Viscous dissipation of a power law fluid in axial flow between isothermal eccentric cylinders

We study the temperature distribution of a power law fluid in a pressure-driven axial flow between isothermal eccentric cylinders in bipolar cylindrical coordinates. We begin our analysis by writing the equation of energy in bipolar cylindrical coordinates. We then obtain a dimensionless algebraic analytic solution for temperature profiles under a steady, laminar, incompressible and fully developed flow [Eq. (64)]. We find that the dimensionless temperature profile depends upon the radius ratio of the inner to outer cylinders, the eccentricity, the angular position, and the power law exponent n. The temperature is a strong function of the gap between the cylinders. The temperature profiles are flat in the middle of the gap and then, near the wall, suddenly drop to the wall temperature.     

Sidewall flow stability in the MHD channel flows

The velocity distribution between two sidewalls is M-shaped for the MHD channel, flows with rectangular cross section and thin conducting walls in a strong transverse magnetic field. Assume that the dimensionless numbersR _m ?1,M, N? 1, and σ* and that the distance between two perpendicular walls is very long in comparison with the distance between two sidewalls. First, the equation for steady flow is established, and the solution of M-shaped velocity distribution is given. Then, an equation for stability of small disturbances is derived based on the velocity distribution obtained. Finally, it is proved that the stability equation for sidewall flow can be transformed into the famous Orr-Sommerfeld equation, in addition, the following theorems are also proved, namely, the analogy theorem, the generalized Rayleigh's theorem, the generalized Fjørtoft's theorem and the generalized Joseph's theorems.     

An Analytic Solution for the Frontal Flow Period in 1D Counter-Current Spontaneous Imbibition into Fractured Porous Media Including Gravity and Wettability Effects

Including gravity and wettability effects, a full analytical solution for the frontal flow period for 1D counter-current spontaneous imbibition of a wetting phase into a porous medium saturated initially with non-wetting phase at initial wetting phase saturation is presented. The analytical solution applicable for liquid–liquid and liquid–gas systems is essentially valid for the cases when the gravity forces are relatively large and before the wetting phase front hits the no-flow boundary in the capillary-dominated regime. The new analytical solution free of any arbitrary parameters can also be utilized for predicting non-wetting phase recovery by spontaneous imbibition. In addition, a new dimensionless time equation for predicting dimensionless distances travelled by the wetting phase front versus dimensionless time is presented. Dimensionless distance travelled by the waterfront versus time was calculated varying the non-wetting phase viscosity between 1 and 100 mPas. The new dimensionless time expression was able to perfectly scale all these calculated dimensionless distance versus time responses into one single curve confirming the ability for the new scaling equation to properly account for variations in non-wetting phase viscosities. The dimensionless stabilization time, defined as the time at which the capillary forces are balanced by the gravity forces, was calculated to be approximately 0.6. The full analytical solution was finally used to derive a new transfer function with application to dual-porosity simulation.     

The interaction of gas bubbles in an anisotropic elastic solid

The problem of finding the stress field induced in the neighbourhood of two spherical gas bubbles or voids in an anisotropic matrix is formulated in terms of an integral equation for the “transformation stress” in equivalent homogeneous inclusions. An iterative method of solution is outlined, involving the solution of a class of problems for a single spherical inclusion perturbing a polynomial field of stress. Explicit solutions are obtained for polynomials up to second degree. Estimates of the energy of interaction between gas bubbles in α-U and between voids in Mo are deduced as examples, and the results are discussed in relation to earlier calculations and to observations.     

Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability简

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.     

Nonlinear waves in electromigration dispersion in a capillary

We construct exact solutions to an unusual nonlinear advection–diffusion equation arising in the study of Taylor–Aris (also known as shear) dispersion due to electroosmotic flow during electromigration in a capillary. An exact reduction to a Darboux equation is found under a traveling-wave ansatz. The equilibria of this ordinary differential equation are analyzed, showing that their stability is determined solely by the (dimensionless) wave speed without regard to any (dimensionless) physical parameters. Integral curves, connecting the appropriate equilibria of the Darboux equation that governs traveling waves, are constructed, which in turn are shown to be asymmetric kink solutions (i.e., non-Taylor shocks). Furthermore, it is shown that the governing Darboux equation exhibits bistability, which leads to two coexisting non-negative kink solutions for (dimensionless) wave speeds greater than unity. Finally, we give some remarks on other types of traveling-wave solutions and a discussion of some approximations of the governing partial differential equation of electromigration dispersion.     

Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity

This paper concerns the Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid in their stream function formulations are used. The hydrodynamic drag force acting on each porous cylindrical particle in a cell and permeability of membrane built up by cylindrical particles with a porous shell are evaluated. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified. Variation of the drag coefficient and dimensionless hydrodynamic permeability with permeability parameter σ, particle volume fraction γ has been studied and some new results are reported. The flow patterns through the regions have been analyzed by stream lines. Effect of particle volume fraction γ and permeability parameter σ on flow pattern is also discussed. In our opinion, these results will have significant contributions in studying, Stokes flow through cylindrical swarms.     

Motion of a strong shock front in a two-dimensional gas layer with moving internal boundary

We examine the problem of planar one-dimensional motion of a strong shock wave with moving internal boundary in which the initial position of the front, its intensity, the mass of the gas involved in the motion, and the energy contained in this gas are known. The problem is not self-similar and its exact solution, which involves working with partial differential equations, presents serious difficulties. In the following we determine the law of shock-front motion in this problem via the method of [1], which makes it possible to find a system of ordinary differential equations for the problem. The method is based on an initial specification of the power-law coupling between the dimensionless Lagrangian and Eulerian variables and replacement of the energy equation by this coupling and the energy integral. The solution is sought in the first approximation.     

Radiative-convective heat transfer in a plane layer of a selectively absorbing mist

This paper deals with the thermal field in a plane layer of selectively absorbing gas which has been injected into a steady turbulent stream of high-temperature gas flowing around a porous plate. The boundary-value problem in terms of the energy equation reduces to a nonlinear integral equation in terms of a dimensionless temperature, and this equation is solved numerically by the Newton-Kantorovich method. The results are presented on graphs of temperature and thermal flux in the absorbing gas layer as functions of the space coordinate. Such a problem has been analyzed in [1] for the case of an injected gray gas.Translated from Zhurnal Prikladnoi Mekhaniki i Technicheskoi Fiziki, No. 3, pp. 179–182, May–June, 1972.     

An implicit characteristic-flux-averaging method for the euler equations for real gases

A formulation of an implicit characteristic-flux-averaging method for the unsteady Euler equations with real gas effects is presented. Incorporation of a real gas into a general equation of state is achieved by considering the pressure as a function of density and specific internal energy. The Ricmann solver as well as the flux-split algorithm are modified by introducing the pressure derivatives with respect to density and internal energy. Expressions for calculating the values of the flow variables for a real gas at the cell faces are derived. The Jacobian matrices and the eigenvectors are defined for a general equation of state. The solution of the system of equations is obtained by using a mesh-sequencing method for acceleration of the convergence. Finally, a test case for a simple form of equation of state displays the differences from the corresponding solution for an ideal gas.     

Generation of tones due to flow past a deep cavity: Effect of streamwise length

Shear flow past a deep cavity can generate self-sustained oscillations, including locked-on flow tones, due to coupling between the inherent instability of the separated shear layer and an acoustic mode of the cavity resonator. This investigation focuses on the dimensionless pressure amplitude response within a deep cavity, as a function of the streamwise length of the cavity opening; for each length, the pressure response is characterized over a wide range of dimensionless inflow velocity. Criteria for locked-on flow tones are assessed. They include a measure of the strength of lock-on, SoL and the quality factor Q. All self-excited oscillations are assessed using both of these criteria, in order to interpret dimensionless forms of the fluctuation pressure amplitude. The dimensionless pressure amplitude response of the cavity involves several successive regimes, due to variations of streamwise length L of the cavity opening. These regimes are defined in relation to L/θ, where θ is the momentum thickness of the inflow boundary layer. Below a minimum value of L/θ, flow tones cannot be generated. Furthermore, these regimes are defined in terms of the possible hydrodynamic modes (stages) of the unsteady shear layer and the acoustic modes of the deep cavity.     

