The structure of wind-sand flow under different total sand transport rates was measured with field vertical anemometer and sand trap on the crest of typical coastal transverse ridge in Changli Gold Coast of Hebei Prov...The structure of wind-sand flow under different total sand transport rates was measured with field vertical anemometer and sand trap on the crest of typical coastal transverse ridge in Changli Gold Coast of Hebei Province, which is one of the most typical coastal aeolian distribution regions in China and famous for the tall and typical coastal transverse ridges. The measurement results show that, on the conditions of approximate wind velocities and same surface materials and environments, some changes happen to the structure of wind-sand flow with the increase of total sand transport rate on the crest of coastal transverse ridge. First, the sand transport rates of layers at different heights in the wind-sand flow increase, with the maximum increase at the height layer of 4-8cm. Second, the ratios of sand trans-port rates of layers at different heights to total sand transport rate decrease at the low height layer (0-4cm), but increase at the high height layer (4-60cm). Third, the distribution of the sand transport rate in the wind-sand flow can be expressed by an exponential function at the height layer of 0-40cm, but it changes from power function model to ex-ponential function model in the whole height layer (0-60cm) and changes into polynomial function model at the height layer of 40-60cm with the increase of total sand transport rate. Those changes have a close relationship with the limit of sand grain size of wind flow transporting and composition of sand grain size in the wind-sand flow.展开更多
Against the background of the sand-flow foundation treatment engineering of Guangzhou Zhoutouzui variable cross-section immersed tunnel, a kind of sand deposit-detecting method was devised on the basis of full-scale m...Against the background of the sand-flow foundation treatment engineering of Guangzhou Zhoutouzui variable cross-section immersed tunnel, a kind of sand deposit-detecting method was devised on the basis of full-scale model test of sand-flow method. The real-time data of sand-deposit height and radius were obtained by the self-developed sand-deposit detectors. The test results show that the detecting method is simple and has high precision. In the use of sand-flow method, the sand-carrying capability of fluid is limited, and sand particles are all transported to the sand-deposit periphery through crater, gap and chutes after the sand deposit formed. The diffusion range of the particles outside the sand-deposit does not exceed 2.0 m. Severe sorting of sand particles is not observed because of the unique oblique-layered depositing process. The temporal and spatial distributions of gap and chutes directly affect the sand-deposit expansion, and the expansion trend of the average sand-deposit radius accords with quadratic time-history curve.展开更多
Darcy’s law is widely used to describe the flow in porous media in which there is a linear relationship between fluid velocity and pressure gradient. However, it has been found that for high numbers of Reynolds this ...Darcy’s law is widely used to describe the flow in porous media in which there is a linear relationship between fluid velocity and pressure gradient. However, it has been found that for high numbers of Reynolds this law ceases to be valid. In this work, the Ergun equation is employed to consider the non-linearity of air velocity with the pressure gradient in casting sands. The contribution of non-linearity to the total flow in terms of a variable defined as a non-Darcy flow fraction is numerically quantified. In addition, the influence of the shape factor of the sand grains on the non-linear flow fraction is analyzed. It is found that for values of the Reynolds number less or equal than 1, the contribution of non-linearity for spherical particles is around 1.15%.展开更多
Pulsatory characteristics of wind velocity in sand flow over Gobi and mobile sand surface have been investigated experimentally in the wind tunnel. The primary goal of this paper is to reveal the relation- ship betwee...Pulsatory characteristics of wind velocity in sand flow over Gobi and mobile sand surface have been investigated experimentally in the wind tunnel. The primary goal of this paper is to reveal the relation- ship between pulsatory characteristics of instantaneous wind speed in sand flow and the motion state of sand grains. For a given underlying surface, pulsation of wind velocities in sand flow on different heights has a good correlation. As the space distance among different heights increases, fluctuation of instantaneous wind speed presents a decreasing trend and its amplitude is closely related to the mo- tion state of sand grains and their transport. Pulsatory intensity increases with the indicated wind speed, but its relative value does not depend on it, only agrees with height.展开更多
