Objective:To investigate the effect of Guangdong Shenqu(GSQ)on intestinal flora structure in mice with food stagnation through 16S rDNA sequencing.Methods: Mice were randomly assigned to control,model,GSQ low-dose(GSQ...Objective:To investigate the effect of Guangdong Shenqu(GSQ)on intestinal flora structure in mice with food stagnation through 16S rDNA sequencing.Methods: Mice were randomly assigned to control,model,GSQ low-dose(GSQL),GSQ medium-dose(GSQM),GSQ high-dose(GSQH),and lacidophilin tablets(LAB)groups,with each group containing 10 mice.A food stagnation and internal heat mouse model was established through intragastric administration of a mixture of beeswax and olive oil(1:15).The control group was administered normal saline,and the model group was administered beeswax and olive oil to maintain a state.The GSQL(2 g/kg),GSQM(4 g/kg),GSQH(8 g/kg),and LAB groups(0.625 g/kg)were administered corresponding drugs for 5 d.After administration,16S rDNA sequencing was performed to assess gut microbiota in mouse fecal samples.Results: The model group exhibited significant intestinal flora changes.Following GSQ administration,the abundance and diversity index of the intestinal flora increased significantly,the number of bacterial species was regulated,andαandβdiversity were improved.GSQ administration increased the abundance of probiotics,including Clostridia,Lachnospirales,and Lactobacillus,whereas the abundance of conditional pathogenic bacteria,such as Allobaculum,Erysipelotrichaceae,and Bacteroides decreased.Functional prediction analysis indicated that the pathogenesis of food stagnation and GSQ intervention were primarily associated with carbohydrate,lipid,and amino acid metabolism,among other metabolic pathways.Conclusion: The digestive mechanism of GSQ may be attributed to its role in restoring diversity and abundance within the intestinal flora,thereby improving the composition and structure of the intestinal flora in mice and subsequently influencing the regulation of metabolic pathways.展开更多
Objective To explore the metabolic differences between dry eye patients with Qi-Yin defi-ciency and heat stagnation in liver meridian patterns,and clarify their metabolic characteris-tics.Methods Patients with dry eye...Objective To explore the metabolic differences between dry eye patients with Qi-Yin defi-ciency and heat stagnation in liver meridian patterns,and clarify their metabolic characteris-tics.Methods Patients with dry eye who were treated in the Ophthalmology Ward and Outpatient Department of the First Hospital of Hunan University of Chinese Medicine from October 1,2020,to October 30,2021 were enrolled as the research participants in the study.They were assigned to two groups based on traditional Chinese medicine(TCM)syndrome types:heat stagnation in liver meridian pattern group and Qi-Yin deficiency pattern group.Healthy vol-unteers who underwent health check-ups in the Health Management Department were in-cluded as healthy group following the random number table method.The tears of the pa-tients and the healthy volunteer participants were tested by high-performance liquid chro-matography-mass spectrometry(LC-MS).The differential metabolites were screened out by multivariate statistical analysis,and Kyoto Encyclopedia of Genes and Genomes(KEGG)pathway enrichment was performed on the differential metabolites.Finally,the association analysis of differential proteins and metabolites was conducted to verify and supplement the metabolites.Results A total of 32 dry eye patients were enrolled,including 16 cases with heat stagnation in liver meridian pattern and 16 cases with Qi-Yin deficiency pattern.Fourteen healthy volun-teers were included as healthy group.There were no significant differences in baseline char-acteristics among the three groups(P>0.05).A total of 412 biomarkers were determined in Qi-Yin deficiency pattern group,mainly including lipids,lipid-like molecules,organic acids and their derivatives,organic heterocyclic compounds,and nucleosides and their analogues.For heat stagnation in liver meridian pattern group,112 metabolites were determined,main-ly including organic acids and their derivatives,lipids,and lipid-like molecules.The KEGG enrichment results of pathways and the relative content analysis of differential markers de-monstrate that purine metabolism and caffeine metabolism pathways are common metabol-ic characteristics of all dry eyes.Among them,deoxyinosine monophosphate(dIMP)and 2-(formamido)-N1-(5-phospha-D-ribosyl)acetamidine can serve as their biomarkers.The main characteristics of Qi-Yin deficiency syndrome pattern were the significant enhancement of metabolic pathways such as lysine degradation,ovarian steroidogenesis,cholesterol metabolism,pyrimidine metabolism,and bile secretion(P<0.05).Dry eye associated with the heat stagnation in liver meridian pattern is mainly characterized by inhibition of the valine,leucine,and isoleucine biosynthesis pathways(P<0.05).Conclusion Metabolomics can be used as an effective basis for TCM syndrome classification.Different patterns of dry eye syndrome exhibit typical characteristics in the types and concen-trations of metabolites,which correspond to the syndrome classification in TCM.This study initially confirms the rationality of TCM syndrome classification and provides significant ref-erence for the mechanism of dry eye and drug development.展开更多
This paper proposes a mathematical modeling approach to examine the two-dimensional flow stagnates at x=0 over a heated stretchable sheet in a porous medium influenced by nonlinear thermal radiation,variable viscosity...This paper proposes a mathematical modeling approach to examine the two-dimensional flow stagnates at x=0 over a heated stretchable sheet in a porous medium influenced by nonlinear thermal radiation,variable viscosity,and MHD.This study’s main purpose is to examine how thermal radiation and varying viscosity affect fluid flow motion.Additionally,we consider the convective boundary conditions and incorporate the gyrotactic microorganisms equation,which describes microorganism behavior in response to fluid flow.The partial differential equations(PDEs)that represent the conservation equations for mass,momentum,energy,and microorganisms are then converted into a system of coupled ordinary differential equations(ODEs)through the inclusion of nonsimilarity variables.Using MATLAB’s built-in solver bvp4c,the resulting ODEs are numerically solved.The model’s complexity is assessed by plotting two-dimensional graphics of the solution profiles at various physical parameter values.The physical parameters considered in this study include skin friction coefficient,local Nusselt number,local Sherwood number,and density of motile microorganisms.These parameters measure,respectively,the roughness of the sheet,the transformation rate of heat,the rate at which mass is transferred to it,and the rate at which microorganisms are transferred to it.Our study shows that,depending on the magnetic parameter M,the presence of a porous medium causes a significant increase in fluid velocity,ranging from about 25%to 45%.Furthermore,with an increase in the Prandtl number Pr,we have seen a notable improvement of about 6%in fluid thermal conductivity.Additionally,our latest findings are in good agreement with published research for particular values.This study provides valuable insights into the behavior of fluid flow under various physical conditions and can be useful in designing and optimizing industrial processes.展开更多
