Centers and Isochronous Centers for Two Classes of Generalized Seventh and Ninth Systems

We classify new classes of centers and of isochronous centers for polynomial differential systems in \mathbb R²{\mathbb R^2} of arbitrary odd degree d ≥ 7 that in complex notation z = x + i y can be written as

Similarities Between Rotation and Stratification Effects on Homogeneous Shear Flow

Linear theory is applied to examine rotation and buoyancy effects on homogeneous turbulent shear flows with given vertical velocity shear, S=d/dx ₃. In the rotating shear case (where the rotation vector is perpendicular to the plane of the mean flow, Ω _i =Ωδ _{i 2}), general solutions for the Fourier components of the fluctuating velocity are proposed. These solutions are compared with those proposed in the literature for the Fourier components of the fluctuating velocity and density in the case of a homogeneous stratified shear flow with vertical density gradient, S _ρ=d/dx ₃. It is shown that from the normal mode stability stand point the Bradshaw parameter B=2Ω/S(1+2Ω/S) (in the rotating shear case) and the Richardson number R _i (in the statified shear case) play similar roles in identifying the stability for all the wave components except in the case where Ω·k=0, for which rotation has no effects on the flow. Analysis of the long-time behavior of the non-dimensional spectral density of energy, e ^g , is carried out. In the stable case, e ^g has decaying oscillations or undergoes a power law decay in time. Analytical solutions for the streamwise two-dimensional energy ℰ _ii ¹/2 (i.e. the limit at k ₁=0 of the one-dimensional energy spectra) are proposed. At large time, ℰ _ii ¹(t)/ℰ _ii ¹(0) oscillates around the value (3R _i +1)/(4R _i ) except at R _i =1 it stays constant in time. Similar behavior for ℰ _ii ¹(t)/ℰ _ii ¹(0) is also observed in the rotating shear case (ℰ _ii ¹(t)/ℰ _ii ¹(0) oscillates around the value (1+4B)/(4B)). Due to the behavior of the dimensionless spectral density of energy in both flow cases, the turbulent kinetic energy, /2, the production rate, ?, and the rate due to the buoyancy forces, ℬ, are split into two parts, , ?=?₁+?₂, ℬ=ℬ₁+ℬ₂ (in the stratified shear case, both ?₁ and ℬ₁ vanish when R _i >?, while in the rotating shear case one has ℬ=0). It is shown that when rotation is “cyclonic” (i.e. Ω/S>0), part reaches maximum magnitudes at St ≈2, independent of the B value, and the first time to a zero crossing of ?₂ occurs at this particular value. When rotation is “anticyclonic” (i.e. Ω/S<0) one finds St ≈1.6 instead of St ≈2. In the stratified shear case, both ?₂ and ℬ₂ cross zero at Nt=St ≈2, and part reaches maximum magnitudes at this particular value. These results and in particular those for the turbulent kinetic energy are compared with previous direct numerical simulation (DNS) results in homogeneous stratified shear flows. Received 30 July 2001 and accepted 19 February 2002     

Carleman Estimate for Elliptic Operators with Coefficients with Jumps at an Interface in Arbitrary Dimension and Application to the Null Controllability of Linear Parabolic Equations

In a bounded domain of R ⁿ⁺¹, n ≧ 2, we consider a second-order elliptic operator, ${A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}In a bounded domain of R ⁿ⁺¹, n ≧ 2, we consider a second-order elliptic operator, A=-?_x0² - ?_x ·(c(x) ?_x){A=-{\partial_{x_0}^2} - \nabla_x \cdot (c(x) \nabla_x)}, where the (scalar) coefficient c(x) is piecewise smooth yet discontinuous across a smooth interface S. We prove a local Carleman estimate for A in the neighborhood of any point of the interface. The “observation” region can be chosen independently of the sign of the jump of the coefficient c at the considered point. The derivation of this estimate relies on the separation of the problem into three microlocal regions and the Calderón projector technique. Following the method of Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) we then prove the null controllability for the linear parabolic initial problem with Dirichlet boundary conditions associated with the operator ?_t - ?_x ·(c(x) ?_x){{\partial_t - \nabla_x \cdot (c(x) \nabla_x)}} .     