The high-flowing sand-concrete(HFSC) containing natural sands as aggregate was carried out.The high fluidity and stability of HFSC can be achieved by tailoring the mix design parameters,such as fine to coarse sand rat...The high-flowing sand-concrete(HFSC) containing natural sands as aggregate was carried out.The high fluidity and stability of HFSC can be achieved by tailoring the mix design parameters,such as fine to coarse sand ratio,dosage of additions,water to binder ratio and dosage of admixtures.Mini-cone slump test,v-funnel time test and viscosity model parameters were used to characterize the behaviour of HFSC in fresh state.The mechanical compressive strength in 28 d was also determined.A factorial design approach was used to establish models highlighting the effect of each mix-parameter on measured properties of HFSC.The derived models are valid for mixtures made with 0 to 0.3 of dune sand to total sand ratio,82 to 418 kg/m3 of marble powder,0.42 to 0.46 of water/binder ratio and 1.3% to 1.9% of superplasticizer high water-reducer.The results show that the derived models constitute very efficient means for understanding the influence of key mix-parameters on HFSC properties and are useful in selecting the optimum mix proportions,by simulating their impact on fluidity,stability and compressive strength.展开更多
The flow characteristics of foundation soils subjected to train loads can present engineering hazards in highspeed railways. In order to verify the feasibility of blending coarse sand in modifying soft subsoil, undrai...The flow characteristics of foundation soils subjected to train loads can present engineering hazards in highspeed railways. In order to verify the feasibility of blending coarse sand in modifying soft subsoil, undrained pulling sphere tests were carried out and the train loads were simulated through localized and cyclic vibration at various frequencies. Laboratory testing results indicate that the fl ow characteristics of soft soil can be signifi cantly enhanced by high-frequency vibration;meanwhile the continuous increase in fl ow characteristics caused by cyclic vibration may be an important reason for the long-term settlement of soft subsoil. The infl uence of sand content on fl ow characteristics is also studied in detail, and it is shown that the addition of coarse sand can weaken the fl ow characteristics of soft soil induced by sudden vibration at lower than 50 Hz. Under the condition of cyclic vibration, the growth of the fl ow characteristics of sand-clay mixtures is mainly caused by the fi rst-time vibration in the cycle, and the increase in sand content can make the fl ow characteristics present a faster convergent tendency.展开更多
In the flow on a mobile bed in an open channel, sand ripple often appears after the sediment begins to move. Different scholars have different views on the formation of sand ripples. This paper holds that as the rippl...In the flow on a mobile bed in an open channel, sand ripple often appears after the sediment begins to move. Different scholars have different views on the formation of sand ripples. This paper holds that as the ripple in general is very small, its formation is due to the instability of the laminar flow or the evolution of the small-scale coherent structures in the sublayer adjacent to the wall of the open channel. When the shear stresses caused by the disturbing waves or the coherent structure near the bed surface boundary and the water flow itself are greater than the shields stresses, responses on the bed surface appear and the sand ripple forms. If the frequency of the shear stress caused by the disturbance is close to the natural frequency of the sand grains that produced resonance, such a phenomenon is called the 'detection property' of the sediment. It is at this point that the maximum resonance appears and the sand ripple develops rapidly.展开更多
Using the self-developed visualization test apparatus, centrifuge model tests at 20 g were carried out to research the macro and microscopic formation mechanism of coarse sand debris flows. The formation mode and soil...Using the self-developed visualization test apparatus, centrifuge model tests at 20 g were carried out to research the macro and microscopic formation mechanism of coarse sand debris flows. The formation mode and soil-water interaction mechanism of the debris flows were analyzed from both macroscopic and microscopic points of view respectively using high digital imaging equipment and micro-structure analysis software Geodip. The test results indicate that the forming process of debris flow mainly consists of three stages, namely the infiltration and softening stage, the overall slide stage, and debris flow stage. The essence of simulated coarse sand slope forming debris flow is that local fluidization cause slope to wholly slide. The movement of small particles forms a transient stagnant layer with increasing saturation, causing soil shear strength lost and local fluidization. When the driving force of the saturated soil exceeds the resisting force, debris flow happens on the coarse sand slope immediately.展开更多
