scholarly journals On the Stability of Convection in a Non-Newtonian Vertical Fluid Layer in the Presence of Gold Nanoparticles: Drug Agent for Thermotherapy

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1302
Author(s):  
Khaled S. Mekheimer ◽  
Bangalore M. Shankar ◽  
Shaimaa F. Ramadan ◽  
Hosahalli E. Mallik ◽  
Mohamed S. Mohamed
Keyword(s):  
Gold Nanoparticles ◽  
Newtonian Fluid ◽  
Fluid Layer ◽  
Critical Parameters ◽  
Second Grade ◽  
Brownian Diffusion ◽  
Complex Eigenvalue ◽  
Neutral Stability ◽  
Eighth Order ◽  
The Stability

We consider the effect of gold nanoparticles on the stability properties of convection in a vertical fluid layer saturated by a Jeffreys fluid. The vertical boundaries are rigid and hold at uniform but different temperatures. Brownian diffusion and thermophoresis effects are considered. Due to numerous applications in the biomedical industry, such a study is essential. The linear stability is investigated through the normal mode disturbances. The resulting stability problem is an eighth-order ordinary differential complex eigenvalue problem that is solved numerically using the Chebyshev collection method. Its solution provides the neutral stability curves, defining the threshold of linear instability, and the critical parameters at the onset of instability are determined for various values of control parameters. The results for Newtonian fluid and second-grade fluid are delineated as particular cases from the present study. It is shown that the Newtonian fluid has a more stabilizing effect than the second-grade and the Jeffreys fluids in the presence of gold nanoparticles and, Jeffreys fluid is the least stable.

Instability of a gravity-modulated fluid layer with surface tension variation

2001 ◽  
Vol 434 ◽  
pp. 243-271 ◽  
Author(s):  
J. RAYMOND LEE SKARDA
Keyword(s):  
Surface Tension ◽  
Fluid Layer ◽  
Modulation Frequency ◽  
Galerkin Approximation ◽  
Neutral Stability ◽  
Modulation Amplitude ◽  
Neutral Stability Curve ◽  
Stability Curve ◽  
The Stability ◽  
Rayleigh Benard

Gravity modulation of an unbounded fluid layer with surface tension variations along its free surface is investigated. The stability of such systems is often characterized in terms of the wavenumber, α and the Marangoni number, Ma. In (α, Ma) parameter space, modulation has a destabilizing effect on the unmodulated neutral stability curve for large Prandtl number, Pr, and small modulation frequency, Ω, while a stabilizing effect is observed for small Pr and large Ω. As Ω → ∞ the modulated neutral stability curves approach the unmodulated neutral stability curve. At certain values of Pr and Ω, multiple minima are observed and the neutral stability curves become highly distorted. Closed regions of subharmonic instability are also observed. In (1/Ω, g1Ra)-space, where g1 is the relative modulation amplitude, and Ra is the Rayleigh number, alternating regions of synchronous and subharmonic instability separated by thin stable regions are observed. However, fundamental differences between the stability boundaries occur when comparing the modulated Marangoni–Bénard and Rayleigh–Bénard problems. Modulation amplitudes at which instability tongues occur are strongly influenced by Pr, while the fundamental instability region is weakly affected by Pr. For large modulation frequency and small amplitude, empirical relations are derived to determine modulation effects. A one-term Galerkin approximation was also used to reduce the modulated Marangoni–Bénard problem to a Mathieu equation, allowing qualitative stability behaviour to be deduced from existing tables or charts, such as Strutt diagrams. In addition, this reduces the parameter dependence of the problem from seven transport parameters to three Mathieu parameters, analogous to parameter reductions of previous modulated Rayleigh–Bénard studies. Simple stability criteria, valid for small parameter values (amplitude and damping coefficients), were obtained from the one-term equations using classical method of averaging results.