The Decay Rate and Higher Approximation of Mild Solutions to the Navier�CStokes Equations

In this paper, we consider v(t) = u(t) − e ^tΔ u ₀, where u(t) is the mild solution of the Navier–Stokes equations with the initial data u₀ ? L²(\mathbb Rⁿ)?Lⁿ(\mathbb Rⁿ){u_0\in L^2({\mathbb R}^n)\cap L^n({\mathbb R}^n)} . We shall show that the L ² norm of D ^β v(t) decays like t^{-\frac |b|-1 2-\frac n4}{t^{-\frac {|\beta|-1} {2}-\frac n4}} for |β| ≥ 0. Moreover, we will find the asymptotic profile u ₁(t) such that the L ² norm of D ^β (v(t) − u ₁(t)) decays faster for 3 ≤ n ≤ 5 and |β| ≥ 0. Besides, higher-order asymptotics of v(t) are deduced under some assumptions.     

The Critical Dimension for a Fourth Order Elliptic Problem with Singular Nonlinearity

We study the regularity of the extremal solution of the semilinear biharmonic equation ${{\Delta^2} u=\frac{\lambda}{(1-u)^2}}We study the regularity of the extremal solution of the semilinear biharmonic equation D² u=\fracl(1-u)²{{\Delta^2} u=\frac{\lambda}{(1-u)^2}}, which models a simple micro-electromechanical system (MEMS) device on a ball B ì \mathbbR^N{B\subset{\mathbb{R}}^N}, under Dirichlet boundary conditions u=?_n u=0{u=\partial_\nu u=0} on ?B{\partial B}. We complete here the results of Lin and Yang [14] regarding the identification of a “pull-in voltage” λ* > 0 such that a stable classical solution u _λ with 0 < u _λ < 1 exists for l ? (0,l^*){\lambda\in (0,\lambda^*)}, while there is none of any kind when λ > λ*. Our main result asserts that the extremal solution u_l^*{u_{\lambda^*}} is regular (sup_B u_l^* < 1 ){({\rm sup}_B u_{\lambda^*} <1 )} provided N \leqq 8{N \leqq 8} while u_l^*{u_{\lambda^*}} is singular (sup_B u_l^* = 1){({\rm sup}_B u_{\lambda^*} =1)} for N \geqq 9{N \geqq 9}, in which case 1-C₀|x|^4/3 \leqq u_l^* (x) \leqq 1-|x|^4/3{1-C_0|x|^{4/3} \leqq u_{\lambda^*} (x) \leqq 1-|x|^{4/3}} on the unit ball, where C₀:=(\fracl^*[`(l)])<sup>\frac</sup>13{C_0:=\left(\frac{\lambda^*}{\overline{\lambda}}\right)^\frac{1}{3}} and [`(l)]: = \frac89(N-\frac23)(N- \frac83){\bar{\lambda}:= \frac{8}{9}\left(N-\frac{2}{3}\right)\left(N- \frac{8}{3}\right)}.     

Positive Least Energy Solutions and Phase Separation for Coupled Schr?dinger Equations with Critical Exponent

In this paper we study the following coupled Schr?dinger system, which can be seen as a critically coupled perturbed Brezis–Nirenberg problem: {ll-Du +l₁ u = m₁ u³+buv²,     x ? W,-Dv +l₂ v = m₂ v³+bvu²,     x ? W,u\geqq 0, v\geqq 0 in W,    u=v=0     on ?W.\left\{\begin{array}{ll}-\Delta u +\lambda_1 u = \mu_1 u^3+\beta uv^2, \quad x\in \Omega,\\-\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega,\\u\geqq 0, v\geqq 0\, {\rm in}\, \Omega,\quad u=v=0 \quad {\rm on}\, \partial\Omega.\end{array}\right.     

