Semicircular Region Research Articles

On the limitations of the complex wave velocity for the heterogeneous swirling flows

Abstract The hydrodynamic instability of variable density swirling flows under gravity between two infinite coaxial cylinders is investigated for axisymmetric disturbances. It is shown that the complex wave velocity of any arbitrary unstable axisymmetric mode must lie within the semi-elliptical region whose minor axis depends on the density stratification parameter J(r). The stabilizing effect of density stratification is shown by reducing the semi-circular instability region of [G. K. Batchelor and A. E. Gill, “Analysis of the stability of axisymmetric jets,” J. Fluid Mech., vol. 14, p. 529, 1962]. Furthermore, we have obtained two parabolic instability regions which intersect and reduce the semi-elliptical instability region for density-stratified flows. These parabolic instability regions are uniformly valid for both variable density and density homogeneous flows also.

Viscous shear flow and heating of impact-extruded composite energetic materials

Investigating the macroscopic deformation and localized heating behaviors of high-ductility composite energetic material (CEM) under impact extrusion is crucial to understand the accidental ignition phenomenon of CEM charged weapon systems with structural defects and design the advanced safe munitions. In this study, the rapid viscous shear flow and ignition responses of impact-extruded CEM are first experimentally studied and then simulated using a physically based thermomechanical model. The experimental results show that CEM sample impacted by a 6 kg and ∼2 m/s falling drop-weight could be extruded into the narrow crack with a high flow speed (∼20 m/s) and ignited. A thermomechanical model is developed to describe the continuum viscoelastic-plastic deformation and localized heating due to viscoplastic deformation in solid material, viscous flow in melted liquid and chemical reaction. By simulation, a semicircular extrusion region with a shear concentration is formed in CEM sample above the crack, and high shear stress and strain rate at this region contributes to the localized viscous shear heating and ignition of CEM. The effects of the diameter ratio between crack and sample (0.1∼0.4) on dynamic responses of CEM are further investigated. The simulated main features of CEM sample, including flow history into the crack, ignition time and location, are consistent with the experimental observations.

Exudate identification in retinal fundus images using precise textural verifications

One of the most salient diseases of retina is Diabetic Retinopathy (DR) which may lead to irreparable damages to eye vision in the advanced phases. A large number of the people infected with diabetes experience DR. The early identification of DR signs facilitates the treatment process and prevents from blindness. Hard Exudates (HE) are bright lesions appeared in retinal fundus images of DR patients. Thus, the detection of HEs is an important task preventing the progress of DR. However, the detection of HEs is a challenging process due to their different appearance features. In this paper, an automatic method for the identification of HEs with various sizes and shapes is proposed. The method works based on a pixel-wise approach. It considers several semi-circular regions around each pixel. For each semi-circular region, the intensity changes around several directions and non-necessarily equal radiuses are computed. All pixels for which several semi-circular regions include considerable intensity changes are considered as the pixels located in HEs. In order to reduce false positives, an optic disc localization method is proposed in the post-processing phase. The performance of the proposed method has been evaluated on DIARETDB0 and DIARETDB1 datasets. The experimental results confirm the improved performance of the suggested method in term of accuracy.

Heat transfer characteristics of jet impingement onto the concave surface of a cone

