Uniform Aperture Research Articles

Supramolecular Polymer Frameworks with Controlled and Uniform Pore Apertures.

Porous frameworks with controlled pore structure and tunable aperture are greatly demanded. However, precise synthesis of this kind of materials is a formidable challenge. Herein, we report the fabrication of two-dimensional (2D) supramolecular polymer frameworks using a precisely synthesized rod-like helical polyisocyanide as link. Four three-arm star-shaped polyisocyanides with the degree of the polymerization of 10, 20, 30 and 40, and having 2-ureido-4[1H]-pyrimidinone (UPy) terminals were synthesized. 2D-Crystalline polymer frameworks with apertures of 5.3, 10.1, 13.9, and 19.1 nm were respectively obtained through intermolecular hydrogen bonding interaction between the terminal Upy units. The pore aperture is dependent on the length of polyisocyanide backbone. Thus, well-defined supramolecular polymer frameworks with controlled and uniform hexagonal pores were obtained, as proved by small-angle X-ray scattering (synchrotron radiation facility), atomic force microscopy, and Brunauer-Emmett-Teller analyses. The frameworks with uniform large pore aperture were used to purify nanomaterials and immobilize biomacromolecules. For instance, the membranes of the polymer frameworks could size-fractionation of silver nanoparticles into uniform nanoparticles with very low dispersity. The frameworks with large aperture facilitated the inclusion of myoglobin and enhanced the stability and catalytic activity.

REV and Three-Dimensional Permeability Tensor of Fractured Rock Masses with Heterogeneous Aperture Distributions

This study performed a representative elementary volume (REV) and 3D equivalent continuum study of rock fractures based on fluid simulations of 3D discrete fracture networks (DFNs). A series of 3D DFNs with heterogeneous aperture distributions (the DFN-H model) and uniform apertures (the DFN-I model) were established, in which the fractures were oriented according to the geological field mapping of a high-level radioactive waste candidate site in China. The 3D DFNs of the different model sizes were extracted and rotated in a number of directions to check whether there was a tensor quality of the permeability at a certain scale. The results show that aperture heterogeneity increases the REV size and results in a necessarily larger model size to reach an equivalent continuum behavior, and this effect is more obvious when the fracture density is smaller. The shape of the 2D permeability contour is irregular, with some breaks when the model size is small. As the model size increases, its shape gradually tends to become smooth and approaches an ellipse. The shape of the permeability contours of the DFN-H model is slender compared to the DFN-I model, indicating a larger difference between the minimum and maximum values of the permeability. For the DFN-H model, there is no appropriate approximation for the equivalent permeability tensor over the studied model size range, whereas a good fit of the permeability ellipsoid is obtained for the DFN-I model, and the 3D directional permeability is calculated at this model scale. The corresponding magnitude and direction of the principal permeability are obtained, which can be viewed as the equivalent permeability tensor for the approximated continuum medium.

Supramolecular Polymer Frameworks with Controlled and Uniform Pore Apertures

AbstractPorous frameworks with controlled pore structure and tunable aperture are greatly demanded. However, precise synthesis of this kind of materials is a formidable challenge. Herein, we report the fabrication of two‐dimensional (2D) supramolecular polymer frameworks using a precisely synthesized rod‐like helical polyisocyanide as link. Four three‐arm star‐shaped polyisocyanides with the degree of the polymerization of 10, 20, 30 and 40, and having 2‐ureido‐4[1H]‐pyrimidinone (UPy) terminals were synthesized. 2D‐Crystalline polymer frameworks with apertures of 5.3, 10.1, 13.9, and 19.1 nm were respectively obtained through intermolecular hydrogen bonding interaction between the terminal Upy units. The pore aperture is dependent on the length of polyisocyanide backbone. Thus, well‐defined supramolecular polymer frameworks with controlled and uniform hexagonal pores were obtained, as proved by small‐angle X‐ray scattering (synchrotron radiation facility), atomic force microscopy, and Brunauer–Emmett–Teller analyses. The frameworks with uniform large pore aperture were used to purify nanomaterials and immobilize biomacromolecules. For instance, the membranes of the polymer frameworks could size‐fractionation of silver nanoparticles into uniform nanoparticles with very low dispersity. The frameworks with large aperture facilitated the inclusion of myoglobin and enhanced the stability and catalytic activity.