Background: Although a number of studies have reported that the hot and humid compress from traditional Chinese medicine (TCM) is effective in treating lumbar disc herniation (LDH) with qi stagnation and blood stasis,...Background: Although a number of studies have reported that the hot and humid compress from traditional Chinese medicine (TCM) is effective in treating lumbar disc herniation (LDH) with qi stagnation and blood stasis, clinical evidence is limited. Objective: The purpose of this study is to provide high-quality evidence to support the effectiveness of the traditional Chinese hot and humid compress in the treatment of LDH with qi stagnation and blood stasis. Methods: From October 2021 to November 2023, 86 patients with LDH of qi stagnation and blood stasis type were recruited in our hospital and divided into a control (n = 43) and an observation group (n = 43) according to the random number table method. The control group was given routine clinical treatment, and the observation group was treated with the hot and humid compress therapy for two weeks. The visual analogue scale (VAS) score, Japanese Orthopaedic Association (JOA) score, TCM syndrome score, serum interleukin-6 (IL-6), serum interleukin-1β (IL-1β), and tumor necrosis factor-α (TNF-α) were observed and compared between the two groups before and after treatment, and the clinical efficacy of the two groups was evaluated. Results: After treatment, the VAS score, TCM symptom score, and serum IL-6, IL-1β, and TNF-α levels decreased in both groups (P P P P P Conclusions: The hot and humid compress of traditional Chinese medicine can effectively relieve pain, restore lumbar function, improve TCM syndromes, reduce the level of inflammatory factors, and have a curative effect in treating LDH.展开更多
Objective:To Discuss the correlation between Hypersensitivity C-reactive Protein(Hs-CRP),Total Cholesterol(TC),Triglyceride(TG),negative emotion scale and TCM syndrome scores in“Double Heart Disease”patients with Qi...Objective:To Discuss the correlation between Hypersensitivity C-reactive Protein(Hs-CRP),Total Cholesterol(TC),Triglyceride(TG),negative emotion scale and TCM syndrome scores in“Double Heart Disease”patients with Qi stagnation blood stasis and heart gallbladder heat stagnation.Method:Fifty-two patients in Western Medicine Diagnosis of Double Heart Disease,in TCM syndrome identified as Qi stagnation blood stasis Heart gallbladder heat stagnation syndrome,detects it Hs-CRP,TC,TG levels by ELISA,use TCM Syndromes Scale to evaluate TCM Syndrome,use the Pittsburgh Sleep Quality Index(PSQI)and Self-rating symptom scale(SCL-90)to assess anxiety and sleep levels,analyze the correlation between TCM syndrome scores and Hs-CRP,TC,TG level,PSQI index,SCL-90 index.Result:There was a significant positive correlation between Hs-CRP,TG level and TCM Syndrome scores(P<0.05);TC level was postively correlated with TCM Syndrome scores,but there was no statistical significance(P>0.05);There was a significant positive correlation between PSQI index,SCL-90 index and TCM Syndrome scores(P<0.05).Conclusion:In“Double Heart Disease”patients with Qi stagnation blood stasis and heart gallbladder heat stagnation,there have characteristic syndrome changes in terms of inflammatory factor level,blood lipid level and negative emotion score;Which the above indexes can reflect the severity of TCM syndromes to a certain extent and provide the basis for the effective intervention treatment of TCM.展开更多
When perforating with an abrasive water jet, it is possible that the pressure in the hole (perforation) will be higher than that in the annulus because of water jet blasting against the hole wall, which also is the ...When perforating with an abrasive water jet, it is possible that the pressure in the hole (perforation) will be higher than that in the annulus because of water jet blasting against the hole wall, which also is the theoretical basis for the technology of hydro-jet fracturing. This paper analyzes the mechanism of generating pressure stagnation in water jet hole, and puts forward a new concept of hydroseal. Then, the distribution of pressure in the hole was simulated with the finite element method. The simulation results showed that the pressure in the hole was higher than that in the annulus. Also, the lower the annular pressure (confining pressure) and the higher the blasting pressure, the greater the pressure difference. An experiment indicated that the cement sample was lifted up under the pressure stagnation in the hole, which proved the finite element simulation results obviously.展开更多
This paper concerns the characteristics of heat and mass transfer in upper convected Maxwell fluid flow over a linear stretching sheet with solar radiation,viscous desperation and temperature based viscosity.After bou...This paper concerns the characteristics of heat and mass transfer in upper convected Maxwell fluid flow over a linear stretching sheet with solar radiation,viscous desperation and temperature based viscosity.After boundary layer approximation,the governing equations are achieved(namely Maxwell,upper convected material derivative,thermal and concentration diffusions).By using the self-similarity transformations the governing PDEs are converted into nonlinear ODEs and solved by RK-4 method in combination with Newton Raphson(shooting technique).The effects of developed parameters on velocity,temperature,concentration,fraction factor,heat and mass diffusions are exemplified through graphs and tabular form and are deliberated in detail.Numerical values of fraction factor,heat and mass transfer rates with several parameters are computed and examined.It is noticed that the temperature is more impactable for higher values of radiative heat transport,thermal conductivity and viscous dissipation.The comparison data for some limiting case are acquired and are originated to be in good agreement with previously published articles.展开更多
Objective: To evaluate the clinical effect and safety of Safflower Yellow injection (SYI) in treating coronary heart disease angina pectoris (OHD-AP) with Xin-blood stagnation syndrome (XBSS). Methods: Adopted...Objective: To evaluate the clinical effect and safety of Safflower Yellow injection (SYI) in treating coronary heart disease angina pectoris (OHD-AP) with Xin-blood stagnation syndrome (XBSS). Methods: Adopted was the multi-centered, randomized, positive parallel controlled method, 448 patients with CHD-AP-XBSS were enrolled and divided into two groups, 336 in the tested group treated with SYI and 112 in the control group treated with Salvia injection by intravenous dripping once a day for 14 days, so as to observe the conditions of angina, electrocardiogram, and therapeutic effect on traditinal Chinese medicine (TCM) symptoms as well as the safety of the treatment. Results: The significantly effective rate and total effective rate in the tested group were 60.06% (194/323) and 91.02 % (294/323) respectively; those in improvement of TOM symptoms were 40. 18% (129/321) and 75.23% (243/323) respectively, which were better than those in the control group (P〈0.01). Conclusion: SYI Injection is effective and safe in treating OHD-AP-XBSS.展开更多