scholarly journals Structure and Branching of Unstable Modes in a Swirling Flow

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 99
Author(s):  
Vadim Akhmetov
Keyword(s):  
Vortex Breakdown ◽  
Swirling Flow ◽  
Maximum Growth ◽  
Critical Parameters ◽  
Return Flow ◽  
Neutral Stability ◽  
Wave Disturbances ◽  
Branching Points ◽  
Hydrodynamic Stability Theory ◽  
The Stability

Swirling has a significant effect on the main characteristics of flow and can lead to its fundamental restructuring. On the flow axis, a stagnation point with zero velocity is possible, behind which a return flow zone is formed. The apparent instability leads to the formation of secondary vortex motions and can also be the cause of vortex breakdown. In the paper, a swirling flow with a velocity profile of the Batchelor vortex type has been studied on the basis of the linear hydrodynamic stability theory. An effective numerical method for solving the spectral problem has been developed. This method includes the asymptotic solutions at artificial and irregular singular points. The stability of flows was considered for the values of the Reynolds number in the range 10≤Re≤5×106. The calculations were carried out for the value of the azimuthal wavenumber parameter n=−1. As a result of the analysis of the solutions, the existence of up to eight simultaneously occurring unstable modes has been shown. The paper presents a classification of the detected modes. The critical parameters are calculated for each mode. For fixed values of the Reynolds numbers 60≤Re≤5000, the curves of neutral stability are plotted. Branching points of unstable modes are found. The maximum growth rates for each mode are determined. A new viscous instability mode is found. The performed calculations reveal the instability of the Batchelor vortex at large values of the swirl parameter for long-wave disturbances.

scholarly journals An Arnoldi-Frontal Approach for the Stability Analysis of Flows in a Collapsible Channel

2016 ◽  
Vol 08 (06) ◽  
pp. 1650073
Author(s):  
Yujue Hao ◽  
Zongxi Cai ◽  
Steven Roper ◽  
Xiaoyu Luo
Keyword(s):  
Finite Element ◽  
Stability Analysis ◽  
Krylov Subspace ◽  
Eigenvalue Problems ◽  
Complex Eigenvalue ◽  
Neutral Stability ◽  
Arnoldi Method ◽  
New Approach ◽  
Generalized Complex ◽  
The Stability

In this paper, we present a new approach based on a combination of the Arnoldi and frontal methods for solving large sparse asymmetric and generalized complex eigenvalue problems. The new eigensolver seeks the most unstable eigensolution in the Krylov subspace and makes use of the efficiency of the frontal solver developed for the finite element methods. The approach is used for a stability analysis of flows in a collapsible channel and is found to significantly improve the computational efficiency compared to the traditionally used QZ solver or a standard Arnoldi method. With the new approach, we are able to validate the previous results obtained either on a much coarser mesh or estimated from unsteady simulations. New neutral stability solutions of the system have been obtained which are beyond the limits of previously used methods.

scholarly journals Elastohydrodynamical instabilities of active filaments, arrays and carpets analyzed using slender body theory

2020 ◽  
Author(s):  
Ashok S. Sangani ◽  
Arvind Gopinath
Keyword(s):  
Critical Force ◽  
Slender Body ◽  
Critical Parameters ◽  
Neutral Stability ◽  
Slender Body Theory ◽  
Follower Forces ◽  
Freely Suspended ◽  
Filament Spacing ◽  
The Stability ◽  
Body Theory

