The transformational analytical solution for nonlinear convection in the presence of two-way rotation

• Autorzy: El-Kholy S. A. , Ismail I. A.
• Języki publikacji: EN

Abstrakty

EN

Steady two-dimensional finite amplitude solutions are obtained for the problem of convection in a horizontal fluid layer heated from below and rotated about its vertical axis. The flow is assumed to be laminar and two-dimensional. The density variation is taken into account by the Boussinesq approximation. Different boundaries with prescribed constant temperature are assumed and the solutions are obtained. The transform for summing the variables, which reduce the nonlinear partial differential equation into ordinary differential equation of the high order, is used. The existence of steady subcritical finite amplitude solutions is demonstrated for different Prandtl numbers. A strong reduction in the domain of stable rolls that occurs as the rotation rate is increasing. Convection driven by thermal buoyancy in the presence of the Coriolis force occurs in planetary atmospheres and interiors. Asymptotic expressions for the onset of convection in a horizontal fluid layer of finite extent heated from below and rotating about a vertical axis are derived in the limit of large rotation rates in the case of stress-free upper and lower boundaries. In the presence of the vertical sidewalls the critical Rayleigh number R is much lower than the classical value of the infinity extended layer.

Słowa kluczowe

EN

convection; rigid; stress-free; mixed boundaries; Boussinesq approximation; nonlinear; analytical solution; Navier-Stocks; PDE; two-way rotation

PL

konwekcja; aproksymacja; sztywność; nieliniowość; rozwiązanie analityczne; obieg dwukierunkowy

• Wydawca: Wydawnictwo Politechniki Łódzkiej
• Czasopismo: Mechanics and Mechanical Engineering
• Rocznik: 2004
• Tom: Vol. 7, nr 1
• Strony: 119--130
• Opis fizyczny: Bibliogr. 18 poz.

Twórcy

autor

El-Kholy S. A.

autor

Ismail I. A.

• Department of Mathematics Faculty of Science, Menoufia University, Shebin El-kom, Egypt

Bibliografia

• [1] Bolton, FW and Busse, FH: Stability of convection rolls in layers with stressfree boundary J. fluid mech. Vol. ISO, pp. 987-998, (1985). 2- .
• [2] Buell, JC and Catton, I: Effect of rotation on the stability of a bounded cylindrical layer of fluid heated from below, Phys. Fluids, (1983), 26, pp.892-896.
• [3] Busse, FH: Non-linear interaction of magnetic field and convection, 1. Fluid Mech., (1995), 71, pp.193-206.
• [4] Busse, FH and Clever, RM: Three-dimensional convection in the presence of strong vertical magnetic fields, Eur. J. Mech. B/Fluids., (1996), 15, (1), pp. 1-15.
• [5] Chandrasekhar, S: Hydrodynamic and Hydromagnetic Stability, (1961), Oxford, Clarendon Press.
• [6] Clever, RM and Busse, FH: Non-linear oscillatory convection in the presence of a vertical magnetic field, J. Fluid Mech., (1989), 201, pp. 507-523.
• [7] Derrick and Grossman: Introduction to Differential Equation with Boundary Value Problems, Third Edition, by University of Montana, New York, Los Angeles, San Francisco (1987).
• [8] Ecke, ER, Zhong, F and Knobloch, E: Hopf bifurcation with broken reflection symmetry in rotating Rayleigh-Benard convection, Europhys. Lett., (1992), 19, pp. 177-182.
• [9] Goldstein, HF, Knobloch, E, Mercader, I and Net, M: Conveaion in a rotating cylinder, Part I. linear theory for moderate Prandtl numbers, J. Fluid Mech., (1993).
• [10] Hadid, H, Henry, D and Kaddech, S: Numerical study of convection in the horizontal Bridgman configuration under the action of a constant magnetic field, Part I, J. Fluid Mech., (1997), 333, pp. 23.
• [11] Herrman, J and Busse, FH: Asymptotic theory of wall-attached convection in a rotating fluid layer, J. Fluid Mech., (1993), 255, pp. 183-194.
• [12] Holman, JP: Heat Transfer, (1996).
• [13] Lucas, PGJ, Pfotenhauer, JM and Donnelly, RJ: Stability and heat transfer of rotating cryogens, Part 1, Influence of rotation on the onset of convection in liquid He, (1983), J. Fluid Mech., 129, pp. 251-264.
• [14] Pfotenhauer, JM, Niemela, JJ and Donnelly, RJ: Stability and heat transfer of rotating cryogens. Part 3: Effects of finite cylindrical geometry and rotation on the onset of convection, (1985), J. Fluid Mech. 175, pp. 85-96.
• [15] Rossby, HT: A study of Benard convection with and without rotation, 1. Fluid Mech., (1969), 36, pp. 309-335.
• [16] Vasseur, P and Roillard, L: The Brinkman model for natural convection in a porous layer - effect of non-uniform thermal quarclient, Int J. Heat Mass Transfer, (1993), 36, (17), pp. 4199-4206.
• [17] Wolfram, S: Mathematica: A system for doing mathematics by computer, Bonn, New York, (1996).
• [18] Zhong, F, Ecke, ER and Steinberg, V: Asymmetric modes and the transition to vortex structures in rotating Rayleigh-Benard convection, Phys. Rev. Lett., (1991), 67, pp.2473-2476.