Some Aspects of the Asymptotic Dynamics of Solutions of the Homogeneous Navier–Stokes Equations in General Domains

In the first part of the paper we study decays of solutions of the Navier–Stokes equations on short time intervals. We show, for example, that if w is a global strong nonzero solution of homogeneous Navier–Stokes equations in a sufficiently smooth (unbounded) domain Ω ⊆ R³ and β ∈[1/2, 1) , then there exist C₀ > 1 and δ₀ ∈ (0, 1) such that

On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Consider the class of C ^r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracp_nq_n| ￡ C exp (-q^r+1+k_n)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracP_nq_n{\frac{P_n}{q_n}} is some sequence of irreducible fractions.     

Effects of Large Scales Motion in Unstable Stratified Shear Flows

Homogeneous turbulence under unstable uniform stratification (N ² < 0) and vertical shear is investigated by using the linear theory (or the so-called rapid distortion theory, RDT) for an initial isotropic turbulence over a range −∞ ≤ R _i =N ²/S ² ≤ 0. The initial potential energy is zero and P _r =1 (i.e. the molecular Prandtl number).One-dimensional (streamwise) k ₁−spectra, especially Θ₃₃(k ₁) (i.e., that of the vertical kinetic energy, are investigated. In agreement with previous experiments, it is found that the unstable stratification affects the turbulence quantities at all scales. A significant increase of the vertical kinetic energy is observed at low wavenumbers k ₁ (i.e. large scales) due to an increase of the stratification . The effect of the shear (S) is appreciable only at high wavenumbers k ₁ (i.e. small scales).Based on the importance of the spectral components with k ₁ = 0, the asymptotic forms of Θ _ij (k ₁ = 0) or equivalently the so-called “two-dimensional” energy components (2DEC) are analyzed in detail. The asymptotic form for the ratio of 2DEC is compared to the long-time limit of the ratio of real energies. In the unstable stratified shearless case (S=0,N ² ≠ 0) the variances and the covariances of the velocity and the density are derived analytically in terms of the Weber functions, while when S ≠ 0 and N ² ≠ 0 they are obtained numerically (−100 ≤ R _i <0 and . The results are discussed in connection to previous experimental results in unstable stratified open channel flows cooled from above by Komori et al. Phy Fluids 25, 1539–1546 (1982).It is shown that the Richardson number dependence of the long-time limit of the ratios of real energies is well described by this “simple” model (i.e. the dependence of the long-time limit of 2DEC on R _i ). For example, the ratio of the potential energy to the kinetic energy (q ²/2), approaches −R _i /(1−R _i ), the ratio of turbulent energy production by buoyancy forces to production by shearing forces (i.e. the flux Richardson number, R _if ), approaches R _i . Also, the Richardson number dependence of the principal angle (β) of the Reynolds stress tensor and the angle (β_ρ) of the scalar flux vector is fairly predicted by this model .On the other hand, it is found that the above ratios are insensitive to viscosity, while the ratios ɛ /q ² and , depend on the viscosity and they evolve asymptotically like t ⁻¹. The turbulent Froude number, F _rt =(L _Oz /L _E )^2/3, where L _Oz and L _E are the Ozmidov length scale and the Ellison length scale, respectively, evolves asymptotically like t ^−1/3.     

On the Navier�CStokes equations with rotating effect and prescribed outflow velocity

We consider the equations of Navier–Stokes modeling viscous fluid flow past a moving or rotating obstacle in \mathbb R^d{\mathbb R^d} subject to a prescribed velocity condition at infinity. In contrast to previously known results, where the prescribed velocity vector is assumed to be parallel to the axis of rotation, in this paper we are interested in a general outflow velocity. In order to use L ^p -techniques we introduce a new coordinate system, in which we obtain a non-autonomous partial differential equation with an unbounded drift term. We prove that the linearized problem in \mathbb R^d{\mathbb R^d} is solved by an evolution system on L^p_s(\mathbb R^d){L^p_{\sigma}(\mathbb R^d)} for 1 < p < ∞. For this we use results about time-dependent Ornstein–Uhlenbeck operators. Finally, we prove, for p ≥ d and initial data u₀ ? L^p_s(\mathbb R^d){u_0\in L^p_{\sigma}(\mathbb R^d)}, the existence of a unique mild solution to the full Navier–Stokes system.     