ABSTRACT Heat transfer coefficient measurements for jet impingement onto the concave surface of a cone with apex angles equal to 30° and 70° are reported in this study. Static pressure measurements along the cone surface are also reported which aid in explaining the results from the heat transfer measurements. The conical geometry finds application for the heating of aircraft gas turbine engine nose bullet region which could experience ice formation whenever aircraft operates at higher altitudes. The concave surface of the cone is heated by impinging hot air tapped from the compressor. The heat transfer coefficients, which are significantly different compared to flat surface impingement, and are scarcely available, are reported in this study. The influence of the jet axis coincident with the cone axis and impinging into apex region was studied first. The methodology where a thin stainless-steel foil is heated with a known heat flux by passing electric current through it and then measuring the detailed surface temperature using an infrared thermal camera, was used to compute the wall Nusselt distribution. The Nusselt number variation is presented as a function of nondimensional apex to nozzle spacing (L/D) and the nondimensional axis displacement (O/D). Three different diameter (D equal to 10 mm,14 mm and 20 mm) pipes were used for the concentric impingement case and the Nusselt number distribution was observed not depend on the injection diameter. The jet Reynolds number was varied between 25000 and 82000. The peak Nusselt number values are similar to those for flat plate impingement and do not occur at the cone apex, but at a nondimensional distance between 3.2 and 3.9 for 4.25 < L/D < 10.25 for the 30° cone. The peak Nusselt number values increase by almost 30% with decrease in the L/D from 10.2 to 3.0. Wall static pressure measurements indicated that the peak Nusselt number values occur in regions where the interaction between the incoming and outgoing streams is strongest – these regions occur at distances downstream from the stagnation region toward cone exit. The peak heat transfer coefficient location could not be obtained for the 70° cone, but the measured maximum values are higher by 30–50% compared to those for the 30° case. The jet is able to penetrate further toward the apex of the cone for 70° cone compared to 30° cone shifting the peak heat transfer locations closer to the apex for larger apex angle case. There is insignificant influence of jet diameter on Nusselt number variation for the three diameter values investigated in this study. The influence of displacing the jet axis by a distance equal to half the jet diameter in a direction perpendicular to the cone axis was also studied but detailed measurements were possible only for the 30° cone. Two semi-circular regions, one close to the impingement tube and the other away from the tube, displaying similar heat transfer coefficient behavior were observed. A peak in the Nusselt number occurs in each semi-circular region and the magnitude of the peaks are nearly same and they lie on either side of the peak location observed for the concentric impingement case when the flow geometric parameters are similar for the two cases. The measurements for the 70° cone exhibited all features of the 30° cone but the maximum values were higher by nearly 40% for this case.

Application of scaling methods to foster ground development of active shielding concepts for space exploration

Space radiation in the form of Solar Energetic Protons (SEPs) and Galactic Cosmic Rays (GCR) poses a substantial risk for long-duration space exploration. Active shielding concepts, i.e. shielding a spacecraft with electric and or magnetic fields, has been a topic of interest for more than 50 years. Mass and power requirements have yet to be fulfilled to enable active shielding technology. We present results for a single electric dipole to show the utility of scale-invariance and universal scaling for advancing active shielding concepts. Scattering patterns downstream from the dipole of 2 MeV to 6 MeV electrons and protons show a semi-circular region devoid of incident particles that increases in size with increasing dipole voltage. In particular, results are presented that show how scale-invariant approaches can be leveraged to enable small-scale shields to be built and tested on the ground and then scaled up for in-space use.

Radial interpolation of GPS and leveling data of ground deformation in a resurgent caldera: application to Campi Flegrei (Italy)

This study presents a new method, called the Radial Interpolation Method, to interpolate data characterized by an approximately radial pattern around a relatively constrained central zone, such as the ground deformation patterns shown in many active volcanic areas. The method enables the fast production of short-term deformation maps on the base of spatially sparse ground deformation measurements and can provide uncertainty quantification on the interpolated values, fundamental for hazard assessment purposes and deformation source reconstruction. The presented approach is not dependent on a priori assumptions about the geometry, location and physical properties of the source, except for the requirement of a locally radial pattern, i.e., allowing multiple centers of symmetry. We test the new method on a synthetic point source example, and then, we apply the method to selected time intervals of real geodetic data collected at the Campi Flegrei caldera during the last 39 years, including examples of leveling, Geodetic Precise Traversing measurements and Global Positioning System. The maps of horizontal displacement, calculated inland, show maximum values lying along a semicircular annular region with a radius of about 2–3 km in size. This semi-annular area is marked by mesoscale structures such as faults, sand dikes and fractures. The maps of vertical displacement describe a linear relation between the maximum vertical uplift measured and the volume variation. The multiplicative factor in the linear relation is about 0.3 × 106 m3/cm if we estimate the proportion of the ΔV that is captured by the GPS network onland and we use this to estimate the full ΔV. In this case, the 95% confidence interval on K because of linear regression is ± 5%. Finally, we briefly discuss how the new method could be used for the production of short-term vent opening maps on the base of real-time geodetic measurements of the horizontal and vertical displacements.