The wavelet approach is introduced to study the influence of the natural convection stagnation point flow of the Williamson fluid in the presence of thermophysical and Brownian motion effects. The thermal radiation ef...The wavelet approach is introduced to study the influence of the natural convection stagnation point flow of the Williamson fluid in the presence of thermophysical and Brownian motion effects. The thermal radiation effects are considered along a permeable stretching surface. The nonlinear problem is simulated numerically by using a novel algorithm based upon the Chebyshev wavelets. It is noticed that the velocity of the Williamson fluid increases for assisting flow cases while decreases for opposing flow cases when the unsteadiness and suction parameters increase, and the magnetic effect on the velocity increases for opposing flow cases while decreases for assisting flow cases. When the thermal radiation parameter, the Dufour number, and Williamson’s fluid parameter increase, the temperature increases for both assisting and opposing flow cases. Meanwhile, the temperature decreases when the Prandtl number increases. The concentration decreases when the Soret parameter increases, while increases when the Schmidt number increases. It is perceived that the assisting force decreases more than the opposing force. The findings endorse the credibility of the proposed algorithm, and could be extended to other nonlinear problems with complex nature.展开更多
An analytical solution of the governing equations of the interacting shear flows for unsteady oblique stagnation point flow is obtained. It has the same form as that of the exact solution obtained from the complete NS...An analytical solution of the governing equations of the interacting shear flows for unsteady oblique stagnation point flow is obtained. It has the same form as that of the exact solution obtained from the complete NS equations and physical analysis and relevant discussions are then presented.展开更多
The present article deals with thermally stratified stagnation-point flow saturated in porous medium on surface of variable thickness along with more convincing and reliable surface condition termed as melting heat tr...The present article deals with thermally stratified stagnation-point flow saturated in porous medium on surface of variable thickness along with more convincing and reliable surface condition termed as melting heat transfer.Homogeneous–heterogeneous reaction and radiative effects have been further taken into account to reconnoiterproperties of heat transfer.Melting heat transfer and phenomenon of homogeneous–heterogeneous reaction have engrossed widespread utilization in purification of metals,welding process,electroslag melting,biochemical systems,catalysis and several industrial developments.Suitable transformations are utilized to attain a scheme of ordinary differential equations possessing exceedingly nonlinear nature.Homotopic process is employed to develop convergent solutions of the resulting problem.Discussion regarding velocity,thermal field and concentration distribution for several involved parameters is pivotal part.Graphical behaviors of skin friction coefficient and Nusselt number are also portrayed.Concentration of the reactants is found to depreciate as a result of strength of both heterogeneous and homogeneous reaction parameters.With existence of melting phenomenon,declining attitude of fluid temperature is observed for higher radiation parameter.展开更多
The self-excited DC air circuit breaker(SE-DCCB)has been widely used in urban rail transit due to its excellent stability.It can realize forward and reverse interruption,but has difficulty interrupting small currents ...The self-excited DC air circuit breaker(SE-DCCB)has been widely used in urban rail transit due to its excellent stability.It can realize forward and reverse interruption,but has difficulty interrupting small currents due to the phenomenon of arc root sticking at the entrance of the arc chamber in the splitting process,which is known as arc root stagnation.A coupling model of the self-excited magnetic field and magnetohydrodynamics is established for the SE-DCCB with the traditional structure.The magnetic field,temperature and airflow distribution in the arc chamber are investigated with an interrupting current of 150 A.The simulation results show that the direction and magnitude of the magnetic blowout force are the dominant factors in the arc root stagnation.The local high temperature of the arc chamber due to arc root stagnation increases the obstruction effect of the airflow vortex on the arc root movement,which significantly increases the arc duration time of small current interruption.Based on the research,the structure of the magnetic conductance plate of the actual product is improved,which can improve the direction and magnitude of the magnetic blowout force at the arc root so as to restrain the development of the airflow vortex effectively and solve the problem of arc root stagnation when the small current is interrupted.The simulation results show that the circuit breaker with improved structure has a better performance for a small current interruption range from 100 A to 350 A.展开更多
The effect of variable viscosity and thermal conductivity on steady magnetohydrodynamic(MHD) heat and mass transfer flow of viscous and incompressible fluid near a stagnation point towards a permeable stretching she...The effect of variable viscosity and thermal conductivity on steady magnetohydrodynamic(MHD) heat and mass transfer flow of viscous and incompressible fluid near a stagnation point towards a permeable stretching sheet embedded in a porous medium are presented,taking into account thermal radiation and internal heat genberation/absorbtion.The stretching velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point.The Rosseland approximation is used to describe the radiative heat flux in the energy equation.The governing fundamental equations are first transformed into a system of ordinary differential equations using a scaling group of transformations and are solved numerically by using the fourth-order Rung-Kutta method with the shooting technique.A comparison with previously published work has been carried out and the results are found to be in good agreement.The results are analyzed for the effect of different physical parameters,such as the variable viscosity and thermal conductivity,the ratio of free stream velocity to stretching velocity,the magnetic field,the porosity,the radiation and suction/injection on the flow,and the heat and mass transfer characteristics.The results indicate that the inclusion of variable viscosity and thermal conductivity into the fluids of light and medium molecular weight is able to change the boundary-layer behavior for all values of the velocity ratio parameter λ except for λ = 1.In addition,the imposition of fluid suction increases both the rate of heat and mass transfer,whereas fluid injection shows the opposite effect.展开更多
The effect of melting heat transfer on the two dimensional boundary layer flow of a micropolar fluid near a stagnation point embedded in a porous medium in the presence of internal heat generation/absorption is invest...The effect of melting heat transfer on the two dimensional boundary layer flow of a micropolar fluid near a stagnation point embedded in a porous medium in the presence of internal heat generation/absorption is investigated. The governing non-linear partial differential equations describing the problem are reduced to a system of non-linear ordinary differential equations using similarity transformations solved numerically using the Chebyshev spectral method. Numerical results for velocity, angular velocity and temperature profiles are shown graphically and discussed for different values of the inverse Darcy number, the heat generation/absorption parameter, and the melting parameter. The effects of the pertinent parameters on the local skin-friction coefficient, the wall couple stress, and the local Nusselt number are tabulated and discussed. The results show that the inverse Darcy number has the effect of enhancing both velocity and temperature and suppressing angular velocity. It is also found that the local skin-friction coefficient decreases, while the local Nusselt number increases as the melting parameter increases.展开更多
