Laminar flow in symmetrical channels with slightly curved walls, I. On the Jeffery-Hamel solutions for flow between plane walls

1962 ◽  
Vol 267 (1328) ◽  
pp. 119-138 ◽  
Keyword(s):  
Series Solution ◽  
Stokes Equations ◽  
Asymptotic Series ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Navier Stokes Equations ◽  
Laminar Separation ◽  
Single Region ◽  
Leading Term ◽  
Asymptotic Series Solution

The Jeffery-Hamel solutions for plane, viscous, source or sink flow between straight walls are not unique. In this paper these solutions are regarded as providing the leading term of a series solution for a class of channels with walls that are nearly straight in a certain sense, but are such that the fluid is not required to emerge from, or converge on, a point. This approach suggests a further condition which the appropriate solution must satisfy, and hence leads to uniqueness in a limited domain of the physical parameters. The resulting velocity profiles include, at one extreme, the parabolic one of Poiseuille flow, and, at the other, profiles with a single region of flow reversal at each wall. The way is thus opened to an asymptotic series solution of the Navier-Stokes equations which shows laminar separation

INTRODUCING OF INTERNAL VISCOUS FRICTION IN EQUATIONS OF BEAM OSCILLATION AS LONG RECTANGULAR STRIP

2021 ◽  
pp. 54-58
Author(s):  
E.M. Zveriaev ◽  
Keyword(s):  
Stokes Equations ◽  
Asymptotic Series ◽  
Viscous Friction ◽  
Theory Of Elasticity ◽  
Navier Stokes ◽  
Dimensional Theory ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
The One ◽  
Beam Oscillation

Abstract. On the base of the method of simple iterations generalising methods of semi-inverse one of Saint-Venant, Reissner and Timoshenko the one-dimensional theory is constructed using the example of dynamic equations of a plane problem of elasticity theory for a long elastic strip. The resolving equation of that one-dimensional theory coincides with the equation of beam vibrations. The other problems with unknowns are determined without integration by direct calculations. In the initial equations of the theory of elasticity the terms corresponding to the viscous friction in the Navier-Stokes equations are introduced. The asymptotic characteristics of the unknowns obtained by the method of simple iterations allow to search for a solution in the form of expansions of the unknowns into asymptotic series. The resolving equation contains a term that depends on the coefficient of viscous friction.

Modeling and computation of the cavitating flow in injection nozzle holes

2015 ◽  
Vol 06 (01) ◽  
pp. 1550003 ◽  
Author(s):  
Hatem Kanfoudi ◽  
Ridha Zgolli
Keyword(s):  
Transport Equation ◽  
Source Term ◽  
Stokes Equations ◽  
Volume Fraction ◽  
Variable Density ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Cavitating Flows ◽  
Navier Stokes Equations ◽  
Injection Nozzle

Cavitating flows inside a diesel injection nozzle hole were simulated using a mixture model. A two-dimensional (2D) numerical model is proposed in this paper to simulate steady cavitating flows. The Reynolds-averaged Navier–Stokes equations are solved for the liquid and vapor mixture, which is considered as a single fluid with variable density and expressed as a function of the vapor volume fraction. The closure of this variable is provided by the transport equation with a source term Transport-equation based methods (TEM). The processes of evaporation and condensation are governed by changes in pressure within the flow. The source term is implanted in the CFD code ANSYS CFX. The influence of numerical and physical parameters is presented in detail. The numerical simulations are in good agreement with the experimental data for steady flow.

scholarly journals Inertial dynamics of air bubbles crossing a horizontal fluid–fluid interface

2012 ◽  
Vol 707 ◽  
pp. 405-443 ◽  
Author(s):  
Romain Bonhomme ◽  
Jacques Magnaudet ◽  
Fabien Duval ◽  
Bruno Piar
Keyword(s):  
High Speed ◽  
Stokes Equations ◽  
Complex Dynamics ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Air Bubbles ◽  
Fluid Interface ◽  
Navier Stokes Equations ◽  
Film Drainage ◽  
Two Fluids

AbstractThe dynamics of isolated air bubbles crossing the horizontal interface separating two Newtonian immiscible liquids initially at rest are studied both experimentally and computationally. High-speed video imaging is used to obtain a detailed evolution of the various interfaces involved in the system. The size of the bubbles and the viscosity contrast between the two liquids are varied by more than one and four orders of magnitude, respectively, making it possible to obtain bubble shapes ranging from spherical to toroidal. A variety of flow regimes is observed, including that of small bubbles remaining trapped at the fluid–fluid interface in a film-drainage configuration. In most cases, the bubble succeeds in crossing the interface without being stopped near its undisturbed position and, during a certain period of time, tows a significant column of lower fluid which sometimes exhibits a complex dynamics as it lengthens in the upper fluid. Direct numerical simulations of several selected experimental situations are performed with a code employing a volume-of-fluid type formulation of the incompressible Navier–Stokes equations. Comparisons between experimental and numerical results confirm the reliability of the computational approach in most situations but also points out the need for improvements to capture some subtle but important physical processes, most notably those related to film drainage. Influence of the physical parameters highlighted by experiments and computations, especially that of the density and viscosity contrasts between the two fluids and of the various interfacial tensions, is discussed and analysed in the light of simple models and available theories.

scholarly journals Asymptotic solution of the low Reynolds-number flow between two co-axial cones of common apex

1984 ◽  
Vol 7 (4) ◽  
pp. 765-784 ◽  
Author(s):  
M. A. Serag-Eldin ◽  
Y. K. Gayed
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Asymptotic Series ◽  
Creeping Flow ◽  
Navier Stokes ◽  
Cross Sectional ◽  
Navier Stokes Equations ◽  
Spherical Coordinate ◽  
Flow Solution ◽  
Reynolds Number Flow

The paper is concerned with the axi-symmetrlc, incompressible, steady, laminar and Newtonian flow between two, stationary, conical-boundaries, which exhibit a common apex but may include arbitrary angles. The flow pattern and pressure field are obtained by solving the pertinent Navier-Stokes' equations in the spherical coordinate system. The solution is presented in the form of an asymptotic series, which converges towards the creeping flow solution as a cross-sectional Reynolds-number tends to zero. The first term in the series, namely the creeping flow solution, is given in closed form; whereas, higher order terms contain functions which generally could only be expressed in infinite series form, or else evaluated numerically. Some of the results obtained for converging and diverging flows are displayed and they are demonstrated to be plausible and informative.

