Analyzing the longitudinal effect of hypersonic flow past a conical cone via the perturbation method

To analyze the hypersonic flow past a conical cone, the variations of gasdynamic properties subjected to the longitudinal curvature effect by using the perturbation method. An outer perturbation expansion has been carried out by recent researchers, but a problem occurred, the outer expansion solutions are not uniformly valid in the shock layer, however, the outcome near the conical body surface called vortical layer remains deflective. This study intends to discover uniformly valid analytical solutions in the shock layer by applying the inner perturbation expansions matching with the out expansions to analyze the characteristics in the whole region including shock layer and vortical layer. Starting from the zero-order approximate solutions for hypersonic conical flow is then applied as the basic solutions for the outer perturbation expansions of a flow field. The governing equations and boundary conditions are also expanded via outer perturbations. Using an approximate analytical scheme in the derivation process, first-order perturbation equations can be simplified and the approximate closed-form solutions are obtained; furthermore, the various flow field quantities, including the normal force coefficient on the cone surface, have been calculated. According to the variations of gasdynamic properties, the longitudinal curvature effect for the hypersonic flow past a conical cone can be determined. Thicknesses of shock layer and vortical layer can be predicted as well. The physical phenomena inside both layers can be investigated carefully, the conditions for an elliptic cone with longitudinal curvature, m = 1 and n = 2 and other conditions of parameters; the perturbation parameter, ε_m2 = 0.1, semi-vertex angle of the unperturbed cone, δ = 10°, and hypersonic similarity parameter, K_δ = M_∞δ = 1.0, the thickness of vortical layer, η_VL, can be calculated at the position angle of conical cone body, ? = 30° was demonstrated here. Results show how very thin the vortical layer is approximately only 10% of the shock layer close to the body, the pressure in the whole shock layer is verified to be uniformly valid which agrees with previous studies. Large gradient changes in entropy and density are found when the flow approaches the cone surface, the most important is, this method provides a benchmark solution to the hypersonic flow past a conical cone and to assist the grids and numerics for numerical computation should be fashioned to accommodate the whole flow field region including the vortical layer of rapid adjustment, and let the analysis become more effective and low cost.     

Numerical study of pressure distribution in entrance pipe flow

This article describes the computation of pipe flow in the entrance region. The pressure distribution and flow characteristics, particularly the effect of vorticity in the vicinity of the wall, were analyzed for moderate Reynolds numbers (Re) ranging from 500 to 10,000. It was found, for the first time, that a large pressure gradient in the radial direction exists near the pipe inlet. The pressure gradient is caused by the radial component of the curl of vorticity, which decreases as Re increases. The pressure at the wall is lower than that at the central core for Re ≤$\le$ 5000. This result is beyond the scope of the boundary-layer assumption for pressure, although it applies to flows at high Reynolds numbers.     

Analysis of a conducting fluid in a thin annulus with rotating insulated walls under radial magnetic effect

This work considers an electrically conducting fluid filled between two concentric cylindrical walls relatively close to each other. A theoretical solution for the steady Taylor–Couette flow between these two electrically insulated rotating cylinders under the influence of a radial magnetic field is provided in this work. By solving the appropriate set of governing equations simultaneously, the profiles of fluid tangential velocity component and induced magnetic field were obtained as complicated functions involving the modified Bessel functions of the first and second kinds of the first-order in terms of radial coordinates and Hartmann number. A computational study was also performed to validate the present theoretical solution. The analytical and computational results are identical when Ha = 1 while these results only slightly deviate from each other as Ha increases. Current results show that, the presence of the external magnetic field causes the flow close to the slower cylinder to accelerate while that close to the faster cylinder to decelerate. This has clearly implied the fact that an external magnetic field tends to make the velocity distribution across the inner and outer cylinders more uniform.     

On the diameters of commuting graphs

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let D be a division ring and n ? 3. In this paper we investigate the diameters of Γ(M_n(D)) and determine the diameters of some induced subgraphs of Γ(M_n(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in M_n(D). For every field F, it is shown that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 6. We conjecture that if Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) ? 5. We show that if F is an algebraically closed field or n is a prime number and Γ(M_n(F)) is a connected graph, then diam Γ(M_n(F)) = 4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.     