ABSTRACTThe rhythmic motions and wave-like planar oscillations in filamentous soft structures are ubiquitous in biology. Inspired by these, recent work has focused on the creation of synthetic colloid-based active mimics that can be used to move, transport cargo, and generate fluid flows. Underlying the functionality of these mimics is the coupling between elasticity, geometry, dissipation due to the fluid, and active force or moment generated by the system. Here, we use slender body theory to analyze the linear stability of a subset of these - active elastic filaments, filament arrays and filament carpets - animated by follower forces. Follower forces can be external or internal forces that always act along the filament contour. The application of slender body theory enables the accurate inclusion of hydrodynamic effects, screening due to boundaries, and interactions between filaments. We first study the stability of fixed and freely suspended sphere-filament assemblies, calculate neutral stability curves separating stable oscillatory states from stable straight states, and quantify the frequency of emergent oscillations. When shadowing effects due to the physical presence of the spherical boundary are taken into account, the results from the slender body theory differ from that obtained using local resistivity theory. Next, we examine the onset of instabilities in a small cluster of filaments attached to a wall and examine how the critical force for onset of instability and the frequency of sustained oscillations depend on the number of filaments and the spacing between the filaments. Our results emphasize the role of hydrodynamic interactions in driving the system towards perfectly in-phase or perfectly out of phase responses depending on the nature of the instability. Specifically, the first bifurcation corresponds to filaments oscillating in-phase with each other. We then extend our analysis to filamentous (line) array and (square) carpets of filaments and investigate the variation of the critical parameters for the onset of oscillations and the frequency of oscillations on the inter-filament spacing. The square carpet also produces a uniform flow at infinity and we determine the ratio of the mean-squared flow at infinity to the energy input by active forces. We conclude by analyzing the bending and buckling instabilities of a straight passive filament attached to a wall and placed in a viscous stagnant flow - a problem related to the growth of biofilms, and also to mechanosensing in passive cilia and microvilli. Taken together, our results provide the foundation for more detailed non-linear analyses of spatiotemporal patterns in active filament systems.

Exact and approximate solutions to the double-diffusive Marangoni–Bénard problem with cross-diffusive terms

1998 ◽  
Vol 366 ◽  
pp. 109-133 ◽  
Author(s):  
J. R. L. SKARDA ◽  
D. JACQMIN ◽  
F. E. McCAUGHAN
Keyword(s):  
Exact Solutions ◽  
Fluid Layer ◽  
Galerkin Approximation ◽  
Relevant Parameter ◽  
Neutral Stability ◽  
Zero Gravity ◽  
Concentration Difference ◽  
Stability Behaviour ◽  
First Case ◽  
The Stability

We discuss the linear stability of a cross-doubly-diffusive fluid layer with surface tension variation along the free surface. Two limiting cases of the mass flux basic state are considered in the presence of non-zero Soret and Dufour diffusivities. The first case, which has remained largely unexplored, is one where a temperature difference, ΔT¯, and a concentration difference, ΔC¯, are both imposed across the layer. The second case, which has greater significance to thermosolutal systems, is that where the imposed ΔT¯ across the layer induces a ΔC¯. We rescale the problem in the absence of buoyancy, which leads to a more concise representation of neutral stability results in and near the limit of zero gravity. We obtain exact solutions for stationary stability in both cases. One important consequence of the exact solutions is the validation of recently published numerical results in the limit of zero gravity. Moreover, the precise location of asymptotes in relevant parameter (Smc, Mac) space are computed from exact solutions. Both numerical and exact solutions are used to further examine stability behaviour. We also derive algebraic expressions for stationary stability, oscillatory stability, frequency, and codimension two point from a one-term Galerkin approximation. The one-term solutions qualitatively reflect the stability behaviour of the system over the parameter ranges in our investigation. A practical consequence is that the nature of the stability (oscillatory or stationary) for a given set of parameter values can be determined approximately, without solving the numerical eigenvalue problem.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

2019 ◽  
Vol 14 (1) ◽  
pp. 52-58 ◽  
Author(s):  
A.D. Nizamova ◽  
V.N. Kireev ◽  
S.F. Urmancheev
Keyword(s):  
Reynolds Number ◽  
Spectral Characteristics ◽  
Wave Number ◽  
Critical Parameters ◽  
Fluid Viscosity ◽  
Flat Channel ◽  
Stability Equation ◽  
The Stability ◽  
Thermoviscous Fluid ◽  
Sommerfeld Equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

Systems with Condensates and Pairing

2018 ◽  
Author(s):  
Klaus Morawetz
Keyword(s):  
Critical Parameters ◽  
Einstein Condensation ◽  
Cooper Pairs ◽  
Pair Breaking ◽  
Systematic Theory ◽  
Bose Einstein Condensation ◽  
T Matrix ◽  
The Stability ◽  
Bose Einstein

