Derivation of a Thin Film Equation by a Direct Approach

1996 ◽  
Vol 63 (2) ◽  
pp. 467-473
Author(s):  
F. Y. Huang ◽  
C. D. Mote
Keyword(s):  
Thin Film ◽  
Velocity Field ◽  
Viscous Fluid ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Film Structure ◽  
Navier Stokes ◽  
Slip Boundary ◽  
Thin Film Equations ◽  
Thin Film Equation

A new model of the thin viscous fluid film, constrained between two translating, flexible surfaces, is presented in this paper: The unsteady inertia of the film is included in the model. The derivation starts with the reduced three-dimensional Navier-Stokes equations for an incompressible viscous fluid with a small Reynolds number. By introduction of an approximate velocity field, which satisfies the continuity equation and the no-slip boundary conditions exactly, into weighted integrals of the three-dimensional equations over the film thickness, a two-dimensional thin film equation is obtained explicitly in a closed form. The 1th thin film equation is obtained when the velocity field is approximated by 21th order polynominals, and the three-dimensional viscous film is described with increasing accuracy by thin film equations of increasing order. Two cases are used to illustrate the coupling of the film to the vibration of the structure and to show that the second thin film equation can be applied successfully to the prediction of a coupled film-structure response in the range of most applications. A reduced thin film equation is derived through approximation of the second thin film equation that relates the film pressure to transverse accelerations and velocities, and to slopes and slope rates of the two translating surfaces.

Attractors for the modifications of the three-dimensional Navier-Stokes equations

1994 ◽  
Vol 346 (1679) ◽  
pp. 173-190 ◽  
Keyword(s):  
Viscous Fluid ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Fluid Flows ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Initial Boundary Value ◽  
Dimensional Domain

The modifications of the three-dimensional Navier-Stokes equations, which I suggested earlier for the description of viscous fluid flows with large gradients of velocities, are considered. It is proved that the first initial-boundary value problem for these equations in any bounded three-dimensional domain has a compact minimal global B-attractor. Some properties of the attractor are established.

MHD FLOWS DUE TO NON-COAXIAL ROTATIONS OF POROUS DISK AND A VISCOUS FLUID AT INFINITY: GRAPHICAL SOLUTIONS USING MATLAB

2020 ◽  
Vol 9 (11) ◽  
pp. 9287-9301
Author(s):  
R. Lakshmi ◽  
Santhakumari
Keyword(s):  
Velocity Field ◽  
Analytical Solution ◽  
Viscous Fluid ◽  
Stokes Equations ◽  
Daily Life ◽  
Vital Role ◽  
Navier Stokes ◽  
Porous Disk ◽  
Navier Stokes Equations ◽  
Complex Phenomena

Fluids play a vital role in many aspects of our daily life. We drink water, breath air, fluids run through our bodies and it controls the weather. The study of motion of fluids is a complex phenomena. The equations which govern the flows of Newtonian fluids are Navier-Stokes equations. In this paper, the flows which are due to non – coaxial rotations of porous disk and a fluid at infinity are considered. Analytical solution for the velocity field using Laplace transform is derived. MATLAB coding is written to get the graphical solutions. The results are compared with the existing results. MATLAB software provides accurate results depending on the solution we obtained.

Formation of columnar baroclinic vortices in thermally stratified nonlinear spin-up

2012 ◽  
Vol 702 ◽  
pp. 265-285 ◽  
Author(s):  
J. R. Pacheco ◽  
R. Verzicco
Keyword(s):  
Vortex Core ◽  
Stokes Equations ◽  
Fluid Layer ◽  
Three Dimensional ◽  
Temperature Gradients ◽  
Navier Stokes ◽  
Central Vortex ◽  
Slip Boundary ◽  
Aspect Ratios ◽  
Vorticity Production

AbstractWe investigate the mechanisms that affect the formation of columnar vortices for spin-up in a cylinder where the temperatures at the horizontal walls are prescribed. Numerical results from the three-dimensional Navier–Stokes equations show that a short-lived instability, suppressed by the combined effect of rotation and stratification, generates small temperature variations in the azimuthal direction. Temperature-gradient anomalies produce vorticity, and these vortices stir the fluid at the interface of the central vortex core thus reinforcing the temperature gradients. For sufficiently strong temperature gradients, the central vortex core breaks up into several columnar vortices. It is found, in particular, that small aspect ratios (height over radius of the cylindrical fluid layer) $\Gamma = 1, 2$ tend to inhibit the instability, while larger ones, $\Gamma = 3. 3$, have the opposite effect. The main source of instability is the baroclinic vorticity production and not the presence of a solid sidewall since, counter-intuitively, the flow is more unstable for a free-slip boundary than for a no-slip one. Finally the effect of the temperature boundary conditions (isothermal versus adiabatic) on the horizontal boundaries has been investigated. The adiabatic boundaries help to preserve for longer times the sloping density interfaces that feed, with their potential energy, the baroclinic vorticity production; this results in more unstable flows.

Calculations of Two and Three-Dimensional Transonic Cascade Flow Fields Using the Navier-Stokes Equations

1986 ◽  
Vol 108 (1) ◽  
pp. 93-102 ◽  
Author(s):  
B. C. Weinberg ◽  
R.-J. Yang ◽  
H. McDonald ◽  
S. J. Shamroth
Keyword(s):  
Flow Field ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Solid Surfaces ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Slip Boundary ◽  
Cascade Calculation ◽  
Transonic Cascade ◽  
Good Agreement

The multidimensional, ensemble-averaged, compressible, time-dependent Navier-Stokes equations have been used to study the turbulent flow field in two and three-dimensional turbine cascades. The viscous regions of the flow were resolved and no-slip boundary conditions were utilized on solid surfaces. The calculations were performed in a constructive ‘O’-type grid which allows representation of the blade rounded trailing edge. Converged solutions were obtained in relatively few time steps (∼ 80–150) and comparisons for both surface pressure and heat transfer showed good agreement with data. The three-dimensional turbine cascade calculation showed many of the expected flow-field features.