Analysis of flow and thermal field in nanofluid using a single phase thermal dispersion model

Flow and thermal field in nanofluid is analyzed using single phase thermal dispersion model proposed by Xuan and Roetzel [Y. Xuan, W. Roetzel, Conceptions for heat transfer correlation of nanofluids, Int. J. Heat Mass Transfer 43 (2000) 3701–3707]. The non-dimensional form of the transport equations involving the thermal dispersion effect is solved numerically using semi-explicit finite volume solver in a collocated grid. Heat transfer augmentation for copper–water nanofluid is estimated in a thermally driven two-dimensional cavity. The thermo-physical properties of nanofluid are calculated involving contributions due to the base fluid and nanoparticles. The flow and heat transfer process in the cavity is analyzed using different thermo-physical models for the nanofluid available in literature. The influence of controlling parameters on convective recirculation and heat transfer augmentation induced in buoyancy driven cavity is estimated in detail. The controlling parameters considered for this study are Grashof number (10³ < Gr < 10⁵), solid volume fraction (0 < ? < 0.2) and empirical shape factor (0.5 < n < 6). Simulations carried out with various thermo-physical models of the nanofluid show significant influence on thermal boundary layer thickness when the model incorporates the contribution of nanoparticles in the density as well as viscosity of nanofluid. Simulations incorporating the thermal dispersion model show increment in local thermal conductivity at locations with maximum velocity. The suspended particles increase the surface area and the heat transfer capacity of the fluid. As solid volume fraction increases, the effect is more pronounced. The average Nusselt number from the hot wall increases with the solid volume fraction. The boundary surface of nanoparticles and their chaotic movement greatly enhances the fluid heat conduction contribution. Considerable improvement in thermal conductivity is observed as a result of increase in the shape factor.     

A non-Newtonian fluid flow model for blood flow through a catheterized artery—Steady flow

In this paper the effects catheterization and non-Newtonian nature of blood in small arteries of diameter less than 100 μm, on velocity, flow resistance and wall shear stress are analyzed mathematically by modeling blood as a Herschel–Bulkley fluid with parameters n and θ and the artery and catheter by coaxial rigid circular cylinders. The influence of the catheter radius and the yield stress of the fluid on the yield plane locations, velocity distributions, flow rate, wall shear stress and frictional resistance are investigated assuming the flow to be steady. It is shown that the velocity decreases as the yield stress increases for given values of other parameters. The frictional resistance as well as the wall shear stress increases with increasing yield stress, whereas the frictional resistance increases and the wall shear stress decreases with increasing catheter radius ratio k (catheter radius to vessel radius). For the range of catheter radius ratio 0.3–0.6, in smaller arteries where blood is modeled by Herschel–Bulkley fluid with yield stress θ = 0.1, the resistance increases by a factor 3.98–21.12 for n = 0.95 and by a factor 4.35–25.09 for n = 1.05. When θ = 0.3, these factors are 7.47–124.6 when n = 0.95 and 8.97–247.76 when n = 1.05.     

Numerical computation of end plate effect on Taylor vortices between rotating conical cylinders

End plate effect on Taylor vortices between rotating conical cylinders is studied by numerical method in this paper. We suppose that the inner cone rotates together with the end plate at the top and the outer one as well as the end plate at the bottom remains at rest. It is found that the instability sets in at a critical Reynolds number about Re = 80. Increase Re to about Re = 200 the first single clockwise vortex is formed near the top of the flow system. Further increase Re to about Re = 440 another clockwise vortex is formed under the first one. At about Re = 448 the third vortex is formed which rotates in counterclockwise direction between the first two vortices. With increasing of Re the process continues. Finally, a configuration is obtained with an odd number of vortices in the annulus at about Re = 700, which confirms the experimental observation. Moreover, the local extreme values of pressure and velocity are achieved at the adjacent lines between neighboring vortices or at the medium lines of vortices. The effect of gap size on vortices is also considered. It is shown that the number of vortices increases with decreasing of the gap size and the end plates play an important role in the parity of the number of the vortices.     

k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A − λB ∈ Γ_n if and only if ?(A) − λ?(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that ? ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k−1 = 1 such that either ?(A) = cPAP⁻¹ for every A ∈ M_n, or ?(A) = cPA^TP⁻¹ for every A ∈ M_n.     

