Linear Stability Analysis Of Buoyancy-Thermocapillary Convection Driven By Bidirectional Temperature Gradients

Temperature-driven buoyancy-thermocapllary convection research has always been a focus in fluid mechanics and heat and mass transfer.Buoyancy-thermocapllary convection under horizontal temperature gradient and vertical temperature gradient has been extensively studied,and the results are abundant.For systems with both horizontal temperature gradient and vertical temperature gradient,that is,buoyancy-thermocapllary convection under bidirectional temperature gradient,there is yet not much research currently.Whether in the natural world or in practical industrial applications,the strict one-way temperature gradient is not much,and the existence of bidirectional temperature gradient is difficult to avoid.Therefore,studying the buoyancy-thermocapllary convection under the bidirectional temperature gradient has both theoretical and practical significance.Since there is no systematic linear stability analysis of the buoyancy-thermocapllary convection under bidirectional temperature gradient,this paper establishes a model of horizontally infinite liquid layer under the bidirectional temperature gradient.Both the single layer model and gas-liquid two-layer model are considered,and a systematic linear stability analysis is conducted on them.The influence of temperature gradient inclination?ratio of vertical temperature gradient to horizontal temperature gradient b?,buoyancy?Bo_D number?,surface heat exchange?Bi number?and fluid property parameters?Pr number?on flow stability are studied.By analyzing the two modes of transverse rolls instability and longitudinal rolls instability,the critical Marangoni number,critical wave number and critical frequency of buoyancy-thermocapllary convection under different control parameters are obtained,and the results are summarized through analysis and comparison.The parameters that define liquid layer being dominated by the horizontal temperature gradient or the vertical temperature gradient for buoyancy-thermoapillary convection stability under bidirectional temperature gradient are determined,and the characteristics of different flow instability states are obtained.The results are as follows:?1?For transverse rolls instability,when the temperature gradient gradient?b?and buoyancy?Bo_D number?of the system increase,the liquid layer will change from being dominated by the horizontal temperature gradient to be dominated by the vertical temperature gradient.When the horizontal temperature gradient dominates,the critical Marangoni number of the liquid layer increases with the number of Bo_D and b,where the buoyancy has a positive effect on the stability of the flow,and when the liquid layers are dominated by vertical temperature gradients,it gradually increases the instability of the flow.In addition,for longitudinal rolls instability,its critical value of instability is not affected by buoyancy.?2?When the vertical temperature gradient inside the liquid layer exceeds a certain critical value,we observe that the critical Marangoni number will have a"sudden drop"phenomenon,at which time the critical Marangoni number will be reduced to a minimum value,such as in the transverse rolls instability.?3?For transverse rolls instability,when the temperature gradient gradient?b?and the free surface heat transfer coefficient?Bi number?of the system increase,the liquid layer will change from being dominated by the horizontal temperature gradient to the vertical temperature gradient.In both cases,the effect of Bi number on fluid stability is the same as that of Bo_D,while the longitudinal rolls instability mode is relatively less affected by Bi number.Comparing the normal gravity with the microgravity condition,we find that the node that transforms the fluid into a vertical temperature gradient under normal gravity is pushed back,but has no effect on the longitudinal rolls instability.?4?For transverse rolls instability,the effect of Pr number on fluid stability is similar to that of Bi number and Bo_D number.When horizontal temperature gradient dominates,the critical Marangoni number of liquid layer increases with the increase of Pr number.Different from the other two influencing factors,the influence of the Pr number on the longitudinal rolls instability has a greater influence than the Bi number and the Bo_D number.When the horizontal temperature gradient dominates,the critical Marangoni number of the liquid layer increases as the Pr number increases,while the microgravity condition is substantially the same as the critical value under normal gravity.?5?For transverse rolls instability,when the vertical temperature gradient dominates,the critical frequency of the unstable rolls is significantly higher than that of the horizontal temperature gradient,which can be as high as 3-4 times,except for the critical Marangoni number.When the“sudden drop”phenomenon occurs,the corresponding critical frequency drops to about 10^-1.?6?Under the single-layer model,the critical Marangoni number is smaller than that of the transverse rolls instability mode for longitudinal rolls instability,and it is more prone to the critical Marangoni number"sudden drop"phenomenon.In addition,we found that the critical frequency of the"sudden drop"phenomenon corresponding to the critical Marangoni number in the longitudinal rolls instability mode is zero,indicating that static rolls appears at this time.?7?The influence of temperature gradient gradient,buoyancy and surface heat transfer?gas-liquid layer thickness ratio a?on flow stability in a two-layer model is similar to that of a single-layer model,and the instability threshold i.e.the critical Marangoni number under the two-layer model is also similar to that calculated by the single-layer model.The big difference between the calculation result of the two-layer model and the single-layer model is the critical frequency of the unstable rolls calculated by the double-layer model is more by an order of magnitude higher than that of the single-layer model,except for case when the“sudden drop”of the critical Marangoni number happen,then the critical frequency corresponding to the interval is consistent with the single-layer model.