Computational and experimental investigations of two-dimensional nonlinear peristaltic flows

1977 ◽  
Vol 83 (2) ◽  
pp. 249-272 ◽  
Author(s):  
Thomas D. Brown ◽  
Tin-Kan Hung
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Periodic Wave ◽  
Peristaltic Flow ◽  
Navier Stokes ◽  
Experimental Investigations ◽  
Navier Stokes Equations ◽  
Flow Processes ◽  
Wave Curvature

An implicit finite-difference technique employing orthogonal curvilinear co-ordinates is used to solve the Navier–Stokes equations for peristaltic flows in which both the wall-wave curvature and the Reynolds number are finite (§2). The numerical solutions agree closely with experimental flow visualizations. The kinematic characteristics of both extensible and inextensible walls (§3) are found to have a distinct influence on the flow processes only near the wall. Without vorticity, peristaltic flow observed from a reference frame moving with the wave will be equivalent to steady potential flow through a stationary wavy channel of similar geometry (§4). Solutions for steady viscous flow (§5) are obtained from simulation of unsteady flow processes beginning from an initial condition of potential peristaltic flow. For nonlinear flows due to a single peristaltic wave of dilatation, the highest stresses and energy exchange rates (§6) occur along the wall and in two instantaneous stagnation regions in the bolus core. A series of computations for periodic wave trains reveals that increasing the Reynolds number from 2[sdot ]3 to 251 yields a modest augmentation in the ratio of flow rate to Reynolds number but induces a much greater increase in the shear stress (§7.1). The transport effectiveness is markedly reduced for pumping against a mild adverse pressure drop (§7.2). Increasing the wave amplitude will lead to the development of travelling vortices within the core region of the peristaltic flow (§7.3).

Numerical and asymptotic methods for certain viscous free-surface flows

1992 ◽  
Vol 340 (1656) ◽  
pp. 1-45 ◽  
Keyword(s):  
Reynolds Number ◽  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Free Surface Flows ◽  
Navier Stokes ◽  
Lubrication Approximation ◽  
Navier Stokes Equations ◽  
Flow Rates

This paper concerns the two-dimensional motion of a viscous liquid down a perturbed inclined plane under the influence of gravity, and the main goal is the prediction of the surface height as the fluid flows over the perturbations. The specific perturbations chosen for the present study were two humps stretching laterally across an otherwise uniform plane, with the flow being confined in the lateral direction by the walls of a channel. Theoretical predictions of the flow have been obtained by finite-element approximations to the Navier-Stokes equations and also by a variety of lubrication approximations. The predictions from the various models are compared with experimental measurements of the free-surface profiles. The principal aim of this study is the establishment and assessment of certain numerical and asymptotic models for the description of a class of free-surface flows, exemplified by the particular case of flow over a perturbed inclined plane. The laboratory experiments were made over a range of flow rates such that the Reynolds number, based on the volume flux per unit width and the kinematical viscosity of the fluid, ranged between 0.369 and 36.6. It was found that, at the smaller Reynolds numbers, a standard lubrication approximation provided a very good representation of the experimental measurements but, as the flow rate was increased, the standard model did not capture several important features of the flow. On the other hand, a lubrication approximation allowing for surface tension and inertial effects expanded the range of applicability of the basic theory by almost an order of magnitude, up to Reynolds numbers approaching 10. At larger flow rates, numerical solutions to the full equations of motion provided a description of the experimental results to within about 4% , up to a Reynolds number of 25, beyond which we were unable to obtain numerical solutions. It is not known why numerical solutions were not possible at larger flow rates, but it is possible that there is a bifurcation of the Navier-Stokes equations to a branch of unsteady motions near a Reynolds number of 25.

Capsular flow in pipelines

1972 ◽  
Vol 56 (1) ◽  
pp. 49-59 ◽  
Author(s):  
A. E. Vardy ◽  
M. I. G. Bloor ◽  
J. A. Fox
Keyword(s):  
Reynolds Number ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Steady Motion ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Hydraulic Pressure ◽  
Central Difference ◽  
Navier Stokes Equations

The problem considered is that of the steady motion of a series of neutrally buoyant, flat-faced, rigid, cylindrical capsules along the axis of a pipeline under the influence of a hydraulic pressure gradient. The Navier-Stokes equations are non-dimensionalized and expressed in central-difference form. Numerical solutions are found by the method of relaxation for Reynolds numbers up to 20 000 and a close agreement is obtained with readings from a laboratory apparatus for Reynolds numbers up to 2200.The flow is examined in detail and the existence of toroidal vortices between successive capsules is demonstrated. Their shape is shown to be increasingly influenced by inertial forces as the Reynolds number increases, but the overall pressure gradient is not greatly dependent on the Reynolds number.