AIM To investigate the pathophysiologic basis of syndrome of Liver Qi stagnation and parameters for clinical differentiation. METHODS Plasma L ENK, AVP, ANP and serum gastrin were determined by RIA in 84 patients...AIM To investigate the pathophysiologic basis of syndrome of Liver Qi stagnation and parameters for clinical differentiation. METHODS Plasma L ENK, AVP, ANP and serum gastrin were determined by RIA in 84 patients with neurasthenia, mastodynia, chronic gastritis, and chronic cholecystitis presenting the same syndrome of Liver Qi stagnation in traditional Chinese medicine (TCM). Healthy subjects served as controls in comparison with patients having the same syndrome but with different diseases. RESULTS Among the patients with Liver Qi stagnation, the plasma L ENK, ANP and gastrin levels were 38 83ng/L ± 6 32ng/L , 104 11ng/L ± 29 01ng/L and 32 20ng/L ± 6 68ng/L , being significantly lower than those in the healthy controls ( t =3 34, 6 17, 4 48; P <0 01). The plasma AVP of the patient group ( 52 82ng/L ± 19 09ng/L ) was significantly higher than that of the healthy controls ( t =5 79, P <0 01). The above changes in patients having the same symptom complex but different diseases entities showed no significant differences, P >0 05. CONCLUSION The syndrome of Liver Qi stagnation is closely related to the emotional modulatory abnormality of the brain, with decrease of plasma L ENK, ANP and gastrin, and increase of plasma AVP as the important pathophysiologic basis.展开更多
Combined effects of Soret(thermal-diffusion) and Dufour(diffusion-thermo) in MHD stagnation point flow by a permeable stretching cylinder were studied. Analysis was examined in the presence of heat generation/absorpti...Combined effects of Soret(thermal-diffusion) and Dufour(diffusion-thermo) in MHD stagnation point flow by a permeable stretching cylinder were studied. Analysis was examined in the presence of heat generation/absorption and chemical reaction. The laws of conservation of mass, momentum, energy and concentration are found to lead to the mathematical development of the problem. Suitable transformations were used to convert the nonlinear partial differential equations into the ordinary differential equations. The series solutions of boundary layer equations through momentum, energy and concentration equations were obtained.Convergence of the developed series solutions was discussed via plots and numerical values. The behaviors of different physical parameters on the velocity components, temperature and concentration were obtained. Numerical values of Nusselt number, skin friction and Sherwood number with different parameters were computed and analyzed. It is found that Dufour and Soret numbers result in the enhancement of temperature and concentration distributions, respectively.展开更多
In this research,the three-dimensional(3D)steady and incompressible laminar Homann stagnation point nanofluid flow over a porous moving surface is addressed.The disturbance in the porous medium has been characterized ...In this research,the three-dimensional(3D)steady and incompressible laminar Homann stagnation point nanofluid flow over a porous moving surface is addressed.The disturbance in the porous medium has been characterized by the Darcy-Forchheimer relation.The slip for viscous fluid is considered.The energy equation is organized in view of radiative heat flux which plays an important role in the heat transfer rate.The governing flow expressions are first altered into first-order ordinary ones and then solved numerically by the shooting method.Dual solutions are obtained for the velocity,skin friction coefficient,temperature,and Nusselt number subject to sundry flow parameters,magnetic parameter,Darcy-Forchheimer number,thermal radiation parameter,suction parameter,and dimensionless slip parameter.In this research,the main consideration is given to the engineering interest like skin friction coefficient(velocity gradient or surface drag force)and Nusselt number(temperature gradient or heat transfer rate)and discussed numerically through tables.In conclusion,it is noticed from the stability results that the upper branch solution(UBS)is more reliable and physically stable than the lower branch solution(LBS).展开更多
The present study inspects the non-aligned stagnation point nano fluid over a convective surface in the presence of partial slip. Two types of base fluids namely water and kerosene are selected with Cu nanoparticles. ...The present study inspects the non-aligned stagnation point nano fluid over a convective surface in the presence of partial slip. Two types of base fluids namely water and kerosene are selected with Cu nanoparticles. The governing physical problem is presented and transformed into a system of coupled nonlinear differential equations using suitable similarity transformations. These equations are then solved numerically using midpoint integration scheme along with Richardson extrapolation via Maple. Impact of relevant physical parameters on the dimensionless velocity and temperature profiles are portrayed through graphs. Physical quantities such as local skin frictions co-efficient and Nusselt numbers are tabularized.It is detected from numerical computations that kerosene-based nano fluids have better heat transfer capability compared with water-based nanofluids. Moreover it is found that water-based nanofluids offer less resistance in terms of skin friction than kerosene-based fluid. In order to authenticate our present study, the calculated results are compared with the prevailing literature and a considerable agreement is perceived for the limiting case.展开更多
In this study,the stagnation point transport of second grade fluid with linear stretching under the effects of variable thermal conductivity is considered.Induced magnetic field impact is also incorporated.The nonline...In this study,the stagnation point transport of second grade fluid with linear stretching under the effects of variable thermal conductivity is considered.Induced magnetic field impact is also incorporated.The nonlinear set of particle differential equations is converted into set of ordinary differential equations through appropriate transformation.The resulting equations are then resolved by optimal homotopy analysis method.The effect of pertinent parameters of interest on skin friction coefficient,temperature,induced magnetic field,velocity and local Nusselt number is inspected by generating appropriate plots.For numerical results,the built-in bvp4 c technique in computational software MATLAB is used for the convergence and residual errors of obtained series solution.It is perceived that the induced magnetic field is intensified by increasing β.It can also be observed that skin friction coefficient enhances with increasing value of magnetic parameter depending on the stretching ratio a/c.For the validness of the obtained results,a comparison has been made and an excellent agreement of current study with existing literature is found.展开更多
This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy ...This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.展开更多
基金supported by the National Natural Science Foundation of China(81872995).
