On the Stability of Penetrative Convection in a Couple-Stress Fluid

Mahajan, Amit; Nandal, Reena

doi:10.1007/s40819-017-0324-6

Cited by 11 publications

(6 citation statements)

References 23 publications

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“…The form of this system of partial differential equations is exactly the same as what one finds from Navier-Stokes theory except that there is now present the higher derivative term −ν 2 v i . Devi and Mahajan [22] and Mahajan and Nandal [23] develop linear instability and nonlinear stability analyses for (9) and for an analogous system with a heat source, respectively. Both sets of writers prescribe boundary conditions on v i and T , but no discussion is included initially on further boundary conditions, a topic we return to in Sect.…”

Section: 3mentioning

confidence: 99%

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Thermal convection in a higher-gradient Navier–Stokes fluid

Straughan

2023

Eur. Phys. J. Plus

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We discuss models for flow in a class of generalized Navier–Stokes equations. The work concentrates on producing models for thermal convection, analysing these in detail, and deriving critical Rayleigh and wave numbers for the onset of convective fluid motion. In addition to linear instability theory we present a careful analysis of fully nonlinear stability theory. The theories analysed all possess a bi-Laplacian term in addition to the normal spatial derivative term. The theories discussed are Stokes couple stress theory, dipolar fluid theory, Green–Naghdi theory, Fried–Gurtin–Musesti theory, and a second theory of Fried and Gurtin. We show that the Stokes couple stress theory and the Fried–Gurtin–Musesti theory involve the same partial differential equations while those of Green–Naghdi and dipolar theory are similar. However, we concentrate on boundary conditions which are crucial to understand all five theories and their differences.

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Section: 3mentioning

confidence: 99%

“…These results are useful in our nonlinear stability analysis. One might proceed directly from (23) by multiplying by u i and integrating over to find…”

Section: Boundary Conditionsmentioning

confidence: 99%

Thermal convection in a higher-gradient Navier–Stokes fluid

Straughan

2023

Eur. Phys. J. Plus

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“…Now, we apply the transformation

θ = \frac{\hat{θ}}{\sqrt{λ_{1}}}, ϕ = \frac{\hat{ϕ}}{\sqrt{λ_{2}}}

into Equation (47) and use the calculus of variation (see Mahajan and Nandal 40 ), and then Euler–Lagrange equation takes to the following system of equations:

\begin{matrix} trueleft \\ 2 true[μ (z) \nabla^{2} q + (D μ (z)) true{true(\frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} true) i + true(\frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} true) j + 2 \frac{\partial w}{\partial z} k true} true] + R true(\frac{1 + λ_{1}}{\sqrt{λ_{1}}} true) θ k \\ - R s true(\frac{1 - H_{1} λ_{2}}{\sqrt{λ_{2}}} true) ϕ k = - Δ τ, \end{matrix}

2 \nabla^{2} θ infixitalic+ italicR ((), \frac{1 + λ_{1}}{\sqrt{λ_{1}}}) w + \sqrt{\frac{λ_{2}}{λ_{1}}} h ϕ = 0,

2 (\nabla^{2} - η) ϕ - R s ((), \frac{1 - H_{1} λ_{2}}{\sqrt{λ_{2}}}) w + \sqrt{\frac{λ_{2}}{λ_{1}}} h θ = 0,

where

τ

is the Lagrange multiplier. Applying the double curl in Equation (…”

Section: Stability Analysismentioning

confidence: 99%

Unconditional nonlinear stability for double‐diffusive convection with temperature‐ and pressure‐dependent viscosity

Mahajan

Tripathi

2020

Heat Trans

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A linear and nonlinear stability analyses are carried out for a double‐diffusive chemically reactive fluid layer with viscosity being a function of temperature and pressure. The linear stability analysis is studied when the stabilizing salt gradient acts against the destabilizing thermal gradient. The effect of reaction parameters and variable viscosity on the stability of the system is studied for heated below, salted above, and the heated and salted below models with Rigid–Rigid boundary conditions. Chebyshev pseudospectral method is applied to determine the numerical solutions.

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“…For analyzing the linear instability results, we return to the perturbed equations (7)-(10), neglecting the nonlinear terms. We again perform the standard stationary normal mode analysis and look for the solution of these equations in the form (38). The boundary conditions in the present case are same, i.e.…”

Section: Variational Problemmentioning

confidence: 99%

“…The driving force for many studies in double-diffusive or multicomponent convection is largely physical applications. The double-diffusive convection problems have been studied by many authors [13][14][29][30][31][32][33][34][35][36][37][38][39]. In porous media, an alternative to Darcy's equation is what is known as Brinkman's equation [28].…”

Section: Introductionmentioning

confidence: 99%

Global Stability For Double-Diffusive Convection In A Couple-Stress Fluid Saturating A Porous Medium

Choudhary

Sunil

2019

Studia Geotechnica Et Mechanica

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We show that the global non-linear stability threshold for convection in a double-diffusive couple-stress fluid saturating a porous medium is exactly the same as the linear instability boundary. The optimal result is important because it shows that linearized instability theory has captured completely the physics of the onset of convection. It is also found that couple-stress fluid saturating a porous medium is thermally more stable than the ordinary viscous fluid, and the effects of couple-stress parameter (F ) , solute gradient ( S f ) and Brinkman number ( D a ) on the onset of convection is also analyzed.

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On the Stability of Penetrative Convection in a Couple-Stress Fluid

Cited by 11 publications

References 23 publications

Thermal convection in a higher-gradient Navier–Stokes fluid

Thermal convection in a higher-gradient Navier–Stokes fluid

Unconditional nonlinear stability for double‐diffusive convection with temperature‐ and pressure‐dependent viscosity

Global Stability For Double-Diffusive Convection In A Couple-Stress Fluid Saturating A Porous Medium

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