The Bose–Einstein condensation and appearance of superfluidity and superconductivity are introduced from basic phenomena. A systematic theory based on the asymmetric expansion of chapter 11 is shown to correct the T-matrix from unphysical multiple-scattering events. The resulting generalised Soven scheme provides the Beliaev equations for Boson’s and the Nambu–Gorkov equations for fermions without the usage of anomalous and non-conserving propagators. This systematic theory allows calculating the fluctuations above and below the critical parameters. Gap equations and Bogoliubov–DeGennes equations are derived from this theory. Interacting Bose systems with finite temperatures are discussed with successively better approximations ranging from Bogoliubov and Popov up to corrected T-matrices. For superconductivity, the asymmetric theory leading to the corrected T-matrix allows for establishing the stability of the condensate and decides correctly about the pair-breaking mechanisms in contrast to conventional approaches. The relation between the correlated density from nonlocal kinetic theory and the density of Cooper pairs is shown.

scholarly journals On a degenerate parabolic equation with Newtonian fluid∼non-Newtonian fluid mixed type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Sujun Weng
Keyword(s):  
Parabolic Equation ◽  
Newtonian Fluid ◽  
Weak Solutions ◽  
Mixed Type ◽  
Degenerate Parabolic Equation ◽  
Type Equation ◽  
Existence Of Weak Solutions ◽  
Degenerate Parabolic ◽  
The Stability ◽  
Increasing Function

AbstractWe study the existence of weak solutions to a Newtonian fluid∼non-Newtonian fluid mixed-type equation $$ {u_{t}}= \operatorname{div} \bigl(b(x,t){ \bigl\vert {\nabla A(u)} \bigr\vert ^{p(x) - 2}}\nabla A(u)+\alpha (x,t)\nabla A(u) \bigr)+f(u,x,t). $$ u t = div ( b ( x , t ) | ∇ A ( u ) | p ( x ) − 2 ∇ A ( u ) + α ( x , t ) ∇ A ( u ) ) + f ( u , x , t ) . We assume that $A'(s)=a(s)\geq 0$ A ′ ( s ) = a ( s ) ≥ 0 , $A(s)$ A ( s ) is a strictly increasing function, $A(0)=0$ A ( 0 ) = 0 , $b(x,t)\geq 0$ b ( x , t ) ≥ 0 , and $\alpha (x,t)\geq 0$ α ( x , t ) ≥ 0 . If $$ b(x,t)=\alpha (x,t)=0,\quad (x,t)\in \partial \Omega \times [0,T], $$ b ( x , t ) = α ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × [ 0 , T ] , then we prove the stability of weak solutions without the boundary value condition.

An Operationally Optimized Seven Link Biped Robot Moving on Slopes

2013 ◽  
Author(s):  
Peiman Naseradinmousavi
Keyword(s):  
Simulated Annealing Algorithm ◽  
Nonlinear Modeling ◽  
Biped Robot ◽  
Gait Pattern ◽  
Critical Parameters ◽  
Modeling Process ◽  
Operational Optimization ◽  
Ankle Joints ◽  
Annealing Algorithm ◽  
The Stability

In this paper, we discuss operational optimization of a seven link biped robot using the well-known “Simulated Annealing” algorithm. Some critical parameters affecting the robot gait pattern are selected to be optimized reducing the total energy used. Nonlinear modeling process we published elsewhere is shown here for completeness. The trajectories of both the hip and ankle joints are used to plan the robot gait on slopes and undoubtedly those parameters would be the target ones for the optimization process. The results we obtained reveal considerable amounts of the energy saved for both the ascending and descending surfaces while keeping the robot stable. The stability criterion we utilized for both the modeling and then optimization is “Zero Moment Point”. A comparative study of human evolutionary gait and the operationally optimized robot is also presented.

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