文摘Objective:To investigate the effect of Guangdong Shenqu(GSQ)on intestinal flora structure in mice with food stagnation through 16S rDNA sequencing.Methods: Mice were randomly assigned to control,model,GSQ low-dose(GSQL),GSQ medium-dose(GSQM),GSQ high-dose(GSQH),and lacidophilin tablets(LAB)groups,with each group containing 10 mice.A food stagnation and internal heat mouse model was established through intragastric administration of a mixture of beeswax and olive oil(1:15).The control group was administered normal saline,and the model group was administered beeswax and olive oil to maintain a state.The GSQL(2 g/kg),GSQM(4 g/kg),GSQH(8 g/kg),and LAB groups(0.625 g/kg)were administered corresponding drugs for 5 d.After administration,16S rDNA sequencing was performed to assess gut microbiota in mouse fecal samples.Results: The model group exhibited significant intestinal flora changes.Following GSQ administration,the abundance and diversity index of the intestinal flora increased significantly,the number of bacterial species was regulated,andαandβdiversity were improved.GSQ administration increased the abundance of probiotics,including Clostridia,Lachnospirales,and Lactobacillus,whereas the abundance of conditional pathogenic bacteria,such as Allobaculum,Erysipelotrichaceae,and Bacteroides decreased.Functional prediction analysis indicated that the pathogenesis of food stagnation and GSQ intervention were primarily associated with carbohydrate,lipid,and amino acid metabolism,among other metabolic pathways.Conclusion: The digestive mechanism of GSQ may be attributed to its role in restoring diversity and abundance within the intestinal flora,thereby improving the composition and structure of the intestinal flora in mice and subsequently influencing the regulation of metabolic pathways.
基金National Natural Science Foundation of China Project(82305037)Guiding Project for Innovative Clinical Medical Technology in Hunan Province(2021SK50809)Scientific Research Project of Hunan Administration of Traditional Chinese Medicine(D2022053).
文摘Objective To explore the metabolic differences between dry eye patients with Qi-Yin defi-ciency and heat stagnation in liver meridian patterns,and clarify their metabolic characteris-tics.Methods Patients with dry eye who were treated in the Ophthalmology Ward and Outpatient Department of the First Hospital of Hunan University of Chinese Medicine from October 1,2020,to October 30,2021 were enrolled as the research participants in the study.They were assigned to two groups based on traditional Chinese medicine(TCM)syndrome types:heat stagnation in liver meridian pattern group and Qi-Yin deficiency pattern group.Healthy vol-unteers who underwent health check-ups in the Health Management Department were in-cluded as healthy group following the random number table method.The tears of the pa-tients and the healthy volunteer participants were tested by high-performance liquid chro-matography-mass spectrometry(LC-MS).The differential metabolites were screened out by multivariate statistical analysis,and Kyoto Encyclopedia of Genes and Genomes(KEGG)pathway enrichment was performed on the differential metabolites.Finally,the association analysis of differential proteins and metabolites was conducted to verify and supplement the metabolites.Results A total of 32 dry eye patients were enrolled,including 16 cases with heat stagnation in liver meridian pattern and 16 cases with Qi-Yin deficiency pattern.Fourteen healthy volun-teers were included as healthy group.There were no significant differences in baseline char-acteristics among the three groups(P>0.05).A total of 412 biomarkers were determined in Qi-Yin deficiency pattern group,mainly including lipids,lipid-like molecules,organic acids and their derivatives,organic heterocyclic compounds,and nucleosides and their analogues.For heat stagnation in liver meridian pattern group,112 metabolites were determined,main-ly including organic acids and their derivatives,lipids,and lipid-like molecules.The KEGG enrichment results of pathways and the relative content analysis of differential markers de-monstrate that purine metabolism and caffeine metabolism pathways are common metabol-ic characteristics of all dry eyes.Among them,deoxyinosine monophosphate(dIMP)and 2-(formamido)-N1-(5-phospha-D-ribosyl)acetamidine can serve as their biomarkers.The main characteristics of Qi-Yin deficiency syndrome pattern were the significant enhancement of metabolic pathways such as lysine degradation,ovarian steroidogenesis,cholesterol metabolism,pyrimidine metabolism,and bile secretion(P<0.05).Dry eye associated with the heat stagnation in liver meridian pattern is mainly characterized by inhibition of the valine,leucine,and isoleucine biosynthesis pathways(P<0.05).Conclusion Metabolomics can be used as an effective basis for TCM syndrome classification.Different patterns of dry eye syndrome exhibit typical characteristics in the types and concen-trations of metabolites,which correspond to the syndrome classification in TCM.This study initially confirms the rationality of TCM syndrome classification and provides significant ref-erence for the mechanism of dry eye and drug development.