Adaptive complementary neighboring sub-aperture beamforming for thermoacoustic imaging.

When applied to thermoacoustic imaging (TAI), the delay-and-sum (DAS) algorithm produces strong sidelobes due to its disadvantages of uniform aperture weighting. As a result, the quality of TAI images recovered by DAS is often severely degraded by strong non-coherent clutter, which restricts the development and application of TAI. To address this issue, we propose an adaptive complementary neighboring sub-aperture (NSA) beamforming algorithm for TAI. In NSA, we introduce a coordinate system transformation when calculating the normalized cross-correlation (NCC) matrix. This approach enables the computation of the NCC coefficient within the specified kernel without complex coordinate calculations. We first conducted the numerical simulation experiment to validate NSA using a tree branch phantom. In addition, we also conducted phantom (five sauce tubes), ex vivo (ablation needle in ex vivo porcine liver), and in vivo (human arm) TAI experiments using our TAI system with a center frequency of 3GHz. In the numerical simulation experiment, the structural similarity index (SSIM) value for NSA is increased from 0.37828 for DAS to 0.75492. In the point target phantom TAI experiment, the generalized contrast-to-noise ratio (gCNR) value for NSA is increased from 0.936 for DAS to 0.962. The experimental results show that NSA can recover clearer thermoacoustic images compared to DAS. In the ex vivo TAI experiment, the full width at half maxima (FWHM) of an ablation needle (diameter=1.5mm) for coherence factor (CF) weighted DAS and NSA are 0.9 and 1.3mm, respectively. Furthermore, in the in vivo TAI experiment, CF reduces the signals within the arm compared to NSA. Therefore, compared with CF, NSA can maintain the integrity of target information in TAI while effectively suppressing non-coherent background clutter. NSA can effectively reduce non-coherent background noise while ensuring the completeness of the target information. So, NSA offers the potential to provide high-quality thermoacoustic images and further advance their clinical application.

Doped Porous Carbon Spheres with Controllable Vesicle Structure: Preparation and the Effects of Pore Size on Electromagnetic Wave Absorption Properties.

This work reports on the preparation of uniform vesicle-structural carbon spheres doped with heteroatoms of N, P, and S, with the pore sizes strictly controlled by the hard templates of monodisperse submicron SiO2 spheres. The uniformly doped vesicular carbon microspheres are obtained in three steps: Stöber hydrolysis for the SiO2; in situ polymerization for the immobilization; and alkaline etching after carbonization. The size of the vesicles can be easily adjusted by regulating the particle size of the submicron SiO2 spheres, which has a significant effect on its electromagnetic wave (EMW) absorption performance. Compared with microspheres with pore sizes of 180 and 480nm, when the vesicle aperture is 327nm, with only 5.5wt.% filling load and 1.9mm thickness, the material shows the best EMW absorption behavior with the effective absorption bandwidth (EAB) covers the entire Ku band (6.32GHz) and the minimum reflection loss (RLmin) of -36.10dB, suggesting the optimized pore size of the microspheres can significantly improve the overall impedance matching of the material and achieve broadband wave absorption. This work paves the way for the enhancement of EMW absorption properties of porous material by optimizing the pore size of uniform apertures while maintaining their composition.

Recent Progress and Challenges in the Field of Metal–Organic Framework-Based Membranes for Gas Separation

Metal–organic frameworks (MOFs) represent the largest class of materials among crystalline porous materials ever developed, and have attracted attention as core materials for separation technology. Their extremely uniform pore aperture and nearly unlimited structural and chemical characteristics have attracted great interest and promise for applying MOFs to adsorptive and membrane-based separations. This paper reviews the recent research into and development of MOF membranes for gas separation. Strategies for polycrystalline membranes and mixed-matrix membranes are discussed, with a focus on separation systems involving hydrocarbon separation, CO2 capture, and H2 purification. Challenges to and opportunities for the industrial deployment of MOF membranes are also discussed, providing guidance for the design and fabrication of future high-performance membranes. The contributions of the underlying mechanism to separation performance and adopted strategies and membrane-processing technologies for breaking the selectivity/permeability trade-off are discussed.

