($\lambda$), membrane stretching coefficient (φ) and bending coefficient (γ). The transition between tank-treading (TT) and swinging (SW) to tumbling (TU) motions was observed and the lateral equilibrium positions of the capsules varied according to the conditions. The initial behavior of the elastic capsule was influenced by variation in the initial lateral position (y₀), but the equilibrium position and dynamic motion of the capsule were not affected by such variation. The capsules had a stronger tendency toward TU motion at higher values of Re, φ and γ, whereas the capsules underwent TT or SW motion as the values of ɛ and $\lambda$ increased. Under moderate Re Poiseuille flows, capsules tended to migrate across streamlines to a specific equilibrium position. The lateral equilibrium position shifted toward the centerline at larger $\lambda$ and migrated toward the wall at larger $\epsilon \text{,}\varphi \text{and}\gamma$. As Re increased, the equilibrium position first shifted toward the bottom wall, then toward the channel center. However, different equilibrium position trends were obtained around the SW–TU transition. The capsule undergoing TU motion tended to migrate downward toward the bottom wall more than the capsule undergoing SW motion, all other conditions being similar.     

Thermal instability of a viscoelastic fluid in a fluid-porous system with a plane Poiseuille flow

The thermal convection of a Jeffreys fluid subjected to a plane Poiseuille flow in a fluid-porous system composed of a fluid layer and a porous layer is studied in the paper. A linear stability analysis and a Chebyshev τ-QZ algorithm are employed to solve the thermal mixed convection. Unlike the case in a single layer, the neutral curves of the two-layer system may be bi-modal in the proper depth ratio of the two layers. We find that the longitudinal rolls(LRs) only depend on the depth ratio. Wi...     

Unstable manifold computations for the two-dimensional plane Poiseuille flow

We follow the unstable manifold of periodic and quasi-periodic solutions in time for the Poiseuille problem, using two formulations: holding a constant flux or mean pressure gradient. By means of a numerical integrator of the Navier–Stokes equations, we let the fluid evolve from an initially perturbed unstable solution until the fluid reaches an attracting state. Thus, we detect several connections among different configurations of the flow such as laminar, periodic, quasi-periodic with two or three basic frequencies, and more complex sets that we have not been able to classify. These connections make possible the location of new families of solutions, usually hard to find by means of numerical continuation of curves, and show the richness of the dynamics of the Poiseuille flow. PACS 05.45.-a, 47.11.+j, 47.20.-k, 47.20.Ft     

The linear stability of plane Poiseuille flow under unsteady distortion

This paper investigates the linear stability behaviour of plane Poiseuille flow underunsteady distortion by multiscale perturbation method and discusses further the problemproposed by paper[1].The results show that in the initial period of disturbancedevelopment,the distortion profiles presented by paper[1]will make the disturbances growup,thus augmenting the possibility of instability.     

Finite difference analysis of plane Poiseuille and Couette flow developments

Summary The problem of flow development from an initially flat velocity profile in the plane Poiseuille and Couette flow geometry is investigated for a viscous fluid. The basic governing momentum and continuity equations are expressed in finite difference form and solved numerically on a high speed digital computer for a mesh network superimposed on the flow field. Results are obtained for the variations of velocity, pressure and resistance coefficient throughout the development region. A characteristic development length is defined and evaluated for both types of flow.Nomenclature h width of channel - L ratio of development length to channel width - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, p/ ² - P ₀ dimensionless pressure at channel mouth - P pressure defect, P ₀–P - (P)₀ pressure defect neglecting inertia - Re Reynolds number, uh/ - u fluid velocity in x-direction - mean u velocity across channel - u ₀ wall velocity - U dimensionles u velocity u/ - U _c dimensionless centreline velocity - U ₀ dimensionless wall velocity - v fluid velocity in y-direction - V dimensionless v velocity, hv/ - x coordinate along channel - X dimensionless x-coordinate, x/h ² - y coordinate across channel - Y dimensionless y-coordinate, y/h - resistance coefficient, - ₀ resistance coefficient neglecting inertia - fluid density - fluid viscosity     

Stabilization of small perturbations of Poiseuille flow in a channel with elastic walls

Arbitrary three-dimensional perturbations are considered. It is established that as the compliance of the walls increases, oblique waves become the most dangerous, which essentially differentiates the system in question from Poiseuille flow in a rigid channel. The flow stability is analyzed over a broad interval of values of the elasticity parameter overlapping the values for real materials.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 67–72, May–June, 1988.     