文摘This paper proposes a mathematical modeling approach to examine the two-dimensional flow stagnates at x=0 over a heated stretchable sheet in a porous medium influenced by nonlinear thermal radiation,variable viscosity,and MHD.This study’s main purpose is to examine how thermal radiation and varying viscosity affect fluid flow motion.Additionally,we consider the convective boundary conditions and incorporate the gyrotactic microorganisms equation,which describes microorganism behavior in response to fluid flow.The partial differential equations(PDEs)that represent the conservation equations for mass,momentum,energy,and microorganisms are then converted into a system of coupled ordinary differential equations(ODEs)through the inclusion of nonsimilarity variables.Using MATLAB’s built-in solver bvp4c,the resulting ODEs are numerically solved.The model’s complexity is assessed by plotting two-dimensional graphics of the solution profiles at various physical parameter values.The physical parameters considered in this study include skin friction coefficient,local Nusselt number,local Sherwood number,and density of motile microorganisms.These parameters measure,respectively,the roughness of the sheet,the transformation rate of heat,the rate at which mass is transferred to it,and the rate at which microorganisms are transferred to it.Our study shows that,depending on the magnetic parameter M,the presence of a porous medium causes a significant increase in fluid velocity,ranging from about 25%to 45%.Furthermore,with an increase in the Prandtl number Pr,we have seen a notable improvement of about 6%in fluid thermal conductivity.Additionally,our latest findings are in good agreement with published research for particular values.This study provides valuable insights into the behavior of fluid flow under various physical conditions and can be useful in designing and optimizing industrial processes.
文摘Background: Although a number of studies have reported that the hot and humid compress from traditional Chinese medicine (TCM) is effective in treating lumbar disc herniation (LDH) with qi stagnation and blood stasis, clinical evidence is limited. Objective: The purpose of this study is to provide high-quality evidence to support the effectiveness of the traditional Chinese hot and humid compress in the treatment of LDH with qi stagnation and blood stasis. Methods: From October 2021 to November 2023, 86 patients with LDH of qi stagnation and blood stasis type were recruited in our hospital and divided into a control (n = 43) and an observation group (n = 43) according to the random number table method. The control group was given routine clinical treatment, and the observation group was treated with the hot and humid compress therapy for two weeks. The visual analogue scale (VAS) score, Japanese Orthopaedic Association (JOA) score, TCM syndrome score, serum interleukin-6 (IL-6), serum interleukin-1β (IL-1β), and tumor necrosis factor-α (TNF-α) were observed and compared between the two groups before and after treatment, and the clinical efficacy of the two groups was evaluated. Results: After treatment, the VAS score, TCM symptom score, and serum IL-6, IL-1β, and TNF-α levels decreased in both groups (P P P P P Conclusions: The hot and humid compress of traditional Chinese medicine can effectively relieve pain, restore lumbar function, improve TCM syndromes, reduce the level of inflammatory factors, and have a curative effect in treating LDH.
基金Beijing University of Traditional Chinese Medicine Fundamental Research Fund Project(No.2019-JYB-JS-174)Beijing Municipal Science and Technology Commission"Capital Characteristic Clinical Application Research"Project(No.Z161100000516136)。
文摘Objective:To Discuss the correlation between Hypersensitivity C-reactive Protein(Hs-CRP),Total Cholesterol(TC),Triglyceride(TG),negative emotion scale and TCM syndrome scores in“Double Heart Disease”patients with Qi stagnation blood stasis and heart gallbladder heat stagnation.Method:Fifty-two patients in Western Medicine Diagnosis of Double Heart Disease,in TCM syndrome identified as Qi stagnation blood stasis Heart gallbladder heat stagnation syndrome,detects it Hs-CRP,TC,TG levels by ELISA,use TCM Syndromes Scale to evaluate TCM Syndrome,use the Pittsburgh Sleep Quality Index(PSQI)and Self-rating symptom scale(SCL-90)to assess anxiety and sleep levels,analyze the correlation between TCM syndrome scores and Hs-CRP,TC,TG level,PSQI index,SCL-90 index.Result:There was a significant positive correlation between Hs-CRP,TG level and TCM Syndrome scores(P<0.05);TC level was postively correlated with TCM Syndrome scores,but there was no statistical significance(P>0.05);There was a significant positive correlation between PSQI index,SCL-90 index and TCM Syndrome scores(P<0.05).Conclusion:In“Double Heart Disease”patients with Qi stagnation blood stasis and heart gallbladder heat stagnation,there have characteristic syndrome changes in terms of inflammatory factor level,blood lipid level and negative emotion score;Which the above indexes can reflect the severity of TCM syndromes to a certain extent and provide the basis for the effective intervention treatment of TCM.
文摘When perforating with an abrasive water jet, it is possible that the pressure in the hole (perforation) will be higher than that in the annulus because of water jet blasting against the hole wall, which also is the theoretical basis for the technology of hydro-jet fracturing. This paper analyzes the mechanism of generating pressure stagnation in water jet hole, and puts forward a new concept of hydroseal. Then, the distribution of pressure in the hole was simulated with the finite element method. The simulation results showed that the pressure in the hole was higher than that in the annulus. Also, the lower the annular pressure (confining pressure) and the higher the blasting pressure, the greater the pressure difference. An experiment indicated that the cement sample was lifted up under the pressure stagnation in the hole, which proved the finite element simulation results obviously.
基金funding this work through research groups program under grant number R.G.P-59/40.
文摘This paper concerns the characteristics of heat and mass transfer in upper convected Maxwell fluid flow over a linear stretching sheet with solar radiation,viscous desperation and temperature based viscosity.After boundary layer approximation,the governing equations are achieved(namely Maxwell,upper convected material derivative,thermal and concentration diffusions).By using the self-similarity transformations the governing PDEs are converted into nonlinear ODEs and solved by RK-4 method in combination with Newton Raphson(shooting technique).The effects of developed parameters on velocity,temperature,concentration,fraction factor,heat and mass diffusions are exemplified through graphs and tabular form and are deliberated in detail.Numerical values of fraction factor,heat and mass transfer rates with several parameters are computed and examined.It is noticed that the temperature is more impactable for higher values of radiative heat transport,thermal conductivity and viscous dissipation.The comparison data for some limiting case are acquired and are originated to be in good agreement with previously published articles.
