On the existence of the derivative of the volume average

A general theorem on the derivative of the volume average is formulated and proved. Conditions for the existence of the derivative are presented and discussed. This is done in order to give a better base to the theory of spatial averaging.Latin Letters E ₃ three-dimensional vector space over the field of real numbers - K, K(x) averaging domain - G, G _w, G_s open sets in E ₃; components of the two-phase system - C ¹(G) the set of functions 1-times continuously differentiable in G - W1/2(G) Sobolev space - V volume of the domain K - f function defined in G, G _w - K _infi ^sup* (x), K _infi ^sup– (x) special parts of K(x) Greek Letters boundary of G, G _w, G_s; w-s interface - _ij Kronecker delta - v unit outward normal of G, G _w - _j j-dimensional Lebesgue measure Other M closure of a set M in the metric space E ₃ - f phase average of f for the w-phase - (u, v) scalar product of u, v in E ₃ - one-sided derivatives     

Qualitative Convergence in the Discrete Approximation of the Euler Problem∗

ABSTRACT

There are two mathematically rigorous ways to derive Euler's differential equation of the elastica. The first is to start from integral rules and use variational principles, whereas the second is to regard the continuous rod as a limit of a discrete sequence of elastically connected rigid elements when the length of the elements decreases to zero. Discrete models of the Euler buckling problem are investigated. The global number s of solutions of the boundary-value problem is expressed as a function of the number of elements in the discrete model, s = s(n), at constant loading P. The functions s (n) are described by the characteristic parameters n ₁ and n ₂, and graphs of n ₁(P) and n ₂(P) are plotted. Observations related to these diagrams reveal interesting features in the behavior of the discrete model, from the point of view of both theory and application.     

On the Nonexistence of Some Special Eigenfunctions for the Dirichlet Laplacian and the Lamé System

Let Ω be a bounded Lipschitz domain in ℝ ⁿ with n ≥ 3. We prove that the Dirichlet Laplacian does not admit any eigenfunction of the form u(x) =ϕ(x′)+ψ(x _n) with x′=(x1, ..., x _n−1). The result is sharp since there are 2-d polygonal domains in which this kind of eigenfunctions does exist. These special eigenfunctions for the Dirichlet Laplacian are related to the existence of uniaxial eigenvibrations for the Lamé system with Dirichlet boundary conditions. Thus, as a corollary of this result, we deduce that there is no bounded Lipschitz domain in 3-d for which the Lamé system with Dirichlet boundary conditions admits uniaxial eigenvibrations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Permeability evolution induced by the precipitation of an advected solute: Effect of the microscopic and the macroscopic scales competition on the clogging mechanism

