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This paper presents a series of numerical simulations on the effect of heat dissipation on the thermocapillary convection of a low Prandtl number (Pr = 0.011) fluid in an annular pool heated from its inner cylinder. The radius ratio and the aspect ratio of the annular pool set as η = 0.5 and e = 0.1, respectively. The results show that with the increase of Biot number, the isotherms of the two-dimensional basic flow are seriously compressed near the hot inner cylinder and the thermocapillary flow pattern will be impaired. With the increase of Marangoni number, the thermocapillary convection first evolves from the basic flow to a three-dimensional stationary flow with longitudinal stationary stripes on the free surface, and then to a three-dimensional oscillatory flow featured with the combination of stationary stripes and azimuthal waves, which is caused by the mutual effect of the radial convection, azimuthal local flow and heat dissipation. After the flow destabilization, the amplitude of temperature fluctuation increases and the wave number decreases with the increase of Marangoni number when the free surface is adiabatic. Furthermore, the temperature fluctuation pattern is changed due to the surface heat dissipation. In addition, with the increase of Biot number, the wave number almost keeps constant when Marangoni number is small, but it increases significantly at a large Marangoni number.

This paper presented a set of direct numerical simulations on the thermocapillary flow instability of moderate-Prandtl number fluids in annular pools heated from inner cylinder. The emphasis is on the aspect ratio dependence of the critical Marangoni number of flow destabilization and the formation mechanism of the different flow patterns. It was found that the two-dimensional axisymmetric steady flow will bifurcate into three types of oscillatory flows, including the oscillatory concentric rings, the alternating straight spoke and the rotating wave patterns. The bifurcation process depends mainly on the aspect ratio of the annular pool when the radius ratio is fixed. With the increase of the aspect ratio, the first instability flow pattern is oscillatory concentric rings and the corresponding critical Marangoni number and the oscillatory frequency decrease first, and then increase slightly. When the flow pattern transits to the alternating straight spoke pattern at the aspect ratio of 0.415, there are an abrupt increase of the critical Marangoni number and a sudden decrease of the oscillatory frequency. Furthermore, when the rotating wave pattern appears, the critical Marangoni number decreases gradually, but the corresponding oscillatory frequency has only a slight decrease.

In order to understand the effect of surface heat dissipation on thermocapillary convection of moderate Prandtl number fluid in an annular shallow pool, we performed a series of three-dimensional numerical simulations by using the finite volume method. An annular shallow pool is full of 0.65cSt silicone oil with Prandtl number of 6.7. The radius ratio and the aspect ratio of the pool are respectively fixed at 0.5 and 0.1. Surface heat dissipation Biot number is varied from 0 to 50. Results indicate that the critical Marangoni number of flow destabilization mainly depends on the coupling effect of the thermocapillary force and surface heat dissipation on the free surface, which decreases first, and then increases gradually. When Biot number is small, the steady axisymmetric flow after the flow destabilization will bifurcate to the hydrothermal wave. With the increase of Marangoni number, the flow pattern evolution at a small Biot number is similar to that of an adiabatic free surface, and the fundamental oscillatory frequency increases gradually. With the increase of Biot number, the radial roll cells near the outer cylindrical wall gradually extend to the inner wall, and eventually occupy the whole liquid pool. When Biot number is large, the flow pattern after the flow destabilization is a radial rolling cell pattern with alternating azimuthal direction. Then it gradually evolves into azimuthal waves, and the fundamental oscillatory frequency has a slight decrease.

In order to understand the effect of Marangoni convection on the evaporation rate and the flow pattern, we performed a series of three-dimensional numerical simulations on evaporation process of sessile water droplet by introducing a kinetic model. Substrate temperature and the ratio of vapor pressure varied from 299 K to 308 K and from 0.92 to 0.99, respectively. The variation range of contact angle θ was between 9.15° and 120°. Results show that the axisymmetric ring-shape temperature distribution on droplet free surface transforms into a serrated temperature distribution because of the enhancing Marangoni convection with the increase of substrate temperature. In addition, with the decrease of vapor pressure, the total evaporation rate on free surface will be increased and the Marangoni effect will be enhanced. The flow pattern is axisymmetric at 90o ≤ θ ≤ 120o when the substrate temperature is fixed. However, it shifts to a multi-cellular pattern with the decrease of contact angle as a result of the competition between Marangoni flow and the evaporation. Additionally, the total evaporation rate increases with the increase of contact angle when contact angle is above 60o, but when the contact angle is below 60o, the trend is reverted. The temperature distribution becomes distinct at a small contact angle.