The sand-driven flow is studied from the continuum viewpoint in this paper. The crux of this work is how to model the stresses of the particle phase properly. By analysing the two-fluid model which usually, works in s...The sand-driven flow is studied from the continuum viewpoint in this paper. The crux of this work is how to model the stresses of the particle phase properly. By analysing the two-fluid model which usually, works in solving gas-particle two-phase .flow,. we find that this model has many. deficiencies for studying the sand-driven flow,even for the simplest case- the steady, two-dimensional fully-developed flow.Considering this, we have proposed the three-fluid model in which the upward particles and the downward-particles ore regarded as two kinds of fluids respectively.It is shown that the three-fluid model is better than the two-fluid model in reflecting the internal structure of the flow, region and the influence of the boundary situations on the flow. and it is advantageous to find an approximate solution in that the main components of the particle-phase stresses can be explicitly expressed by those variables in the three-fluid model.In the end, the governing equations as well as the boundary. conditions for the three-fluid model are provided with a discussion.展开更多
Sand ripples are common bedforms. The formation of sand ripples is related to flow conditions; different flow conditions cause different ripple geometries. The main aim of this study was to assess the relationship bet...Sand ripples are common bedforms. The formation of sand ripples is related to flow conditions; different flow conditions cause different ripple geometries. The main aim of this study was to assess the relationship between flow intensity and two-dimensional ripple geometry characteristics. The experiments were carried out in a laboratory flume with natural sand whose bulk density Ps was 2 650 kg/m3 and median diameter D50 was 0.41 mm. The Froude number (Fr), a flow intensity parameter, varied from 0.16 to 0.53, entirely within the subcritical range. Two-dimensional sand ripple geometry was measured and processed via statistical methods. The probability distributions of ripple length and height were obtained with different flow conditions. Through dimensionless analysis, the relationship between the flow intensity parameter (grain size Reynolds number Re. ) and the sand ripple geometry characteristic length ( ∧ ) and height ( △ ) was analyzed, and two formulas were obtained: ∧/D50 = 191.76Re 0.3 and △/D50 = 1.97Re 1.3, which are consistent with previous research results.展开更多
The influence of core sand properties on flow dynamics was investigated synchronously with various core sands, transparent core-box and high-speed camera. To confirm whether the core shooting process has significant t...The influence of core sand properties on flow dynamics was investigated synchronously with various core sands, transparent core-box and high-speed camera. To confirm whether the core shooting process has significant turbulence, the flow pattern of sand particles in the shooting head and core box was reproduced with colored core sands. By incorporating the kinetic theory of granular flow(KTGF), kinetic-frictional constitutive correlation and turbulence model, a two-fluid model(TFM) was established to study the flow dynamics of the core shooting process. Two-fluid model(TFM) simulations were then performed and a areasonable agreement was achieved between the simulation and experimental results. Based on the experimental and simulation results, the effects of turbulence, sand density, sand diameter and binder ratio were analyzed in terms of filling process, sand volume fraction(αs) and sand velocity(Vs).展开更多
Subsurface water flow velocity influences the hydrodynamic characteristics of soil seepage and the interaction between subsurface water flow and surface runoff during soil erosion and sediment transport.A visualized m...Subsurface water flow velocity influences the hydrodynamic characteristics of soil seepage and the interaction between subsurface water flow and surface runoff during soil erosion and sediment transport.A visualized method and equipment was adopted in this study to observe the subsurface water flow.Quartz sand was used as the test material of subsurface water flow and fluorescent dye was used as the indicator for tracing subsurface water flow.Water was supplied at the same flow discharge to the three parts at the bottom of the test flume,and the subsurface water flow were determined with four slope gradients(4°,8°,10°,and 12°).The results showed that the seepage velocity gradually increased with increasing slope gradient.The pore water velocity at different depths of sand layer profile increased with increasing slope gradient,whereas the thickness of the flow front gradually decreased.For the same slope gradient,the pore water velocity in the lower layer was the largest,whereas the thickness of the flow front was the smallest.Comparative analysis of the relationship between seepage velocity and pore water velocity at different depths of sand layer profile showed that the maximum relative difference between the measured pore water velocity and the computational pore water velocity at different depths of sand profile in the experiment was 4.38%.Thus,the test method for measuring the subsurface water flow velocity of sand layer profile adopted in this study was effective and feasible.The development of this experiment and the exploration of research methods would lay a good test foundation for future studies on the variation law of subsurface water flow velocity and the determination of flow velocity in purple soils,thus contributing to the improvement of the hydrodynamic mechanism of purple soils.展开更多