A model is proposed for coupling the one-dimensional transport of solute with surface precipitation kinetics which induces the clogging of an initially homogeneous porous medium. The aim is to focus the non-linear feedback effect between the transport and the chemical reaction through the permeability of the medium. A Lagrangian formulation, used to solve the coupled differential equations, gives semi-analytical expressions of the hydrodynamic quantities. A detailed analysis reveals that the competition between the microscopic and macroscopic scales controls the clogging mechanism, which differs depends on whether short or long times are considered. In order to illustrate this analysis, more quantitative results were obtained in the case of a second and zeroth order kinetic. It was necessary to circumvent the semi-analytic character of the solutions problem by successive approximation. A comparison with results obtained by simulations displays a good agreement during the most part of the clogging time.Nomenclature a(x, t) Capillary tube radius (L) - A _(aq) Chemical species in the aqueous phase - A _n(s) Chemical species of the solid phase - C(x, t) Aqueous concentration in a capillary tube ([mole/L³] in the case of a permanent injection - [mole/L³/L] in the case of an instantaneous injection) - C(x, t) C(x, t)/C ₀ Dimensionless aqueous concentration in a capillary tube - C ₀ Aqueous concentration imposed at the inlet and also initial concentration in an elementary volume of fluid (mole/L³) - C _i(t) Concentration in a fluid element i (mole/L³) - C(_R) (t, C_o) Aqueous concentration in a stirred reactor (mole/L³) - d_ij (t) Length belonging to the volume, inside a fluid element i, which interacts with a precipitate element j (L) - dM _ij(t) Mass exchange between a fluid element i and a precipitate element j (M) - dN ₀ Number of molecules in an elementary volume of fluid injected at the inlet of a capillary tube during dt ₀ - dN(x, t, C₀) Number of molecules in an elementary volume of fluid - dt ₀ Time injection of an elementary volume of fluid (T) - D(x, t) Dispersion coefficient (L²/T) - Da(t, x) Damköhler number - D _m Molecular diffusion coefficient (L²/T) - F(x, t) Advective flux (mole/L²/T) - k ₁ Kinetic constant of dissolution (mole/L³/T) - k ₂ Kinetic constant of precipitation ([mole/L³]^{1 - n} /T) - k ₂ Kinetic constant of precipitation in the case of a zeroth order kinetics (mole/L³/T) - K(x, t) Permeability in a capillary tube (L²) - K(x, t) K (x, t)/K₀ Dimensionless permeability - k ₀ Permeability of a capillary tube at t = 0 (L²) - L Length of a capillary tube (L) - m Molecular weight of the reactive species (M/mole) - n Stochiometry of the chemical reaction and kinetic order of the precipitation reaction - P(x, t) Precipitate concentration in a capillary tube (mole/L³) - P _j(t) Concentration in a precipitate element j (mole/L³) - P(_r) (t, C_o) Precipitate concentration in a stirred reactor (mole/L³) - Pr(x, t) Local pressure in a capillary tube - (M/T²/L³) Pr(x, t) Pr(x, t)/Pr(x, 0) Dimensionless local pressure in a capillary tube - Q(t) Flow rate (L³/T) - Q(t) Q(t)U ₀/S₀ Dimensionless flow rate - R(x, t) Chemical flux between the aqueous and the solid phase in a capillary tube (mole/L³/T) - R _i(t) Chemical flux between an aqueous element i and the solid phase (mole/L³/T) - R _(R)[t, C₀] Chemical flux between the aqueous and the solid phase in a stirred reactor (mole/L³/T) - S(x, t) Cross sectional area of a capillary tube accessible to the aqueous phase (L²) - S(x, t) S(x, t)/S₀ Dimensionless cross-sectional area - S ₀ Cross-sectional area of a capillary tube at t = 0 (L²) - tlim(x) Time at which the precipitation front concentration vanishes in the case of zeroth order kinetics (T) - t _max Time of maximum propagation of the precipitation front in the case of zeroth order kinetics (T) - tmin(x) Time at which the precipitation front arrives at x (T) - t _p L/U ₀. Time necessary for an elementary volume of fluid, moving with the velocity U ₀, to reach the oulet of the medium - t _U max Time of maximum value of the velocity field in the case of zeroth order kinetics (T) - t ₀ Time at which an elementary volume of fluid has left the inlet of a capillary tube (T) - t _0m (x, t) Time at which the last elementary volume of fluid has left the inlet of a capillary tube to reach x at a time lower or equal to t (T) - U(x, t) Fluid velocity (L/T) - U(x, t) U(x, t)/U ₀. Dimensionless fluid velocity - U _j(x, t) Fluid velocity defined from the precipitate element j (L/T) - U _l (t₀, t) Lagrangian fluid velocity (L/T) - U _l (t ₀, t) U _l (x, t)/U ₀. Dimensionless lagrangian fluid velocity - U ₀ Velocity of the fluid at t = 0 (L/T) - V _ij(t) Volume, inside a fluid element i, which interacts with a precipitate element j (L³) - x _i(t) Front position of the fluid element i (L) - x _j Front position of the precipitate element j (L) - X _front(t) Position of the precipitation front (L) - x _lim(t) Position of the precipitation front when the value of its concentration is zero (L) - xmax Position of the maximum propagation of the precipitation front in the case of zeroth order kinetics and for high value of C ₀ (L) - Xmin (t) Position of the precipitation front (L) - x _inf* ^supmax Position of the maximum propagation of the precipitation front in the case of zeroth order kinetics and for small value of C_o (L) Greek Symbols t Time step used during the numerical computation (T) - Pro Imposed pressure drop (M/L/T²) - Injection time of reactive species (T) - Density of the precipitate (M/L³) - Dynamic viscosity (M/L/T) - <Ri(t)> _infi ^supj Mean chemical flux between a precipitate element j and all the fluid elements i susceptible to interact with the precipitate element j (mole/L³/T)     

