scholarly journals Instability thresholds for thermal convection in a Kelvin–Voigt fluid of variable order

2021 ◽  
Author(s):  
B. Straughan
Keyword(s):  
Thermal Convection ◽  
Navier Stokes ◽  
Memory Term ◽  
Variable Order ◽  
Fading Memory ◽  
Oscillatory Convection ◽  
Numerical Techniques ◽  
Order Zero ◽  
Neutral Curves ◽  
Voigt Fluid

AbstractWe present numerical techniques for calculating instability thresholds in a model for thermal convection in a complex viscoelastic fluid of Kelvin–Voigt type. The theory presented is valid for various orders of an exponential fading memory term, and the strategy for obtaining the neutral curves and instability thresholds is discussed in the general case. Specific numerical results are presented for a fluid of order zero, also known as a Navier–Stokes–Voigt fluid, and fluids of order 1 and 2. For the latter cases it is shown that oscillatory convection may occur, and the nature of the stationary and oscillatory convection branches is investigated in detail, including where the transition from one to the other takes place.

scholarly journals Thermosolutal Convection with a Navier–Stokes–Voigt Fluid

2020 ◽  
Author(s):  
Brian Straughan
Keyword(s):  
Viscoelastic Fluid ◽  
Nonlinear Stability ◽  
Thermal Convection ◽  
Stability Limit ◽  
Thermosolutal Convection ◽  
Navier Stokes ◽  
Critical Instability ◽  
Convection Problem ◽  
Voigt Fluid ◽  
Voigt Parameter

Abstract We present a model for convection in a Navier–Stokes–Voigt fluid when the layer is heated from below and simultaneously salted from below, the thermosolutal convection problem. Instability thresholds are calculated for thermal convection with a dissolved salt field in a complex viscoelastic fluid of Navier–Stokes–Voigt type. The Kelvin–Voigt parameter is seen to play a very important role in acting as a stabilizing agent when the convection is of oscillatory type. The quantitative size of this effect is displayed. Nonlinear stability is also discussed, and it is briefly indicated how the global nonlinear stability limit may be increased, although there still remains a region of potential sub-critical instability, especially when the Kelvin–Voigt parameter increases.

Local solvability of an inverse problem to the Navier–Stokes equation with memory term

2020 ◽  
Vol 36 (6) ◽  
pp. 065007
Author(s):  
Yu Jiang ◽  
Jishan Fan ◽  
Sei Nagayasu ◽  
Gen Nakamura
Keyword(s):  
Inverse Problem ◽  
Stokes Equation ◽  
Local Solvability ◽  
Navier Stokes ◽  
Memory Term ◽  
Navier Stokes Equation ◽  
With Memory

scholarly journals Reflections on dissipation associated with thermal convection

2013 ◽  
Vol 725 ◽  
Author(s):  
Thierry Alboussière ◽  
Yanick Ricard
Keyword(s):  
Heat Capacity ◽  
Steady State ◽  
Energy Dissipation ◽  
Thermal Convection ◽  
Constant Pressure ◽  
Reference State ◽  
Navier Stokes ◽  
Volume Integral ◽  
Anelastic Equations ◽  
Buoyancy Driven Convection

AbstractBuoyancy-driven convection is modelled using the Navier–Stokes and entropy equations. It is first shown that the coefficient of heat capacity at constant pressure, ${c}_{p} $, must in general depend explicitly on pressure (i.e. is not a function of temperature alone) in order to resolve a dissipation inconsistency. It is shown that energy dissipation in a statistically steady state is the time-averaged volume integral of $- \mathrm{D} P/ \mathrm{D} t$ and not that of $- \alpha T(\mathrm{D} P/ \mathrm{D} t)$. Secondly, in the framework of the anelastic equations derived with respect to the adiabatic reference state, we obtain a condition when the anelastic liquid approximation can be made, $\gamma - 1\ll 1$, independent of the dissipation number.

HOMOGENIZATION AND TWO-SCALE CONVERGENCE FOR A STOKES OR NAVIER-STOKES FLOW IN AN ELASTIC THIN POROUS MEDIUM

1996 ◽  
Vol 06 (07) ◽  
pp. 941-955 ◽  
Author(s):  
IOANA-ANDREEA ENE ◽  
JEANNINE SAINT JEAN PAULIN
Keyword(s):  
Stokes Flow ◽  
Viscoelastic Medium ◽  
General Framework ◽  
Homogenization Method ◽  
The Other ◽  
Navier Stokes ◽  
Fading Memory ◽  
Convergence Method ◽  
Solid Part ◽  
The One

In the general framework of the homogenization method we study the behavior of a thin elastic periodic structure immersed in a viscous fluid. After the proof of the convergence of the homogenization process by using the two-scale convergence method it is possible to take the limit as δ→0, the small parameter which represents the thickness of the solid part. We obtain a viscoelastic medium with fading memory. We consider two cases: the one in which we have a Stokes flow in the fluid part and the other one when we have a Navier-Stokes flow in the fluid part.