Fine-scale structure of thin vortical layers

1998 ◽  
Vol 364 ◽  
pp. 297-318
Author(s):  
TAKASHI ISHIHARA ◽  
YUKIO KANEDA
Keyword(s):  
Velocity Field ◽  
Fourier Spectrum ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Scale Structure ◽  
Navier Stokes ◽  
Small Scale ◽  
Planar Jet ◽  
Exponential Decay Rate ◽  
Vortical Layer

A class of exact solutions of the Navier–Stokes equations is derived. Each of them represents the velocity field v=U+u of a thin vortical layer (a planar jet) under a uniform strain velocity field U in three-dimensional infinite space, and provides a simple flow model in which nonlinear coupling between small eddies plays a key role in small-scale vortex dynamics. The small-scale structure of the velocity field is studied by numerically analysing the Fourier spectrum of u. It is shown that the Fourier spectrum of u falls off exponentially with wavenumber k for large k. The Taylor expansion in powers of the coordinate (say y) in the direction perpendicular to the vortical layer suggests that the solution may be well approximated by a function with certain poles in the complex y-plane. The Fourier spectrum based on the singularities is in good agreement with that obtained numerically, where the exponential decay rate is given by the distance of the poles from the real axis of y.

scholarly journals On a class of unsteady three-dimensional Navier—Stokes solutions relevant to rotating disc flows: threshold amplitudes and finite-time singularities

1992 ◽  
Vol 238 ◽  
pp. 297-323 ◽  
Author(s):  
Philip Hall ◽  
P. Balakumar ◽  
D. Papageorgiu
Keyword(s):  
Exact Solution ◽  
Velocity Field ◽  
Continuous Spectrum ◽  
Stagnation Point ◽  
Finite Time ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Rotating Disc

A class of ‘exact’ steady and unsteady solutions of the Navier—Stokes equations in cylindrical polar coordinates is given. The flows correspond to the motion induced by an infinite disc rotating in the (x, y)-plane with constant angular velocity about the z-axis in a fluid occupying a semi-infinite region which, at large distances from the disc, has velocity field proportional to (x, — y,O) with respect to a Cartesian coordinate system. It is shown that when the rate of rotation is large Kármán's exact solution for a disc rotating in an otherwise motionless fluid is recovered. In the limit of zero rotation rate a particular form of Howarth's exact solution for three-dimensional stagnation-point flow is obtained. The unsteady form of the partial differential system describing this class of flow may be generalized to time-periodic flows. In addition the unsteady equations are shown to describe a strongly nonlinear instability of Kármán's rotating disc flow. It is shown that sufficiently large perturbations lead to a finite-time breakdown of that flow whilst smaller disturbances decay to zero. If the stagnation point flow at infinity is sufficiently strong the steady basic states become linearly unstable. In fact there is then a continuous spectrum of unstable eigenvalues of the stability equations but, if the initial-value problem is considered, it is found that, at large values of time, the continuous spectrum leads to a velocity field growing exponentially in time with an amplitude decaying algebraically in time.

scholarly journals Bifurcations in a quasi-two-dimensional Kolmogorov-like flow

2017 ◽  
Vol 828 ◽  
pp. 837-866 ◽  
Author(s):  
Jeffrey Tithof ◽  
Balachandra Suri ◽  
Ravi Kumar Pallantla ◽  
Roman O. Grigoriev ◽  
Michael F. Schatz
Keyword(s):  
Boundary Conditions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Quantitative Agreement ◽  
Navier Stokes ◽  
Dielectric Fluid ◽  
Two Dimensional ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
2D Model

We present a combined experimental and theoretical study of the primary and secondary instabilities in a Kolmogorov-like flow. The experiment uses electromagnetic forcing with an approximately sinusoidal spatial profile to drive a quasi-two-dimensional (Q2D) shear flow in a thin layer of electrolyte suspended on a thin lubricating layer of a dielectric fluid. Theoretical analysis is based on a two-dimensional (2D) model (Suri et al., Phys. Fluids, vol. 26 (5), 2014, 053601), derived from first principles by depth-averaging the full three-dimensional Navier–Stokes equations. As the strength of the forcing is increased, the Q2D flow in the experiment undergoes a series of bifurcations, which is compared with results from direct numerical simulations of the 2D model. The effects of confinement and the forcing profile are studied by performing simulations that assume spatial periodicity and strictly sinusoidal forcing, as well as simulations with realistic no-slip boundary conditions and an experimentally validated forcing profile. We find that only the simulation subject to physical no-slip boundary conditions and a realistic forcing profile provides close, quantitative agreement with the experiment. Our analysis offers additional validation of the 2D model as well as a demonstration of the importance of properly modelling the forcing and boundary conditions.

The initial value problem for the non-homogeneous Navier—Stokes equations with general slip boundary condition

2000 ◽  
Vol 130 (4) ◽  
pp. 827-835 ◽  
Author(s):  
S. Itoh ◽  
A. Tani
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Slip Boundary Condition ◽  
Navier Stokes ◽  
Solid Boundary ◽  
Dimensional Problem ◽  
Navier Stokes Equations ◽  
Slip Boundary

The initial-boundary value problem for the non-homogeneous Navier-Stokes equations including the slipping on the solid boundary is considered. The unique solvability is established locally in time for the three-dimensional problem and globally in time for the two-dimensional problem without so-called smallness restrictions.

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