Improvement of misalignment tolerance in free-space optical interconnects

In this paper, the use of micro lenses with non-uniform transmittance apertures as an alternative to those with uniform transmittance apertures in optical communication systems is proposed. In particular, we consider the use of micro lenses with tapered Gaussian transmittance profiles to improve the misalignment tolerance in optical interconnects. We study the effects of utilizing Gaussian transmittance profiles on the propagation of light beams and the signal to crosstalk ratio of misaligned optical systems. Moreover, we consider the use of uniform transmittance profiles in optical systems for the sake of comparison. To this end, the crosstalk optical noise is modeled at the plane of the detectors array considering the two scenarios of uniform and Gaussian apertures. This was possible after finding the optical field for both scenarios at the of the detectors array. Numerical results clearly demonstrate the significant improvement in decreasing the crosstalk and increasing the signal to crosstalk ratio in the considered optical systems upon utilizing the Gaussian profiles.

Radiation source personalization for nanoparticle-enhanced radiotherapy using dynamic contrast-enhanced MRI in the treatment planning process

Nanoparticle-enhanced radiotherapy offers the potential to selectively increase the radiation dose imparted to the tumor while at the same time sparing the healthy structures around it. Among the recommendations of an interdisciplinary group for the clinical translation of this treatment modality, is the developing of methods to quantify the effects that the nanoparticle concentration has on the radiation dosimetry and incorporate these effects into the treatment planning process. In this work, using Monte Carlo simulations and dynamic contrast-enhanced MRI images, treatment plans for nanoparticle-enhanced radiotherapy are calculated in order to evaluate the effects that realistic distributions of the nanoparticles have on the resultant plans and to devise treatment strategies to account for these effects, including the selection of the proper x-ray source configuration in terms of energy and collimation. Dynamic contrast-enhanced MRI studies were obtained for two treatment sites, namely brain and breast. A model to convert the MRI signal to contrast agent concentration was applied to each set of images. Two treatment modalities, 3D conformal radiotherapy and Stereotactic Body Radiation Therapy, were evaluated at three different x-ray spectra, namely 6 MV from a linear accelerator, 110 kVp and 220 kVp from a tungsten target. For the breast patient, as the nanoparticle distribution varies markedly with time, the treatment plans were obtained at two different times after administration. It was determined that maximum doses to the healthy structures around the tumor are mostly determined by the minimum nanoparticle concentration in the tumor. The presence of highly hypoxic or necrotic tissue, which fails to accumulate the nanoparticles, or leakage of the contrast agent into the surrounding healthy tissue, make irradiation with conventional conformal radiotherapy unfeasible for kilovoltage beam energies, as the uniform beam apertures lack the ability to compensate for the non-uniform distribution of the nanoparticles. Therefore, proper quantification of the nanoparticle distribution not only in the target volume but also in the surrounding tissues and structures is crucial for the proper planning of nanoparticle-enhanced radiotherapy and a treatment delivery with a high-degree of freedom, such as small-field stereotactic body radiotherapy, should be the method of choice for this treatment modality.

Multifocal meta-fiber based on the fractional Talbot effect.

Multi-focusing of light is a crucial capability for photonic devices that can be effectively achieved by precisely modulating the phase delay on the incident wavefront. However, integrating functional structures into optical fibers for remote light focusing remains challenging due to the complex device design and limited fabrication approaches. Here, we present the design and fabrication of metalens array on the end-face of a tailored single-mode step-index fiber for focusing light field into closely packed focal spot array. The metalenses are configured based on the fractional Talbot effect and benefit a modular design capability. Light passing through the optical fiber can be focused into different focal planes. With a synergistic 3D laser nanoprinting technique based on two-photon polymerization, high-quality meta-fibers are demonstrated for focusing light parallelly with a uniform numerical aperture (NA) as high as approximately 0.77. This may facilitate various applications such as optical trapping, generation of sophisticated beam profiles, and boosting light coupling efficiencies.

