Low-frequency free vibrations of viscoelastic cylindrical shells

Vitebsk State Pedagogical Institute, Belarus. Translated from Prikladnaya Mekhanika, Vol. 28, No. 9, pp. 50–55, September, 1992.     

Electrohydrodynamic instability of two superposed Walters B´ viscoelastic fluids in relative motion through porous medium

Summary  The electrohydrodynamic Kelvin–Helmholtz instability of the interface between two uniform superposed viscoelastic (B′ model) dielectric fluids streaming through a porous medium is investigated. The considered system is influenced by applied electric fields acting normally to the interface between the two media, at which there are no surface charges present. In the absence of surface tension, perturbations transverse to the direction of streaming are found to be unaffected by either streaming and applied electric fields for the potentially unstable configuration, or streaming only for the potentially stable configuration, as long as perturbations in the direction of streaming are ignored. For perturbations in all other directions, there exists instability for a certain wavenumber range. The instability of this system can be enhanced (increased) by normal electric fields. In the presence of surface tension, it is found also that the normal electric fields have destabilizing effects, and that the surface tension is able to suppress the Kelvin–Helmholtz instability for small wavelength perturbations, and the medium porosity reduces the stability range given in terms of the velocities difference and the electric fields effect. Finally, it is shown that the presence of surface tension enhances the stabilizing effect played by the fluid velocities, and that the kinematic viscoelasticity has a stabilizing as well as a destabilizing effect on the considered system under certain conditions. Graphics have been plotted by giving numerical values to the parameters, to depict the stability characteristics. Received 27 March 2000; accepted for publication 3 May 2001     

Coriolis effect on thermal convective instability of viscoelastic fluids in a rotating porous cylindrical annulus

Linear stability analysis of thermal convection is studied for a viscoelastic fluid in a rotating porous cylindrical annulus. The modified Darcy–Jeffrey model with the addition of the Coriolis term in a rotating frame of reference is applied to characterize the non-Newtonian rheology in porous media. We investigate how the interaction among the Coriolis force, viscoelasticity, and bounded sidewalls affects the preferred mode at the onset of convection. The results show that for a slowly rotating case, the oscillatory mode is always preferred for any considered cylindrical radii. However, for a moderately rotating case, the oscillatory preferred mode only arises intermittently as the outer cylindrical radius gradually increases. This result is quite different from the case for viscoelastic fluids in a rotating porous layer or in a porous cylinder without rotation. Further, we discover that for a pair of given cylindrical radii when the Taylor number exceeds a critical value depending on the viscoelastic parameters, the oscillatory convection does not occur. We also examine how the variations of the Taylor number and the viscoelastic parameters affect the patterns of temperature disturbance at the onset of convection.     

Linear dynamic model for porous media saturated by two immiscible fluids

A linear isothermal dynamic model for a porous medium saturated by two immiscible fluids is developed in the paper. In contrast to the mixture theory, phase separation is avoided by introducing one energy for the porous medium. It is an important advantage of the model based on one energy approach that it can account for the couplings between the phases. The volume fraction of each phase is characterized by the porosity of the porous medium and the saturation of the wetting phase. The mass and momentum balance equations are constructed according to the generalized mixture theory. Constitutive relations for the stress, pore pressure are derived from the free energy function. A capillary pressure relaxation model characterizing one attenuation mechanism of the two-fluid saturated porous medium is introduced under the constraint of the entropy inequality. In order to describe the momentum interaction between the fluids and the solid, a frequency independent drag force model is introduced. The details of parameter estimation are discussed in the paper. It is demonstrated that all the material parameters in our model can be calculated by the phenomenological parameters, which are measurable. The equations of motion in the frequency domain are obtained in terms of the Fourier transformation. In terms of the equations of motion in the frequency domain, the wave velocities and the attenuations for three P waves and one S wave are calculated. The influences of the capillary pressure relaxation coefficient and the saturation of the wetting phase on the velocities and attenuation coefficients for the four wave modes are discussed in the numerical examples.     