Steady magnetohydrodynamic flow in a diverging channel with suction or blowing

An analysis is made of steady two-dimensional divergent flow of an electrically conducting incompressible viscous fluid in a channel formed by two non-parallel walls, the flow being caused by a source of fluid volume at the intersection of the walls. The fluid is permeated by a magnetic field produced by an electric current along the line of intersection of the channel walls. The walls are porous and subjected to either suction (k > 0) or blowing (k < 0) of equal magnitude on both the walls. It is found that when the Reynolds number for the flow is large and the magnetic Reynolds number is very small, boundary layers are formed on the channel walls such that a sufficient condition for the existence of a unique boundary layer solution (without separation) in the case of suction is N > 2, N being the magnetic parameter. When k = 0, boundary layer exists without separation only when N > 2. Further, it is found that the necessary and sufficient condition for the existence of a unique solution for boundary layer flow (without separation) even in the presence of blowing (k < 0) is N > 2. For given value of k, velocity at a point increases with increase in N. It is also shown that when N > 2, blowing makes the boundary layer thinner. A similarity solution for steady temperature distribution in the divergent flow is also presented when the channel walls are held at variable temperature. It is found that for fixed value of wall suction, temperature at a point decreases with increase in N. It is further shown that when N > 2, steady distribution of temperature exists even in the case of blowing at the walls.     

Numerical modelling of aeroelastic behaviour of an airfoil in viscous incompressible flow

In this paper, the problem of the numerical approximation of a two-dimensional incompressible viscous fluid flow interacting with a flexible structure is considered. Due to high Reynolds numbers in the range 10⁴ − 10⁶ the turbulent character of the flow is considered and modelled with the aid of Reynolds equations coupled with the k − ω turbulence model. The structure motion is described by a system of ordinary differential equations for three degrees of freedom: vertical displacement, rotation and rotation of the aileron. The problem is discretized in space by the Galerkin Least-Squares stabilized finite element method and the computational domain is treated with the aid of Arbitrary Lagrangian Eulerian method.     

Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations

We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three: a suitable weak solution is regular near an interior point z if the scaled -norm of the velocity with 1?3/p+2/q?2, 1?q?∞ is sufficiently small near z and if the scaled -norm of the magnetic field with 1?3/l+2/m?2, 1?m?∞ is bounded near z. Similar results are also obtained for the vorticity and for the gradient of the vorticity. Furthermore, with the aid of the regularity criteria, we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations.     

Inverse heat conduction problem in a thin circular plate and its thermal deflection

An inverse problem of transient heat conduction in a thin finite circular plate with the given temperature distribution on the interior surface of a thin circular plate being a function of both time and position has been solved with the help of integral transform technique and also determine the thermal deflection on the outer curved surface of a thin circular plate defined as 0 ? r ? a, 0 ? z ? h. The results, obtained in the series form in terms of Bessel’s functions, are illustrated numerically.     

Optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy

The main purpose of this paper is to investigate the optimal retailer’s replenishment decisions under two levels of trade credit policy within the economic production quantity (EPQ) framework. We assume that the supplier would offer the retailer a delay period and the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer’s replenishment model under the replenishment rate is finite. Furthermore, we assume that the retailer’s trade credit period offered by supplier M is not shorter than the customer’s trade credit period offered by retailer N (M ? N). Since the retailer cannot earn any interest in this situation, M < N.     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

Stagnation-point flow and heat transfer over an exponentially shrinking sheet

An analysis is carried out to investigate the stagnation-point flow and heat transfer over an exponentially shrinking sheet. Using the boundary layer approximation and a similarity transformation in exponential form, the governing mathematical equations are transformed into coupled, nonlinear ordinary differential equations which are then solved numerically by a shooting method with fourth order Runge-Kutta integration scheme. The analysis reveals that a solution exists only when the velocity ratio parameter satisfies the inequality −1.487068 ? c/a. Also, the numerical calculations exhibit the existence of dual solutions for the velocity and the temperature fields; and it is observed that their boundary layers are thinner for the first solution (in comparison with the second). Moreover, the heat transfer from the sheet increases with an increase in c/a for the first solution, while the heat transfer decreases with increasing c/a for the second solution, and ultimately heat absorption occurs.     

Smoothness of the trajectories of ideal fluid particles with Yudovich vorticities in a planar bounded domain

We consider the incompressible Euler equations in a (possibly multiply connected) bounded domain Ω of R², for flows with bounded vorticity, for which Yudovich (1963) proved in [29] global existence and uniqueness of the solution. We prove that if the boundary ∂Ω of the domain is C^∞ (respectively Gevrey of order M?1) then the trajectories of the fluid particles are C^∞ (respectively Gevrey of order M+2). Our results also cover the case of “slightly unbounded” vorticities for which Yudovich (1995) extended his analysis in [30]. Moreover if in addition the initial vorticity is Hölder continuous on a part of Ω then this Hölder regularity propagates smoothly along the flow lines. Finally we observe that if the vorticity is constant in a neighborhood of the boundary, the smoothness of the boundary is not necessary for these results to hold.     