In order to understand surface heat dissipation dependence of thermocapillary convection for moderate Prandtl number fluid in a deep annular pool, a series of three-dimensional numerical simulations have been carried out by using the finite volume method. The radius ratio and the aspect ratio of an annular pool are fixed at 0.5 and 1.0, respectively. The working fluid is 0.65cSt silicone oil with Prandtl number of 6.7. Surface heat dissipation Biot (Bi) number is varied from 0 to 50. Results indicate that with the increase of Biot number, the radial temperature gradient near the inner cylindrical wall decreases, and near the outer cylindrical wall it increases, so the flow is enhanced. When 0

This study aims to improve the thermo-economic performance of the subcritical organic Rankine cycle (ORC). Accordingly, an economical evaluation and the simultaneous multiparametric optimization on the subcritical ORC system for recovering low-grade waste heat of flue gas were conducted using electricity production cost (EPC) as the evaluation indicator. Results show that the optimum evaporating and condensing temperatures are mainly influenced by the inlet temperatures of flue gas and cooling fluid, respectively. High critical temperature of working fluids indicates high thermal-economic performance of the ORC system for 19 kinds of selected working fluids. Furthermore, the optimum pinch point temperature difference of evaporator decreases and that of condenser increases with the variation in the critical temperature of working fluids, but the optimum condensing temperature is unchanged. On the basis of a large number of calculation data, the correlations among the optimum cycle and external operating parameters of the subcritical ORC are proposed. These correlations can guide the engineering design of the subcritical ORC system.

This paper presented a series of numerical simulations on thermocapillary convection for mixed toluene/hexane solution with mass fraction of $c_{0}= 26.27$ % in a shallow annular pool. The Prandtl number of the binary solution is 5.54. Results indicate that when the annular pool subjects to a radial temperature gradient, the solute shifts toward the inner wall under the Soret effect, which leads to a concentration gradient with the opposite direction to the temperature gradient. With the increase of surface heat dissipation, thermocapillary convection is enhanced and the flow is more prone to destabilization. Therefore, the critical thermocapillary Reynolds number of the flow destabilization and the corresponding critical oscillation frequency all decrease, but the critical wave number increases. After flow destabilization, the concentration fluctuation similar to the temperature fluctuation on the free surface appears. No matter the free surface is adiabatic or not, the flow always undergoes a transition from two-dimensional steady axisymmetric flow to the hydrothermal waves, and then to chaos with the increase of thermocapillary Reynolds number.

In order to understand the characteristics of Rayleigh-Benard convection of nanofluids in a rectangular cavity, a series of threedimensional numerical simulations were carried out by using the finite volume method. The working mediums were Al2O3-water and CuO-water nanofluids, and Rayleigh number varied from 1708 to 1.5×105. Results show that the critical Rayleigh number of the Rayleigh-Benard convection onset increases with the increase of the volume fraction of nanoparticles, which hints that nanoparticles make the flow become more stable. Furthermore, the critical Rayleigh number decreases with the increase of the aspect ratio of the cavity. After convection appears, the average Nusselt number at the bottom wall increases, but the flow intensity and the oscillation frequency decrease with the increase of the volume fraction of nanoparticles. With the increase of Rayleigh number, there exist four bifurcation sequences, which are Steady flow (SF) → Periodic oscillatory flow (POF) → Chaotic flow (CF), SF → POF → SF → POF →CF, SF → POF → Multiple periodic oscillatory flow (MPOF) → POF →CF and SF → POF → MPOF →CF.

