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.

scholarly journals MHD convection ow in a constricted channel

2018 ◽  
Vol 26 (2) ◽  
pp. 267-283
Author(s):  
M. Tezer-Sezgin ◽  
Merve Gürbüz
Keyword(s):  
Magnetic Field ◽  
Applied Magnetic Field ◽  
Hartmann Number ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Particular Solution ◽  
Laminar Convection ◽  
Long Channel ◽  
Velocity Pressure

Abstract We consider the steady, laminar, convection ow in a long channel of 2D rectangular constricted cross-section under the inuence of an applied magnetic field. The Navier-Stokes equations including Lorentz and buoyancy forces are coupled with the temperature equation and are solved by using linear radial basis function (RBF) approximations in terms of the velocity, pressure and the temperature of the fluid. RBFs are used in the approximation of the particular solution which becomes also the approximate solution of the problem. Results are obtained for several values of Grashof number (Gr), Hartmann number (M) and the constriction ratios (CR) to see the effects on the ow and isotherms for fixed values of Reynolds number and Prandtl number. As M increases, the ow is flattened. An increase in Gr increases the magnitude of the ow in the channel. Isolines undergo an inversion at the center of the channel indicating convection dominance due to the strong buoyancy force, but this inversion is retarded with the increase in the strength of the applied magnetic field. When both Hartmann number and constriction ratio are increased, ow is divided into more loops symmetrically with respect to the axes.

Some numerical experiments on developing laminar flow in circular-sectioned bends

1985 ◽  
Vol 154 ◽  
pp. 357-375 ◽  
Author(s):  
J. A. C. Humphrey ◽  
H. Iacovides ◽  
B. E. Launder
Keyword(s):  
Shear Stress ◽  
Laminar Flow ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Streamwise Velocity ◽  
Navier Stokes ◽  
Dean Number ◽  
Navier Stokes Equations ◽  
Numerical Predictions

The paper reports numerical solutions to a semi-elliptic truncation of the Navier–Stokes equations for the case of developing laminar flow in circular-sectioned bends over a range of Dean numbers. The ratios of bend radius to pipe radius are 7:1 and 20:1, corresponding with the configurations examined experimentally by Talbot and his co-workers in recent years. The semi-elliptic treatment facilitates a much finer grid than has been possible in earlier studies. Numerical accuracy has been further improved by assuming radial equilibrium over a thin sublayer immediately adjacent to the wall and by re-formulating the boundary conditions at the pipe centre.Streamwise velocity profiles at Dean numbers of 183 and 565 are in excellent agreement with laser-Doppler measurements by Agrawal, Talbot & Gong (1978). Good, albeit less complete, accord is found with the secondary velocities, though the differences that exist may be mainly due to the difficulty of making these measurements. The paper provides new information on the behaviour of the streamwise shear stress around the inner line of symmetry. Upstream of the point of minimum shear stress, our numerical predictions display a progressive shift towards the result of Stewartson, Cebici & Chang (1980) as the Dean number is successively raised. Downstream of the minimum, however, in contrast with the monotonic approach to an asymptotic level reported by Stewartson, the numerical solutions display a damped oscillatory behaviour reminiscent of those from Hawthorne's (1951) inviscid-flow calculations. The amplitude of the oscillation grows as the Dean number is raised.

Simulations of the formation of an axisymmetric vortex ring

1997 ◽  
Vol 339 ◽  
pp. 199-211 ◽  
Author(s):  
R. S. HEEG ◽  
N. RILEY
Keyword(s):  
Vortex Ring ◽  
Reasonable Agreement ◽  
Stokes Equations ◽  
Circular Tube ◽  
Inviscid Flow ◽  
High Rate ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Axisymmetric Vortex ◽  
High Reynolds

In this paper we present the results from numerical calculations, based upon the Navier–Stokes equations at relatively high Reynolds number, of the formation of a vortex ring when fluid is ejected from a circular tube. Our results are compared with the experiments of Didden (1979), and the inviscid flow calculations of Nitsche & Krasny (1994). Reasonable agreement is achieved except for the rate of shedding of circulation during the initial stages of ring formation. The theoretically predicted rate of shedding is substantially higher than that predicted by Didden. By contrast the inviscid theory predicts an anomalously high rate of initial shedding. We offer explanations for both of these apparent discrepancies.

Viscous oscillatory flow about a circular cylinder at small to moderate Strouhal number

1995 ◽  
Vol 303 ◽  
pp. 215-232 ◽  
Author(s):  
H. M. Badr ◽  
S. C. R. Dennis ◽  
S. Kocabiyik ◽  
P. Nguyen
Keyword(s):  
Circular Cylinder ◽  
Strouhal Number ◽  
Transient Flow ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Navier Stokes ◽  
Mean Flow ◽  
Navier Stokes Equations ◽  
Method Of Solution ◽  
High Reynolds

The transient flow field caused by an infinitely long circular cylinder placed in an unbounded viscous fluid oscillating in a direction normal to the cylinder axis, which is at rest, is considered. The flow is assumed to be started suddenly from rest and to remain symmetrical about the direction of motion. The method of solution is based on an accurate procedure for integrating the unsteady Navier–Stokes equations numerically. The numerical method has been carried out for large values of time for both moderate and high Reynolds numbers. The effects of the Reynolds number and of the Strouhal number on the laminar symmetric wake evolution are studied and compared with previous numerical and experimental results. The time variation of the drag coefficients is also presented and compared with an inviscid flow solution for the same problem. The comparison between viscous and inviscid flow results shows a better agreement for higher values of Reynolds and a Strouhal numbers. The mean flow for large times is calculated and is found to be in good agreement with previous predictions based on boundary-layer theory.