Note on Instability Regions in the Circular Rayleigh Problem of Hydrodynamic Stability

Batchelor and Gill’s semicircular bound on the range of the complex wave velocity of an arbitrary unstable mode in the stability problem of inviscid incompressible homogeneous axial flows with respect to axisymmetric disturbances is further reduced. Significance In the circular Rayleigh problem of hydrodynamic stability, we have obtained a parabolic instability region. The significance of this result is that it depends on the discriminant $$\Psi (r)$$ which plays a vital role in deciding whether a basic axial flow is stable or not to axisymmetric disturbances. However, our parabolic instability region is an unbounded one unlike the semicircular instability region of Batchelor and Gill, which has the weakness that it does not depend on $$\Psi (r)$$ . So we have demonstrated that our parabola intersects the semicircle of Batchelor and Gill for two classes of basic flows, and consequently the semicircular instability region of Batchelor and Gill is further reduced. Moreover, we have stated the stability problem consisting of a second-order ODE and associated boundary conditions in three different but related variables. The significance of this is that further analytical results can be obtained for this problem, and some of these are presented in our subsequent paper.

Comparison of Lipid and Water Contents by Time-domain Diffuse Optical Spectroscopy and Dual-energy Computed Tomography in Breast Cancer Patients

We previously compared time-domain diffuse optical spectroscopy (TD-DOS) with magnetic resonance imaging (MRI) using various water/lipid phantoms. However, it is difficult to conduct similar comparisons in the breast, because of measurement differences due to modality-dependent differences in posture. Dual-energy computed tomography (DECT) examination is performed in the same supine position as a TD-DOS measurement. Therefore, we first verified the accuracy of the measured fat fraction of fibroglandular tissue in the normal breast on DECT by comparing it with MRI in breast cancer patients (n = 28). Then, we compared lipid and water signals obtained in TD-DOS and DECT from normal and tumor-tissue regions (n = 16). The TD-DOS breast measurements were carried out using reflectance geometry with a source–detector separation of 3 cm. A semicircular region of interest (ROI), with a transverse diameter of 3 cm and a depth of 2 cm that included the breast surface, was set on the DECT image. Although the measurement area differed between the modalities, the correlation coefficients of lipid and water signals between TD-DOS and DECT were rs = 0.58 (p &lt; 0.01) and rs = 0.90 (p &lt; 0.01), respectively. These results indicate that TD-DOS captures the characteristics of the lipid and water contents of the breast.

An improved instability region for the extended Rayleigh problem of hydrodynamic stability

For the extended Rayleigh problem of hydrodynamics stability dealing with homogeneous shear flows with variable cross section, we have obtained a parabolic instability region. This improved parabolic instability region intersects with the semicircular instability region under certain condition. The validity of the result is illustrated with an example of basic flows. Furthermore, we have obtained a bound for the complex part of the phase velocity.

18F-Fluoride Positron Emission Tomographic Imaging of Penile Arteries and Erectile Dysfunction

BackgroundFluorine-18 sodium fluoride (NaF), a bone-seeking radiopharmaceutical used to detect osseous metastases, localizes in regions of microcalcification in atherosclerosis. ObjectivesTo determine if atherosclerosis of penile arteries plays a role in erectile dysfunction (ED), this study analyzed NaF images in prostate cancer patients. MethodsNaF positron emission tomography–computed tomography bone scans were evaluated in 437 prostate cancer patients (age 66.6 ± 8.7 years). Their urologic histories were reviewed for prevalent ED (diagnosed before the scan date) or incident ED (no ED at first scan, but developed during 1-year follow-up); patients with no ED (neither before the scan nor during follow-up) were included as a control group. A semicircular region of interest was set on the dorsal one-half of the penis (to avoid residual excreted activity in the urethra) on 5 contiguous slices at the base of the penis on positron emission tomography–computed tomography coronal reconstructions, and the average standardized uptake value (SUVmax) was described as NaF uptake. ResultsOf 437 patients, 336 (76.9%) had prevalent ED, 60 incident ED (13.7%), and 41 had no ED (9.4%). SUVmax in patients with prevalent (median 1.88; interquartile range [IQR]: 1.67 to 2.16) or incident (median 1.86; IQR: 1.72 to 2.08) ED was significantly higher than no ED (median 1.42; IQR: 1.25 to 1.54) patients (p < 0.001). After adjustment for other risk factors, the odds ratio of prevalent or incident ED was 25.2 (95% confidence interval: 9.5 to 67.0) for every 0.5-U increment in SUVmax with receptor operating characteristic area of 0.91 (95% confidence interval: 0.88 to 0.94). ConclusionsNaF uptake in penile vessels suggests that atherosclerosis is associated with ED in prostate cancer patients. The importance of NaF uptake needs to be tested in noncancer subjects and cause-effect relationship needs to be established.