文摘Objective: To evaluate the clinical effect and safety of Safflower Yellow injection (SYI) in treating coronary heart disease angina pectoris (OHD-AP) with Xin-blood stagnation syndrome (XBSS). Methods: Adopted was the multi-centered, randomized, positive parallel controlled method, 448 patients with CHD-AP-XBSS were enrolled and divided into two groups, 336 in the tested group treated with SYI and 112 in the control group treated with Salvia injection by intravenous dripping once a day for 14 days, so as to observe the conditions of angina, electrocardiogram, and therapeutic effect on traditinal Chinese medicine (TCM) symptoms as well as the safety of the treatment. Results: The significantly effective rate and total effective rate in the tested group were 60.06% (194/323) and 91.02 % (294/323) respectively; those in improvement of TOM symptoms were 40. 18% (129/321) and 75.23% (243/323) respectively, which were better than those in the control group (P〈0.01). Conclusion: SYI Injection is effective and safe in treating OHD-AP-XBSS.
基金Project supported by the National Natural Science Foundation of China(Nos.51709191,51706149,and 51606130)the Key Laboratory of Advanced Reactor Engineering and Safety,Ministry of Education of China(No.ARES-2018-10)the State Key Laboratory of Hydraulics and Mountain River Engineering of Sichuan University of China(No.Skhl1803)
文摘The wavelet approach is introduced to study the influence of the natural convection stagnation point flow of the Williamson fluid in the presence of thermophysical and Brownian motion effects. The thermal radiation effects are considered along a permeable stretching surface. The nonlinear problem is simulated numerically by using a novel algorithm based upon the Chebyshev wavelets. It is noticed that the velocity of the Williamson fluid increases for assisting flow cases while decreases for opposing flow cases when the unsteadiness and suction parameters increase, and the magnetic effect on the velocity increases for opposing flow cases while decreases for assisting flow cases. When the thermal radiation parameter, the Dufour number, and Williamson’s fluid parameter increase, the temperature increases for both assisting and opposing flow cases. Meanwhile, the temperature decreases when the Prandtl number increases. The concentration decreases when the Soret parameter increases, while increases when the Schmidt number increases. It is perceived that the assisting force decreases more than the opposing force. The findings endorse the credibility of the proposed algorithm, and could be extended to other nonlinear problems with complex nature.
文摘An analytical solution of the governing equations of the interacting shear flows for unsteady oblique stagnation point flow is obtained. It has the same form as that of the exact solution obtained from the complete NS equations and physical analysis and relevant discussions are then presented.
文摘The present article deals with thermally stratified stagnation-point flow saturated in porous medium on surface of variable thickness along with more convincing and reliable surface condition termed as melting heat transfer.Homogeneous–heterogeneous reaction and radiative effects have been further taken into account to reconnoiterproperties of heat transfer.Melting heat transfer and phenomenon of homogeneous–heterogeneous reaction have engrossed widespread utilization in purification of metals,welding process,electroslag melting,biochemical systems,catalysis and several industrial developments.Suitable transformations are utilized to attain a scheme of ordinary differential equations possessing exceedingly nonlinear nature.Homotopic process is employed to develop convergent solutions of the resulting problem.Discussion regarding velocity,thermal field and concentration distribution for several involved parameters is pivotal part.Graphical behaviors of skin friction coefficient and Nusselt number are also portrayed.Concentration of the reactants is found to depreciate as a result of strength of both heterogeneous and homogeneous reaction parameters.With existence of melting phenomenon,declining attitude of fluid temperature is observed for higher radiation parameter.
基金supported by National Natural Science Foundation of China(No.51977132)the Key Special Science and Technology Project of Liaoning Province(No.2020JH1/10100012)the General Program of the Education Department of Liaoning Province(No.LJKZ0126).
文摘The self-excited DC air circuit breaker(SE-DCCB)has been widely used in urban rail transit due to its excellent stability.It can realize forward and reverse interruption,but has difficulty interrupting small currents due to the phenomenon of arc root sticking at the entrance of the arc chamber in the splitting process,which is known as arc root stagnation.A coupling model of the self-excited magnetic field and magnetohydrodynamics is established for the SE-DCCB with the traditional structure.The magnetic field,temperature and airflow distribution in the arc chamber are investigated with an interrupting current of 150 A.The simulation results show that the direction and magnitude of the magnetic blowout force are the dominant factors in the arc root stagnation.The local high temperature of the arc chamber due to arc root stagnation increases the obstruction effect of the airflow vortex on the arc root movement,which significantly increases the arc duration time of small current interruption.Based on the research,the structure of the magnetic conductance plate of the actual product is improved,which can improve the direction and magnitude of the magnetic blowout force at the arc root so as to restrain the development of the airflow vortex effectively and solve the problem of arc root stagnation when the small current is interrupted.The simulation results show that the circuit breaker with improved structure has a better performance for a small current interruption range from 100 A to 350 A.
文摘The effect of variable viscosity and thermal conductivity on steady magnetohydrodynamic(MHD) heat and mass transfer flow of viscous and incompressible fluid near a stagnation point towards a permeable stretching sheet embedded in a porous medium are presented,taking into account thermal radiation and internal heat genberation/absorbtion.The stretching velocity and the ambient fluid velocity are assumed to vary linearly with the distance from the stagnation point.The Rosseland approximation is used to describe the radiative heat flux in the energy equation.The governing fundamental equations are first transformed into a system of ordinary differential equations using a scaling group of transformations and are solved numerically by using the fourth-order Rung-Kutta method with the shooting technique.A comparison with previously published work has been carried out and the results are found to be in good agreement.The results are analyzed for the effect of different physical parameters,such as the variable viscosity and thermal conductivity,the ratio of free stream velocity to stretching velocity,the magnetic field,the porosity,the radiation and suction/injection on the flow,and the heat and mass transfer characteristics.The results indicate that the inclusion of variable viscosity and thermal conductivity into the fluids of light and medium molecular weight is able to change the boundary-layer behavior for all values of the velocity ratio parameter λ except for λ = 1.In addition,the imposition of fluid suction increases both the rate of heat and mass transfer,whereas fluid injection shows the opposite effect.