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.

Orifice Contraction Coefficient for Inviscid Incompressible Flow

1985 ◽  
Vol 107 (1) ◽  
pp. 36-43 ◽  
Author(s):  
R. D. Grose
Keyword(s):  
Stokes Equations ◽  
Inviscid Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Contraction Coefficient ◽  
Two Dimensional Flow ◽  
Conservation Of Momentum ◽  
Orifice Flow ◽  
Contraction Effect ◽  
Control Volumes

The theory for steady flow of an incompressible fluid through an orifice has been semi-empirically established for only certain flow conditions. In this paper, the development of a more rigorous theory for the prediction of the orifice flow contraction effect is presented. This theory is based on the conservation of momentum and mass principles applied to global control volumes for continuum flow. The control volumes are chosen to have a particular geometric construction which is based on certain characteristics of the Navier-Stokes equations for incompressible and, in the limit, inviscid flow. The treatment is restricted to steady incompressible, single phase, single component, inviscid Newtonian flow, but the principles that are developed hold for more general conditions. The resultant equations predict the orifice contraction coefficient as a function of the upstream geometry ratio for both axisymmetric and two-dimensional flow fields. The predicted contraction coefficient values agree with experimental orifice discharge coefficient data without the need for empirical adjustment.

Inertia Effects in Thin Film Flow With a Corrugated Boundary

1991 ◽  
Vol 58 (1) ◽  
pp. 272-277 ◽  
Author(s):  
Ilter Serbetci ◽  
John A. Tichy
Keyword(s):  
Pressure Distribution ◽  
Film Flow ◽  
Stokes Equations ◽  
Lubrication Theory ◽  
Navier Stokes ◽  
Thin Film Flow ◽  
Perturbation Expansions ◽  
Navier Stokes Equations ◽  
Inertia Effects ◽  
Grooved Surface

An analytical solution is presented for two-dimensional, incompressible film flow between a sinusoidally grooved (or rough) surface and a flat surface. The upper grooved surface is stationary whereas the lower, smooth surface moves with a constant speed, The Navier-Stokes equations were solved employing both mapping techniques and perturbation expansions. Due to the inclusion of the inertia effects, a different pressure distribution is obtained than predicted by the classical lubrication theory. In particular, the amplitude of the pressure distribution of the classical lubrication theory is found to be in error by over 100 percent (for modified Reynolds number of 3–4).

Re-entrant corner flows of Oldroyd-B fluids

2005 ◽  
Vol 461 (2060) ◽  
pp. 2573-2603 ◽  
Author(s):  
J.D Evans
Keyword(s):  
Boundary Layers ◽  
Newtonian Fluid ◽  
Similarity Solution ◽  
Stokes Equations ◽  
Inviscid Flow ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Outer Solution ◽  
The Core ◽  
Retardation Parameter

The method of matched asymptotic expansions is used to construct solutions for the planar steady flow of Oldroyd-B fluids around re-entrant corners of angles π / α (1/2≤ α <1). Two types of similarity solutions are described for the core flow away from the walls. These correspond to the two main dominant balances of the constitutive equation, where the upper convected derivative of stress either dominates or is balanced by the upper convected derivative of the rate of strain. The former balance gives the incompressible Euler or inviscid flow equations and the latter balance the incompressible Navier–Stokes equations. The inviscid flow similarity solution for the core is that first derived by Hinch (Hinch 1993 J. Non-Newtonian Fluid Mech. 50 , 161–171) with a core stress singularity that depends upon the corner angle and radial distance as O ( r −2(1− α ) ) and a velocity behaviour that vanishes as O ( r α (3− α )−1 ). Extending the analysis of Renardy (Renardy 1995 J. Non-Newtonian Fluid Mech. 58 , 83–39), this outer solution is matched to viscometric wall behaviour for both upstream and downstream boundary layers. This structure is shown to hold for the majority of the retardation parameter range. In contrast, the similarity solution associated with the Navier–Stokes equations has a velocity behaviour O ( r λ ) where λ ∈(0,1) satisfies a nonlinear eigenvalue problem, dependent upon the corner angle and an associated Reynolds number defined in terms of the ratio of the retardation and relaxation times. This similarity solution is shown to hold as an outer solution and is matched into stress boundary layers at the walls which recover viscometric behaviour. However, the matching is restricted to values of the retardation parameter close to the relaxation parameter. In this case the leading order core stress is Newtonian with behaviour O ( r −(1− λ ) ).

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