In order to understand the onset of Rayleigh-Benard convection of cold water near its density maximum in a vertical annular container heated from below and cooled from above, the instability of the static diffusive state was investigated by linear stability analysis. Two kinds of thermal boundary conditions, perfect insulating and conducting sidewalls, were used. The aspect ratio and radius ratio of the annular container varied from 0.5 to 4.5 and 0.1 to 0.8, respectively. Results indicate that the critical Rayleigh number and the critical wave number increase with the increase of density inversion parameter. There are some new flow modes with large azimuthal wave number near the convection onset for cold water with density maximum, which are not observed in Rayleigh-Benard convection of the Boussinesq fluids. The stratified flow phenomenon in cold water with density maximum is certified. The larger the density inversion parameter is, the more obvious the stratified flow phenomenon becomes. With the increase of the aspect ratio, the critical Rayleigh number decreases, but the critical wave number increases. However, with the increase of the radius ratio, both the critical Rayleigh number and the critical wave number increase continuously. Furthermore, the effect of thermal boundary conditions at the sidewalls on the convection stability is also validated.

In order to understand the effect of surface evaporation on thermocapillary convection in an annular pool, a series of numerical simulation on thermocapillary convection of the fluids with Prandtl number from 0.01 to 50 in the pure vapor environment were carried out. The results show that thermocapillary convection is always coupled with the evaporation process on the free surface. With the increase of evaporation Biot number, the surface temperature decreases, and the evaporation mass flux near the hot wall increases obviously. However, near the cold wall, the evaporation mass flux increases first, and then decreases. When Marangoni number is small, the total evaporation mass rate at free surface increases with the increase of evaporation Biot number; when Marangoni number is larger, it increases first and then approaches a constant value. The aspect ratio of the annular pool has a positive influence on the thermocapillary convection strength and the total evaporation mass rate. With the increase of Prandtl number, the surface temperature rises gradually and the evaporative mass flux increases, and the thermocapillary convection cell moves gradually toward the outer wall and the free surface. This effect decreases with the increase of evaporation Biot number When evaporation Biot number is smaller, the total evaporation mass rate increases with the Prandtl number; when Biot number is larger, Prandtl number has little impact on the total evaporation mass rate.

In order to understand the characteristics of thermocapillary flow of a toluene/n-hexane mixture with the Soret effect in a shallow annular pool, a series of three-dimensional numerical simulations were carried out. The shallow annular pool was heated from the outer cylinder and cooled at the inner cylinder. The initial toluene concentration in the toluene/n-hexane mixture varied from 0 to 0.4467. Results indicate that the flow undergoes two transitions from the axisymmetric steady flow to the hydrothermal waves, and then to chaos with the increase of the thermocapillary Reynolds number. The critical thermocapillary Reynolds number for the incipience of the oscillatory flow decreases with the increase of the initial solute concentration. When the thermocapillary flow transits to a three-dimensional oscillatory flow, a concentration fluctuation is observed on the free surface, which is similar to the hydrothermal waves. However, compared with that of the temperature, the dimensionless fluctuation amplitude of the concentration is relatively weak. Furthermore, the fundamental oscillation frequency increases linearly with the initial solute concentration, but the wave number of the hydrothermal waves is almost unchangeable.

In order to understand the characteristics of pure solutocapillary flow in a shallow annular pool subjected to a constant radial solutal gradient, a series of three-dimensional numerical simulations were performed. The annular pool was filled with the toluene/n-hexane mixture fluid with the Schmidt number of 142.8. The inner and outer cylinders were respectively maintained at low and high solutal concentrations. Aspect ratio of the annular pool is fixed at e = 0.15 or 0.05. Results indicate that the solutocapillary flow is steady and axisymmetric at a small solutal capillary Reynolds number. The surface fluid flows radially from the inner cylinder toward the outer cylinder and a return flow exists near the bottom. With the increase of the solutal capillary Reynolds number, an axisymmetric oscillatory flow firstly appears and then becomes a three-dimensional oscillatory flow at e = 0.15. Whereas at e = 0.05 a direct transition from the steady and axisymmetric flow to the three-dimensional oscillatory flow is observed. Three types of the flow instabilities are the standing wave, hydrosolutal wave and source/sink type wave instabilities. Furthermore, the physical mechanism of the flow destabilization is analyzed.