Experimental investigation of the wave-induced flow around a surface-touching cylinder

The wave-induced flow around a circular cylinder near both a rigid wall and an erodible bed is experimentally investigated using Particle Tracking Velocimetry (PTV). The aim of this study is to gain quantitative information on the local mean flow, the vorticity dynamics and the evolution of the erodible bed. The flow is characterized in terms of the Keulegan–Carpenter (KC), Reynolds (Re) and Ursell (U_r) numbers. The effects of changing these parameters over the ranges 1<KC<31, 3×10³<Re<2.6×10⁴ and 1.5<U_r<152 are investigated. For KC<1.1 the flow does not separate. When KC increases, the flow becomes unstable and large-scale vortical structures develop. The dimensionless intensity (|Γ^⁎|) depends non-monotonically on KC, with a local maximum at KC=17, and the dimensionless area of the same macrovortex (A^⁎) follows a somewhat similar law. Although the dimensionless boundary layer thickness (δ^⁎) exhibits some discontinuities between KC regimes, it decreases with KC at x/D=0.5, as x/D=1 weakly depends on KC and can be regarded as constant (δ^⁎=0.7) and then, increases with KC when moving away from the cylinder. These findings are used to interpret the physics governing the flow around a cylinder touching a wall and are compared with available results from the literature (Sumer et al., 1991). The evolution of the scour mechanism occurring over an erodible sandy bed is also investigated. The validity of some empirical formulas in the literature is also tested on the basis of the available dataset. The empirical relationships of Cevik and Yuksel (1999) and Sumer and Fredsøe (1990) for the dimensionless scour depth (S/D) agree well with our results. The dimensionless scour width (W_s/D) is predicted well by Sumer and Fredsøe's (2002) empirical equation for KC<23, whereas Catano-Lopera and Garcia's (2007) formula is more accurate for higher values of KC.     

Transient response of fluid pressure in a poroelastic material under uniaxial cyclic loading

Poroelasticity is a theory that quantifies the time-dependent mechanical behavior of a fluid-saturated porous medium induced by the interaction between matrix deformation and interstitial fluid flow. Based on this theory, we present an analytical solution of interstitial fluid pressure in poroelastic materials under uniaxial cyclic loading. The solution contains transient and steady-state responses. Both responses depend on two dimensionless parameters: the dimensionless frequency Ω that stands for the ratio of the characteristic time of the fluid pressure relaxation to that of applied forces, and the dimensionless stress coefficient H governing the solid-fluid coupling behavior in poroelastic materials. When the phase shift between the applied cyclic loading and the corresponding fluid pressure evolution in steady-state is pronounced, the transient response is comparable in magnitude to the steady-state one and an increase in the rate of change of fluid pressure is observed immediately after loading. The transient response of fluid pressure may have a significant effect on the mechanical behavior of poroelastic materials in various fields.     

Theoretical and Numerical Investigation of Gas Lubrication Processes on the Basis of Aerodynamic Equations

The dimensionless parameters of the complete system of Navier-Stokes equations of a compressible gas are estimated with reference to a typical gas bearing. It is found that the three-dimensional compressible boundary layer equations should be used as the determining equations for describing gas lubrication processes. After introducing certain assumptions with respect to the dimensionless parameters in the determining equations, an equation for the pressure, the generalized Reynolds equation, is obtained.Use of the spectral method of analysis makes it possible to transform the generalized Reynolds equation into a system of ordinary differential equations. An analytic solution of the entire boundary value problem is obtained for a journal bearing with fairly small eccentricity. By comparing the numerical results obtained using both the solution of the generalized Reynolds equation and the traditional theory it is possible to estimate the effect of the inertia forces, dissipation processes, and heat transfer.     

Peristaltic flow of a heated Jeffrey fluid inside an elliptic duct:streamline analysis

The peristaltic flow of a heated Jeffrey fluid inside a duct with an elliptic cross-section is studied. A thorough heat transfer mechanism is interpreted by analyzing the viscous effects in the energy equation. The governing mathematical equations give dimensionless partial differential equations after simplification. The final simplified form of the mathematical equations is evaluated with respect to the relevant boundary conditions, and the exact solution is attained. The results are further i...     