Algorithm for transient growth of perturbations in channel Poiseuille flow

This study develops a direct optimal growth algorithm for three-dimensional transient growth analysis of perturbations in channel flows which are globally stable but locally unstable. Different from traditional non-modal methods based on the OrrSommerfeld and Squire(OSS) equations that assume simple base flows, this algorithm can be applied to arbitrarily complex base flows. In the proposed algorithm, a reorthogonalization Arnoldi method is used to improve orthogonality of the orthogonal basis of the Krylov subspace generated by solving the linearized forward and adjoint Navier-Stokes(N-S) equations. The linearized adjoint N-S equations with the specific boundary conditions for the channel are derived, and a new convergence criterion is proposed. The algorithm is then applied to a one-dimensional base flow(the plane Poiseuille flow) and a two-dimensional base flow(the plane Poiseuille flow with a low-speed streak)in a channel. For one-dimensional cases, the effects of the spanwise width of the channel and the Reynolds number on the transient growth of perturbations are studied. For two-dimensional cases, the effect of strength of initial low-speed streak is discussed. The presence of the streak in the plane Poiseuille flow leads to a larger and quicker growth of the perturbations than that in the one-dimensional case. For both cases, the results show that an optimal flow field leading to the largest growth of perturbations is characterized by high-and low-speed streaks and the corresponding streamwise vortical structures.The lift-up mechanism that induces the transient growth of perturbations is discussed.The performance of the re-orthogonalization Arnoldi technique in the algorithm for both one-and two-dimensional base flows is demonstrated, and the algorithm is validated by comparing the results with those obtained from the OSS equations method and the crosscheck method.     

On the dispersion of a solute in Poiseuille flow of a third-grade fluid in a channel

Gill and Sankarasubramanian's analysis of the dispersion of Newtonian fluids in laminar flow between two parallel walls are extended to the flow of non-Newtonian viscoelastic fluid (known as third-grade fluid) using a generalized dispersion model which is valid for all times after the solute injection. The exact expression is obtained for longitudinal convective coefficient K₁(Γ), which shows the effect of the added viscosity coefficient Γ on the convective coefficient. It is seen that the value of the K₁(Γ) for Γ≠0 is always smaller than the corresponding value for a Newtonian fluid. Also, the effect of the added viscosity coefficient on the K₂(t,Γ) (which is a measure of the longitudinal dispersion coefficient of the solute) is explored numerically. Finally, the axial distribution of the average concentration C_m of the solute over the channel cross-section is determined at a fixed instant after the solute injection for several values of the added viscosity coefficient.     

The stability of the modified plane Poiseuille flow in the presence of a transverse magnetic field

The stability against small disturbances of the pressure-driven plane laminar motion of an electrically conducting fluid under a transverse magnetic field is investigated. Assuming that the outer regions adjacent to the fluid layer are electrically non-conducting and not ferromagnetic, the appropriate boundary conditions on the magnetic field perturbations are presented. The Chebyshev collocation method is adopted to obtain the eigenvalue equation, which is then solved numerically. The critical Reynolds number R_c, the critical wave number α_c, and the critical wave speed c_c are obtained for wide ranges of the magnetic Prandtl number P_m and the Hartmann number M. It is found that except for the case when P_m is sufficiently small, the magnetic field has both stabilizing and destabilizing effects on the fluid flow, and that for a fixed value of M the fluid flow becomes more unstable as P_m increases.     

On the accuracy of the calculation of transient growth in plane Poiseuille flow

This paper addresses the accuracy of numerical methods to compute the transient energy growth of plane Poiseuille flow. We show that using the Chebyshev collocation method to discretize the linearized governing equations in the wall‐normal direction can introduce numerical problems when computing the energy evolution of the flow. We demonstrate that spurious eigenmodes of the discretized linear operator and numerical integration errors are the possible sources of the numerical problems, and we also show that spurious eigenvalues with negative real parts of large magnitude can affect the calculation of energy growth. These difficulties can be avoided by using a spectral Galerkin method where the basis functions satisfy the boundary conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonsteady-state nonisothermal flow of a magnetic liquid in a plane channel

Magnetic liquids are finding wider and wider use in various fields of technology [1]. Such liquids can be used as heat exchange fluids in equipment which generates a magnetic field under conditions of weightlessness [2] and in a number of other applications. The efficiency of heat exchange equipment is determined to a significant degree by the temperature of the magnetic liquid. In connection with this fact, it is of interest to examine nonisothermal flows at a temperature near the Curie point, where the dependence of volume magnetization M on temperature is expressed most clearly. In this case the character of the liquid flow will be affected not only by the dependence of saturation volume magnetization on temperature, but also by temperature inhomogeneity caused by development of external heat sources and sinks produced by the magnetocaloric effect. We note that although this is a weak effect [3], the temperature redistribution over channel section which it produces may be significant. With a high gradient in the external magnetic field H even a small change in temperature can significantly change the force acting on a magnetic liquid element. The unique features of magnetic liquid flow at a temperature close to the Curie point can be investigated by simultaneously solving the equations of motion and thermal conductivity.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 1, pp. 93–96, January–February, 1984.The author expresses his gratitude to the participants in K. B. Pavlov's scientific seminar for their evaluation of the study.