Improvement of current crowding effect in VCSEL arrays with non-uniform oxidation aperture design

A compact electro-opto-thermal model of 2-D vertical cavity surface emitting laser (VCSEL) arrays considering the current crowding effect in each array cell is established to study the impact of oxidation aperture on the device performance. Simulated results shows that increasing oxidation aperture of array cell is helpful to improve the uniformity of current density distribution. With careful design of non-uniform oxidation aperture layout, both the uniformity of the temperature distribution and the current distribution is improved by 36.52% and 42.08%, respectively. Furthermore, 3×3 VCSEL arrays with uniform oxidation aperture (array-1) and non-uniform oxidation aperture (array-2) are fabricated and the L-I-V curves of two types of VCSEL arrays at different biases are also measured. The peak output optical power of array-2 is enhanced to 1.83mW with an improvement of 8.91% when compared with that of array-1. Moreover, the total optical output power of array-2 is always superior to that of array-1 over a wide bias current range.

Reconstruction of Cardiac Images using the Confocal Laser Scanning Microscope

We suggested a cardioid aperture in the processing of cardiac images using the confocal laser scanning microscope (CSLM). We calculated the Point Spread Function (PSF) corresponding to the cardioid aperture and the cardiac images by using the FFT algorithm. Then, we studied the effect of the image rotation on the PSF. In addition, we computed the autocorrelation corresponding to the cardiac images and cardiac aperture and compared them with the autocorrelation of the uniform circular aperture. Also, we computed the cross-correlation of aligned and rotated cardiac images. Finally, we imaged the cardiac images using the confocal scanning microscope provided with the circular and cardioid shapes of apertures. The MATLAB code is used in the formation of all images and plots.

Near-Field Gain Expression for Tapered Circular Aperture Antennas

Near-field electromagnetic wave, which is radiated by tapered circular aperture antennas, has found more engineering applications in many diverse fields. Through the simulation of a reflector antenna whose efficiency is up to 51.4%, it can be found that the aperture distribution cannot be regarded as uniform distribution but as tapered distribution. Thus, expressions for near fields of uniform circular apertures cannot be applied to tapered circular aperture antennas in practical engineering. To alleviate such problem, near-field calculation formulas are proposed based on the Sommerfeld integrals with steepest descents technique, which ensures the accurate and efficient analysis of the near field of the tapered circular aperture antenna. And near-field characteristics of tapered circular aperture antennas are displayed. The maximum difference of normalized near-field amplitudes between tapered and uniform distribution is 0.35V/m. Three-dimensional near-field gain of circular aperture antennas is developed from the near-field radiation pattern. The proposed expression extends the range of calculation from the axial to more directions compared with the previous expression. Also, this expression is valid in both the near field and the far field. Using full-wave simulations, results obtained from the proposed expression are compared with the simulated results. And the deviation in near-field gain is less than 0.3dB, which indicates the convenience and effectiveness of our expression. The proposed near-field gain expression can be applicable to the near-field measurement and design of millimeter or terahertz radar systems that work in the near-field region.

A single layered broadband circularly polarized MIMO antenna based on metasurface with aperture CPW feed

ABSTRACT In this paper, a single-layered broadband circularly polarized MIMO antenna is presented for a 5 GHz wireless local area network (WLAN). The Metasurface is designed with an array of patches and these patches are excited by four uniform aperture CPW feed MIMO antennas. The stubs are added across the aperture and central patches at respective apertures are increased to achieve broad bandwidth. Truncated corner square patches are used for the metasurface to achieve circular polarization (CP). The CP MIMO antenna has achieved an impedance bandwidth of 33% (4.35 GHz-6.08 GHz), a 3 dB axial ratio (AR) bandwidth of 11.95% (5.18 GHz-5.8 GHz) and a gain of 8dBi.

Smart bifunctional covalent organic frameworks for water treatment: Reversible colorimetric pH detection and photocatalytic removal of pollutants

Covalent Organic Frameworks (COFs) with uniform aperture size, high surface area and tunable structure have been proposed as a potential material in water treatment. In this work, we utilized methoxy based covalent organic frameworks (COF–OMe) at room temperature as a dual-role smart material. For one thing, it can respond to the pH of solutions by visual, colorimetric change. Due to the protonation effect of methoxy groups in its pore skeleton, the fluorescent intensity of COF–OMe went down and the color changed from yellow to black as the pH decreased reversibly. For another, the abundant pores and ordered conjugated structure endowed it with a large surface area and light absorption ability. Therefore, the light-triggered oxidase-mimic porous COF–OMe can be applied in the synergistic removal of organic pollutants with synergistic physical absorption and light degradation. The outstanding and sensitive response to pH, great removal efficiency and dual function made it applied in water treatment with great potential, achieving killing two birds with one stone.