Plane-parallel flow around a gas cavity

The stationary motion of a gas cavity in an ideal incompressible fluid is studied taking account of surface tension by using a variational equation. Approximate analytical dependences of the dimensionless parameters on the degree of cavity deformation are obtained. It is shown that the variational equation admits of an exact analytical solution. The stability of motion corresponding to the exact solution is proved relative to arbitrary perturbations in the cavity shape. A solution is given for the problem of stationary motion of an elliptical cavity in a gravity viscous fluid and the stability problem is investigated. Dependences are found for the velocity of cavity rise, the Reynolds number, and the Froude number as a function of the cavity size.     

Combined natural convection heat and mass transfer in an enclosure having finite thickness walls

Unsteady three-dimensional conjugate heat and mass transfer in an enclosure having finite thickness heat-conducting walls has been analyzed numerically. The governing unsteady, three-dimensional flow, energy and contaminant transport equations for the gas cavity and unsteady heat conduction equation for solid walls, written in dimensionless terms of the vector potential functions, the vorticity vector, the temperature and the concentration, have been solved using an iterative implicit finite-difference method. Main attention was paid to the effects of the Rayleigh number, buoyancy ratio and the dimensionless time on the flow structure and heat and mass transfer regimes. It should be noted that the dominant cause of the oscillations in the dimensionless time dependences of the average Nusselt number on the heat source surface and the average Sherwood number on the contaminant source surface at Ra>5?10⁵ is the mutual influence of the analyzed object geometry and the thermo-diffusivity impact on the flow. The change in the buoyancy ratio can lead to the essential modifications of the flow, temperature and concentration fields owing to the significant influence of the concentration gradient.     

Energy dissipation mechanisms in ductile fracture

The objective is to investigate energy dissipation mechanisms that operate at different length scales during fracture in ductile materials. A dimensional analysis is performed to identify the sets of dimensionless parameters which contribute to energy dissipation via dislocation-mediated plastic deformation at a crack tip. However, rather than using phenomenological variables such as yield stress and hardening modulus in the analysis, physical variables such as dislocation density, Burgers vector and Peierls stress are used. It is then shown via elementary arguments that the resulting dimensionless parameters can be interpreted in terms of competitions between various energy dissipation mechanisms at different length scales from the crack tip; the energy dissipations mechanisms are cleavage, crack tip dislocation nucleation and also dislocation nucleation from a Frank-Read source. Therefore, the material behavior is classified into three groups. The first two groups are the well-known intrinsic brittle and intrinsic ductile behavior. The third group is designated to be extrinsic ductile behavior for which Frank-Read dislocation nucleation is the initial energy dissipation mechanism. It is shown that a material is predicted to exhibit extrinsic ductility if the dimensionless parameter bρ_disl^1/2 (b is Burgers vector, ρ_disl is dislocation density) is within a certain range defined by other dimensionless parameters, irrespective of the competition between cleavage and crack tip dislocation nucleation. The predictions compare favorably to the documented behavior of a number of different classes of materials.     

Couette flow of a Bingham plastic in a channel with equally porous parallel walls

In the present paper the flow of a Bingham fluid between two parallel porous walls is studied. One of the walls moves with constant velocity parallel to the other, which is fixed, while a longitudinal pressure gradient exists, as well as a transverse flow field due the porosity of the walls. An exact analytical solution is given for the u-velocity field, which has four different forms depending on the values of the three dimensionless parameters, which are the Bingham, Couette and Reynolds numbers.     

Modelling gas dynamics in 1D ducts with abrupt area change

Most gas dynamic computations in industrial ducts are done in one dimension with cross-section-averaged Euler equations. This poses a fundamental difficulty as soon as geometrical discontinuities are present. The momentum equation contains a non-conservative term involving a surface pressure integral, responsible for momentum loss. Definition of this integral is very difficult from a mathematical standpoint as the flow may contain other discontinuities (shocks, contact discontinuities). From a physical standpoint, geometrical discontinuities induce multidimensional vortices that modify the surface pressure integral. In the present paper, an improved 1D flow model is proposed. An extra energy (or entropy) equation is added to the Euler equations expressing the energy and turbulent pressure stored in the vortices generated by the abrupt area variation. The turbulent energy created by the flow–area change interaction is determined by a specific estimate of the surface pressure integral. Model’s predictions are compared with 2D-averaged results from numerical solution of the Euler equations. Comparison with shock tube experiments is also presented. The new 1D-averaged model improves the conventional cross-section-averaged Euler equations and is able to reproduce the main flow features.