We present a novel coupled organic Rankine cycle (CORC) system driven by the low-grade waste heat, which couples a transcritical organic Rankine cycle with a subcritical organic Rankine cycle. Based on classical thermodynamic theory, a detailed performance analysis on the novel CORC system was performed. The results show that the pressure ratio of the expander is decreased in the CORC and the selection of the working fluids becomes more flexible and abundant. With the increase of the pinch point temperature difference of the internal heat exchanger, the net power output and thermal efficiency of the CORC all decrease. With the increase of the critical temperature of the working fluid, the system performance of the CORC is improved. The net power output and thermal efficiency of the CORC with isentropic working fluids are higher than those with dry working fluids.

This paper gives a special focus on the role of outlet temperature of flue gas (Tgo) in organic Rankine cycle (ORC) system for low tem- perature flue gas waste heat recovery. The variations of performance indicators: net work (Wnet), exergy efficiency (ηex) and levelized energy cost (LEC) versus Tgo are discussed. Considering the corrosion of low temperature flue gas, the necessity and reasonability of limiting Tgo at its minimum allowed discharge temperature (355.15 K) are analyzed. Results show that there exist optimal Tgo (Tgo,opt) for Wnet and LEC, while Tgo,opt for ηex does not appear under the investigated range of Tgo. Moreover, the Tgo,opt for Wnet is always lower than 355.15 K, the Tgo,opt for LEC, despite being greater than the one for Wnet, is just slightly higher than 355.15 K when the inlet temperature of flue gas varies from 408.15 K to 463.15 K. For the waste heat recovery of low temperature flue gas, it is reasonable to fix Tgo at 355.15 K if Wnet or LEC is selected as primary performance indicator under the pinch point temperature difference of evaporator (ΔTe) below 20 K.

In this paper, the net power output, exergy efficiency and levelized energy cost of system were selected as performance indicators for assessing Organic Rankine Cycle (ORC). Firstly, the turbine inlet temperature and pressure meeting the requirement of pinch point temperature difference of evaporator in transcritical ORC (trans-ORC) were determined based on performance optimization. Subsequently, the thermo-economic performance of a subcritical ORC (sub-ORC) using R601 as working fluid and a trans-ORC using R134a as working fluid were compared under different heat source temperatures and a fixed outlet temperature of flue gas. Results show that for trans-ORC, when the pinch point temperature difference of evaporator lies between the inlet and outlet of evaporator, a lower inlet pressure of turbine is favorable; when the pinch point temperature difference of evaporator is located at the outlet of evaporator, there exists an optimal inlet pressure of turbine. Either for sub-ORC or trans-ORC, the net power output increases and levelized energy cost decreases with the increase in heat source temperature. For sub-ORC, exergy efficiency of system increases monotonously with heat source temperature, while for trans-ORC, exergy efficiency of system grows up firstly and then reduces (or keeps constant) with the increasing of heat source temperature. Moreover, for net power output and exergy efficiency of system, there exist a range of heat source temperatures making trans-ORC better than sub-ORC, and the heat source temperature region extends with the increase in pinch point temperature difference of evaporator. For levelized energy cost of system, the sub-ORC is always superior to trans-ORC.

To explore the influence of Chinese medicine constitution type on the susceptibility of hypertensive patients to intracerebral haemorrhage. Primary hypertensive patients were studied and divided into the hypertension and the hypertensive intracerebral haemorrhage groups, depending on whether or not the patients had intracerebral haemorrhage. The demographic characteristics, physiological characteristics, living habits, biochemical tests, other chronic diseases, Chinese medicine constitution type, etc. were collected and compared between the two groups. The neurological deficit in the hypertensive intracerebral haemorrhage group was also compared among the different constitution types. A total of 304 patients participated in this investigation, including 213 cases in the hypertension group and 91 cases in the hypertensive intracerebral haemorrhage group. The percentages of dampness-heat and qi-depression types in the intracerebral haemorrhage group were greater than those with the same types in the hypertension group (20.9% vs 6.1%, 22.0% vs 8.0%; P