Temperature-compensated refractive index sensor based on bent-fiber interference

In this letter, we present a hybrid single-mode fiber (SMF) structure for refractive index (RI) measurement with the temperature compensation capability. The interferometer has simple structure, consisting of a bare fiber semicircular bending region (SBR) and a fiber Bragg grating (FBG). The section of SBR serves as a refractometer with a temperature compensator using a simple FBG. The sensing mechanism of this device has been investigated and experimentally demonstrated. The FBG structure is insensitive towards the change in the ambient RI but is sensitive towards the changes in the ambient temperature whereas the SBR structure is sensitive towards both of RI and temperature. In the RI range of 1.3288–1.3730, the corresponding RI sensitivity for the bare fiber bending region is 160 nm/RIU (refractive index unit) and the RI resolution is about 8.5 × 10−5 RIU.

The initiation of a planar fluid plume beneath a rigid lid

The unsteady growth of a viscous fluid plume beneath a rigid upper lid is investigated. Two-dimensional (planar) flow is assumed, through a fissure in the horizontal lower boundary. Initially, the fluid exiting the bottom is assumed to form a semi-circular region, but rises as time progresses, and spreads across the upper boundary. The problem is modelled using Boussinesq theory, and solved using a time-dependent spectral method. These numerical solutions are also compared with the results of a simpler inviscid asymptotic solution. Results are indicated for different input fluid speeds and fissure widths.

Electrochemical characteristics of Ti-6Al-4V after plasma electrolytic oxidation in solutions containing Ca, P, and Zn ions

The electrochemical characteristics of Ti-6Al-4V after plasma electrolytic oxidation (PEO) in solutions containing Ca, P, and Zn ions were studied using various experimental techniques. A Ti-6Al-4V ELI disk was used as a substrate for plasma electrolytic oxidation. A pulsed DC power supply was used to apply a potential of 280V in the electrolyte for 3min. To analyze the electrochemical properties of the PEO film as a function of the Zn content, potentiodynamic and AC impedance tests were carried out with a potentiostat (Model 2273, EG&G Co., USA) using a 0.9% NaCl solution. To examine the bioactivity, the PEO films formed on the specimens and the untreated Ti-6Al-4V specimens were immersed in a simulated body fluid (SBF) for 12h. The PEO-treated and SBF-immersed surfaces were observed by X-ray diffraction, field-emission scanning electron microscopy, energy-dispersive X-ray spectroscopy, and attenuated total reflectance Fourier transform infrared spectroscopy.The number of pores in the PEO films increased, whereas the pore size decreased slightly, as the Zn ion concentration increased. The corrosion potential of the PEO-treated films decreased, while the corrosion current density increased slightly, as the Zn content increased. As shown by the Nyquist plots, the semicircular region in the profile of the Ti-6Al-4V specimens increased compared to that of the other specimens. The films obtained after PEO treatment in the solution containing Zn showed Warburg impedance characteristics. Bone-like apatite was well-formed in specimens of PEO films treated in solutions containing Zn ions at concentrations of 0, 5, and 10mM%. However, formation of the bone-like apatite decreased in specimens of the PEO films treated in a solution containing Zn ions at a concentration of 20mM%.