In this paper, we are concerned with computation of a mathematical model of sand dune formation in a water of surface to incompressible out-flows in two space dimensions by using Chebyshev projection scheme. The mathe...In this paper, we are concerned with computation of a mathematical model of sand dune formation in a water of surface to incompressible out-flows in two space dimensions by using Chebyshev projection scheme. The mathematical model is formulate by coupling Navier-Stokes equations for the incompressible out-flows in 2D fluid domain and Prigozhin’s equation which describes the dynamic of sand dune in strong parameterized domain in such a way which is a subset of the fluid domain. In order to verify consistency of our approach, a relevant test problem is considered which will be compared with the numerical results given by our method.展开更多
基金Under the auspices of National Natural Science Foundation of China (No 40571019)
文摘The structure of wind-sand flow under different total sand transport rates was measured with field vertical anemometer and sand trap on the crest of typical coastal transverse ridge in Changli Gold Coast of Hebei Province, which is one of the most typical coastal aeolian distribution regions in China and famous for the tall and typical coastal transverse ridges. The measurement results show that, on the conditions of approximate wind velocities and same surface materials and environments, some changes happen to the structure of wind-sand flow with the increase of total sand transport rate on the crest of coastal transverse ridge. First, the sand transport rates of layers at different heights in the wind-sand flow increase, with the maximum increase at the height layer of 4-8cm. Second, the ratios of sand trans-port rates of layers at different heights to total sand transport rate decrease at the low height layer (0-4cm), but increase at the high height layer (4-60cm). Third, the distribution of the sand transport rate in the wind-sand flow can be expressed by an exponential function at the height layer of 0-40cm, but it changes from power function model to ex-ponential function model in the whole height layer (0-60cm) and changes into polynomial function model at the height layer of 40-60cm with the increase of total sand transport rate. Those changes have a close relationship with the limit of sand grain size of wind flow transporting and composition of sand grain size in the wind-sand flow.
基金Project(51108190) supported by the National Natural Science Foundation of ChinaProject(2012ZC27) supported by the Independence Research Subject from State Key Laboratory of Subtropical Building Science,ChinaProject(GTCC 2008-253) supported by the Research Subject from Guangzhou City,China
文摘Against the background of the sand-flow foundation treatment engineering of Guangzhou Zhoutouzui variable cross-section immersed tunnel, a kind of sand deposit-detecting method was devised on the basis of full-scale model test of sand-flow method. The real-time data of sand-deposit height and radius were obtained by the self-developed sand-deposit detectors. The test results show that the detecting method is simple and has high precision. In the use of sand-flow method, the sand-carrying capability of fluid is limited, and sand particles are all transported to the sand-deposit periphery through crater, gap and chutes after the sand deposit formed. The diffusion range of the particles outside the sand-deposit does not exceed 2.0 m. Severe sorting of sand particles is not observed because of the unique oblique-layered depositing process. The temporal and spatial distributions of gap and chutes directly affect the sand-deposit expansion, and the expansion trend of the average sand-deposit radius accords with quadratic time-history curve.
文摘Darcy’s law is widely used to describe the flow in porous media in which there is a linear relationship between fluid velocity and pressure gradient. However, it has been found that for high numbers of Reynolds this law ceases to be valid. In this work, the Ergun equation is employed to consider the non-linearity of air velocity with the pressure gradient in casting sands. The contribution of non-linearity to the total flow in terms of a variable defined as a non-Darcy flow fraction is numerically quantified. In addition, the influence of the shape factor of the sand grains on the non-linear flow fraction is analyzed. It is found that for values of the Reynolds number less or equal than 1, the contribution of non-linearity for spherical particles is around 1.15%.
基金Supported by the National Key Basic Research Project (Grant No. G1999043504) the National Science and Technology Key Task Project (Grant No. 2002BA517A11)
文摘Pulsatory characteristics of wind velocity in sand flow over Gobi and mobile sand surface have been investigated experimentally in the wind tunnel. The primary goal of this paper is to reveal the relation- ship between pulsatory characteristics of instantaneous wind speed in sand flow and the motion state of sand grains. For a given underlying surface, pulsation of wind velocities in sand flow on different heights has a good correlation. As the space distance among different heights increases, fluctuation of instantaneous wind speed presents a decreasing trend and its amplitude is closely related to the mo- tion state of sand grains and their transport. Pulsatory intensity increases with the indicated wind speed, but its relative value does not depend on it, only agrees with height.
