scholarly journals Droplet motion on inclined heterogeneous substrates

2013 ◽  
Vol 725 ◽  
pp. 462-491 ◽  
Author(s):  
Nikos Savva ◽  
Serafim Kalliadasis
Keyword(s):  
Evolution Equations ◽  
Dynamic Behaviour ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Stick Slip ◽  
Droplet Motion ◽  
Chemical Gradients ◽  
Long Wave Approximation ◽  
Analytical Expressions ◽  
Quantities Of Interest

AbstractWe consider the static and dynamic behaviour of two-dimensional droplets on inclined heterogeneous substrates. We utilize an evolution equation for the droplet thickness based on the long-wave approximation of the Stokes equations in the presence of slip. Through a singular perturbation procedure, evolution equations for the location of the two moving fronts are obtained under the assumption of quasi-static dynamics. The deduced equations, which are verified by direct comparisons with numerical solutions to the governing equation, are scrutinized in a variety of dynamic and equilibrium settings. For example, we demonstrate the possibility for stick–slip dynamics, substrate-induced hysteresis, the uphill motion of the droplet for sufficiently strong chemical gradients and the existence of a critical inclination angle beyond which the droplet can no longer be supported at equilibrium. Where possible, analytical expressions are obtained for various quantities of interest, which are also verified by appropriate numerical experiments.

A general theory for the dynamics of thin viscous sheets

2002 ◽  
Vol 457 ◽  
pp. 255-283 ◽  
Author(s):  
N. M. RIBE
Keyword(s):  
Evolution Equations ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Extensional Flow ◽  
Constant Thickness ◽  
Scaling Analysis ◽  
Two Dimensional ◽  
Principal Curvatures ◽  
Kinematic Evolution ◽  
Starting Point

A model for the deformation of thin viscous sheets of arbitrary shape subject to arbitrary loading is presented. The starting point is a scaling analysis based on an analytical solution of the Stokes equations for the flow in a shallow (nearly planar) sheet with constant thickness T0 and principal curvatures k1 and k2, loaded by an harmonic normal stress with wavenumbers q1 and q2 in the directions of principal curvature. Two distinct types of deformation can occur: an ‘inextensional’ (bending) mode when [mid ]L3(k1q22 + k2q21)[mid ] [Lt ] ε, and a ‘membrane’ (stretching) mode when [mid ]L3(k1q22 + k2q21)[mid ] [Gt ] ε, where L ≡ (q21 + q22)−1/2 and ε = T0/L [Lt ] 1. The scales revealed by the shallow-sheet solution together with asympotic expansions in powers of ε are used to reduce the three-dimensional equations for the flow in the sheet to a set of equivalent two-dimensional equations, valid in both the inextensional and membrane limits, for the velocity U of the sheet midsurface. Finally, kinematic evolution equations for the sheet shape (metric and curvature tensors) and thickness are derived. Illustrative numerical solutions of the equations are presented for a variety of buoyancy-driven deformations that exhibit buckling instabilities. A collapsing hemispherical dome with radius L deforms initially in a compressional membrane mode, except in bending boundary layers of width ∼ (εL)1/2 near a clamped equatorial edge, and is unstable to a buckling mode which propagates into the dome from that edge. Buckling instabilities are suppressed by the extensional flow in a sagging inverted dome (pendant drop), which consequently evolves entirely in the membrane mode. A two-dimensional viscous jet falling onto a rigid plate exhibits steady periodic folding, the frequency of which varies with the jet height and extrusion rate in a way similar to that observed experimentally.

Computations of the Numerical Solutions of Higher Class of Navier-Stokes Equations: 2D Newtonian Fluid Flow

2001 ◽  
Author(s):  
K. S. Surana ◽  
H. Vijayendra Nayak
Keyword(s):  
Boundary Condition ◽  
Newtonian Fluid ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Model Problem ◽  
Free Field ◽  
Navier Stokes ◽  
Stick Slip ◽  
Navier Stokes Equations ◽  
Slip Point