Numerical Solutions of the Navier-Stokes Equations in Inlet Regions

1972 ◽  
Vol 39 (4) ◽  
pp. 873-878 ◽  
Author(s):  
J. W. McDonald ◽  
V. E. Denny ◽  
A. F. Mills
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Improved Method ◽  
Navier Stokes Equations ◽  
Two Dimensional Flow ◽  
Vorticity Equations

Numerical solutions of the Navier-Stokes equations are obtained for steady two-dimensional flow in the inlet region of both a tube and a channel. The entering flow is considered to be either uniform (u = constant, v = 0) or irrotational (u = constant, ω = 0). Values of Reynolds number Re = u0a/ν range from 0.75 to 2 × 106. An improved method for solving the stream function-vorticity equations of hydrodynamics has been developed. The method is stable at all Reynolds numbers and appears to be computationally superior to previous methods.

Low Reynolds-Number Flow in the Vicinity of Axisymmetric Constrictions

1979 ◽  
Vol 21 (2) ◽  
pp. 73-84 ◽  
Author(s):  
N. S. Vlachos ◽  
J. H. Whitelaw
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
Local Shear ◽  
Number Flow ◽  
Recirculation Length

Numerical solutions of the two-dimensional, Navier-Stokes equations are presented for boundary conditions corresponding to the laminar flow of Newtonian and non-Newtonian fluids in a round tube with axisymmetric constrictions. The influence of Reynolds number, blockage diameter ratio and length on the velocity components, streamlines, local shear stress and pressure drop are quantified and, in the case of the first two, shown to be large. The non-Newtonian stress-strain relationship corresponds to that for blood flowing in venules and results in an increased recirculation length and larger regions of high shear stress.

Low Reynolds-Number Pipe Flow in the Vicinity of Three-Dimensional Obstacles

1979 ◽  
Vol 21 (5) ◽  
pp. 335-343 ◽  
Author(s):  
A. D. Gosman ◽  
N. S. Vlachos ◽  
J. H. Whitelaw
Keyword(s):  
Reynolds Number ◽  
Turbulent Flows ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Recirculating Flow ◽  
Local Shear ◽  
Sector Angle

Numerical solutions of the three-dimensional Navier-Stokes equations are presented for boundary conditions corresponding to the laminar flow of Newtonian and non-Newtonian fluids in a round pipe with truncated sector-shaped obstacles. The influences of Reynolds number and sector angle on the velocity distributions, local shear stress and pressure drop are quantified and shown to be large. The results are complementary to those previously reported by Vlachos and Whitelaw (1)§ for axisymmetric obstacles, where related two-dimensional effects were quantified. They provide new information on three-dimensional, recirculating flow in ducts and form a basis for future calculations of corresponding turbulent flows.

Effects of Distance Between Pads on the Inlet Pressure Build-Up on Pad Bearings

2002 ◽  
Vol 124 (3) ◽  
pp. 506-514 ◽  
Author(s):  
Jong-Soo Kim ◽  
Kyung-Woong Kim
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Numerical Results ◽  
Loss Coefficient ◽  
Inlet Pressure ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Lubricating Film ◽  
Wide Range

Full Navier-Stokes equations are solved numerically for a cavity region between two consecutive pads and a parallel lubricating film. Numerical solutions are obtained for a wide range of Reynolds number and various values of a distance between pads. Numerical results show that the inlet pressure build-up is significantly affected by Reynolds number and the distance between two adjacent pads. A new formula is derived of loss coefficient with Reynolds number and a distance factor, for using it in an extended Bernoulli equation, on the basis of numerical results. Experiments are conducted to investigate the validity of the formula of loss coefficient proposed by authors.

A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure

1973 ◽  
Vol 59 (2) ◽  
pp. 391-396 ◽  
Author(s):  
N. C. Freeman ◽  
S. Kumar
Keyword(s):  
Shock Wave ◽  
Reynolds Number ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Low Pressure ◽  
Navier Stokes ◽  
Spherically Symmetric ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Type Flow

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.

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.

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