A conjecture on the least stable mode for the energy stability of plane parallel flows

2019 ◽  
Vol 881 ◽  
pp. 794-814 ◽  
Author(s):  
Xiangming Xiong ◽  
Zhi-Min Chen
Keyword(s):  
Critical Reynolds Number ◽  
Normal Modes ◽  
Slip Boundary Condition ◽  
Energy Stability ◽  
Parallel Plates ◽  
Stable Mode ◽  
Slip Boundary ◽  
Parallel Flows ◽  
Parallel Shear Flows ◽  
Decreasing Functions

In the energy stability theory, the critical Reynolds number is usually defined as the minimum of the first positive eigenvalue $R_{1}$ of an eigenvalue equation for all wavenumber pairs $(\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD})$, where $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ are the streamwise and spanwise wavenumbers of the normal mode. We prove that $(\cos \unicode[STIX]{x1D703}\pm 1)R_{1}$ are decreasing functions of $\unicode[STIX]{x1D703}=\arctan (\unicode[STIX]{x1D6FD}/\unicode[STIX]{x1D6FC})$ for the parallel flows between no-slip or slip parallel plates with or without variations in temperature. Numerical results inspire us to conjecture that $R_{1}$ is also a decreasing function of $\unicode[STIX]{x1D703}$ for the parallel shear flows under the no-slip boundary condition and without variations in temperature. If the conjecture is correct, the least stable normal modes for the energy stability will be streamwise vortices for these base flows.

scholarly journals Tensorial hydrodynamic slip

2008 ◽  
Vol 613 ◽  
pp. 125-134 ◽  
Author(s):  
MARTIN Z. BAZANT ◽  
OLGA I. VINOGRADOVA
Keyword(s):  
Slip Boundary Condition ◽  
Parallel Plates ◽  
Textured Surfaces ◽  
Navier Slip ◽  
Slip Boundary ◽  
Hydrodynamic Slip ◽  
Simple Matrix ◽  
Local Slip ◽  
Pressure Driven Flow ◽  
Interfacial Mobility

We describe a tensorial generalization of the Navier slip boundary condition and illustrate its use in solving for flows around anisotropic textured surfaces. Tensorial slip can be derived from molecular or microstructural theories or simply postulated as a constitutive relation, subject to certain general constraints on the interfacial mobility. The power of the tensor formalism is to capture complicated effects of surface anisotropy, while preserving a simple fluid domain. This is demonstrated by exact solutions for laminar shear flow and pressure-driven flow between parallel plates of arbitrary and different textures. From such solutions, the effects of rotating a texture follow from simple matrix algebra. Our results may be useful for extracting local slip tensors from global measurements, such as the permeability of a textured channel or the force required to move a patterned surface, in experiments or simulations.

scholarly journals Application of Daftardar Jafari Method to First Grade MHD Squeezing Fluid Flow in a Porous Medium with Slip Boundary Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Inayat Ullah ◽  
M. T. Rahim ◽  
Hamid Khan
Keyword(s):  
Boundary Condition ◽  
Porous Medium ◽  
Velocity Profile ◽  
Nonlinear Boundary ◽  
Slip Boundary Condition ◽  
First Grade ◽  
Squeezing Flow ◽  
Parallel Plates ◽  
Governing Equations ◽  
Slip Boundary

In the present work, in the presence of magnetic field and with slip boundary condition, squeezing flow of a Newtonian fluid in a porous medium between two large parallel plates is investigated. The governing equations are transformed to a single nonlinear boundary value problem. Daftardar Jafari Method (DJM) is used to solve the problem in order to obtain the velocity profile of the fluid. By using residual of the problem, the validity of solution is established. The velocity profile is argued through graphs for various values of parameters.

Numerical study on the rotating electro-osmotic flow of third grade fluid with slip boundary condition

2020 ◽  
Vol 75 (7) ◽  
pp. 649-655
Author(s):  
Juan Song ◽  
Shaowei Wang ◽  
Moli Zhao ◽  
Ning Li
Keyword(s):  
Boundary Condition ◽  
Velocity Profile ◽  
Slip Boundary Condition ◽  
Third Grade ◽  
Parallel Plates ◽  
Third Grade Fluid ◽  
Osmotic Flow ◽  
Slip Boundary ◽  
Grade Fluid ◽  
Electro Osmotic Flow

AbstractConsidering the slip boundary condition, the rotating electro-osmotic flow of a third grade fluid in a channel formed by two parallel plates is investigated in the present study. The charge distribution is treated with the Debye–Hückel approximation analytically. Based on the finite difference method, the velocity profile for rotating electro-osmotic flow of third grade fluid is obtained numerically. It is shown that the non-Newtonian parameter of third grade fluid and the velocity slip factor play the important roles for the rotating electro-osmotic flow. The increasing non-Newtonian parameter slows down the flow and decreases the velocity magnitude, and the increasing slip parameter β has the similar influence on the velocity profile. Furthermore, the effect of the inclusion of third grade on the velocity profile is more conspicuous in the area near the walls.

Rarefied gas flow between parallel plates

1969 ◽  
Vol 66 (1) ◽  
pp. 189-196 ◽  
Author(s):  
M. M. R. Williams
Keyword(s):  
Boundary Condition ◽  
Integral Equation ◽  
Gas Flow ◽  
Rarefied Gas ◽  
Volumetric Flow Rate ◽  
Slip Boundary Condition ◽  
General Boundary Condition ◽  
Boltzmann Transport ◽  
Parallel Plates ◽  
Slip Boundary

AbstractThe flow of a rarefied gas between parallel plates has been studied via the linearized Boltzmann transport equation. Using a general boundary condition, which includes an arbitrary ratio of specular to diffuse reflection from the wall, we have derived an integral equation for the mass flow velocity. The integral equation is solved by using a replication property of the kernel and application of the method of Muskelishvili.The total volumetric flow rate is obtained and a slip boundary condition is deduced for use with the hydrodynamic equations.Certain aspects of the eigenvalue spectrum associated with the Boltzmann equation are discussed.

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.

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