(Invited) Cantilever-Based Resonant Chemical Microsensors

This paper reviews cantilever-based resonant chemical microsensors, which detect an analyte of interest via a change of the resonance frequency of a characteristic vibration mode, caused by the added mass upon sorption of the analyte into a sensing film deposited on the cantilever surface. Besides ease of fabrication, such mass-sensitive chemical microsensors benefit from a well-understood transduction mechanism and the fact that frequencies and frequency changes can be sensed with high accuracy. While the focus of this paper is on silicon-based, hammer-like resonant sensors that comprise a head region suspended by a cantilever support structure, the results can be easily applied to other material systems and resonator geometries as well. The cantilever vibrations are excited electrothermally and sensed using four piezoresistors arranged in a Wheatstone bridge. The location of the piezoresistors is chosen to sense the vibration mode of interest, while rejecting signals stemming from other possible vibration modes. The particular cantilever sensors with semi-circular head region discussed in the paper are operated at their fundamental in-plane vibration mode at frequencies between 300-800 kHz and exhibit Q-factors in air as high as 5,000, resulting in short-term frequency stabilities in the 10-8 range. Approaches to address the three ‘S’ of chemical sensors, i.e., sensitivity, selectivity, and stability are highlighted. To increase the sensor sensitivity, designs that maximize the Q-factor of the chosen vibration mode and, thus, the short-term frequency stability of the resonator coated with the sensing film are explored. Along these lines, in-plane vibration modes, in which the cantilever slices through the surrounding air (or liquid) are preferred over the more conventional out-of-plane vibration modes. In addition, deposition of the sensing film away from region that are stressed during cantilever vibration minimizes the impact of the (polymeric) sensing film on the Q-factor and frequency stability. This enables larger sensing film volumes, which in combination with minimizing non-active sensor mass, yields higher sensor sensitivities. Designs featuring a recessed head region enable ink jetting of the sensing film while minimizing the silicon mass. To improve the microsensor selectivity, signal transients are generated on-chip using embedded heating resistors. These heaters uniformly heat the head portion of the cantilever with millisecond thermal time constants, enabling to monitor analyte desorption and re-absorption into the polymeric sensing films in real time and without the need of a complex gas mixing system. The generated sorption transients depend on the polymer/analyte-specific diffusion coefficients and, thus, help to distinguish between analyte of interest and possible interferents. Finally, differential setups combining a coated resonator with an uncoated reference resonator are explored to improve the long-term stability of the sensing system. Similarly, the thermally induced analyte desorption from the sensing film can be used to monitor a baseline frequency to improve system stability.

A New Approach to Thermal Analysis of a Multilayered Cylindrical Structure With Imperfect Bonds and Internal Heat Source

In the present research, a new and straightforward mathematical model, named augmented state-space method, is introduced to solve the heat conduction equation for a multilayered orthotropic hollow cylinder with bonding imperfection in the presence of heat source. Since such problems including heat source are inherently inhomogeneous and complex, augmented state-space method converts these inhomogeneous equations into homogeneous ones. The transient solution will be achieved by present method based on laminate approximation theory in the Laplace domain, and then the solutions obtained are retrieved into the time domain by applying the numerical Laplace transform inversion. All material properties can be considered to vary continuously within the cylinder along the radial direction with arbitrary grading pattern. Based on the proposed method, the solution of heat conduction problem can be also obtained for general boundary conditions which may be included various combinations of arbitrary temperature, flux, or convection. Due to lack of any data on the transient thermal analysis corresponding to problems with imperfect bonds in the cylindrical coordinate system (r,θ), comparison is carried out with the available results for the three-layer semi-circular annular region with perfect bonds in the literature. Finally, the influence of orthotropy and interface imperfection on the distribution of the temperature field for three-layer hollow cylinder, in which the second layer is made of orthotropic functionally graded material (FGM), will be visualized.

Optimization of long-period grating-based refractive index sensor by bent-fiber interference.

In this paper, we propose and demonstrate a novel approach to enhance the refractive index (RI) sensitivity and eliminate the temperature cross-sensitivity of a long-period grating (LPG) -based refractive index sensor by bent-fiber interference. The approach is based on a hybrid structure composed of an LPG and a bent-fiber intermodal interferometer. The bent-fiber intermodal interferometer has a simple structure, which consists of a bare fiber semi-circular bending region with a 5 mm bending radius. As the RI increases, the resonance wavelength of the LPG moves toward a shorter wavelength, while the resonance wavelength of the bent-fiber intermodal interferometer shifts to a longer wavelength. The separation of two resonance dips increases with the RI; using two resonance dips allows us to measure an RI with a higher sensitivity than if we had only used one resonance dip. However, as the temperature increases, the separation of the two resonance dips is constant. This approach can effectively enhance the RI sensitivity and eliminate temperature cross-sensitivity.

Bent Fiber Interferometer

We report a novel fiber-optic interferometer based on Mach–Zehnder interferometer (MZI) from a bent single-mode fiber (SMF) and experimentally demonstrated that it can be used as a fiber-optic refractometer. The interferometer has a simple structure including a standard SMF with a bare fiber semicircular bending region of a given diameter. A fiber MZI is formed by the different optical paths taken by the core and cladding modes from the designed bending fiber structure. We also realize a dual-channel interferometer by using a 9 and 11 mm bending diameter fiber structure and measure the refractive index of the ambient environment by monitoring the wavelength shift of the resonance dips in its transmission spectrum. Additionally, we investigate the relationship between the wavelength shifts of the resonant dips with temperature changes. This dual-channel interferometer can be potentially used for simultaneous measurement of dual chemical parameters.