文摘The high-flowing sand-concrete(HFSC) containing natural sands as aggregate was carried out.The high fluidity and stability of HFSC can be achieved by tailoring the mix design parameters,such as fine to coarse sand ratio,dosage of additions,water to binder ratio and dosage of admixtures.Mini-cone slump test,v-funnel time test and viscosity model parameters were used to characterize the behaviour of HFSC in fresh state.The mechanical compressive strength in 28 d was also determined.A factorial design approach was used to establish models highlighting the effect of each mix-parameter on measured properties of HFSC.The derived models are valid for mixtures made with 0 to 0.3 of dune sand to total sand ratio,82 to 418 kg/m3 of marble powder,0.42 to 0.46 of water/binder ratio and 1.3% to 1.9% of superplasticizer high water-reducer.The results show that the derived models constitute very efficient means for understanding the influence of key mix-parameters on HFSC properties and are useful in selecting the optimum mix proportions,by simulating their impact on fluidity,stability and compressive strength.
基金Natural Science Foundation of Jiangsu Province of China under Grant No.BK2012810
文摘The flow characteristics of foundation soils subjected to train loads can present engineering hazards in highspeed railways. In order to verify the feasibility of blending coarse sand in modifying soft subsoil, undrained pulling sphere tests were carried out and the train loads were simulated through localized and cyclic vibration at various frequencies. Laboratory testing results indicate that the fl ow characteristics of soft soil can be signifi cantly enhanced by high-frequency vibration;meanwhile the continuous increase in fl ow characteristics caused by cyclic vibration may be an important reason for the long-term settlement of soft subsoil. The infl uence of sand content on fl ow characteristics is also studied in detail, and it is shown that the addition of coarse sand can weaken the fl ow characteristics of soft soil induced by sudden vibration at lower than 50 Hz. Under the condition of cyclic vibration, the growth of the fl ow characteristics of sand-clay mixtures is mainly caused by the fi rst-time vibration in the cycle, and the increase in sand content can make the fl ow characteristics present a faster convergent tendency.
文摘In the flow on a mobile bed in an open channel, sand ripple often appears after the sediment begins to move. Different scholars have different views on the formation of sand ripples. This paper holds that as the ripple in general is very small, its formation is due to the instability of the laminar flow or the evolution of the small-scale coherent structures in the sublayer adjacent to the wall of the open channel. When the shear stresses caused by the disturbing waves or the coherent structure near the bed surface boundary and the water flow itself are greater than the shields stresses, responses on the bed surface appear and the sand ripple forms. If the frequency of the shear stress caused by the disturbance is close to the natural frequency of the sand grains that produced resonance, such a phenomenon is called the 'detection property' of the sediment. It is at this point that the maximum resonance appears and the sand ripple develops rapidly.
基金Funded by National Natural Science Foundation of China(Grant No.41272296)
文摘Using the self-developed visualization test apparatus, centrifuge model tests at 20 g were carried out to research the macro and microscopic formation mechanism of coarse sand debris flows. The formation mode and soil-water interaction mechanism of the debris flows were analyzed from both macroscopic and microscopic points of view respectively using high digital imaging equipment and micro-structure analysis software Geodip. The test results indicate that the forming process of debris flow mainly consists of three stages, namely the infiltration and softening stage, the overall slide stage, and debris flow stage. The essence of simulated coarse sand slope forming debris flow is that local fluidization cause slope to wholly slide. The movement of small particles forms a transient stagnant layer with increasing saturation, causing soil shear strength lost and local fluidization. When the driving force of the saturated soil exceeds the resisting force, debris flow happens on the coarse sand slope immediately.
文摘The sand-driven flow is studied from the continuum viewpoint in this paper. The crux of this work is how to model the stresses of the particle phase properly. By analysing the two-fluid model which usually, works in solving gas-particle two-phase .flow,. we find that this model has many. deficiencies for studying the sand-driven flow,even for the simplest case- the steady, two-dimensional fully-developed flow.Considering this, we have proposed the three-fluid model in which the upward particles and the downward-particles ore regarded as two kinds of fluids respectively.It is shown that the three-fluid model is better than the two-fluid model in reflecting the internal structure of the flow, region and the influence of the boundary situations on the flow. and it is advantageous to find an approximate solution in that the main components of the particle-phase stresses can be explicitly expressed by those variables in the three-fluid model.In the end, the governing equations as well as the boundary. conditions for the three-fluid model are provided with a discussion.