Magnetohydrodynamics instability of interfacial waves between two immiscible incompressible cylindrical fluids

The problem of nonlinear instability of interfacial waves between two immiscible conducting cylindrical fluids of a weak Oldroyd 3-constant kind is studied. The system is assumed to be influenced by an axial magnetic field, where the effect of surface tension is taken into account. The analysis, based on the method of multiple scale in both space and time, includes the linear as well as the nonlinear effects. This scheme leads to imposing of two levels of the solvability conditions, which are used to construct like-nonlinear Schr6dinger equations （1-NLS） with complex coefficients. These equations generally describe the competition between nonlinearity and dispersion. The stability criteria are theoret- ically discussed and thereby stability diagrams are obtained for different sets of physical parameters. Proceeding to the nonlinear step of the problem, the results show the appearance of dual role of some physical parameters. Moreover, these effects depend on the wave kind, short or long, except for the ordinary viscosity parameter. The effect of the field on the system stability depends on the existence of viscosity and differs in the linear case of the problem from the nonlinear one. There is an obvious difference between the effect of the three Oldroyd constants on the system stability. New instability regions in the parameter space, which appear due to nonlinear effects, are shown.     

Conjugate free convection of non-Newtonian fluids about a vertical cylindrical fin in porous media

Conjugate free convection along a vertical cylindrical fin in a non-Newtonian fluid-saturated porous medium has been investigated theoretically. The boundary layer equations based on the power lay model appropriate for the Darcy flows are solved numerically exploiting a very efficient finite difference method. Effects of the power-law index, conjugate convection-conduction parameter and the surface curvature parameter on the fin temperature distribution, local heat transfer-coefficient and local heat flux are studied and presented in graphical and tabular form.     

Wave propagation through an interface between viscoelastic media in the presence of defects

The propagation of plane harmonic waves through an interface between viscoelastic media is considered using the equations of field theory of defects, the kinematic identities for an elastic continuum with defects, and the dynamic equations of gauge theory. The reflection and refraction coefficients of elastic displacement waves and the waves of a defect field characterized by a dislocation density tensor and a defect flux tensor are determined. Dependences of the obtained quantities on the parameters of the interfacing media are analyzed.     

Low-frequency dilatational wave propagation through unsaturated porous media containing two immiscible fluids

An analytical theory is presented for the low-frequency behavior of dilatational waves propagating through a homogeneous elastic porous medium containing two immiscible fluids. The theory is based on the Berryman–Thigpen–Chin (BTC) model, in which capillary pressure effects are neglected. We show that the BTC model equations in the frequency domain can be transformed, at sufficiently low frequencies, into a dissipative wave equation (telegraph equation) and a propagating wave equation in the time domain. These partial differential equations describe two independent modes of dilatational wave motion that are analogous to the Biot fast and slow compressional waves in a single-fluid system. The equations can be solved analytically under a variety of initial and boundary conditions. The stipulation of “low frequency” underlying the derivation of our equations in the time domain is shown to require that the excitation frequency of wave motions be much smaller than a critical frequency. This frequency is shown to be the inverse of an intrinsic time scale that depends on an effective kinematic shear viscosity of the interstitial fluids and the intrinsic permeability of the porous medium. Numerical calculations indicate that the critical frequency in both unconsolidated and consolidated materials containing water and a nonaqueous phase liquid ranges typically from kHz to MHz. Thus engineering problems involving the dynamic response of an unsaturated porous medium to low excitation frequencies (e.g., seismic wave stimulation) should be accurately modeled by our equations after suitable initial and boundary conditions are imposed.     

Structural instability in the sedimentation of particulate suspensions through viscoelastic fluids

A theory is developed to describe a structural instability that has been observed during the sedimentation of particulate suspensions through viscoelastic fluids. The theory is based on the assumption that the influence of hydrodynamic interactions in viscoelastic fluids, which tend to cause particles to aggregate, is in competition with hydrodynamic dispersion, which acts to maintain a homogeneous microstructure. In keeping with the experimental observations, it predicts that the suspension structure will stratify into vertical columns when a dimensionless stability parameter exceeds a critical value. The column-to-column separation, measured in particle radii, is predicted to be proportional to the square root of the ratio of the dimensionless dispersion coefficient to the product of the particle volume fraction and the Deborah number. The time for the formation of the columns is predicted to scale with the inverse of the average volume fraction. These predictions are in agreement with experimental data reported in the literature.     