Abstract This paper presents formulations, computations, investigations and consequences of the various aspects of the numerical solutions of classes C00 and C11 of the two dimensional Navier-Stokes equations in primitive variables u, v, p, τxx, τxy and τyy for incompressible, isothermal and laminar Newtonian fluid flows using p-version Least Squares Finite Element Formulations (LSFEF). The stick-slip problem is used as a model problem in all investigations since this model problem is typical of many other flow situations like contraction, expansion etc. The major thrust of the work presented is to attempt to resolve the local behavior of the solutions in the immediate vicinity of the stick-slip point. The investigations reveal the following: a) The manner in which the stresses are non-dimensionalized in the governing differential equations (GDEs) influences the performance of the iterative procedure of solving non-linear algebraic equations and thus, computational efficiency. b) Solutions of the class C00 are always the wrong class of solutions and thus are always spurious. c) In the flow domains, containing sharp gradients of dependent variables, conservation of mass is difficult to achieve specially at lower p-levels. d) C11 solutions of the Navier-Stokes equations are in conformity with the continuity considerations in the GDEs. e) An augmented form of the Navier-Stokes equations is proposed that always ensures conservation of mass regardless of mesh, p-levels and the nature of the solution gradients. This approach yields the most desired class of C11 solutions. f) It is mathematically established and numerically demonstrated using stick-slip problem that τij are in fact zero at the stick-slip point and the peak values of τxx and τyy must occur, and in fact do, past the stick-slip point in the free field and that peak values of τxy must occur before the stick-slip point on the no-slip boundary. Thus, there is no singularity of τij in the stick-slip problem at the stick-slip point. A significant finding is that imposition of symmetry boundary condition (necessary based on physics) at the stick-slip point even in C11 interpolations is not possible without deteriorating τij behavior in the vicinity of the stick-slip point. However, with the boundary condition, the peak of τxy does occur before the stick-slip point, while the locations of τxx and τyy remain past the stick-slip point in the free field. h) A significant feature of our research work is that we utilize straightforward p-version LSFEF with C00 and C11 type interpolation without linearizing GDEs and that SUPG, SUPG/DC, SUPG/DC/LS operators are neither needed nor used. All numerical studies are conducted and presented using three different meshes (progressively refined and graded) for two different velocities (0.01 and 100 m/s).

scholarly journals NEWTONIAN FLOW IN CONVERGING-DIVERGING CAPILLARIES

2013 ◽  
Vol 04 (03) ◽  
pp. 1350011 ◽  
Author(s):  
TAHA SOCHI
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Volumetric Flow Rate ◽  
Navier Stokes ◽  
Lubrication Approximation ◽  
Newtonian Flow ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
The One ◽  
Analytical Expressions

The one-dimensional Navier–Stokes equations are used to derive analytical expressions for the relation between pressure and volumetric flow rate in capillaries of five different converging-diverging axisymmetric geometries for Newtonian fluids. The results are compared to previously derived expressions for the same geometries using the lubrication approximation. The results of the one-dimensional Navier–Stokes are identical to those obtained from the lubrication approximation within a nondimensional numerical factor. The derived flow expressions have also been validated by comparison to numerical solutions obtained from discretization with numerical integration. Moreover, they have been certified by testing the convergence of solutions as the converging-diverging geometries approach the limiting straight geometry.

A numerical study of the interaction between unsteay free-stream disturbances and localized variations in surface geometry

1989 ◽  
Vol 209 ◽  
pp. 285-308 ◽  
Author(s):  
R. J. Bodonyi ◽  
W. J. C. Welch ◽  
P. W. Duck ◽  
M. Tadjfar
Keyword(s):  
Numerical Solutions ◽  
Free Stream ◽  
Stokes Equations ◽  
Numerical Study ◽  
Navier Stokes ◽  
Surface Geometry ◽  
Navier Stokes Equations ◽  
S Waves ◽  
Tollmien Schlichting

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

Unit Mobility Ratio Displacement Calculations for Pattern Floods in Homogeneous Medium

1966 ◽  
Vol 6 (03) ◽  
pp. 217-227 ◽  
Author(s):  
Hubert J. Morel-Seytoux
Keyword(s):  
Oil Recovery ◽  
Numerical Solutions ◽  
Mobility Ratio ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Numerical Techniques ◽  
Sweep Efficiency ◽  
Closed Form Solutions ◽  
Regular Patterns ◽  
Quantities Of Interest

