Nonsteady Stagnation-Point Flows Over Permeable Surfaces: Explicit Solutions of the Nevier-Stokes Equations

1995 ◽  
Vol 117 (1) ◽  
pp. 189-191 ◽  
Author(s):  
G. I. Burde
Keyword(s):  
Time Dependence ◽  
Stagnation Point ◽  
Stokes Equations ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Explicit Solutions ◽  
Navier Stokes Equations ◽  
New Approach ◽  
The Common ◽  
Suction Or Injection

Some new explicit solutions of the unsteady two-dimensional Navier-Stokes equations describing nonsteady stagnation-point flows with surface suction or injection are presented. The solutions have been obtained using a new approach for finding explicit similarity solutions of partial differential equations. As distinct from the common Birkhoff’s similarity transformation, which permits only one form of an unsteady potential flow field and only one form of time dependence for suction (or blowing) velocity, the transforms obtained permit consideration of a variety of special solutions differing in the forms of time dependence.

Similarity Solutions for the Interaction of a Potential Vortex With Free Stream Sink Flow and a Stationary Surface

1974 ◽  
Vol 96 (1) ◽  
pp. 49-54 ◽  
Author(s):  
J. A. Hoffmann
Keyword(s):  
Boundary Layer ◽  
Differential Equations ◽  
Free Stream ◽  
Stokes Equations ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Stationary Surface ◽  
Direct Numerical Integration ◽  
Potential Vortex

Similarity equations, using an assumed transformation which reduces the partial differential equations to sets of ordinary differential equations, are obtained from the boundary layer and the complete Navier-Stokes equations for the interaction of vortex flows with free stream sink flows and a stationary surface. Solutions to the boundary layer equations for the case of the potential vortex that satisfy the prescribed boundary conditions are shown to be nonexistent using the assumed transformation. Direct numerical integration is used to obtain solutions to the complete Navier-Stokes equations under a potential vortex with equal values of tangential and radial free stream velocities. Solutions are found for Reynolds numbers up to 2.0.

Similarity solutions for viscous vortex cores

1992 ◽  
Vol 238 ◽  
pp. 487-507 ◽  
Author(s):  
Ernst W. Mayer ◽  
Kenneth G. Powell
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Axial Flow ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Core ◽  
Axial Pressure Gradient ◽  
Self Similar ◽  
Similar Solutions

Results are presented for a class of self-similar solutions of the steady, axisymmetric Navier–Stokes equations, representing the flows in slender (quasi-cylindrical) vortices. Effects of vortex strength, axial gradients and compressibility are studied. The presence of viscosity is shown to couple the parameters describing the core growth rate and the external flow field, and numerical solutions show that the presence of an axial pressure gradient has a strong effect on the axial flow in the core. For the viscous compressible vortex, near-zero densities and pressures and low temperatures are seen on the vortex axis as the strength of the vortex increases. Compressibility is also shown to have a significant influence upon the distribution of vorticity in the vortex core.

Melting From a Horizontal Rotating Disk

1989 ◽  
Vol 56 (1) ◽  
pp. 47-50 ◽  
Author(s):  
C. Y. Wang
Keyword(s):  
Heat Transfer ◽  
Numerical Integration ◽  
Stokes Equations ◽  
Rotating Disk ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Relative Importance ◽  
Governing Equations ◽  
Navier Stokes Equations ◽  
Transfer Rates

Melting of a disk is facilitated by rotation. The problem is governed by a nondimensional parameter α which represents the relative importance of injection (melt) rate and rotation times viscosity. The nonlinear governing equations are solved by perturbations for small α and numerical integration for arbitrary α. Torque and heat transfer rates are found. The solution is one of the rare exact similarity solutions of the Navier-Stokes equations.