On the Euler Equations in the Critical Triebel-Lizorkin Spaces

In this paper we study the initial value problem of the incompressible Euler equations in ⁿ for initial data belonging to the critical Triebel-Lizorkin spaces, i.e., v ₀ F ⁿ⁺¹ _1,q , q[1, ]. We prove the blow-up criterion of solutions in F ⁿ⁺¹ _1,q for n=2,3. For n=2, in particular, we prove global well-posedness of the Euler equations in F ³ _1,q , q[1, ]. For the proof of these results we establish a sharp Moser-type inequality as well as a commutator-type estimate in these spaces. The key methods are the Littlewood-Paley decomposition and the paradifferential calculus by J. M. Bony.     

On the structure of continua and the mathematical properties of algebraic elastodynamic of a triclinic structural system

This paper is neither laudatory nor derogatory but it simply contrasts with what might be called elastostatic (or static topology), a proposition of the famous six equations. The extension strains and the shearing strains which were derived by A.L. Cauchy, are linearly expressed in terms of nine partial derivatives of the displacement function (u _i ,u _j ,u _h )=u(x ⁱ ,x ^j ,x ^k ) and it is impossible for the inverse proposition to sep up a system of the above six equations in expressing the nine components of matrix (∂(u _i ,u _j ,u _h )/∂(x ⁱ ,x ^j ,x ^k )). This is due to the fact that our geometrical representations of deformation at a given point are as yet incomplete^[1]. On the other hand, in more geometrical language this theorem is not true to any triangle, except orthogonal, for “squared length” in space^[2]. The purpose of this paper is to describe some mathematic laws of algebraic elastodynamics and the relationships between the above-mentioned important questions.     

On the derivation of the Forchheimer equation by means of the averaging theorem

The averaging theorem is applied to the microscopic momentum equation to obtain the macroscopic flow equation. By examining some very simple tube models of flow in porous media, it is demonstrated that the averaged microscopic inertial terms cannot lead to a meaningful representation of non-Darcian (Forchheimer) effects. These effects are shown to be due to microscopic inertial effects distorting the velocity and pressure fields, hence leading to changes in the area integrals that result from the averaging process. It is recommended that the non-Darcian flow regime be described by a Forchheimer number, not a Reynolds number, and that the Forchheimer coefficient be more closely examined as it may contain information on tortuosity.English a _i gravitational acceleration (m/s²) - A _fs interfacial area between the fluid and solid phases (m²) - Fo Forchheimer number - k permeability (m²) - k ₀ permeability at zero velocity (m²) - p thermodynamic pressure (Pa) - r _i coordinate on the microscopic scale (m) - Re Reynolds number - t time (s) - u _i ,u bulk velocity (m/s) - V volume (m³) - V _f fluid volume (m³) - w _i ,w microscopic velocity (m/s) - x _i ,x coordinate on the macroscopic scale (m) Greek the Forchheimer coefficient (1/m) - _ij extra (viscous) stress tensor (Pa) - _ij stress tensor (Pa) - Viscosity (Pa. s) - density (kg/m³) - porosity - a general variable Symbols < > phase average - < > ^f intrinsic phase average - the fluctuating part of a variable     

Reconsideration on the role of the specific heat ratio in Arrhenius law applications

Arrhenius law implicates that only those molecules which possess the internal energy greater than the activation energy Ea can react. However, the internal energy will not be proportional to the gas temperature if the specific heat ratio y and the gas constant R vary during chemical reaction processes. The varying y may affect significantly the chemical reaction rate calculated with the Arrhenius law under the constant γ assumption, which has been widely accepted in detonation and combustion simulations for many years. In this paper, the roles of variable γ and R in Arrhenius law applications are reconsidered, and their effects on the chemical reaction rate are demonstrated by simulating one- dimensional C-J and two-dimensional cellular detonations. A new overall one-step detonation model with variable γ and R is proposed to improve the Arrhenius law. Numerical experiments demonstrate that this improved Arrhenius law works well in predicting detonation phenomena with the numerical results being in good agreement with experimental data.     