Triple-Mode Resonant Backfire Feed Antenna With Nearly Equal E-/H-Plane and Inverse Taper Patterns

A design approach to triple-mode resonant feed antenna is advanced for parabolic reflector antenna system. The antenna is composed of a slit-loaded 1.5-wavelength sectorial dipole, a primary corner reflector, and an auxiliary circular reflector. The corner angle of the primary reflector can be employed to properly narrow down the H-plane half-power beam width (HPBW), while with the E-plane one rarely affected, such that equal E-/H-plane characteristic can be yielded accordingly. Unlike traditional backfire antennas, high-order resonant modes are employed to generate inverse taper radiation pattern at the high frequency band. As is numerically demonstrated and experimentally validated, the corner angle of the primary reflector can be eventually determined as 45∘. The antenna exhibits an impedance bandwidth of up to about 79.7%, and a nearly equal E-/H-plane bandwidth of up to about 29.4%. Average HPBW of about 65∘, gain up to about 8.9 dBi with fluctuation of less than 1 dB can be achieved within the equal E-/H-plane bandwidth. The inverse taper radiation pattern at the high frequency band can provide effective edge illumination and uniform aperture field distribution in parabolic reflector antenna systems.

Broadband and high gain circularly polarized mushroom‐loaded nonuniform patch array antenna with aperture CPW feed

AbstractThe circularly polarized antenna with high gain and broad bandwidth in single layer substrate is realized with mushroom‐loaded nonuniform square patches and aperture CPW feed for 5 GHz Wi‐Fi applications. An array of nonuniform square patches are excited by using a uniform aperture CPW feed. Moreover, the mushrooms are connected between the patches and the ground plane to reduce the spurious radiations on the patch. Mushrooms are moved in the diagonal direction of patches to generate circular polarization. The dimensions of the middle patches are increased and the mushrooms of the middle patches are moved further in the diagonal direction to enhance the impedance bandwidth (IBW) and 3 dB axial ratio bandwidth (ARBW). An array of mushroom‐loaded nonuniform square patches with aperture CPW feed has achieved an IBW of 54.2% (4.76–8.3 GHz), an ARBW of 14.3% (5.49–6.29 GHz) and the peak gain of 9.5–9.9 dBi over 5 GHz Wi‐Fi band.

Epitaxial Single-Domain Cu-BTC Metal-Organic Framework Thin Films and Foils by Electrochemical Conversion of Cuprous Oxide.

Metal-organic frameworks (MOFs) are an important class of crystalline porous materials with extensive chemical and structural merits. However, the fabrication of MOF thin films oriented along all crystallographic axes to achieve well-aligned nanopores and nanochannels with uniform apertures remains a challenge. Here, we achieved highly crystalline single-domain MOF thin films with the [111] out-of-plane orientation by electrochemical conversion of cuprous oxide. Copper(II)-benzene-1,3,5-tricarboxylate, Cu3(BTC)2 (referred to as Cu-BTC), is a well-known metal-organic open framework material with a cubic crystal system. Epitaxial Cu-BTC(111) thin films were manufactured by electrochemical oxidation of Cu2O(111) films electrodeposited on single-crystal Au(111). The Cu-BTC(111) shows an in-plane antiparallel relationship with the precursor Cu2O(111) with a -0.91% coincidence site lattice mismatch. A plausible mechanism was proposed for the electrochemical conversion of Cu2O into Cu-BTC, indicating formation of intermediate CuO, growth of Cu-BTC islands, and termination with coalesce into a dense film with a limiting thickness of about 740 nm. The Faradaic efficiency for the electrochemical conversion was 63%. In addition, epitaxial Cu-BTC(111) foils were fabricated by epitaxial lift-off following the electrochemical etching of residual Cu2O underneath the Cu-BTC. It was also demonstrated that Cu-BTC(111) films with two in-plane domains and textured Cu-BTC(111) films can be achieved on a large scale using electrodeposited Au/Si and Au-coated glass as low-cost substrates.

