Viscous Flow Equations Research Articles

Three-Dimensional Viscous Flowfield Computations in a Streamline Coordinate System

A method of predicting the three-dimensional laminar hypersonic viscous flowfield over a body at very high angles of attack has been developed by using the parabolized Navier-Stokes equations in a streamline coordinate system. At a very high angle of attack, the primary flow direction deviates substantially from the marching coordinate direction of the conventional body-generator coordinate direction. Hence, in the present approach the viscous flow equations are parabolized along the surface streamline direction instead of the body-generator coordinate direction, which reduces the restriction on the angle of attack of a flight vehicle. The streamline traces on the body surface are evaluated from a three-dimensio nal inviscid flowfield solution. Computational results for a test case have been obtained by both the present streamline approach (SLPNS) and the conventional body-generator coordinate method (PNS) for the purpose of comparison. The SLPNS results for the test case at moderate angle of attack compare well with conventional PNS results.

Analysis of particulate removal in venturi scrubbers—effect of operating variables on performance

AbstractA theoretical model of venturi scrubber performance consisting of the governing differential equations for momentum, material, and energy exchange is solved using a Milne fourth‐order predictor—Hamming corrector numerical integration procedure. The model provides for specification of the operating variables, including scrubber geometry, throat gas velocity, liquid to gas loading ratio, and collector droplet and particle‐size distributions. Liquid loading ratio and gas velocity are shown to be the two most important operating variables, while the dispersity of the droplet size distribution only slightly affects collection efficiency over the operating range normally encountered. Location of the liquid injection site and length of the throat are also important design considerations. While particle collection in venturi scrubbers has typically been assumed to occur in the potential flow regime, the present results show that collection can also occur under conditions corresponding to viscous flow about the collecting droplets. The transition Reynolds number used to change from the potential to viscous flow equations affects the predicted performance.

The thermodynamic parameters of viscous flow: Part II. Application of the Feakins-Lawrence formulation of the erying rate equation for viscous flow to some aqueous solutions

The Feakins-Lawrence extension of Eyring's transition state treatment of viscosity of solutions to the case of electrolyte solutions has been applied to aqueous solutions of four quaternary ammonium bromides, four aromatic trialkyl ammonium chlorides and four amino acids at 25°C. The values of the thermodynamic parameters of viscous flow Δμ 2 0*, Δ H̄ 2 0*, TΔ S̄ 2 0* have been calculated. Nightingale's method of calculation of the corresponding parameters Δ G 1,2 *, Δ E 1,2 * and Δ S 1,2 * for the same systems has also been used, and therefrom the solute contributions proper to the values of the above thermodynamic parameters have been calculated and compared with the corresponding FL parameters. The generally small discrepancies between the two have been discussed. The solute contributions to the values of the viscous flow parameters have been further reduced into the corresponding cationic contributions. The effect of the cation size on the B and Δμ 2 0* values in the case of the two series of quaternary ammonium halide salts have been comparatively discussed. Further, the change in the values of Δ H̄ 2 0* and TΔ S̄ 2 0* parameters in relation to the cation size has also been studied for such cations.

Calculation of internal viscous flows in axisymmetric ducts at moderate to high reynolds numbers

A formal procedure is presented for parabolizing the Navier-Stokes equations for two dimensional internal flows in artitrary ducts with streamline curvature and streamline divergence. This procedure consists of constructing an orthogonal coordinate system from the potential flow solution and parabolizing the corresponding Navier-Stokes equations using the thin channel approximation. Theoretical arguments based on the fundamental existence theorem and numerical examples are used to demonstrate the validity of this procedure. A new method using conformal mapping based on the Schwartz-Christoffel transformation is presented which obtains the solution of the inverse potential flow problem in a direct manner. A numerical solution algorithm based on the two point box scheme is presented for solving the viscous flow equations. This method is shown to be accurate and stable for flows at moderate to high Reynolds numbers over a wide range of conditions.

Viscous Transverse-Aligned MHD Flows

Basic equations for plane steady viscous flows of an electrically conducting fluid, when the applied magnetic field is transverse-aligned, are recast in a new form by following Martin and Chandna and Garg. Three illustrations of these new form of equations, when the streamlines are straight or involutes of a curve or form an isometric net with their orthogonal trajectories, are studied.

Material models applied to pacific trench flexure

The oceanic lithosphere adjacent to a subduction zone is modelled as an elastic beam with variable flexural rigidity or as a viscoelastic beam incorporating Mohr-Coulomb and tensile failure. The viscous flow equations include temperature and pressure dependent terms. As an oceanic plate approaches the axis of a deep trench, pervasive failure causes a large reduction in flexural rigidity over a distance of roughly 100 km. The same plate suffers little reduction in flexural rigidity in a shallow, low-curvature trench. In approaching the trench axis the upper part of the oceanic lithosphere model undergoes brittle failure, with normal faults extending to depths of up to 55 km. Below the zone of normal faulting elastic deformation prevails through a depth interval of 15 km with maximum differential stresses of 5–10 kb. In the 15 km below the elastic zone stresses drop by an order of magnitude as strain becomes viscous. The upper brittle-elastic zone of the lithospheric model has strength consistent with low-temperature experimental results for olivine. Thermally activated continuum deformation occurs only deep in the plate.