Improvement in the frequency response of the electrochemical wall shear stress meter

One of the very few disadvantages of the mass-transfer transducer when compared with the hot-film sensor, is a slightly diminished frequency-response due to the higher Prandtl number encountered. Mass or thermal balance and transfer equations were solved first by Fortuna and Hanratty (1971) for small fluctuations of the wall shear. The solutions allow to make accurate corrections on the frequency spectra and the power of the fluctuations, but in different time. In this paper, the author deduces the frequency response of split rectangular electrodes and shows how a combination of signals improves the response at higher frequencies and makes it comparable to the thermal transducer with the same size, in the same fluid. Two experimental devices are described and compared. With these devices, the measurement of the wall shear fluctuations is improved in real time. Accurate determinations of turbulent power fluctuations and probability density spectra are feasible and illustrate the subject.List of symbols A total area of the electrode - A _j area of the part j of the electrode - a coefficient - C concentration - C bulk concentration - c fluctuation of concentration - D diffusion coefficient - F Faraday's constant - f(n ⁺) transfer function - g gain of the differential electrode - I _j electrolysis current on the part j of the electrode - K transfert coefficient - k fluctuation of K - l electrode length - n frequency - P _r Prandtl number - S wall shear - s fluctuation of the wall-shear - t time - x direction of the flow - y direction normal to the wall - phase delay - v kinematic viscosity A version of this paper was presented at the 11th Symposium on Turbulence, University of Missouri-Rolla, 17–19 October 1988     

Morawetz's method for the decay of the solution of the exterior initial-boundary value problem for the linearized equation of dynamic elasticity

In this paper, the behavior of the solution of the time-dependent linearized equation of dynamic elasticity is examined.For the homogeneous problem, it is proved that in the exterior of a star-shaped body on the surface of which the displacement field is zero, the solution decays at the rate t ^-1 as the time t tends to infinity.For the non-homogeneous problem with a harmonic forcing term, it is proved that for large times, the elastic material in the exterior of the body, tends to a harmonic motion, with the period of the external force.The convergence to the steady harmonic state solution is at the rate t ^-1/2 as t tends to infinity, and is uniform on bounded sets.     

Opportunities of the elastometry method for investigating the elastic properties of the eyeball shell

On the basis of a maximally simple mechanical model, the possibility to estimate the stiffness of the eyeball shell using the intraocular pressures measured in the presence of different weights applied to the cornea (elastometry method) is investigated. On the basis of general considerations of dimensional theory, it is shown that the pressure difference used in ophthalmology for estimating the rigidity of the eye ball and determined in the elastometry procedure depends on the ratio E/p ₀, where E is a quantity characterizing the shell stiffness and p ₀ is the intraocular pressure in the unloaded eye. An experimental procedure, which can be treated as a development of the elastometry method and makes it possible to estimate a certain parameter dependent only on the eyeball shell stiffness, is proposed. The practical realization of the method proposed needs to be checked experimentally.     

On the use of the rolling-ball viscometer for the measurement of rheological parameters of power law fluids

Summary An attempt has been made to determine the flow curve of fluids obeying the power law rheological model on a rolling ball viscometer. The theoretical analysis is based on the hydrodynamical model developed byLewis in 1956 for the purpose of rolling ball viscometer calibration and extended in 1964 byTurian andBird for the two-parametric power law model. The shear rate variation has been accomplished on varying the tube angle of inclination. Experiments performed on an adapted, commercialHoeppler viscometer with aqueous solutions of carboxymethylcellulose and polyacrylamide reproduced the flow curves obtained simultaneously on rotational and capillary instruments with reasonable accuracy.

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Zusammenfassung Die Möglichkeit der Bestimmung von FließkurvenOstwaldscher Flüssigkeiten in einem Fallkugelviskosimeter wurde überprüft. Die analytische Behandlung des Problems beruht auf dem hydrodynamischen Modell, das 1956Lewis zur Eichung von Fallkugel-viskosimetern entwickelte und das später vonBird undTurian verallgemeinert wurde. Die Änderung der Schergeschwindigkeit wurde durch Neigung des Fallrohrs erzielt. Versuche wurden an einem handelsüblichenHöppler-Viskometer mit wässerigen Lösungen von Karboxymethylzellulose und Polyakrylamid durchgeführt. Vergleiche der gewonnenen Ergebnisse mit Fließkurven, die gleichzeitig an Rotations- und Kapillar-Viskosimetern gemessen wurden, ergaben im Gültigkeitsbereich der Theorie befriedigende Übereinstimmung.