Fabrication of novel thin-film composite membrane based on ultrathin metal-organic framework interlayer for enhancing forward osmosis performance

To improve operation efficiency, an interlayered thin-film composite forward osmosis (iTFC-FO) membrane was designed by introducing an ultrathin and porous interlayer based on aluminum tetra-(4-carboxyphenyl)porphyrin (a stable metal−organic framework nanosheet, Al-MOF). Surface characterization results revealed that Al-MOF spread evenly in the macro-porous substrate, and provided a flat and smooth reaction interface with moderate hydrophilicity and uniform small aperture. The resultant polyamide (PA) layer had a thin base (without intrusion into substrate) and crumpled surface (with abundant leaves). The leaves size and cross-linking degree of PA layer firstly increased and then decreased with the Al-MOF loading. Compared to the original membrane, the iTFC-FO showed an enhanced water permeability and a reduced reverse sodium flux in both modes of active layer facing feed solution (AL-FS) and active layer facing draw solution (AL-DS). To be specific, the specific reverse sodium flux (reverse sodium flux/pure water flux) decreased from 0.27 g/L to 0.04 g/L in the AL-FS mode, while from 1.36 g/L to 0.23 g/L in the AL-DS mode with 2 mol/L NaCl as DS. Moreover, the iTFC-FO maintained high stability and high permeability under high-salinity and contaminated environment. This study offers a new possibility for the rational fabrication of high-performance TFC-FO membranes.

Ka-Band Huygens Antenna Array With Very High Aperture Efficiency and Low Sidelobes

A Ka-band Huygens antenna array with extremely high aperture efficiency (AE) and low sidelobe levels is reported for 5G millimeter-wave (mm-wave) wireless applications. The basic array element is an innovative Huygens subarray consisting of two open rectangular waveguides that form an aperture consisting of a 1 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> 2 set of orthogonal, tightly coupled electric and magnetic elements separated by a virtual gap that are balanced, in- phase, and radiate Huygens cardioid patterns. A larger 8 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times16$ </tex-math></inline-formula> element array is then realized with 64 of these 1 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> 2 subarrays. With excitations of equal amplitude, the full broadside-radiating array achieves an AE up to 97.5% in the operating bandwidth, which is very close to the 100% limit associated with an ideal uniform aperture distribution. Moreover, a highly efficient and compact feed network with tapered-amplitude and in- phase excitations from the array center to its edges is designed to achieve sidelobe and backlobe levels less than −20 dB. The −10 dB impedance bandwidth covers 26.7–29.65 GHz, and the peak realized AE (RAE) reaches 82%. A full-aluminum array prototype was fabricated with standard machining processes and tested. The measured performance characteristics agree well with their simulated values, confirming the efficacy of their designs.

Temperature-dependent viscosity: Relevance to the numerical simulation of enhanced geothermal systems

The primary focus of this paper is to investigate the relevance of temperature-dependent viscosity in the numerical modeling of enhanced geothermal systems (EGS) for thermal performance evaluation and forecasting. The numerical simulation model studied accounts for two situations that may occur during heat extraction from an enhanced geothermal system. First, the viscosity and density of water vary with temperature and pressure. Second is the possibility that the fractures, the main flow conduits, may have asperities that could create channels and alter flow paths, thus affecting the amount and distribution of the surface area available for heat transfer. The study shows that if the fracture aperture is of uniform aperture, there is no significant difference between assuming a constant viscosity in the model over a temperature-dependent viscosity. However, if an enhanced geothermal system is known to be channelized and the temperature difference between the reservoir and the injected fluid is large, then the temperature-dependent viscosity would be necessary for modeling the system to accurately simulate its thermal performance. On the other hand, if the aperture distribution of the enhanced geothermal system is evenly distributed, a constant viscosity may suffice in simulating the process. Moreover, if the temperature difference between the reservoir and the injected fluid is small, a constant viscosity can be used in the model without significantly impacting computational accuracy. Overall a large correlation length leading to an increased area of preferential flow paths presents the most significant effect contributing to the differences seen between modeling with a constant viscosity or a temperature-dependent viscosity.