文摘The effect of melting heat transfer on the two dimensional boundary layer flow of a micropolar fluid near a stagnation point embedded in a porous medium in the presence of internal heat generation/absorption is investigated. The governing non-linear partial differential equations describing the problem are reduced to a system of non-linear ordinary differential equations using similarity transformations solved numerically using the Chebyshev spectral method. Numerical results for velocity, angular velocity and temperature profiles are shown graphically and discussed for different values of the inverse Darcy number, the heat generation/absorption parameter, and the melting parameter. The effects of the pertinent parameters on the local skin-friction coefficient, the wall couple stress, and the local Nusselt number are tabulated and discussed. The results show that the inverse Darcy number has the effect of enhancing both velocity and temperature and suppressing angular velocity. It is also found that the local skin-friction coefficient decreases, while the local Nusselt number increases as the melting parameter increases.
文摘AIM To investigate the pathophysiologic basis of syndrome of Liver Qi stagnation and parameters for clinical differentiation. METHODS Plasma L ENK, AVP, ANP and serum gastrin were determined by RIA in 84 patients with neurasthenia, mastodynia, chronic gastritis, and chronic cholecystitis presenting the same syndrome of Liver Qi stagnation in traditional Chinese medicine (TCM). Healthy subjects served as controls in comparison with patients having the same syndrome but with different diseases. RESULTS Among the patients with Liver Qi stagnation, the plasma L ENK, ANP and gastrin levels were 38 83ng/L ± 6 32ng/L , 104 11ng/L ± 29 01ng/L and 32 20ng/L ± 6 68ng/L , being significantly lower than those in the healthy controls ( t =3 34, 6 17, 4 48; P <0 01). The plasma AVP of the patient group ( 52 82ng/L ± 19 09ng/L ) was significantly higher than that of the healthy controls ( t =5 79, P <0 01). The above changes in patients having the same symptom complex but different diseases entities showed no significant differences, P >0 05. CONCLUSION The syndrome of Liver Qi stagnation is closely related to the emotional modulatory abnormality of the brain, with decrease of plasma L ENK, ANP and gastrin, and increase of plasma AVP as the important pathophysiologic basis.
文摘Combined effects of Soret(thermal-diffusion) and Dufour(diffusion-thermo) in MHD stagnation point flow by a permeable stretching cylinder were studied. Analysis was examined in the presence of heat generation/absorption and chemical reaction. The laws of conservation of mass, momentum, energy and concentration are found to lead to the mathematical development of the problem. Suitable transformations were used to convert the nonlinear partial differential equations into the ordinary differential equations. The series solutions of boundary layer equations through momentum, energy and concentration equations were obtained.Convergence of the developed series solutions was discussed via plots and numerical values. The behaviors of different physical parameters on the velocity components, temperature and concentration were obtained. Numerical values of Nusselt number, skin friction and Sherwood number with different parameters were computed and analyzed. It is found that Dufour and Soret numbers result in the enhancement of temperature and concentration distributions, respectively.
基金Project supported by the National Natural Science Foundation of China(Nos.11971142,11871202,61673169,11701176,11626101,and 11601485)。
文摘In this research,the three-dimensional(3D)steady and incompressible laminar Homann stagnation point nanofluid flow over a porous moving surface is addressed.The disturbance in the porous medium has been characterized by the Darcy-Forchheimer relation.The slip for viscous fluid is considered.The energy equation is organized in view of radiative heat flux which plays an important role in the heat transfer rate.The governing flow expressions are first altered into first-order ordinary ones and then solved numerically by the shooting method.Dual solutions are obtained for the velocity,skin friction coefficient,temperature,and Nusselt number subject to sundry flow parameters,magnetic parameter,Darcy-Forchheimer number,thermal radiation parameter,suction parameter,and dimensionless slip parameter.In this research,the main consideration is given to the engineering interest like skin friction coefficient(velocity gradient or surface drag force)and Nusselt number(temperature gradient or heat transfer rate)and discussed numerically through tables.In conclusion,it is noticed from the stability results that the upper branch solution(UBS)is more reliable and physically stable than the lower branch solution(LBS).
文摘The present study inspects the non-aligned stagnation point nano fluid over a convective surface in the presence of partial slip. Two types of base fluids namely water and kerosene are selected with Cu nanoparticles. The governing physical problem is presented and transformed into a system of coupled nonlinear differential equations using suitable similarity transformations. These equations are then solved numerically using midpoint integration scheme along with Richardson extrapolation via Maple. Impact of relevant physical parameters on the dimensionless velocity and temperature profiles are portrayed through graphs. Physical quantities such as local skin frictions co-efficient and Nusselt numbers are tabularized.It is detected from numerical computations that kerosene-based nano fluids have better heat transfer capability compared with water-based nanofluids. Moreover it is found that water-based nanofluids offer less resistance in terms of skin friction than kerosene-based fluid. In order to authenticate our present study, the calculated results are compared with the prevailing literature and a considerable agreement is perceived for the limiting case.
文摘In this study,the stagnation point transport of second grade fluid with linear stretching under the effects of variable thermal conductivity is considered.Induced magnetic field impact is also incorporated.The nonlinear set of particle differential equations is converted into set of ordinary differential equations through appropriate transformation.The resulting equations are then resolved by optimal homotopy analysis method.The effect of pertinent parameters of interest on skin friction coefficient,temperature,induced magnetic field,velocity and local Nusselt number is inspected by generating appropriate plots.For numerical results,the built-in bvp4 c technique in computational software MATLAB is used for the convergence and residual errors of obtained series solution.It is perceived that the induced magnetic field is intensified by increasing β.It can also be observed that skin friction coefficient enhances with increasing value of magnetic parameter depending on the stretching ratio a/c.For the validness of the obtained results,a comparison has been made and an excellent agreement of current study with existing literature is found.
文摘This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.