Force multipole moments for a spherically symmetric particle in solution

We present a general theorem for the force multipole moments of arbitrary order induced in a spherically symmetric particle immersed in a fluid whose motion satisfies the linear Navier-Stokes equation for steady incompressible viscous flow. The multipole moments are expressed in terms of the unperturbed fluid velocity field. It is shown that for a particle with a finite extension only a few terms give rise to fluid perturbations which are not confined to the interior of the particle. We give explicit results for a polymer satisfying the Debye-Bueche-Brinkman equations and for a hard sphere with mixed slip-stick boundary conditions.

Densification kinetics of a porous material based on molybdenum disulfide in hot pressing

The relation has been established between the relative density of hot-pressed materials based on molybdenum disulfide and temperature, time, and applied pressure. The process of densification of a mixture of molybdenum disulfide and molybdenum powders in hot pressing is described by a volume viscous-flow equation allowing for the nonlinear dependence of the rate of flow of the solid phase of the porous material on stress. The effective rate of creep of the solid phase in hot pressing (1400°C) is inversely proportional to the gas pressure in the closed pores. Reduced rates of steady-state creep have been calculated, and the exponent n indicative of linearity of flow of the solid phase forming the porous solid has been determined.

A completeness observation on the stability equations for stratified viscous shear flows

Abstract

A Derivation of the Field Equations for Slow Viscous Flow through a Porous Medium

A Derivation of the Field Equations for Slow Viscous Flow through a Porous Medium

Viscous flow effects in the periodic pressure cycling of gas-phase catalytic reactions

The previous study of Hamer and Cormack[1], on the increased effectiveness of gas phase catalytic reactions under conditions of oscillating external pressure, is extended to include viscous flow effects. For this purpose, the full, time dependent equations of viscous flow, diffusion and chemical reaction for a first order, isothermal reaction in a porous catalyst are solved numerically. The inclusion of viscous effects results in a new dimensionless parameter, N. It is shown that in general the catalyst effectiveness factor is smaller for smaller N. It is concluded, however, that even though viscous effects might be very important for extremely low permeability catalysts, viscous effects can be neglected for operating conditions such that N > 1. For one example cited this corresponds to a permeability of 0.2 millidarcy.

Numerical Solution of Compressible Viscous Flows

The presented review is concerned with the problem of calculating compressible viscous flows. Basic numerical considerations and problems associated with calculating viscous flows are examined and current numerical approaches toward the solution of the Navier-Stokes equations are discussed. It is pointed out that the numerical solution of the full time-dependent equations for turbulent flow is not practical with present computers. Therefore, turbulence effects must be accounted for by modeling. Developments related to turbulence modeling are described. In connection with a discussion of numerical methods for solving viscous flow equations, attention is given to numerical domains of dependence of typical explicit and implicit methods, the diffusion problem, the convection-diffusion problem, and the split-hybrid method.

Unsteady boundary layer over an impulsively started flat plate using finite elements

Abstract The problem under consideration is that of a suddenly accelerated semi-infinite flat plate in an incompressible viscous fluid. The present analysis is based upon the use of finite elements to idealize the flow over the plate and solving the nonlinear unsteady boundary layer equation by Galerkin's procedure and the results are compared with those obtained previously using a mixed explicit-implicit finite difference scheme. An average number of three iterations are needed to obtain a solution within a maximum tolerance between interates less than 0.01%. The method has successfully demonstrated the exponential decay of the velocity at large times, which has been reported by previous investigators. The transition region of the flow is terminated to reach Blasius state, when the plate has moved a distance equal to six times the distance from the station into consideration to the leading edge of the plate. The methods shows excellent accuracy in dealing with the nonlinear equation of the unsteady viscous flow.

Hot pressing of niobium-zirconium-boron powder mixtures

1. An experimental study was made of the process of densification of zirconium diboride—niobium alloys containing 80, 60, and 40 mole% ZrB2. It was established that the highest density is attained after pressing for 8–10 min at temperatures of 2000–2100°C under a pressure of 18.4 MPa. 2. It is shown that the process of densification of the materials investigated is described by an equation for the volume viscous flow of a porous solid including a steady-state creep equation. 3. Results of calculations with the volume viscous flow equation reveal that the energy of activation for the process corresponds to the energy of activation for the self-diffusion of atoms in the metallic sublattice of the diboride. Raising the amount of the metallic component increases the rate of densification and slightly decreases the energy of activation.

Time dependent compressible flow about a spherically symmetric polymer in solution

We obtain the general solution of the linear Navier-Stokes equation for time dependent compressible viscous flow about a spherically symmetric polymer molecule. The solution is presented in a covariant form valid in a general cartesian coordinate frame. In the course of deriving the solution we obtain a general decomposition of the unperturbed flow in the absence of the polymer. Our solution generalizes the earlier solution derived by Schmitz and Felderhof for the case of creeping flow.

