EFFECT OF STRATIFICATION ON STABILITY OF FLOW AND HEAT TRANSFER IN THE LIQUID FILM FLOWING DOWN AN INCLINED HEATED PLATE

2010 ◽  
Vol 24 (13) ◽  
pp. 1461-1465 ◽  
Author(s):  
YOULIANG CHENG ◽  
XIAOCHAO FAN ◽  
YING TIAN

The stability analysis in a stratified liquid film flowing down an inclined heated plate is investigated. The boundary value problem of the stability differential equation on small perturbation for general density distribution is derived. Then, the boundary value problem is solved and the solution to the problem is obtained for a special case. The result for non-stratified is agreement with the known one. And the effects of stratification and the other factors such as Re, Ma, We, Bi, Pr, the inclined angle β on the stability of the film are analyzed.

2012 ◽  
Vol 516-517 ◽  
pp. 202-207
Author(s):  
Xiao Chao Fan ◽  
Rui Jing Shi ◽  
Bo Wei

Stable analysis of flow and heat transfer in the saturated liquid film of liquid low boiling point gases falling down an inclined heated plate is investigated. Firstly, the boundary value problem of linear stability differential equation (Orr–Sommerfeld equation) on small perturbation is derived representing surface tension by nonlinear relationship on temperature. Then, the expression of the wave velocity is got by solving the boundary value problem of O–S equation using the perturbation method. The effects of the inclined angle and some other factors, such as Reynolds number, wave number, temperature of the plate and the parameter for the physical property, on stability in the saturated liquid film of liquid low boiling point gas N2 are numerically analyzed by MATLAB software. Finally, it is shown and analyzed a new critical Reynolds number which is actually the extension of Yih’s.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.


1990 ◽  
Vol 33 (2) ◽  
pp. 169-180 ◽  
Author(s):  
Juan A. Gatica ◽  
Gaston E. Hernandez ◽  
P. Waltman

The boundary value problemis studied with a view to obtaining the existence of positive solutions in C1([0, 1])∩C2((0, 1)). The function f is assumed to be singular in the second variable, with the singularity modeled after the special case f(x, y) = a(x)y−p, p>0.This boundary value problem arises in the search of positive radially symmetric solutions towhere Ω is the open unit ball in ℝN, centered at the origin, Γ is its boundary and |x| is the Euclidean norm of x.


Author(s):  
Ebrahim Esmailzadeh ◽  
Gholamreza Nakhaie-Jazar ◽  
Bahman Mehri

Abstract The transverse vibrating motion of a simple beam with one end fixed while driven harmonically along its axial direction from the other end is investigated. For a special case of zero value for the rigidity of the beam, the system reduces to that of a vibrating string with the corresponding equation of its motion. The sufficient condition for the periodic solution of the beam is then derived by means of the Green’s function and Schauder’s fixed point theorem. The criteria for the stability of the system is well defined and the condition for which the performance of the beam behaves as a nonlinear function is stated.


2003 ◽  
Vol 475 ◽  
pp. 303-331 ◽  
Author(s):  
E. S. BENILOV

We examine the stability of a quasi-geostrophic vortex in a two-layer ocean with a thin upper layer on the f-plane. It is assumed that the vortex has a sign-definite swirl velocity and is localized in the upper layer, whereas the disturbance is present in both layers. The stability boundary-value problem admits three types of normal modes: fast (upper-layer-dominated) modes, responsible for equivalent-barotropic instability, and two slow baroclinic types (mixed- and lower-layer-dominated modes). Fast modes exist only for unrealistically small vortices (with a radius smaller than half of the deformation radius), and this paper is mainly focused on the slow modes. They are examined by expanding the stability boundary-value problem in powers of the ratio of the upper-layer depth to the lower-layer depth. It is demonstrated that the instability of slow modes, if any, is associated with critical levels, which are located at the periphery of the vortex. The complete (sufficient and necessary) stability criterion with respect to slow modes is derived: the vortex is stable if and only if the potential-vorticity gradient at the critical level and swirl velocity are of the same sign. Several vortex profiles are examined, and it is shown that vortices with a slowly decaying periphery are more unstable baroclinically and less barotropically than those with a fast-decaying periphery, with the Gaussian profile being the most stable overall. The asymptotic results are verified by numerical integration of the exact boundary-value problem, and interpreted using oceanic observations.


Author(s):  
Qun Chen

AbstractLet M, N be Riemannian manifolds, f: M → N a harmonic map with potential H, namely, a smooth critical point of the functional EH(f) = ∫M[e(f)−H(f)], where e(f) is the energy density of f. Some results concerning the stability of these maps between spheres and any Riemannian manifold are given. For a general class of M, this paper also gives a result on the constant boundary-value problem which generalizes the result of Karcher-Wood even in the case of the usual harmonic maps. It can also be applied to the static Landau-Lifshitz equations.


1971 ◽  
Vol 5 (2) ◽  
pp. 275-288 ◽  
Author(s):  
J. F. McKenzie

The hydromagnetic analogue of the Kelvin–Helmholtz problem is extended to include the effects of the Hall term. In contrast to other results in the literature it is shown that, in the case of incompressible fluids, the stability of a shear plane is unaffected by the introduction of the Hall term. The special case of a hot, uninagnetized fluid on one side of the interface and a cold, magnetized fluid on the other is studied in some detail. In this case it is shown that the presence of the Hall term can have either a stabilizing or a destabilizing effect, depending upon whether the sound speed in the hot fluid is very much greater than the Alfvén speed in the cold fluid or vice versa.


2015 ◽  
Vol 15 (1) ◽  
pp. 79-95 ◽  
Author(s):  
Martin Stynes ◽  
José Luis Gracia

AbstractA two-point boundary value problem is considered on the interval $[0,1]$, where the leading term in the differential operator is a Caputo fractional derivative of order δ with $1<\delta <2$. Writing u for the solution of the problem, it is known that typically $u^{\prime \prime }(x)$ blows up as $x\rightarrow 0$. A numerical example demonstrates the possibility of a further phenomenon that imposes difficulties on numerical methods: u may exhibit a boundary layer at x = 1 when δ is near 1. The conditions on the data of the problem under which this layer appears are investigated by first solving the constant-coefficient case using Laplace transforms, determining precisely when a layer is present in this special case, then using this information to enlighten our examination of the general variable-coefficient case (in particular, in the construction of a barrier function for u). This analysis proves that usually no boundary layer can occur in the solution u at x = 0, and that the quantity $M = \max _{x\in [0,1]}b(x)$, where b is the coefficient of the first-order term in the differential operator, is critical: when $M<1$, no boundary layer is present when δ is near 1, but when M ≥ 1 then a boundary layer at x = 1 is possible. Numerical results illustrate the sharpness of most of our results.


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