In this paper, an exergo-economic criterion, i.e. the net profit per unit transferred heat load, is established from the perspective of exergy recovery to evaluate the performance of finned tube used in waste heat recovery. Also, the dimensionless exergy change number is introduced to investigate the effect of the flow (mechanical) exergy loss rate on the recovered thermal exergy. Selecting R245fa as a working fluid and exhaust flue gas as a heat source, the effects of the internal Reynolds number Rei, the external Reynolds number Reo, the unit cost of thermal exergy ɛq, the geometric parameter of finned tube ηoβ and the phase change temperature Tv etc. on the performance of finned tube are discussed in detail. The results show that the higher Tv and ηoβ, and lower Rei may lead to the negligible flow (mechanical) exergy loss rate. There exists an optimal value of Rei where the net profit per unit transferred heat load peaks, while the variations of Reo, ɛq and Tv cause monotonic change of the net profit per unit transferred heat load. The phase change temperature exerts relatively greater influence on the exergo-economic performance of finned tube in comparison with other parameters. And there exists a critical phase change temperature, where the net profit per unit transferred heat load is equal to zero.

In order to understand the transition characteristics of the thermal-solutal capillary convection in an annular pool, a series of two-dimensional numerical simulations were conducted. The bottom of the pool is adiabatic rigid wall and the top is adiabatic and non-deformable free surface. The inner and outer cylindrical walls maintain at constant temperature and solute concentration, respectively. The thermo-capillary force is supposed to equal to the solute-capillary force, but their directions are contrary. Results show that the thermal-solutal capillary convection is steady at a small Reylonds number. When the capillary Reynolds number exceeds a critical value, the steady flow transits into unstable thermal-solutal capillary convection. The transition from the steady to oscillatory flow undergoes a Hopf bifurcation. Furthermore, the effects of the liquid layer aspect ratio, the radius ratio, the Prandtl number and the Lewis number on the onset of flow pattern transition are discussed. The physical mechanism of the unstable thermal-solutal capillary convection is also analyzed.

Using asymptotical analysis, we investigate the characteristics of the coupled thermal and solutal capillary convection with the radial temperature and solute concentration gradients in a shallow annular pool with the free surface. The pool is heated from the outer cylinder with high solutal concentration and cooled at the inner cylinder with low solutal concentration. The asymptotic solution is obtained in the core region in the limit as the aspect ratio, which is defined as the ratio of the depth to the width of the pool, goes to zero. The comparison with the previous work certifies that the asymptotic solution is right and believable. The influences of the solutal capillary force, the buoyant force, the Soret effect and the geometric parameters on the fluid flow are analyzed.

By establishing an axial symmetric invariable temperature phase change interface model of porous wick evaporator based on alkali metal thermal-to-electric converter (AMTEC), a parametric study has been implemented to explore the impact of effective thermal conductivity model, working fluid, wick material, porosity and particle size of porous wick on the flow and heat transfer characteristics. The results indicate that the inner surface temperature profile of wick is closely dependent on effective thermal conductivity models; it is the minimum for series model while the highest for parallel model. The inner surface temperature of wick with Na-K alloy as working fluid is obviously lower than that with sodium or potassium as working fluid. The wick made of refractory ceramics makes the minimum of inner surface temperature, while the one with nickel appears the highest temperature. Increasing the porosity can reduce the pressure drop in the wick, but also increases the temperature of evaporator.

Based on the heat transfer characteristics of absorber plate and the heat transfer effectiveness-number of heat transfer unit method of heat exchanger, a new theoretical method of analyzing the thermal performance of heat pipe flat plate solar collector with cross flow heat exchanger has been put forward and validated by comparisons with the experimental and numerical results in pre-existing literature. The proposed theoretical method can be used to analyze and discuss the influence of relevant parameters on the thermal performance of heat pipe flat plate solar collector.