Creeping flow about a sphere

We present the general solution of the linear Navier-Stokes equation for incompressible viscous flow about a sphere with mixed slip-stick boundary conditions. Exphclt results are obtained in cartesian coordinates. The relation wRh Lamb's method is indicated. We derive expressions for a set of fundamental tensors, introduced by Brenner, describing the perturbed flow. Finally the force density induced on the sphere is evaluated. In this paper we give the solution of the linear Navier-Stokes equation for incompressible viscous flow about a sphere with mixed slip-stick boundary conditions for arbitrary incident flow. A method of solution was presented long ago by Lamb l) in terms of solid spherical harmonics, but his solution left something to be desired from a practical point of view. Lamb's solution is described in some detail in the monograph by Happel and Brenner2). The difficulty with Lamb's solution is that application to particular problems often requires writing the harmonics in terms of spherical coordinates, whereby a specific choice of axes, alien to the problem at hand, is made. For many problems an approach using cartesian coordinates is more convenient. In particular for the problem of hydrodynamic interaction of two spheres this approach has recently proved effective34). In the following we persistently use the cartesian point of view and solve the problem by the so-called method of cartesian ansatz. The solution is a direct generalization of earlier results obtained for constant, linear or quadratic incident flow3-7). At the end we show the relation to Lamb's solution. As a benefit we obtain explicit results for a set of tensors introduced by Brenner 8) in his fundamental paper on Stokes resistance. Brenner only gave explicit

Densification kinetics in the hot pressing of VK6 hard alloy

1. A study was made of the densification kinetics of VK6 hard alloy in hot pressing and of the effects of temperature, pressure, and process time on the relative density of hot-pressed specimens. 2. It is shown that the densification VK6 hard alloy is described by a generalized equation of volume viscous flow and, at solid-phase sintering temperatures, is determined by the process of nonlinear flow of the cobalt phase of the alloy. In liquid-phase sintering the densification process is apparently linked with activation of the plastic flow of the carbide phase under the action of the cobalt phase. Under these conditions, the flow of the concentrated suspension of tungsten carbide in cobalt obeys a linear (Newtonian) law.

Detached flow over a step with the formation of a turbulent wake

A method of calculating the plane turbulent layer behind a step interacting with a free potential flow of incompressible fluid is developed. The method includes consideration of the initial boundary layer and injection (or suction) in the isobaric bottom region. Friction on the wall behind the step is neglected, which corresponds to symmetric quasisteady flow behind the straight edge of a plate. The inviscid flow is represented by the Keldysh-Sedov integral equations; the flow in the wake with a one-parameter velocity profile is represented by three first-order differential equations—the equations of momentum for the wake and motion along its axis and the equation of interaction (through the displacement thickness) of the viscous flow with the external potential flow. The turbulent friction in the wake is given, accurate to the single empirical constant, by the Prandtl equation. The different flow regions — on the plate behind the step, the isobaric bottom region, and the wake region — are joined with the aid of the quasi-one-dimensional momentum equation for viscous flow. The momentum equation for the flow as a whole serves as the closure condition. The obtained integrodifferential system of equations is approximated by a system of nonlinear finite-difference equations, whose solution is obtained on a computer by minimization of the sum of the squares of the discrepancies. The results of the calculations agree satisfactorily with experimental data.

A block iterative finite element algorithm for numerical solution of the steady–state, compressible Navier–Stokes equations

AbstractAn iterative method for numerically solving the time independent Navier–Stokes equations for viscous compressible flows is presented. The method is based upon partial application of the Gauss–Seidel principle in block form to the systems of the non‐linear algebraic equations which arise in construction of finite element (Galerkin) models approximating solutions of fluid dynamic problems. The C0‐cubic element on triangles is employed for function approximation. Computational results for a free shear flow at Re = 1000 indicate significant achievement of economy in iterative convergence rate over finite element and finite difference models which employ the customary time dependent equations and symptotic time marching procedure to steady solution. Numerical results are in excellent agreement with those obtained for the same test problem employing time marching finite element and finite difference solution techniques.

膨張空どう形油圧脈動吸収器の減衰特性に関する研究

This paper deals with the attenuation characteristics of expansion chamber-type hydraulic pulsation absorbers, especially taking into consideration the pulsation of the delivery flow of pumps. In carrying out experiments, a rotary spool-type hydraulic vibration machine capable of producing a sinusoidal pulsating flow of large amplitude up to about 800 Hz was recently developed, which enables accurate experiments covering a wide frequency range from the 1st to the 3rd pulsation components of common pumps. After dividing the components of the absorber into circular pipe elements, the transfer matrix method is used to predict the attenuation characteristics. The following results were obtained: (1) The calculated values, obtained by applying the wave equation of two-dimensional viscous flow to all divided circular pipe elements and inlet and outlet conduits, agree well with the experimental values, except for some in the frequency range for which mechanical vibration cannot be disregarded. (2) It is possible to treat fluid motion in an absorber as a lumped parameter system, since the lengths of the expansion chamber and connecting conduit in the absorber are, in general, fairly short compared with the corresponding wave length of the pulsation frequency of the pump. (3) The approximate equation, which represents effects of the inlet conduit, the outlet conduit and the absorber independeals, . is accurate enough for practical use in the case of high loads.