The deformation of an interface between two fluids in a porous medium

Summary The interface between two moving fluids in a porous medium will, in general, deform under the influence of gravity and drag forces. An example of some importance is the formation of so-called gravity tongues in oil reservoirs. This paper deals with the displacement of oil by water in a homogeneous non-horizontal oil stratum. The deformation of such an interface can be deduced by numerical procedures based upon exact methods. The use of these methods is limited, however, owing to the fact that in oil reservoirs the dip is usually smaller than 10 to 20 degrees. In such cases, where the interface is initially horizontal, the computation of the form of the interface as a function of time becomes so enormous, even when a fast electronic computer is used, that an approximative method is more useful. In this paper two approximate solutions are presented. The first one is obtained by using a simplified form of the dynamic interface condition, in which the flow velocity component perpendicular to the dip direction of the reservoir is neglected. This simplification has previously been used by Dietz, who gave a first-order approximation with respect to time. More complicated results are obtained by using the second approximation where, in accordance with the dynamic boundary condition, this velocity component is more or less taken into account. In both methods, the form of the interface as a function of time is expressed in a parametric representation. Moreover, the amount of water that has passed a given cross-section and the flow of water at this section are obtained as a function of time and the parameter used. Results of both methods are compared with each other and with those obtained by an exact method. Both approximations are found to be good in those cases where the dip of the reservoir is not too high, but this is precisely when exact methods are impracticable.Nomenclature d thickness of the idealised reservoir (see fig. 1) - f function of y as given by (2.7) - f, f, f first, second and third derivative of f with respect to y - F(y, ) function of y and as given in the appendix - G dimensionless quantity - G* dimensionless quantity {= G cos /(1–G sin )} - H(y, ) function of y and as given in the appendix - M dimensionless quantity _{2 1}/ _{1 2} - p pressure - q _w the flow of water at a given cross-section - Q _w the total amount of water that has already passed a given cross-section at a certain time - S ₀ oil saturation in the oil region - S _w water saturation in the water region - r integration variable - s the co-ordinate along the interface (positive direction as given in fig. 1) - t time - t _w time at which water breaks through at a given cross-section - u ₁ mean velocity component of fluid 1 in x-direction in the pores of the porous medium (water) - u ₂ mean velocity component of fluid 2 in x-direction in the pores of the porous medium (oil) - U _r the relative deformation velocity of the interface {=(x _i –W ₀ t)/t} _y - the mean fluid velocity vector in the pores of the porous medium - v ₁ mean velocity component of fluid 1 in y-direction in the pores of the porous medium (water) - v ₂ mean velocity component of fluid 2 in y-direction in the pores of the porous medium (oil) - v _n mean velocity component of the fluids normal to the interface (positive direction from fluid 1 to fluid 2) - W ₀ mean velocity of fluid 1 (water) when x –, where the velocity component in y-direction is equal to zero - x co-ordinate, parallel to the boundaries of the reservoir (see fig. 1) - x _e value of x for a given cross-section - x _i , y _i values of the x and y co-ordinates corresponding to the points of the interface - x ₀(y) initial value of the x co-ordinate of the points of the interface (at t=0) - y co-ordinate, perpendicular to the boundaries of the reservoir (see fig. 1) - y _e (t) time-dependent value of the y co-ordinate of the interface if the value of the x co-ordinate is equal to x _e - y _i , x _i values of the y and x co-ordinates corresponding to the points of the interface - z vertical co-ordinate (positive direction as given in fig. 1) - the angle between the horizon and the boundaries of the reservoir (see fig. 1) - the angle between the x axis and the normal to the interface (see fig. 1) - _e the angle if the value of x _i is equal to x _e - ₀(y) initial value of the angle (at t=0) - effective permeability of the porous medium divided by the product of the porosity and fluid saturation - ₁ effective permeability of the porous medium to fluid 1 divided by the product of the porosity and the saturation of fluid 1 - ₂ effective permeability of the porous medium to fluid 2 divided by the product of the porosity and the saturation of fluid 2 - fluid viscosity - ₁ viscosity of fluid 1 (water) - ₂ viscosity of fluid 2 (oil) - fluid density - ₁ density of fluid 1 (water) - ₂ density of fluid 2 (oil) - porosity of the porous medium Formerly with Koninklijke/Shell Exploratie en Produktie Laboratorium, Rijswijk, The Netherlands.     

柱形和球形壳体内爆界面不稳定性数值分析

为了更好地研究柱形和球形构型下果冻界面不稳定性发展,避免内爆聚心反弹前后直角坐标网格计算导致的误差,提高对流场和界面位置的计算精度,通过应用考虑了MVFT程序的网格适应性,使其能够适用于柱形网格和球形网格下的界面不稳定性数值模拟,特别是能够保证内爆聚心反弹前后流场和界面计算的稳定性。应用改进的计算程序对两种构型下的界面不稳定性进行了数值模拟,并对二者界面演化规律进行了详细讨论和归纳。结果表明:对于内外半径相同的柱形和球形果冻,聚心反弹时前者半径较小,而后者反弹时刻早于前者,其向内聚心和向外运动的速度最大值大于前者,对内部气体的压缩强度强于前者。对于外边界带有正弦扰动情况,除遵循上述规律外,计算还给出了峰谷转换现象。该项研究结果为进一步深入进行复杂构型下界面不稳定性高精度数值模拟研究提供了一种分析工具。     