The Effect of Film Thickness on EHD-Driven Instability of Interface Separating Two Liquids

2009 ◽  
Author(s):  
Payam Sharifi ◽  
Asghar Esmaeeli
Keyword(s):  
Stokes Equations ◽  
Thin Liquid Film ◽  
Front Tracking ◽  
Thin Layers ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Interface Instability ◽  
Navier Stokes Equations ◽  
Dielectric Model ◽  
Leaky Dielectric Model

Most of the studies conducted so far on EHD-driven instability of superimposed fluids have been concerned with liquid layers of modest depths. In many applications, however, the liquid layers can be very thin. Since the dynamics in thin films is generally governed by lubrication equations rather than full Navier-Stokes equations, it is expected that the interface dynamics will be quite different from that of the liquids with modest depths. The objective of this study is to explore the effect of initial liquid thickness on the dynamics of the phase boundary. To do this end, we perform Direct Numerical Simulations (DNS) using a front tracking/finite difference scheme, in conjunction with Taylor’s leaky dielectric model. For the physical parameters used here, it is shown that for sufficiently thick liquid layers, the interface instability leads to formation of liquid columns that merge together to form a big column. However, for thin layers, the interactions between the columns are weaker and lead to a short and a longer column that are connected by a thin liquid film.

scholarly journals A reliable algorithm for time-fractional Navier-Stokes equations via Laplace transform

2019 ◽  
Vol 8 (1) ◽  
pp. 695-701 ◽  
Author(s):  
Amit Prakash ◽  
Doddabhadrappla Gowda Prakasha ◽  
Pundikala Veeresha
Keyword(s):  
Fluid Flow ◽  
Viscous Fluid ◽  
Fractional Derivative ◽  
Series Solution ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Viscous Fluid Flow ◽  
Analysis Scheme ◽  
Homotopy Analysis

Abstract In this paper, numerical solution of fractional order Navier-Stokes equations in unsteady viscous fluid flow is found using q-homotopy analysis transform scheme. Fractional derivative is considered in Caputo sense. The proposed technique is a blend of q-homotopy analysis scheme and transform of Laplace. It executes well in efficiency and provides h-curves that show convergence range of series solution.

scholarly journals The interaction of a contact discontinuity with sound waves according to the linearised Navier-Stokes equations

1970 ◽  
Vol 17 (1) ◽  
pp. 37-46
Author(s):  
W. M. Anderson
Keyword(s):  
Fourier Transform ◽  
Contact Surface ◽  
Stokes Equations ◽  
Asymptotic Series ◽  
Density Perturbation ◽  
Navier Stokes ◽  
Sound Waves ◽  
Navier Stokes Equations ◽  
Initial Discontinuity ◽  
The Fourier Transform

AbstractThe resolution of a small initial discontinuity in a gas is examined using the linearised Navier-Stokes equations. The smoothing of the resultant contact surface and sound waves due to dissipation results in small flows which interact. The problem is solved for arbitrary Prandtl number by using a Fourier transform in space and a Laplace transform in time. The Fourier transform is inverted exactly and the density perturbation is found as two asymptotic series valid for small dissipation near the contact surface and the sound waves respectively. The modifications to the structures of the contact surface and the sound waves are exhibited.

Numerical Simulations of Three-Dimensional Flows in a Cubic Cavity With an Oscillating Lid

1993 ◽  
Vol 115 (4) ◽  
pp. 680-686 ◽  
Author(s):  
Reima Iwatsu ◽  
Jae Min Hyun ◽  
Kunio Kuwahara
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Flow Characteristics ◽  
Time Dependent ◽  
Navier Stokes ◽  
Physical Parameters ◽  
Navier Stokes Equations ◽  
Sinusoidal Oscillations ◽  
Dependent Flow

Numerical studies are made of three-dimensional flow of a viscous fluid in a cubical container. The flow is driven by the top sliding wall, which executes sinusoidal oscillations. Numerical solutions are acquired by solving the time-dependent, three-dimensional incompressible Navier-Stokes equations by employing very fine meshes. Results are presented for wide ranges of two principal physical parameters, i.e., the Reynolds number, Re ≤ 2000 and the frequency parameter of the lid oscillation, ω′ ≤ 10.0. Comprehensive details of the flow structure are analyzed. Attention is focused on the three-dimensionality of the flow field. Extensive numerical flow visualizations have been performed. These yield sequential plots of the main flows as well as the secondary flow patterns. It is found that the previous two-dimensional computational results are adequate in describing the main flow characteristics in the bulk of interior when ω′ is reasonably high. For the cases of high-Re flows, however, the three-dimensional motions exhibit additional complexities especially when ω′ is low. It is asserted that, thanks to the recent development of the supercomputers, calculation of three-dimensional, time-dependent flow problems appears to be feasible at least over limited ranges of Re.

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