Effect of LiCF3SO3 on L-Chitosan/PMMA Blend Polymer Electrolytes

This paper focuses on the effect of lithium triflate (LiCF3SO3) on the structural and conduction properties of lauroyl (L)-chitosan/poly(metylmethacryalate) (PMMA)-based polymer electrolytes. Films of L-chitosan/PMMA blends and its complexes were prepared using a solution-casting technique. The ionic conductivity of the system was measured over a wide range of frequency between 50 Hz-1 MHz. Impedance plot for the samples demonstrates two well-defined regions. The disappearance of the high frequency semicircular region led to a conclusion that the current carriers are ions. Sample with 30 wt% of LiCF3SO3 showed the highest conductivity of 7.59 ± 3.64 × 10−4 Scm−1 at room temperature. This is consistent with the results obtained from infrared spectroscopy.

On the Stability of a Compressible Axial Flow with an Axial Magnetic Field

We consider the stability problem of inviscid compressible axial flows with axial magnetic fields following the work of Dandapat and Gupta (Quarterly of Applied Mathematics, 1975). A numerical study of the stability of some basic flows has been carried out and it is found that an increase in the magnetic field strength has a stabilizing effect on subsonic flows and a destabilizing effect on supersonic flows. An analytical study of the stability problem has also been done in the present paper, but this analytical study is restricted by the approximation <svg style="vertical-align:-0.04095pt;width:49.612499px;" id="M1" height="11.225" version="1.1" viewBox="0 0 49.612499 11.225" width="49.612499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28&#xA;q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g><g transform="matrix(.017,-0,0,-.017,22.127,11.113)"><path id="x226A" d="M512 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58zM780 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58z" /></g><g transform="matrix(.017,-0,0,-.017,41.386,11.113)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g> </svg> and <svg style="vertical-align:-3.3907pt;width:43.037498px;" id="M2" height="15.1375" version="1.1" viewBox="0 0 43.037498 15.1375" width="43.037498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path id="x1D450" d="M383 397q0 -32 -35 -49q-12 -6 -23 8q-37 45 -84 45t-90 -71q-40 -65 -40 -167q0 -57 22 -86t59 -29q38 0 81.5 24.5t69.5 51.5l16 -21q-44 -53 -104 -84t-109 -31q-56 0 -89.5 41t-33.5 117q0 61 30 124t79 105q33 28 81 50.5t86 22.5q34 0 59 -15.5t25 -35.5z" /></g> <g transform="matrix(.012,-0,0,-.012,6.962,14.938)"><path id="x1D456" d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30&#xA;q16 0 76 48z" /></g> <g transform="matrix(.017,-0,0,-.017,15.563,10.862)"><use xlink:href="#x226A"/></g><g transform="matrix(.017,-0,0,-.017,34.805,10.862)"><use xlink:href="#x31"/></g> </svg>, where <svg style="vertical-align:-0.0pt;width:17.475px;" id="M3" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.113)"><use xlink:href="#x1D440"/></g> </svg> is the Mach number and <svg style="vertical-align:-3.3907pt;width:10.9px;" id="M4" height="11.9625" version="1.1" viewBox="0 0 10.9 11.9625" width="10.9" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D450"/></g> <g transform="matrix(.012,-0,0,-.012,6.962,11.762)"><use xlink:href="#x1D456"/></g> </svg> is the imaginary part of the complex phase velocity <svg style="vertical-align:-0.1638pt;width:7.0250001px;" id="M5" height="7.9499998" version="1.1" viewBox="0 0 7.0250001 7.9499998" width="7.0250001" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><use xlink:href="#x1D450"/></g> </svg>. A semicircular region depending on the magnetic field parameter and the Mach number is found for subsonic disturbances and as a consequence it is found that sufficiently strong magnetic field stabilizes all subsonic disturbances. Under a weak magnetic field, it is shown that short subsonic disturbances are stable.

Analytical solutions to time-fractional partial differential equations in a two-dimensional multilayer annulus

In this paper, analytical solutions to time-fractional partial differential equations in a multi-layer annulus are presented. The final solutions are obtained in terms of Mittag-Leffler function by using the finite integral transform technique and Laplace transform technique. In addition, the classical diffusion equation (α=1), the Helmholtz equation (α→0) and the wave equation (α=2) are discussed as special cases. Finally, an illustrative example problem for the three-layer semi-circular annular region is solved and numerical results are presented graphically for various kind of order of fractional derivative.