Numerical simulations of flow in the melt (CdZnTe) with different conditions are conducted using the finite-difference method. When the top surface of the melt is solid wall under microgravity condition, the thermocapillary convection is caused in the melt by the surface tension gradient on the free surface. As the Marangoni number is small, the flow is steady thermocapillary convection. As the Marangoni number exceeds the critical value, the steady flow transits into unstable thermocapillary convection. When the top surface of the melt is free surface under microgravity, two roll cells are observed in the melt, which are driven by both the surface tension gradients on the upper and lower free surfaces. When the top surface of the melt is free surface under gravity condition, the effect of the buoyancy on the flow is little as the Marangoni number is small. With the Marangoni number increasing, the effect of the buoyancy increases, which makes the upper roll cell weaken and the lower roll cell strengthen.

This work is devoted to the study of steady thermocapillary-buoyant convection in a system of two horizontal superimposed immiscible liquid layers filling a lateral heated thin annular pool. The governing equations are solved using an asymptotic theory for the aspect ratios ɛ → 0. Asymptotic solutions of the velocity and temperature fields are obtained in the core region away from the cylinder walls. In order to validate the asymptotic solutions, numerical simulations are also carried out and the results are compared to each other. It is found that the present asymptotic solutions are valid in most of the core region. And the applicability of the obtained asymptotic solutions decreases with the increase of the aspect ratio and the thickness ratio of the two layers. For a system of gallium arsenide (lower layer) and boron oxide (upper layer), the buoyancy slightly weakens the thermocapillary convection in the upper layer and strengthens it in the lower layer.

In order to understand the effect of the vertical heat transfer on thermocapillary convection characteristics in a differentially heated open shallow rectangular cavity, a series of two- and three-dimensional numerical simulations were carried out by means of the finite volume method. The cavity was filled with the 1cSt silicone oil (Prandtl number Pr = 13.9) and the aspect ratio ranged from 12 to 30. Results show that thermocapillary convection is stable at a small Marangoni number. With the increase of the heat flux on the bottom surface, thermocapillary convection transits to the asymmetrical bi-cellular pattern with the opposite rotation direction. The roll near the hot wall shrinks as the Marangoni number increases. At a large Marangoni number, numerical simulations predict two types of the oscillatory thermocapillary flow. One is the hydrothermal wave, which is dominant only in a thin cavity. The other appears in a deeper cavity and is characterized by oscillating multi-cellular flow. The critical Marangoni number for the onset of the oscillatory flow increases first and then decreases with the increase of the vertical heat flux. The three-dimensional numerical simulation can predict the propagating direction of the hydrothermal wave. The velocity and temperature fields obtained by three-dimensional simulation in the meridian plane are very close to those obtained by two-dimensional simulation.

The linear-stability analysis of thermocapillary flow in the annular immiscible two-layer system of 5cSt silicone oil and HT-70 with a radial temperature gradient was carried out. The annular two-layer system is heated at the outer cylindrical wall and cooled at the inner wall, the bottom and top surfaces are bounded by two rigid and heat-insulated walls. The influences of the liquid layer depth and radius ratio between the cold inner wall and the hot outer wall on stability are thoroughly investigated. The critical Marangoni number, critical wave number and critical phase velocity are obtained. In addition, the mode of bifurcation for the hydrothermal wave is predicted at different liquid layer depth, and the temperature disturbance pattern of hydrothermal wave at interface is also exhibited.

Three-dimensional numerical simulation was developed to investigate thermocapillary flow induced by non-uniform evaporation on the meniscus in capillary tubes. Capillary tube radiuses ranging from 0.1 to 1 mm were considered and the working liquid was methanol. The effects of tube size, evaporation heat flux and buoyancy on thermocapillary flow were investigated. The results show that the non-uniform evaporation on the meniscus leads to two opposite temperature gradients along the radial direction, which generate two thermocapillary flow vortexes under the meniscus. For horizontal capillary tubes with r0 ≥ 0.32 mm, the path-lines in the vertical center plane are asymmetrical, which is attributed to the combined buoyancy and thermocapillary effects. For the vertical capillary tube, with increasing average evaporation heat flux, the steady axisymmetrical flow will gradually transit to a steady asymmetrical flow and eventually becomes a three-dimensional oscillatory flow.