Stability of Taylor–Couette and Dean flow: A semi-analytical study

An alternative method is presented for solving the eigenvalue problem that governs the stability of Taylor–Couette and Dean flow. The eigenvalue problems defined by the two-point boundary value problems are converted into initial value problems by applying unit disturbance method developed by Harris and Reid [27] in 1964. Thereafter, the initial value problems are solved by differential transform method in series and the eigenvalues are computed by shooting technique. Critical wave number and Taylor number for Taylor–Couette flow are computed for a wide range of rotation ratio (μ), −4 ? μ ? 1 (first mode) and −2 ? μ ? 1 (second mode). The radial eigenfunction and cell patterns are presented for μ = −1, 0, 1. Also, we have computed critical wave number and Dean number successfully.     

On the security analysis of an image scrambling encryption of pixel bit and its improved scheme based on self-correlation encryption

An image scrambling encryption scheme for pixel bits was presented by Ye [Ye GD. Image scrambling encryption algorithm of pixel bit based on chaos map. Pattern Recognit Lett 2010;31:347-54], which can be seen as one kind of typical binary image scrambling encryption considering from the bit-plain of size M × (8N). However, recently, some defects existing in the original image encryption scheme, i.e., Ye’s scheme, have been observed by Li and Lo [Li CQ, Lo KT. Optimal quantitative cryptanalysis of permutation-only multimedia ciphers against plaintext attacks. Signal Process 2011;91:949-54]. In the attack proposed by Li and Lo at least 3 + ⌈log₂(MN)⌉ plain images of size M × N are used to reveal the permutation matrix W = [w(i, k)] (i ∈ {1, 2, … , M}; k ∈ {1, 2, … , 8N}) which can be applied to recover the exact plain image. In the current paper, at first, one type of special plain image/cipher image is used to analyze the security weakness of the original image scrambling scheme under study. The final encryption vectors TM and TN or the decryption vectors TM′ and TN′ are revealed completely according to our attack. To demonstrate the performance of our attack, a quantified comparison is drawn between our attack and the attack proposed by Li and Lo. Compared with Li and Lo’s attack, our attack is more efficient in the general conditions. In particular, when the sizes of images satisfy the condition M = N or M ? 8N, the number of the used plain images/cipher images is at most 9, which is sharply less than 3 + ⌈log₂(MN)⌉ when M and N are of large size. To overcome the weaknesses of the original scheme, in this paper, an improved image scrambling encryption scheme is proposed. In the improved scheme, the idea of the “self-correlation” method is used to resist the chosen-plaintext attack/known-plaintext attack. The corresponding simulations and analyses illustrate that the improved encryption method has good cryptographic properties, and can overcome the weakness of the original image encryption scheme. Finally, farther improvement is briefly presented for the future work.     

Batch delivery scheduling with batch delivery cost on a single machine

We consider a scheduling problem in which n independent and simultaneously available jobs are to be processed on a single machine. The jobs are delivered in batches and the delivery date of a batch equals the completion time of the last job in the batch. The delivery cost depends on the number of deliveries. The objective is to minimize the sum of the total weighted flow time and delivery cost. We first show that the problem is strongly NP-hard. Then we show that, if the number of batches is B, the problem remains strongly NP-hard when B ? U for a variable U ? 2 or B ? U for any constant U ? 2. For the case of B ? U, we present a dynamic programming algorithm that runs in pseudo-polynomial time for any constant U ? 2. Furthermore, optimal algorithms are provided for two special cases: (i) jobs have a linear precedence constraint, and (ii) jobs satisfy the agreeable ratio assumption, which is valid, for example, when all the weights or all the processing times are equal.     

Numerical investigation of a three-dimensional four field model for collisionless magnetic reconnection

In this paper we present the numerical investigation of a three-dimensional four field model for magnetic reconnection in collisionless regimes. The model describes the evolution of the magnetic flux and vorticity together with the perturbations of the parallel magnetic and velocity fields. We explored the different behavior of vorticity and current density structures in low and high β regimes, β being the ratio between the plasma and magnetic pressure. A detailed analysis of the velocity field advecting the relevant physical quantities is presented. We show that, as the reconnection process evolves, velocity layers develop and become more and more localized. The shear of these layers increases with time ending up with the occurrence of secondary instabilities of the Kelvin-Helmholtz type. We also show how the β parameter influences the different evolution of the current density structures, that preserve for longer time a laminar behavior at smaller β values. A qualitative explanation of the structures formation on the different z-sections is also presented.