The Effect of Wall Conduction on the Stability of a Fluid in a Right Circular Cylinder Heated From Below

1983 ◽  
Vol 105 (2) ◽  
pp. 255-260 ◽  
Author(s):  
J. C. Buell ◽  
I. Catton
Keyword(s):  
Critical Rayleigh Number ◽  
Slip Boundary Condition ◽  
Dimensionless Temperature ◽  
Axisymmetric Mode ◽  
Slip Boundary ◽  
Wall Conductivity ◽  
Stream Functions ◽  
Cylindrical Volume ◽  
The Stability ◽  
The Galerkin Method

The onset of natural convection in a cylindrical volume of fluid bounded above and below by rigid, perfectly conducting surfaces and laterally by a wall of arbitrary thermal conductivity is examined. The critical Rayleigh number (dimensionless temperature difference) is determined as a function of aspect (radius to height) ratio and wall conductivity. The first three asymmetric modes as well as the axisymmetric mode are considered. Two sets of stream functions are employed to represent a velocity field that satisfies the no-slip boundary condition on all surfaces and conservation of mass everywhere. The Galerkin method is then used to reduce the linearized perturbation equations to an eigenvalue problem. The results for perfectly insulating and conducting walls are compared with the work of Charlson and Sani[9].

Effect of Wall Conduction on the Stability of a Fluid in a Rectangular Region Heated from Below

1972 ◽  
Vol 94 (4) ◽  
pp. 446-452 ◽  
Author(s):  
Ivan Catton
Keyword(s):  
Lateral Wall ◽  
Rectangular Region ◽  
Slip Boundary Conditions ◽  
Slip Boundary ◽  
Aspect Ratios ◽  
Usual Manner ◽  
Vertical Walls ◽  
The Stability ◽  
The Galerkin Method ◽  
Good Agreement

The initiation of natural convection in a fluid confined above and below by rigid, perfectly conducting surfaces and laterally by vertical walls of arbitrary thermal conductivity which form a rectangle is examined. The linearized perturbation equations are obtained in the usual manner and reduced to an eigenvalue problem. The Rayleigh number is the eigenvalue and is a function of the lateral-wall conductance and horizontal plan form (aspect ratios). The problem associated with satisfying the no-slip boundary conditions on all surfaces is surmounted by using the Galerkin method. Results are compared with experiments and shown to be in good agreement.

On flow in weakly precessing cylinders: the general asymptotic solution

2012 ◽  
Vol 709 ◽  
pp. 610-621 ◽  
Author(s):  
Xinhao Liao ◽  
Keke Zhang
Keyword(s):  
Numerical Analysis ◽  
Asymptotic Solution ◽  
Symmetry Axis ◽  
Slip Boundary Condition ◽  
Frame Of Reference ◽  
Rotating Frame ◽  
Homogeneous Fluid ◽  
Ekman Number ◽  
Slip Boundary ◽  
The Galerkin Method

AbstractWe investigate, through both asymptotic and numerical analysis, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses slowly about a different axis that is fixed in space. After demonstrating that the inviscid approximation is always divergent even far away from resonance, we derive a general asymptotic solution for an asymptotically small Ekman number in the rotating frame of reference describing the weakly precessing flow that satisfies the no-slip boundary condition and that is valid at or near or away from resonance. Numerical analysis of the same problem using the Galerkin method in terms of a Chebyshev polynomial expansion is also carried out, showing satisfactory agreement between the general asymptotic solution and the corresponding numerical solution at or near or away from resonance.

scholarly journals On the Numerical Simulation of HPDEs Using θ-Weighted Scheme and the Galerkin Method

Mathematics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 78
Author(s):  
Haifa Bin Jebreen ◽  
Fairouz Tchier
Keyword(s):  
Differential Equation ◽  
Galerkin Method ◽  
Linear Ordinary Differential Equation ◽  
Time Interval ◽  
Hyperbolic Partial Differential Equation ◽  
Time Step ◽  
Numerical Examples ◽  
One Dimensional ◽  
The Stability ◽  
The Galerkin Method

Herein, an efficient algorithm is proposed to solve a one-dimensional hyperbolic partial differential equation. To reach an approximate solution, we employ the θ-weighted scheme to discretize the time interval into a finite number of time steps. In each step, we have a linear ordinary differential equation. Applying the Galerkin method based on interpolating scaling functions, we can solve this ODE. Therefore, in each time step, the solution can be found as a continuous function. Stability, consistency, and convergence of the proposed method are investigated. Several numerical examples are devoted to show the accuracy and efficiency of the method and guarantee the validity of the stability, consistency, and convergence analysis.

scholarly journals The appearance of boundary layers and drift flows due to high-frequency surface waves

2012 ◽  
Vol 707 ◽  
pp. 482-495 ◽  
Author(s):  
Ofer Manor ◽  
Leslie Y. Yeo ◽  
James R. Friend
Keyword(s):  
Boundary Layer ◽  
Surface Waves ◽  
High Frequency ◽  
Slip Boundary Condition ◽  
Inertial Effects ◽  
Velocity Amplitude ◽  
Slip Boundary ◽  
Long Period ◽  
Bulk Waves ◽  
Schlichting Streaming

AbstractThe classical Schlichting boundary layer theory is extended to account for the excitation of generalized surface waves in the frequency and velocity amplitude range commonly used in microfluidic applications, including Rayleigh and Sezawa surface waves and Lamb, flexural and surface-skimming bulk waves. These waves possess longitudinal and transverse displacements of similar magnitude along the boundary, often spatiotemporally out of phase, giving rise to a periodic flow shown to consist of a superposition of classical Schlichting streaming and uniaxial flow that have no net influence on the flow over a long period of time. Correcting the velocity field for weak but significant inertial effects results in a non-vanishing steady component, a drift flow, itself sensitive to both the amplitude and phase (prograde or retrograde) of the surface acoustic wave propagating along the boundary. We validate the proposed theory with experimental observations of colloidal pattern assembly in microchannels filled with dilute particle suspensions to show the complexity of the boundary layer, and suggest an asymptotic slip boundary condition for bulk flow in microfluidic applications that are actuated by surface waves.

NEW MODELS IN MICROPOLAR FLUID AND THEIR APPLICATION TO LUBRICATION

2005 ◽  
Vol 15 (03) ◽  
pp. 343-374 ◽  
Author(s):  
GUY BAYADA ◽  
NADIA BENHABOUCHA ◽  
MICHÈLE CHAMBAT
Keyword(s):  
Boundary Condition ◽  
Friction Coefficient ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Micropolar Fluid ◽  
Reynolds Equation ◽  
Slip Boundary Condition ◽  
Slip Boundary ◽  
New Models ◽  
Uniqueness Of The Solution

A thin micropolar fluid with new boundary conditions at the fluid-solid interface, linking the velocity and the microrotation by introducing a so-called "boundary viscosity" is presented. The existence and uniqueness of the solution is proved and, by way of asymptotic analysis, a generalized micropolar Reynolds equation is derived. Numerical results show the influence of the new boundary conditions for the load and the friction coefficient. Comparisons are made with other works retaining a no slip boundary condition.

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