In this study, numerical investigations are conducted for forced convective heat transfer in an annular helicoidal tube under uniform wall temperature condition for laminar flow including developing region. The numerical computations reveal the developments and distributions of heat transfer and flow fields in the annular helicoidal tube when the outer tube wall is heated and the inner tube wall is insulated. The effects of Reynolds number, curvature ratio, and coil pitch on the circumferential average friction factor and Nusselt number at different axial locations, and the non-dimensional entropy generation number of laminar convection in an annular helicoidal tube are investigated. In addition, the differences of flow and heat transfer characteristics between the annular helicoidal tube and circular helicoidal tube are also described.

In order to understand the characteristics of thermocapillary-buoyancy convection in the annular two-layer system of 5cSt silicone oil and HT-70 with a radial temperature gradient, a set of two-dimensional numerical simulations are carried out using the finite-volume method. The annular two-layer system is heated from the outer cylindrical wall and cooled at the inner cylindrical wall. The radius ratio and the aspect ratio of the system are 0.2 and 0.05–0.1, respectively. Results show that the flow is steady for sufficiently small Marangoni number. When the Marangoni number exceeds a critical value, the additional cells appear near the interface and the cold inner wall, and then an unsteady multi-cellular structure is developed. The temperature fluctuation wave propagates from the cold inner wall to the hot outer wall. And the critical Marangoni number decreases with the increase of the aspect ratio.

This paper examines the steady thermocapillarybuoyant convection in a shallow annular pool subjected to a radial temperature gradient. A matched asymptotic theory is used to obtain the asymptotic solutions of the flow and thermal fields in the case of small aspect ratios, which is defined as the ratio of the layer thickness to the gap width. The flow domain is divided into the core region away from the cylinder walls and two end regions near each cylinder wall. Asymptotic solutions are obtained in the core region by solving the core and end flows separately and then joining them through matched asymptotic expansions. For the system of silicon melt, the asymptotic solutions are compared with the results of numerical simulations. It is found that the two kinds of solutions have a good agreement in the core region for a small aspect ratio. With the increase of aspect ratio, the applicability of the present asymptotic solutions decreases gradually.

In order to understand the influence of the vertical heat flux on thermocapillary convection, we conducted a series of unsteady two-dimensional numerical simulations of thermocapillary convection in a differently heated shallow rectangular cavity with vertical heat flux on the bottom by means of the finite volume method. The cavity was filled with the 1cSt silicone oil (Prandtl number Pr = 13.9) and aspect ratio is 30. It is found that a small vertical heat flux has slightly influence on the flow pattern of stable or unstable thermocapillary convection. However, the critical Marangoni number increases first, and then decreases with the increase of the heat flux. And the flow pattern of the oscillatory thermocapillary convection transits from a series of the rolls rotating clockwise and moving from the cold wall to the hot wall to the single roll near the hot wall and a series of rolls near the cold wall, further, two series of rolls moving from the hot wall and cold wall towards the hot spot with the maximum temperature. With the increase of the Marangoni number, the period and the wavelength of the oscillatory thermocapillary convection increase, but the wave speed decreases.

The steady laminar two-dimensional thermocapillary convection of two superposed horizontal liquid layers in a shallow annular cavity was investigated using asymptotical analysis. The liquids were supposed to be immiscible with a nondeformable interface. The cavity was heated from the outer cylindrical wall and cooled at the inner wall. Bottom and top surfaces were rigid and adiabatic. Asymptotic solutions were obtained in the core region in the limit as the aspect ratio, which was defined as the ratio of the lower layer thickness to the gap width, and trended to zero. The numerical experiments were also carried out to compare with the asymptotic solution of the steady two-dimensional thermocapillary convection. It is found that the expressions of velocity and temperature fields in the core region are valid in the limit of the small aspect ratio.