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Nomenclature d ball diameter - D tube diameter - f function ofz defined in eq. [16] - F function ofn defined in eq. [30] - g gravity acceleration - H clearance between the ball and tube - I definite integral defined in eq. [17] - J _L = 3/4J(1) definite integral defined in [4] - J (n) definite integral defined in eq. [21] - K instrument constant defined forNewtonian fluids in eq. [1] - L distance marked on the viscometer tube - m consistency parameter - n flow behaviour index - p pressure - r radial variable - S area - t time - T temperature - u _z velocity component - U velocity of the ball - V flow rate - y, z spatial rate - tube angle of inclination - shear rate - gamma function - azimuthal variable - µ dynamic viscosity - fluid density - _s ball density - shear stress - dimensionless variable defined in eq. [9] - — denotes mean value - w denotes conditions at the solid surface With 9 figures and 2 tables     

Effects of the corner radius on the near wake of a square prism

The near wake of square cylinders with different corner radii was experimentally studied based on particle imaging velocimetry (PIV), laser doppler anemometry (LDA) and hotwire measurements. Four bluff bodies, i.e., r/d=0 (square cylinder), 0.157, 0.236, 0.5 (circular cylinder), where r is corner radius and d is the characteristic dimension of the bluff bodies, were examined. A conditional sampling technique was developed to obtain the phase-averaged PIV data in order to characterize quantitatively the effect of corner radii on the near-wake flow structure. The results show that, as r/d increases from 0 to 0.5, the maximum strength of shed vortices attenuates, the circulation associated with the vortices decreases progressively by 50%, the Strouhal number, St, increases by about 60%, the convection velocity of the vortices increases along with the widening of the wake width by about 25%, the vortex formation length and the wake closure length almost double in size. Meanwhile, both the vortex wavelength, λ _x , and the lateral spacing, λ _y , decrease as r/d increases, but the ratio of λ _y to λ _x is approximately 0.29, irrespective of r/d, which is close to the theoretical value of 0.281 for a stable Karman vortex street. The decrease in wavelength is probably responsible for the change in the flow structure from the approximately circular-shaped vortex at r/d=0 to the laterally stretched vortex at r/d=0.5. The leading edge corner radius is more important than the trailing one in influencing the near wake structure since it determines to a great extent the behavior of the streamlines, the separation angle and the base pressure. It is further found that the ratio of the mean drag coefficient to the total shed circulation, C _d/Γ₀, approaches a constant, about 0.25 for different bluff bodies in the subcritical flow regime. The streamwise evolution of vortices and the streamwise fluctuating velocity along the centerline for rounded cylinders are also discussed.     

On the quadrilateral Q2–P1 element for the Stokes problem

The Q₂ – P₁ approximation is one of the most popular Stokes elements. Two possible choices are given for the definition of the pressure space: one can either use a global pressure approximation (that is on each quadrilateral the finite element space is spanned by 1 and by the global co‐ordinates x and y) or a local approach (consisting in generating the local space by means of the constants and the local curvilinear co‐ordinates on each quadrilateral ξ and η). The former choice is known to provide optimal error estimates on general meshes. This has been shown, as it is standard, by proving a discrete inf–sup condition. In the present paper we check that the latter approach satisfies the inf–sup condition as well. However, recent results on quadrilateral finite elements bring to light a lack in the approximation properties for the space coming out from the local pressure approach. Numerical results actually show that the second choice (local or mapped pressure approximation) is suboptimally convergent. Copyright © 2002 John Wiley & Sons, Ltd.     

Determination of the Boundaries between the Domains of Stability and Instability for the Hill Equation

The stability problem for the Hill equation containing two parameters is analyzed using the Mathematica computer algebra system. The characteristic constant is found as a series expansion in powers of a small parameter e. It is shown that the domains of instability are located only between the curves a = a(e) on the a-e plane crossing the axis e = 0 at the points a = (2k – 1)² / 4, k = 1, 2, 3, ....The corresponding curves are found as power series in e with accuracy O(e ⁶).     