Streamwise evolution of an inclined cylinder wake

The streamwise evolution of an inclined circular cylinder wake was investigated by measuring all three velocity and vorticity components using an eight-hotwire vorticity probe in a wind tunnel at a Reynolds number Re_d of 7,200 based on free stream velocity (U _∞) and cylinder diameter (d). The measurements were conducted at four different inclination angles (α), namely 0°, 15°, 30°, and 45° and at three downstream locations, i.e., x/d = 10, 20, and 40 from the cylinder. At x/d = 10, the effects of α on the three coherent vorticity components are negligibly small for α ≤ 15°. When α increases further to 45°, the maximum of coherent spanwise vorticity reduces by about 50%, while that of the streamwise vorticity increases by about 70%. Similar results are found at x/d = 20, indicating the impaired spanwise vortices and the enhancement of the three-dimensionality of the wake with increasing α. The streamwise decay rate of the coherent spanwise vorticity is smaller for a larger α. This is because the streamwise spacing between the spanwise vortices is bigger for a larger α, resulting in a weak interaction between the vortices and hence slower decaying rate in the streamwise direction. For all tested α, the coherent contribution to [`(v<sup>2</sup>)] \overline{{v^{2}}} is remarkable at x/d = 10 and 20 and significantly larger than that to [`(u²)] \overline{{u^{2}}} and [`(w<sup>2</sup>)]. \overline{{w^{2}}}. This contribution to all three Reynolds normal stresses becomes negligibly small at x/d = 40. The coherent contribution to [`(u²)] \overline{{u^{2}}} and [`(v²)] \overline{{v^{2}}} decays slower as moving downstream for a larger α, consistent with the slow decay of the coherent spanwise vorticity for a larger α.     

Natural Convection Induced by a Centrifugal Force Field in a Horizontal Annular Porous Layer Saturated with a Binary Fluid

The Darcy Model with the Boussinesq approximation is used to study natural convection in a horizontal annular porous layer filled with a binary fluid, under the influence of a centrifugal force field. Neumann boundary conditions for temperature and concentration are applied on the inner and outer boundary of the enclosure. The governing parameters for the problem are the Rayleigh number, Ra, the Lewis number, Le, the buoyancy ratio, j{\varphi } , the radius ratio of the cavity, R, the normalized porosity, e{\varepsilon } , and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in a thin annular layer (R → 1), analytical solutions for the stream function, temperature and concentration fields are obtained using a concentric flow approximation and an integral form of the energy equation. The critical Rayleigh number for the onset of supercritical convection is predicted explicitly by the present model. Also, results are obtained from the analytical model for finite amplitude convection for which the flow and heat and mass transfer are presented in terms of the governing parameters of the problem. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations.     

Stability of Degenerate Stationary Waves for Viscous Gases

This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous gases in the half space. We discuss the following two cases: (1) viscous conservation laws and (2) damped wave equations with nonlinear convection. In each case, we prove that the solution converges to the corresponding degenerate stationary wave at the rate t ^−α/4 as t → ∞, provided that the initial perturbation is in the weighted space L²_a=L²(\mathbb R₊; (1+x)^a dx){L^2_\alpha=L^2({\mathbb R}_+;\,(1+x)^\alpha dx)} . This convergence rate t ^−α/4 is weaker than the one for the non-degenerate case and requires the restriction α < α_*(q), where α_*(q) is the critical value depending only on the degeneracy exponent q. Such a restriction is reasonable because the corresponding linearized operator for viscous conservation laws cannot be dissipative in L²_a{L^2_\alpha} for α > α^*(q) with another critical value α^*(q). Our stability analysis is based on the space–time weighted energy method in which the spatial weight is chosen as a function of the degenerate stationary wave.     