Forced axisymmetric motions of viscoelastic cylindrical shells

The isothermal response of a viscoelastic cylindrical shell, of finite length, to arbitary axisymmetric surface forces, initial conditions, and boundary conditions is considered within the linear theory of thin shells. The problem is formulated with the effects of shear deformation and rotatory inertia included; the viscoelastic properties are assumed to be isotropic and homogeneous. The response is first found formally in terms of a causal Green's function. It is then shown that when Poisson's ratio is constant, the causal Green's function can be expanded in a series of orthonormal spatial eigenfunctions of an associated elastic shell eigenvalue problem. The resulting solution for the general problem is an eigenfunction series with Laplace transformed time-dependent coefficients. The general solution is applied to predicting the motion of a uniform, simply-supported cylindrical shell, initially quiescent, which is subjected to a step pressure moving with constant velocity. For this example, the relaxation function of the shell material in uniaxial extension is taken to be that of a standard linear solid. The motions predicted by simpler shell models, namely, shells with bending only and without bending, are also considered for comparison. Here, the absolute values of the Fourier coefficients in the shell displacement series go to zero faster than the inverse of the first or second power of positive integers when bending is excluded or included, respectively. Numerical results are presented for a moderately long and relatively thick, nearly elastic, cylindrical shell.     

Richtmyer-Meshkov instability of an interface between two media due to passage of two successive shocks

The instability of a free surface of aluminum after passage of two shocks that follow one after the other at a certain time interval is studied numerically. The first shock is rather strong (the postshock pressure is about 75 GPa). It is shown that if at the moment when the second shock arrives at the free surface, the perturbation evolution is nonlinear, then, in contrast to the linear stage, the change in the growth rate of the amplitude depends weakly on the wavelength of the initial perturbation. A formula is proposed which describes the effect of the second shock on the amplitude growth rate and in which the main structure of Richtmyer's formula is preserved. It is demonstrated that the parameters of the second shock that ensure freezing of the instability can be determined using only the growth rate of the amplitude. Computing Center, Russian Academy of Sciences, Moscow 117967. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 28–37, January–February, 2000.     

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Flow of viscous and viscoelastic fluids through and around porous bodies

The slow flow of a viscous fluid through and around porous spheres is considered. The numerical simulation uses a special mixture of computational techniques: quadratic approximation and expansion in power series. The resulting calculations predict the evolution of the main features of the flow if the boundary conditions are varying, particularly if the tangential velocity is neglected or if a viscous filtration velocity is assumed at the sphere surface. The cases of full and hollow spheres with uniform and non uniform permeabilities are considered, the external impermeable walls of the flow being concentric spheres or cylinders. Some influence of viscoelastic properties of the fluid is also given.Nomenclature AA _n , A_n, B_n, b_n, C_n, c_n, D_n constants of integration - C _n (t) Gegenbauer functions with degree n and order –1/2 - e shell thickness - K, K* permeability - P _n (t) Legendre functions - Q _v volumetric rate of flow - p, p ₀, p _e pressure, far away pressure, average pressure - R* sphere radius - r, spherical coordinates - Re Reynolds' number (see equation 37) - s, t sinus and cosinus - V ₀ ^* uniform velocity - v velocity component - We Weissenberg's number (see equation (37)) - permeability coefficient - thickness coefficient - structural coefficient - diameter ratio sphere-cylinder - * dynamic viscosity of the fluid - stream functions - normal stress ( _rr ) - tangential stress ( ) - ₀ ^* relaxation time of the fluid     

Motion of the interface of two fluids in a porous medium

In this study we obtain the Integro-differential equation for the motion of the interface of two incompressible fluids in various well areal arrangement systems. The solution of the equation is presented for a five-point system in the form of a power series with respect to time. Formulas are assumed which describe the motion of the particles belonging to the interface along invariant streamlines for five-point, seven-point, and nine-point well arrangement systems. The stratum sweeping coefficients for the fluid which is displacing the stratum oil are calculated (under conditions of the five-point system) at the instant when the fluid breaks through into the operation wells. The results of the calculations are compared with experimental data [1].The author wishes to thank V. L. Danilov for valuable counsel and comments.