基金supported by the National Natural Science Foundation of China (Grant No. 50879020)the Ph.D.Discipline-New Teachers Foundation of the Ministry Education of China (Grant No. 200802941028)
文摘Sand ripples are common bedforms. The formation of sand ripples is related to flow conditions; different flow conditions cause different ripple geometries. The main aim of this study was to assess the relationship between flow intensity and two-dimensional ripple geometry characteristics. The experiments were carried out in a laboratory flume with natural sand whose bulk density Ps was 2 650 kg/m3 and median diameter D50 was 0.41 mm. The Froude number (Fr), a flow intensity parameter, varied from 0.16 to 0.53, entirely within the subcritical range. Two-dimensional sand ripple geometry was measured and processed via statistical methods. The probability distributions of ripple length and height were obtained with different flow conditions. Through dimensionless analysis, the relationship between the flow intensity parameter (grain size Reynolds number Re. ) and the sand ripple geometry characteristic length ( ∧ ) and height ( △ ) was analyzed, and two formulas were obtained: ∧/D50 = 191.76Re 0.3 and △/D50 = 1.97Re 1.3, which are consistent with previous research results.
基金supported by the National Science Foundation of China(Grant Number 51575304)the National Science and Technology Major Project of the Ministry of Science and Technology of China(Grant Number 2012ZX04012011)
文摘The influence of core sand properties on flow dynamics was investigated synchronously with various core sands, transparent core-box and high-speed camera. To confirm whether the core shooting process has significant turbulence, the flow pattern of sand particles in the shooting head and core box was reproduced with colored core sands. By incorporating the kinetic theory of granular flow(KTGF), kinetic-frictional constitutive correlation and turbulence model, a two-fluid model(TFM) was established to study the flow dynamics of the core shooting process. Two-fluid model(TFM) simulations were then performed and a areasonable agreement was achieved between the simulation and experimental results. Based on the experimental and simulation results, the effects of turbulence, sand density, sand diameter and binder ratio were analyzed in terms of filling process, sand volume fraction(αs) and sand velocity(Vs).
基金This work was supported by the Fundamental Research Funds for the National Natural Science Foundation of China(No.41571265,41971244)the Key Research and Development Project of Social Livelihood in Chongqing(cstc2018jscxmszdX0061)the Foundation of Graduate Research and Innovation in Chongqing under project CYB18089.
文摘Subsurface water flow velocity influences the hydrodynamic characteristics of soil seepage and the interaction between subsurface water flow and surface runoff during soil erosion and sediment transport.A visualized method and equipment was adopted in this study to observe the subsurface water flow.Quartz sand was used as the test material of subsurface water flow and fluorescent dye was used as the indicator for tracing subsurface water flow.Water was supplied at the same flow discharge to the three parts at the bottom of the test flume,and the subsurface water flow were determined with four slope gradients(4°,8°,10°,and 12°).The results showed that the seepage velocity gradually increased with increasing slope gradient.The pore water velocity at different depths of sand layer profile increased with increasing slope gradient,whereas the thickness of the flow front gradually decreased.For the same slope gradient,the pore water velocity in the lower layer was the largest,whereas the thickness of the flow front was the smallest.Comparative analysis of the relationship between seepage velocity and pore water velocity at different depths of sand layer profile showed that the maximum relative difference between the measured pore water velocity and the computational pore water velocity at different depths of sand profile in the experiment was 4.38%.Thus,the test method for measuring the subsurface water flow velocity of sand layer profile adopted in this study was effective and feasible.The development of this experiment and the exploration of research methods would lay a good test foundation for future studies on the variation law of subsurface water flow velocity and the determination of flow velocity in purple soils,thus contributing to the improvement of the hydrodynamic mechanism of purple soils.
文摘In this paper, we are concerned with computation of a mathematical model of sand dune formation in a water of surface to incompressible out-flows in two space dimensions by using Chebyshev projection scheme. The mathematical model is formulate by coupling Navier-Stokes equations for the incompressible out-flows in 2D fluid domain and Prigozhin’s equation which describes the dynamic of sand dune in strong parameterized domain in such a way which is a subset of the fluid domain. In order to verify consistency of our approach, a relevant test problem is considered which will be compared with the numerical results given by our method.