Linear Stability of a Convective Flow in an Annulus With a Nonlinear Heat Source

Iltiņš, Ilmārs; Iltiņa, Marija; Kolyshkin, Andrei A.; Koliskina, Valentina

doi:10.17654/hm018020315

Cited by 5 publications

(9 citation statements)

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“…Consider the boundary value problem ( 7), ( 9). Let T 0 (x) be a solution of (7), (9). Then, T ′′ 0 (x) = −F e T 0 (x) < 0 for every x ∈ [−1, 1] and thus T 0 (x) is a strictly concave function in the interval [−1, 1] and T ′ 0 (1) < 0.…”

Section: Solution Of the Nonlinear Boundary Value Problemmentioning

confidence: 99%

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On the Stability of a Steady Convective Flow in a Vertical Layer of a Chemically Reacting Fluid

Gritsans¹,

Koliskina²,

Kolyshkin³

et al. 2021

9th Edition of the International Conference on Computational Methods for Coupled Problems in Science and Engineering

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Linear stability of a steady flow of a chemically reacting fluid located in a vertical fluid layer bounded by two infinite parallel planes is investigated. Steady convective flow in the vertical direction is initiated due to the combined effect of internal heat generation and the temperature difference between the planes. Imposing small perturbations on the base flow, linearizing equations of thermal convection under the Boussinesq approximation in the neighbourhood of the base flow and using the method of normal modes we obtain an eigenvalue problem for a system of ordinary differential equations. Collocation method is used to discretize the problem. Numerical calculations are performed in Matlab. Fluid velocity, pressure, and temperature are the solutions of a nonlinear boundary value problem. Properties of the nonlinear boundary value problem for the base flow are investigated numerically using bifurcation analysis. It is shown that both the temperature difference between the planes and intensity of internal heat generation have a destabilizing influence on the base flow. The intensity of heat transfer in the direction perpendicular to the main flow can promote instability and leads to more intensive mixing. This fact can be used in design of bioreactors for biomass thermal conversion.

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Section: Solution Of the Nonlinear Boundary Value Problemmentioning

confidence: 99%

“…If T 0 (x) is defined by ( 14), then the equation T 0 (−1) = 1 defines a bifurcation curve Γ in the positive quadrant of the (F, β)-plane which determine all solutions of the boundary value problem (7), (9). By calculations, T 0 (−1) = 1 if and only if G(F, β) = 0, where…”

Section: Solution Of the Nonlinear Boundary Value Problemmentioning

confidence: 99%

On the Stability of a Steady Convective Flow in a Vertical Layer of a Chemically Reacting Fluid

Gritsans¹,

Koliskina²,

Kolyshkin³

et al. 2021

9th Edition of the International Conference on Computational Methods for Coupled Problems in Science and Engineering

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“…where E is the activation energy, R is the universal gas constant, Q 0 is a constant and T is the absolute temperature. Models with internal heat generation given by (1) are used in practice in order to describe processes during biomass thermal conversion [20,21,23] with the objective to obtain cleaner and more efficient sources of energy. Instability is a desirable phenomenon in this case since it enhances fluid mixing, which, in turn, leads to more efficient energy conversion.…”

Section: Mathematical Formulation Of the Problemmentioning

confidence: 99%

“…Linear stability of a steady convective flow in an annulus generated by non-homogeneous heat sources is studied in [20]. The effect of nonlinear internal heat sources on the stability boundary of a vertical flow in an annulus is investigated in [21] for both asymmetric and axisymmetric perturbations for wide range of radius ratios, Prandtl numbers and Frank-Kamenetskii parameters (proportional to the intensity of a chemical reaction). Such models are used to investigate processes in biomass thermal conversion.…”

Section: Introductionmentioning

confidence: 99%

Linear Stability of a Steady Convective Flow between Permeable Cylinders

Zigunovs

Kolyshkin

Iltiņš

2021

Fluids

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Linear stability analysis of a steady convective flow in a tall vertical annulus caused by nonlinear heat sources is conducted in the paper. Heat sources are generated as a result of a chemical reaction. The effect of radial cross-flow through permeable porous walls of the annulus is analyzed. The problem is relevant to biomass thermal conversion. The base flow solution is obtained by solving nonlinear boundary value problem. Linear stability analysis is performed, using collocation method. The calculations show that radial inward or outward flow has a stabilizing effect on the flow, while the increase in the Frank–Kamenetskii parameter (proportional to the intensity of the chemical reaction) destabilizes the flow. The increase in the Reynolds number based on the radial velocity leads to the appearance of the second minimum on the marginal stability curves. The rate of increase in the critical Grashof number with respect to the Reynolds number is different for inward and outward radial flows.

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“…The problem is to detect the number of positive solutions to BVP and trace the change of this number under the influence of built-in parameters. For more details the interested reader can consult the papers [11], [5], [4]. In this paper we consider one of these problems.…”

Section: Introductionmentioning

confidence: 99%