AMPLITUDE EQUATION IN POLYMERIC FLUID CONVECTION

1994 ◽  
Vol 04 (05) ◽  
pp. 1347-1351 ◽  
Author(s):  
J. MARTÍNEZ-MARDONES ◽  
R. TIEMANN ◽  
W. ZELLER ◽  
C. PÉREZ-GARCÍA
Keyword(s):  
Nonlinear Analysis ◽  
Standing Waves ◽  
Amplitude Equation ◽  
Oscillatory Convection ◽  
Amplitude Equations ◽  
Numerical Techniques ◽  
Polymeric Fluids ◽  
Polymeric Fluid ◽  
Convective Instabilities ◽  
Fluid Convection

The convective instabilities in viscoelastic polymeric Oldroyd-B models are studied. First, the nonlinear analysis of the stationary and oscillatory convection is carried out. Then, in the scope of weak nonlinear analysis, the coefficients of the amplitude equations are evaluated, in order to be in condition to estimate the possible behavior of stationary patterns and also travelling and standing waves. The values of these coefficients are determined by means of analytical and numerical techniques for convection in polymeric fluids.

scholarly journals A class of exact Navier–Stokes solutions for homogeneous flat-plate boundary layers and their linear stability

2014 ◽  
Vol 752 ◽  
pp. 462-484 ◽  
Author(s):  
Michael O. John ◽  
Dominik Obrist ◽  
Leonhard Kleiser
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Linear Stability ◽  
Boundary Layers ◽  
Stokes Equations ◽  
Plate Boundary ◽  
Navier Stokes ◽  
Transition Prediction ◽  
Neutral Curves ◽  
Tollmien Schlichting

AbstractWe introduce a new boundary layer formalism on the basis of which a class of exact solutions to the Navier–Stokes equations is derived. These solutions describe laminar boundary layer flows past a flat plate under the assumption of one homogeneous direction, such as the classical swept Hiemenz boundary layer (SHBL), the asymptotic suction boundary layer (ASBL) and the oblique impingement boundary layer. The linear stability of these new solutions is investigated, uncovering new results for the SHBL and the ASBL. Previously, each of these flows had been described with its own formalism and coordinate system, such that the solutions could not be transformed into each other. Using a new compound formalism, we are able to show that the ASBL is the physical limit of the SHBL with wall suction when the chordwise velocity component vanishes while the homogeneous sweep velocity is maintained. A corresponding non-dimensionalization is proposed, which allows conversion of the new Reynolds number definition to the classical ones. Linear stability analysis for the new class of solutions reveals a compound neutral surface which contains the classical neutral curves of the SHBL and the ASBL. It is shown that the linearly most unstable Görtler–Hämmerlin modes of the SHBL smoothly transform into Tollmien–Schlichting modes as the chordwise velocity vanishes. These results are useful for transition prediction of the attachment-line instability, especially concerning the use of suction to stabilize boundary layers of swept-wing aircraft.

scholarly journals A SEMI-IMPLICIT SCHEME FOR SOLVING INCOMPRESSIBLE VISCOUS FREE SURFACE FLOWS

2005 ◽  
Vol 4 (2) ◽  
Author(s):  
C. M. Oishi ◽  
J. A. Cuminato ◽  
V. G. Ferreira ◽  
M. F. Tomé ◽  
A. Castelo ◽  
...  
Keyword(s):  
Free Surface ◽  
Numerical Method ◽  
Stokes Equations ◽  
Free Surface Flows ◽  
Numerical Results ◽  
Navier Stokes ◽  
Variable Order ◽  
Navier Stokes Equations ◽  
Reynolds Number Flow ◽  
The Stability

The present work is concerned with a numerical method for solving the two-dimensional time-dependent incompressible Navier-Stokes equations in the primitive variables formulation. The diffusive terms are treated by Implicit Backward and Crank-Nicolson methods, and the non-linear convection terms are, explicitly, approximated by the high order upwind VONOS (Variable-Order Non-Oscillatory Scheme) scheme. The boundary conditions for the pressure field at the free surface are treated implicitly, and for the velocity field explicitly. The numerical method is then applied to the simulation of free surface and confined flows. The numerical results show that the present technique eliminates the stability restriction in the original explicit method. For low Reynolds number flow dynamics, the method is robust and produces numerical results that compare very well with the analytical solutions.

Modelling the influence of wall roughness on heat transfer in thermal convection

2011 ◽  
Vol 686 ◽  
pp. 568-582 ◽  
Author(s):  
Olga Shishkina ◽  
Claus Wagner
Keyword(s):  
Heat Transfer ◽  
Heat Transport ◽  
Thermal Convection ◽  
Navier Stokes ◽  
Wall Roughness ◽  
Two Dimensional ◽  
Boundary Layer Equations ◽  
Aspect Ratios ◽  
Blasius Boundary Layer ◽  
Rayleigh Numbers

AbstractThe objective of this study is to approximate heat transport in thermal convection enhanced by the roughness of heated/cooled horizontal plates. The roughness is introduced by a set of rectangular heated/cooled obstacles located at the corresponding plates. An analytical model to estimate the Nusselt number deviations caused by the wall roughness is developed. It is based on the two-dimensional Prandtl–Blasius boundary layer equations and therefore is valid for moderate Rayleigh numbers and regular wall roughness, for which the height of the obstacles and the distances between them are significantly larger than the thickness of the thermal boundary layers. To validate this model, the transport of heat and momentum in rectangular convection cells is studied in two-dimensional Navier–Stokes simulations, for different aspect ratios of the obstacles. It is found that the model predicts the heat transport with errors ${\leq }6\hspace{0.167em} \% $ for all investigated cases.

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