Conical turbulent swirling vortex with variable eddy viscosity

1986 ◽  
Vol 403 (1825) ◽  
pp. 235-268 ◽  
Keyword(s):  
Eddy Viscosity ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Turbulent Vortex ◽  
The Core ◽  
Vortex Solutions ◽  
Variable Eddy Viscosity

In the one hundred years since Rankine suggested his well known two-dimensional vortex model with finite core, no one has ever found any exact vortex solutions of the Navier-Stokes equations that can satisfy a complete set of physical boundary conditions. In this paper a variable viscosity is introduced and the existence of conical turbulent vortex solutions of the Navier-Stokes equations is examined. It is found that for a class of deliberately chosen eddy viscosity function a steady turbulent vortex can, for the first time, satisfy both the regularity condition at the core and the adherence condition at the surface, except for a singularity at the origin inherent in all conical similarity solutions. In its asymptotic form, if the eddy viscosity only varies in a boundary layer near the surface or the core, outside the layer the solution given would approach one of the laminar solutions of Yih et al . ( Physics Fluids 25, 2147 (1982)) or that of Serrin ( Phil. Trans. R. Soc. Lond. A 271, 325 (1972)) respectively. These results reveal some remarkable relations between the behaviour, and even the existence, of a vortex and turbulence.

Lattice Boltzmann Method for the Simulating Extrudate Swell of Viscoelastic Fluid

2015 ◽  
Vol 799-800 ◽  
pp. 784-787
Author(s):  
Wen Qin Liu ◽  
Yong Li
Keyword(s):  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Extrudate Swell ◽  
Navier Stokes Equations ◽  
New Approach ◽  
Moving Interface ◽  
Good Accord ◽  
Boltzmann Method

The main objective of this work is to develop a new approach based on the Lattice Boltzmann method (LBM) to simulate the extrudate swell of an Oldroyd B viscoelatic fluid. Two lattice Boltzmann equations are used to solve the Navier-Stokes equations and constitutive equation simultaneously at each time iteration. The single LBM model is used to track the moving interface in this paper. To validate the accuracy and stability of this new scheme, we study the steady 2D Poiseuille flow firstly, finding the numerical results be in good accord with the analytical solution. Then the die-swell phenomenon is solved, we successfully acquire the different swelling state of an Oldroyd B fluid at different time.

The axially symmetric flow of a viscous fluid between two infinite rotating disks

1962 ◽  
Vol 266 (1324) ◽  
pp. 109-121 ◽  
Keyword(s):  
Stokes Equations ◽  
Transverse Velocity ◽  
Complete Solution ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Axially Symmetric ◽  
Rotating Disks ◽  
Symmetric Flow ◽  
Navier Stokes Equations ◽  
High Reynolds

This is a numerical investigation of the similarity solutions of the Navier-Stokes equations describing the steady axially symmetric flow of a viscous incompressible fluid between two infinite rotating disks. Several cases have been examined in detail and the radial and transverse velocity profiles are displayed; value of the torque experienced in these cases are also given. It is found that at high Reynolds numbers, the main core of the fluid is in a state of solid rotation for practically all values of the ratio of angular velocity of the two disks. When the disks are rotating in the same sense, and when one is at rest and the other is rotating, the results show that edge effects must be taken into account in any complete solution to the problem. However, when the disks rotate in opposite directions, the solutions exhibit features which appear unlikely to occur in practice.

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.

Laminar film flow on a cylindrical surface

1976 ◽  
Vol 74 (2) ◽  
pp. 297-315 ◽  
Author(s):  
Ernst Becker
Keyword(s):  
Stagnation Point ◽  
Cylindrical Surface ◽  
Film Flow ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Navier Stokes ◽  
Point Problem ◽  
Navier Stokes Equations ◽  
Laminar Film ◽  
Particular Solution

The paper deals with steady laminar film flow which is set up at the cylindrical surface of an idealized horizontal ‘road’ when homogeneous ‘rain’ is falling onto the road in a vertical downward direction. It is shown that a particular solution of the Navier-Stokes equations is possible for which the depth of the liquid film is constant. In that case the Navier-Stokes equations reduce to the equations governing plane stagnation-point flow. However, the boundary conditions differ from those for the classical stagnation-point problem. Solutions for nearly inviscid flow and predominantly viscous flow are derived analytically. In particular, simple formulae for the depth of the film are found in both cases. Finally, the importance of the particular solution as a member of a whole class of solutions is discussed on the basis of a momentum integral approximation.

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