Abstract The influence of pattern geometry on assisted oil recovery for a particular displacement mechanism is the object of investigation in this paper. The displacement is assumed to be of unit mobility ratio and piston-like. Fluids are assumed incompressible and gravity and capillary effects are neglected. With these assumptions it is possible to calculate by analytical methods the quantities of interest to the reservoir engineer for a great variety of patterns. Specifically, this paper presentsvery briefly, the methods and mathematical derivations required to obtain the results of engineering concern, andtypical results in the form of graphs or formulae that can be used readily without prior study of the methods. Results of this work provide checks for solutions obtained from programmed numerical techniques. They also reveal the effect of pattern geometry and, even though the assumptions of piston-like displacement and of unit mobility ratio are restrictive, they can nevertheless be used for rather crude but quick, cheap estimates. These estimates can be refined to account for non-unit mobility ratio and two-phase flow by correlating analytical results in the case M=1 and the numerical results for non-Piston, non-unit mobility ratio displacements. In an earlier paper1 it was also shown that from the knowledge of closed form solutions for unit mobility ratio, quantities called "scale factors" could be readily calculated, increasing considerably the flexibility of the numerical techniques. Many new closed form solutions are given in this paper. INTRODUCTION BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected. BACKGROUND Pattern geometry is a major factor in making water-flood recovery predictions. For this reason many numerical schemes have been devised to predict oil recovery in either regular patterns or arbitrary configurations. The numerical solutions, based on the method of finite difference approximation, are subject to errors often difficult to evaluate. An estimate of the error is possible by comparison with exact solutions. To provide a variety of checks on numerical solutions, a thorough study of the unit mobility ratio displacement process was undertaken. To calculate quantities of interest to the reservoir engineer (oil recovery, sweep efficiency, etc.), it is necessary to first know the pressure distribution in the pattern. Then analytical procedures are used to calculate, in order of increasing difficulty: injectivity, breakthrough areal sweep efficiency, normalized oil recovery and water-oil ratio as a function of normalized PV injected.

Spin-up from rest of a compressible fluid in a rapidly rotating cylinder

1992 ◽  
Vol 237 ◽  
pp. 413-434 ◽  
Author(s):  
Jae Min Hyun ◽  
Jun Sang Park
Keyword(s):  
Mach Number ◽  
Compressible Fluid ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Direction ◽  
Ekman Layer ◽  
Infinite Cylinder ◽  
Full Time ◽  
Viscous Effects ◽  
Initial State

Spin-up flows of a compressible gas in a finite, closed cylinder from an initial state of rest are studied, The flow is characterized by small reference Ekman numbers, and the peripheral Mach number is O(1). Comprehensive numerical solutions have been obtained for the full, time-dependent compressible Navier-Stokes equations. The details of the flow, temperature, and density evolution are described. In the early phase of spin-up, owing to the thermoacoustic disturbances caused by the compressible Rayleigh effect, the flows are oscillatory, and this oscillatory behaviour is pronounced at higher Mach numbers. The principal dynamical role of the Ekman layer is dominant over moderate times of orders of the homogeneous spin-up timescales. Owing to the density stratification in the radial direction, the Ekman layer is thicker in the central region of the interior. The interior azimuthal flows are mainly uniform in the axial direction. As the Mach number increases, the rate of spin-up in the interior becomes slower, and the propagating shear front is more diffusive. Explicit comparisons with the results for an infinite cylinder are made to ascertain the contributions of the endwall disks. In contrast to the usual incompressible spin-up from rest, the viscous effects are relatively more important for the case of a compressible fluid.

A Study of Effect of Various Normal Force Loading Forms on Frictional Stick-Slip Vibration

2021 ◽  
Author(s):  
Xiaocui Wang ◽  
◽  
Runlan Wang ◽  
Bo Huang ◽  
Jiliang Mo ◽  
...  
Keyword(s):  
Dynamic Behaviour ◽  
Normal Force ◽  
Time Varying ◽  
Stick Slip ◽  
Vibration Signals ◽  
Theoretical Methods ◽  
Normal Forces ◽  
Frequency Domains ◽  
Amplitude Vibration ◽  
Fourier Spectra

In this work, a comparative study is performed to investigate the influence of time-varying normal forces on the friction properties and friction-induced stick-slip vibration by experimental and theoretical methods. In the experiments, constant and harmonic-varying normal forces are applied, respectively. The measured vibration signals under two loading forms are compared in both time and frequency domains. In addition, mathematical tools such as phase space reconstruction and Fourier spectra are used to reveal the science behind the complicated dynamic behaviour. It can be found that the friction system shows steady stick-slip vibration, and the main frequency does not vary with the magnitude of the constant normal force, but the size of limit cycle increases with the magnitude of the constant normal force. In contrast, the friction system harmonic normal force shows complicated behaviour, for example, higher-frequency larger-amplitude vibration occurs as the frequency of the normal force increases. The interesting findings offer a new way for controlling friction-induced stick-slip vibration in engineering applications.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

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