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.

Nonaxisymmetric Three-Dimensional Stagnation-Point Flow and Heat Transfer on a Flat Plate

2009 ◽  
Vol 131 (7) ◽  
Author(s):  
Ali Shokrgozar Abbassi ◽  
Asghar Baradaran Rahimi
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Flat Plate ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Dimensional Flow ◽  
Flow And Heat Transfer

The existing solutions of Navier–Stokes and energy equations in the literature regarding the three-dimensional problem of stagnation-point flow either on a flat plate or on a cylinder are only for the case of axisymmetric formulation. The only exception is the study of three-dimensional stagnation-point flow on a flat plate by Howarth (1951, “The Boundary Layer in Three-Dimensional Flow—Part II: The Flow Near Stagnation Point,” Philos. Mag., 42, pp. 1433–1440), which is based on boundary layer theory approximation and zero pressure assumption in direction of normal to the surface. In our study the nonaxisymmetric three-dimensional steady viscous stagnation-point flow and heat transfer in the vicinity of a flat plate are investigated based on potential flow theory, which is the most general solution. An external fluid, along z-direction, with strain rate a impinges on this flat plate and produces a two-dimensional flow with different components of velocity on the plate. This situation may happen if the flow pattern on the plate is bounded from both sides in one of the directions, for example x-axis, because of any physical limitation. A similarity solution of the Navier–Stokes equations and energy equation is presented in this problem. A reduction in these equations is obtained by the use of appropriate similarity transformations. Velocity profiles and surface stress-tensors and temperature profiles along with pressure profile are presented for different values of velocity ratios, and Prandtl number.

Flow Due to Eccentric Rotating a Porous Disk and a Fluid at Infinity

1976 ◽  
Vol 43 (2) ◽  
pp. 203-204 ◽  
Author(s):  
M. Emin Erdogan
Keyword(s):  
Exact Solution ◽  
Velocity Distribution ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Porous Disk ◽  
Navier Stokes Equations ◽  
Uniform Suction ◽  
Asymptotic Profile

An exact solution of the steady three-dimensional Navier-Stokes equations is obtained for the case of flow due to noncoaxially rotations of a porous disk and a fluid at infinity. It is shown that for uniform suction or uniform blowing at the disk an asymptotic profile exists for the velocity distribution.

Oscillating stagnation point flow

1982 ◽  
Vol 384 (1786) ◽  
pp. 175-190 ◽  
Keyword(s):  
Boundary Layer ◽  
Stagnation Point ◽  
High Frequency ◽  
Stokes Equations ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Analytic Approximations ◽  
Numerical Integrations ◽  
Hiemenz Flow

A solution of the Navier-Stokes equations is given for an incompressible stagnation point flow whose magnitude oscillates in time about a constant, non-zero, value (an unsteady Hiemenz flow). Analytic approximations to the solution in the low and high frequency limits are given and compared with the results of numerical integrations. The application of these results to one aspect of the boundary layer receptivity problem is also discussed.

scholarly journals Unsteady Reversed Stagnation-Point Flow over a Flat Plate

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Vai Kuong Sin ◽  
Chon Kit Chio
Keyword(s):  
Laminar Flow ◽  
Flow Field ◽  
Asymptotic Solution ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Total Flow ◽  
Navier Stokes Equations

This paper investigates the nature of the development of two-dimensional laminar flow of an incompressible fluid at the reversed stagnation-point. Proudman and Johnson (1962) first studied the flow and obtained an asymptotic solution by neglecting the viscous terms. Robins and Howarth (1972) stated that this is not true in neglecting the viscous terms within the total flow field. Viscous terms in this analysis are now included, and a similarity solution of two-dimensional reversed stagnation-point flow is investigated by solving the full Navier-Stokes equations.

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.

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.

Flow and Mass Transfer for an Unsteady Stagnation-Point Flow Over a Moving Wall Considering Blowing Effects

2014 ◽  
Vol 136 (7) ◽  
Author(s):  
Tiegang Fang
Keyword(s):  
Mass Transfer ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Local Maximum ◽  
Wall Stress ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Mass Transfer Equation ◽  
Moving Wall

In this paper, the flow and mass transfer of a two-dimensional unsteady stagnation-point flow over a moving wall, considering the coupled blowing effect from mass transfer, is studied. Similarity equations are derived and solved in a closed form. The flow solution is an exact solution to the two-dimensional unsteady Navier–Stokes equations. An analytical solution of the boundary layer mass transfer equation is obtained together with the momentum solution. The examples demonstrate the significant impacts of the blowing effects on the flow and mass transfer characteristics. A higher blowing parameter results in a lower wall stress and thicker boundary layers with less mass transfer flux at the wall. The higher wall moving parameters produce higher mass transfer flux and blowing velocity. The Schmidt parameters generate a local maximum for the mass transfer flux and blowing velocity under given wall moving and blowing parameters.

Similarity solutions for unsteady stagnation point flow

2012 ◽  
Vol 711 ◽  
pp. 394-410 ◽  
Author(s):  
D. Kolomenskiy ◽  
H. K. Moffatt
Keyword(s):  
Stagnation Point ◽  
Stokes Equations ◽  
Nonlinear Ordinary Differential Equation ◽  
Similarity Solutions ◽  
Parameter Family ◽  
Stagnation Point Flow ◽  
Navier Stokes ◽  
Plane Boundary ◽  
Wide Range ◽  
Flow Situation

AbstractA class of similarity solutions for two-dimensional unsteady flow in the neighbourhood of a front or rear stagnation point on a plane boundary is considered, and a wide range of possible behaviour is revealed, depending on whether the flow in the far field is accelerating or decelerating. The solutions, when they exist, are exact solutions of the Navier–Stokes equations, having a boundary-layer character analogous to that of the classical steady front stagnation point flow. The velocity profiles are obtained by numerical integration of a nonlinear ordinary differential equation. For the front-flow situation, the solution is unique for the accelerating case, but bifurcates for modest deceleration, while for sufficient rapid deceleration there exists a one-parameter family of solutions. For the rear-flow situation, a unique solution exists (remarkably!) for sufficiently strong acceleration, and a one-parameter family again exists for sufficient strong deceleration. Analytic results, which are consistent with the numerical results, are obtained in the limits of strong acceleration or deceleration, and for the asymptotic behaviour far from the boundary.

scholarly journals Streamrate stagnation-point flow of a nanofluid on a stationary cylinder

2015 ◽  
Vol 3 (1) ◽  
pp. 124
Author(s):  
Vahid Amerian ◽  
Hamid Mohammadiun ◽  
Mohammad Mohammadiun ◽  
Iman Khazaee
Keyword(s):  
Velocity Field ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Fluid Velocity ◽  
Constant Strain Rate ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Shear Stresses ◽  
Navier Stokes Equations ◽  
Fluid Velocity Field

<p>The steady-state, viscous flow of Nanofluid in the vicinity of an axisymmetric stagnation point of a stationary cylinder is investigated. The impinging free-stream is steady and with a constant strain rate . Exact solution of the Navier–Stokes equations is derived in this problem. A reduction of these equations is obtained by use of appropriate transformations introduced in this research. The general self-similar solution is obtained when the wall temperature of the cylinder is constant. All the solutions above are presented for Reynolds numbers ranging from 0.1 to 1000 and selected values of particle. For all Reynolds numbers, as the particle fraction increases, the depth of diffusion of the fluid velocity field in radial direction, the depth of the diffusion of the fluid velocity field in -direction, shear-stresses and pressure function decreases.<strong></strong></p>

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