Fully developed flow of a viscoelastic film down a vertical cylindrical or planar wall

The one-dimensional, gravity-driven film flow of a linear (l) or exponential (e) Phan-Thien and Tanner (PTT) liquid, flowing either on the outer or on the inner surface of a vertical cylinder or over a planar wall, is analyzed. Numerical solution of the governing equations is generally possible. Analytical solutions are derived only for: (1) l-PTT model in cylindrical and planar geometries in the absence of solvent, b o [(h)\tilde]_s/([(h)\tilde]_s +[(h)\tilde]_p)=0\beta\equiv {\tilde{\eta}_s}/\left({\tilde{\eta}_s +\tilde{\eta}_p}\right)=0, where [(h)\tilde]_p\widetilde{\eta}_p and [(h)\tilde]_s\widetilde{\eta}_s are the zero-shear polymer and solvent viscosities, respectively, and the affinity parameter set at ξ = 0; (2) l-PTT or e-PTT model in a planar geometry when β = 0 and x 1 0\xi \ne 0; (3) e-PTT model in planar geometry when β = 0 and ξ = 0. The effect of fluid properties, cylinder radius, [(R)\tilde]\tilde{R}, and flow rate on the velocity profile, the stress components, and the film thickness, [(H)\tilde]\tilde{H}, is determined. On the other hand, the relevant dimensionless numbers, which are the Deborah, De=[(l)\tilde][(U)\tilde]/[(H)\tilde]De={\tilde{\lambda}\tilde{U}}/{\tilde{H}}, and Stokes, St=[(r)\tilde][(g)\tilde][(H)\tilde]²/([(h)\tilde]_p +[(h)\tilde]_s )[(U)\tilde]St=\tilde{\rho}\tilde{g}\tilde{\rm H}^{2}/\left({\tilde{\eta}_p +\tilde{\eta}_s} \right)\tilde{U}, numbers, depend on [(H)\tilde]\tilde{H} and the average film velocity, [(U)\tilde]\widetilde{U}. This makes necessary a trial and error procedure to obtain [(H)\tilde]\tilde{H} a posteriori. We find that increasing De, ξ, or the extensibility parameter ε increases shear thinning resulting in a smaller St. The Stokes number decreases as [(R)\tilde]/[(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to zero for a film on the outer cylindrical surface, while it asymptotes to very large values when [(R)\tilde]/[(H)\tilde]{\tilde{R}}/{\tilde{H}} decreases down to unity for a film on the inner surface. When x 1 0\xi \ne 0, an upper limit in De exists above which a solution cannot be computed. This critical value increases with ε and decreases with ξ.     

On a Volume‐Constrained Variational Problem

. Existence of minimizers for a volume-constrained energy $ E(u) := \int_{\Omega} W(\nabla u)\, dx Existence of minimizers for a volume-constrained energy E(u) : = ò_W W(?u) dx E(u) := \int_{\Omega} W(\nabla u)\, dx where L^N({u = z_i}) = a_i, i = 1, ?, P, {\cal L}^N(\{u = z_i\}) = \alpha_i, i = 1, \ldots, P, is proved for the case in which z_iz_i are extremal points of a compact, convex set in \Bbb R^d\Bbb R^d and under suitable assumptions on a class of quasiconvex energy densities W. Optimality properties are studied in the scalar-valued problem where d=1d=1, P=2P=2, W(x)=|x|²W(\xi)=|\xi|^2, and the &-limit as the sum of the measures of the 2 phases tends to \L(W)\L(\Omega) is identified. Minimizers are fully characterized when N=1N=1, and candidates for solutions are studied for the circle and the square in the plane.     

