An analytical solution of the Navier-Stokes equations for unsteady backward stagnation-point flow with injection or suction

Shapiro, Alan

doi:10.1002/zamm.200510241

Cited by 6 publications

(3 citation statements)

References 21 publications

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“…Simulations were performed using a commercial finite element analysis solver (COMSOL Multiphysics). Configuration of the cylindrical reactor, in which we solved the 2D Navier-Stokes equations in cylindrical coordinates [37], is demonstrated in figure 1. Due to the low Reynolds number in the subsonic part of the downstream (Re ∼ 300) the flow is laminar.…”

Section: Simulation Of the Plasma Expansionmentioning

confidence: 99%

Improved size distribution control of silicon nanocrystals in a spatially confined remote plasma

Doğan

Westerman

Sanden

2015

Plasma Sources Sci. Technol.

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This work demonstrates how to improve the size distribution of silicon nanocrystals (Si-NCs) synthesized in a remote plasma, in which the flow dynamics and the particular chemistry initially resulted in the formation of small (2-10 nm) and large (50-120 nm) Si-NCs. Plasma consists of two regions: an axially expanding central plasma beam and a background region around the expansion. Continuum fluid dynamics simulations demonstrate that a significant mass flow occurs from the central beam to the background region. This mass flow can be gradually reduced upon confinement of the central beam, preventing the mass transport to the background region. Transmission electron microscopy and Raman spectroscopy analyses demonstrate that the volume fraction of large Si-NCs decreases from ∼77% to below 45% in parallel with the decrease of mass flow to the background region upon confinement, which indicates that large Si-NCs are synthesized in the background and small Si-NCs are synthesized in the central beam. Spatially resolved ion flux analyses demonstrate that the ions are localized in the central beam despite the mass flow to the background, indicating that the formation of small Si-NCs is governed by ion-assisted growth while the formation of large Si-NCs is governed by radical-neutral-assisted growth in the absence of ions. According to these observations, a better uniformity in the size distribution of Si-NCs can be obtained by creating a more uniform plasma flow and controlling the density of plasma species in the plasma.

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Section: Simulation Of the Plasma Expansionmentioning

confidence: 99%

Improved size distribution control of silicon nanocrystals in a spatially confined remote plasma

Doğan

Westerman

Sanden

2015

Plasma Sources Sci. Technol.

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“…In some cases the unsteady problem leads to some nonlinear ODE as in Shapiro's paper [10] about an unsteady axisymmetric incompressible case with a decelerating backward stagnationpoint flow with uniform injection or suction from a porous boundary (plate). The author arrives at a third order nonlinear ODE which after some work is transformed into a Riccati (tractable) equation.…”

Section: Outline Of Some Contemporary Literaturementioning

confidence: 99%

Unsteady Roto-Translational Viscous Flow: analytical Solution to Navier-Stokes Equations in Cylindrical Geometry

Bocci¹,

Scarpello²,

Ritelli³

2018

JGSP

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We provide a integration of Navier-Stokes equations concerning the unsteady-state laminar flow of an incompressible, isothermal (newtonian) fluid in a cylindrical vessel spinning about its symmetry axis, say z, and inside which the liquid motion starts with an axial velocity component as well. Basic physical assumptions are that the pressure axial gradient keeps itself on its hydrostatic value and that no radial velocity exists. In such a way the Navier-Stokes PDEs become uncoupled and can be faced separately. We succeed in computing both the unsteady speed components, i.e. the axial vz and the circumferential v θ as well, by means of infinite series expansions of Fourier-Bessel type under time exponential damping. Following this, we also find the unsteady surfaces of dynamical equilibrium, the wall shear stress and the Stokes streamlines.

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“…Under the boundary conditions f η (∞, τ ) = 1, the value of C(τ ) should be a constant and equal to 1. If the boundary condition f η (∞, τ ) is restricted not to be a constant,t, following the procedure of [9], a particular time-dependence function C(τ ) may be expressed in the form…”

Section: B Particular Solutionmentioning

confidence: 99%

Unsteady Reversed Stagnation-Point Flow over a Flat Plate

Sin

Chio

2012

ISRN Applied Mathematics

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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.

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An analytical solution of the Navier-Stokes equations for unsteady backward stagnation-point flow with injection or suction

Cited by 6 publications

References 21 publications

Improved size distribution control of silicon nanocrystals in a spatially confined remote plasma

Improved size distribution control of silicon nanocrystals in a spatially confined remote plasma

Unsteady Roto-Translational Viscous Flow: analytical Solution to Navier-Stokes Equations in Cylindrical Geometry

Unsteady Reversed Stagnation-Point Flow over a Flat Plate

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