Extended Lagrangian formalism and the corresponding energy relations

In this paper an extended Lagrangian formalism for the rheonomic systems with the nonstationary constraints is formulated, with the aim to examine more completely the energy relations for such systems in any generalized coordinates, which in this case always refer to some moving frame of reference. Introducing new quantities, which change according to the law τ_a=φ_a(t), it is demonstrated that these quantities determine the position of this moving reference frame with respect to an immobile one. In the transition to the generalized coordinates qⁱ they are taken as the additional generalized coordinates q^a=τ_a, whose dependence on time is given a priori. In this way the position of the considered mechanical system relative to this immobile frame of reference is determined completely.Based on this and using the corresponding d'Alembert–Lagrange's principle, an extended system of the Lagrangian equations is obtained. It is demonstrated that they give the same equations of motion qⁱ=qⁱ(t) as in the usual Lagrangian formulation, but substantially different energy relations. Namely, in this formulation two different types of the energy change law dE/dt and the corresponding conservation laws are obtained, which are more general than in the usual formulation. So, under certain conditions the energy conservation law has the form E=T+U+P=const, where the last term, so-called rheonomic potential expresses the influence of the nonstationary constraints.Afterwards, a detailed analysis of the obtained results and their connection with the usual formulation of mechanics are given. It is demonstrated that so formulated energy relations are in full accordance with the corresponding ones in the usual vector formulation, when they are expressed in terms of the rheonomic potential. Finally, the obtained results are illustrated by several simple, but characteristic examples.     

On the sensitivity of interconversion between relaxation and creep

The interconversion equation of linear viscoelasticity defines implicitly the interrelations between the relaxation and creep functions G(t) and J(t). It is widely utilised in rheology to estimate J(t) from measurements of G(t) and conversely. Because different molecular details can be recovered from G(t) and J(t), it is necessary to work with both. This leads naturally to the need to identify whether it is better to first measure G(t) and then determine J(t) or conversely. This requires an understanding of the stability (sensitivity) of the recovery of J(t) from G(t) compared with that of G(t) from J(t). Although algorithms are available that work adequately in both directions, numerical experimentation strongly suggests that the recovery of J(t) from G(t) measurements is the more stable. An elementary theoretical rationale has been given recently by Anderssen et al. (ANZIAM J 48:C346–C363, 2007) for single exponential models of G(t) and J(t). It explicitly exploits the simple algebra of such functions. In this paper, corresponding bounds are derived that hold for arbitrary sums of exponentials. The paper concludes with a discussion, from a practical rheological perspective, about the implications and implementations of the results.     

Numerical studies of the fingering phenomena for the thin film equation

We present a new interpretation of the fingering phenomena of the thin liquid film layer through numerical investigations. The governing partial differential equation is h_t + (h²?h³)_x = ??·(h³?Δh), which arises in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, y, t) is the liquid film height. A robust and accurate finite difference method is developed for the thin liquid film equation. For the advection part (h²?h³)_x, we use an implicit essentially non‐oscillatory (ENO)‐type scheme and get a good stability property. For the diffusion part ??·(h³?Δh), we use an implicit Euler's method. The resulting nonlinear discrete system is solved by an efficient nonlinear multigrid method. Numerical experiments indicate that higher the film thickness, the faster the film front evolves. The concave front has higher film thickness than the convex front. Therefore, the concave front has higher speed than the convex front and this leads to the fingering phenomena. Copyright © 2010 John Wiley & Sons, Ltd.     

Contributions to the theory of the method of steepest descent

Applying the method of steepest descent to F(x ₁,..., x _n ) one obtains a sequence of points _v . To obtain conditions for convergence of _v , the derived set H of the _v in the case of divergence is studied. In this case H is a continuum on which not only grad F vanishes everywhere, but also the rank of the Hessian of F is everywhere less than n-1.     

The effects of boundary to the ECRH and the energy absorption of ICRH

Electron-cyclotron resonant heating (ECRH) of Tokamak plasma is examined. When plasma is heated by waves, we must consider the distribution of incident wave energy toO andX modes as the wave is incident from vacuum to the surface of plasma as well as the absorption efficiency ofO mode andX mode. Numerical calculation shows that for small incident angle, the incident energy transfers principally intoO mode when the electric fieldE ⁱ of incident wave is parallel to the incident plane, therefore it is efficient to heat the plasma byO mode. WhenE ⁱ is perpendicular to the incident plane, the energy transfers principally intoX mode and heating the plasma byX mode is efficient. Ion-cyclotron resonant heating (ICRH) is also considered, the formula of the energy of ion-cyclotron wave absorbed by plasma is obtained.