Special Fast Diffusion with Slow Asymptotics: Entropy Method and Flow on a Riemannian Manifold

We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation ${u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)}We consider the asymptotic behaviour of positive solutions u(t, x) of the fast diffusion equation u_t=D(u^m/m) = div (u^m-1 ?u){u_t=\Delta (u^{m}/m)= {\rm div}\,(u^{m-1} \nabla u)} posed for x ? \mathbb R^d{x\in\mathbb R^d}, t > 0, with a precise value for the exponent m = (d − 4)/(d − 2). The space dimension is d ≧ 3 so that m < 1, and even m = −1 for d = 3. This case had been left open in the general study (Blanchet et al. in Arch Rat Mech Anal 191:347–385, 2009) since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace– Beltrami operator of a suitable Riemannian Manifold (\mathbb R^d,g){(\mathbb R^d,{\bf g})}, with a metric g which is conformal to the standard \mathbb R^d{\mathbb R^d} metric. Studying the pointwise heat kernel behaviour allows to prove suitable Gagliardo–Nirenberg inequalities associated with the generator. Such inequalities in turn allow one to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker–Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of m.     

Symmetries of center singularities of plane vector fields

Let D² ì \mathbbR² {D^2} \subset {\mathbb{R}^2} be a closed unit 2-disk centered at the origin O ? \mathbbR² O \in {\mathbb{R}^2} and let F be a smooth vector field such that O is a unique singular point of F and all other orbits of F are simple closed curves wrapping once around O. Thus, topologically O is a “center” singularity. Let q:D2\{ O } ? ( 0, + ￥ ) \theta :D2\backslash \left\{ O \right\} \to \left( {0, + \infty } \right) be the function associating with each z ≠ O its period with respect to F. In general, such a function cannot be even continuously defined at O. Let also D⁺ (F) {\mathcal{D}^{+} }(F) be the group of diffeomorphisms of D ² that preserve orientation and leave invariant each orbit of F. It is proved that θ smoothly extends to all of D ² if and only if the 1-jet of F at O is a “rotation,” i.e., j¹F(O) = - y\frac??x + x\frac??y {j^1}F(O) = - y\frac{\partial }{{\partial x}} + x\frac{\partial }{{\partial y}} . Then D⁺ (F) {\mathcal{D}^{+} }(F) is homotopy equivalent to a circle.     

Complex order van der Pol oscillator

In this paper a complex-order van der Pol oscillator is considered. The complex derivative D^a±jbD^{\alpha\pm\jmath\beta}, with α,β∈R ⁺ is a generalization of the concept of integer derivative, where α=1, β=0. By applying the concept of complex derivative, we obtain a high-dimensional parameter space. Amplitude and period values of the periodic solutions of the two versions of the complex-order van der Pol oscillator are studied for variation of these parameters. Fourier transforms of the periodic solutions of the two oscillators are also analyzed.     

Generation of dynamic biochemical signals with a tube mixer: effect of dispersion in an oscillatory flow

The generation of controlled dynamic biochemical signals has many applications in the life sciences. This paper presents an analysis of the dispersion of an oscillatory biochemical signal in an incompressible viscous oscillatory flow in a mixing tube. By using the method of Gill and Sankarasubramanian, the dispersion coefficients $K_i(\tau)\;(i=1, 2,\ldots)The generation of controlled dynamic biochemical signals has many applications in the life sciences. This paper presents an analysis of the dispersion of an oscillatory biochemical signal in an incompressible viscous oscillatory flow in a mixing tube. By using the method of Gill and Sankarasubramanian, the dispersion coefficients K_i(t)  (i=1, 2,?)K_i(\tau)\;(i=1, 2,\ldots) are determined as the functions of dimensionless time τ. With the assumptions of quasi-steady flow and steady flow, the dispersion coefficients K_i(t)  (i=1, 2)K_i(\tau)\;(i=1, 2) are simplified. The effects of the frequencies of the oscillatory flow and the oscillatory biochemical signal, and the length of the mixing tube on the average solute concentrations, θ _m , over the tube cross-section at the outlet of the mixing tube are analyzed by numerical simulations with and without the assumptions of the quasi-steady and steady flows. It is concluded that for the dispersion in an oscillatory flow, an excellent accuracy can be achieved by using quasi-steady flow assumption while the steady flow assumption would lead to inaccurate results. However, if the frequency of oscillatory flow is sufficiently high, the steady flow assumption can be used to further simplify the calculation while still maintaining sufficient accuracy. These results are of practical importance in producing dynamic biochemical signals as the stimuli of biological cells by a tube mixer.