Steady thermocapillary migration of a droplet in a uniform temperature gradient combined with a radiation energy source at large Marangoni numbers
aa r X i v : . [ phy s i c s . f l u - dyn ] A ug Steady thermocapillary migration of a dropletin a uniform temperature gradient combinedwith a radiation energy source at largeMarangoni numbers
Zuo-Bing Wu , ∗ State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics,Chinese Academy of Sciences, Beijing 100190, China School of Engineering Science,University of Chinese Academy of Sciences, Beijing 100049, ChinaAugust 17, 2018 Corresponding author. Tel:. +86-10-82543955; fax.: +86-10-82543977.Email addresses: [email protected] (Z.-B. Wu) bstract The steady thermocapillary droplet migration in a uniform tem-perature gradient combined with a radiation energy source at largeReynolds and Marangoni numbers is studied. To reach a terminalquasi-steady process, the magnitude of the radiation energy source isrequired to preserve the conservative integral thermal flux across thesurface. Under the quasi-steady state assumption, an analytical resultfor the steady thermocapillary migration of a droplet at large Reynoldsand Marangoni numbers is derived by using the method of matchedasymptotic expansions. It is shown that the thermocapillary dropletmigration speed increases as Marangoni number increases, while theradiation energy source with the sine square dependence is provided.
Keywords
Interfacial tension; Thermocapillary migration of droplet;Quasi-steady state assumption; Large Marangoni number; Micrograv-ity Introduction
A droplet in an external fluid or on a solid substrate can be driven by bodyforces generated in the gravitational, electric, magnetic, and ultrasonic fields[1]. Even in the absence of the body forces, the variable surface tension alongthe interface can also drive the droplet migration in the external fluid or thesolid substrate. Thermocapillary migration of a droplet in microgravity en-vironment is a very interesting topic on both fundamental hydrodynamictheory and engineering application [2]. Young et al [3] carried out an initialstudy on thermocapillary migration of a droplet in a uniform temperaturegradient in the limits of zero Reynolds(Re) and Marangoni(Ma) numbers(YGB model). Subramanian [4] proposed the quasi-steady state assumptionand obtained analytical results with high order expansions at small Ma num-bers. The thermocapillary droplet migration processes at small Ma numbersare understood very well in the series of theoretical analyses, numerical sim-ulations and experimental investigations [5, 6]. However, the physical behav-iors at large Ma numbers appear rather complicated due to the momentumand energy transfer though the interface of two-phase fluids. Meanwhile,to perform a feasible numerical simulation of thermocapillary migration ofa droplet at large Ma number is still a challenge due to very thin thermalboundary [ O ( M a − / )] and very long migration time [ O ( M a )]. Under theassumption of the quasi-steady state, Balasubramaniam & Subramanian re-ported [7] that the migration speed of a droplet increases with the increasingof Ma number, as is in qualitative agreement with the corresponding numer-ical simulation [8]. The experimental investigation carried out by Hadlandet al [9] and Xie et al [10] shown that the droplet migration speed decreasesas Ma number increases, which is qualitatively discrepant from the abovetheoretical and numerical results. Wu & Hu [11] and Wu [12] identified a3onconservative integral thermal flux across the surface in the steady thermo-capillary droplet migration at large Ma(Re) numbers, which indicates thatthe thermocapillary droplet migration at large Ma(Re) numbers is an un-steady process. To preserve a conservative integral thermal flux across thesurface, two methods, i.e., the adding thermal source inside the droplet orat the surface, were also suggested. With a thermal source added inside thedroplet, an analytical result of the steady thermocapillary migration of thedroplet at large Ma(Re) numbers was determined [13]. Therefore, the ther-mocapillary droplet migration at large Ma numbers remains a topic to bestudied with respect to its physical mechanism.In the above studies, the variable surface tension exerted on the interfaceof two-phases is generated by adding a non-uniform temperature field. Onthe other hand, a radiative heating in contrary to the direction of movement,which provides a thermal source at the surface through the absorption, canalso form the variable surface tension exerted on the interface. Oliver &Dewitt [14] firstly analyzed thermocapillary migration of a droplet causedby a thermal radiation in microgravity environment in the zero Re and Manumber limits. Rednikov & Ryzzantsev [15] independently derived the sim-ilar results and determined the deformation of the droplet. Zhang & Kho-dadadi [16] and Khodadadi & Zhang [17] numerically studied the effects ofthermocapillary convection on melting of droplets at a short-duration andan uniform heat pulses under zero gravity conditions at large Ma numbers,respectively. Lopez et al [18] experimentally observed the thermocapillarymigration of a droplet caused by a laser beam heating due to the absorptionof the laser radiation in making a strongly non-homogeneous distribution oftemperature inside the droplet as well as at its surface.In this paper, a radiation energy source in contrary to the direction of4ovement is placed to preserve the conservative integral thermal flux acrossthe surface, thermocapillary droplet migration at large Re and Ma numberscan thus reach a quasi-steady process. The steady thermocapillary dropletmigration in the uniform temperature gradient combined with the radiationenergy source at large Re and Ma numbers is studied. In comparing withthe previous method to preserve a conservative integral thermal flux acrossthe surface [13], the current method, i.e., placing the radiation energy sourceat the outside of the droplet, is easier to carry out a real space experiment.In the principle, the previous method adds a thermal source in the energyequation within the droplet, but the current method adds a heat flux atthe interface of the droplet. The paper is organized as follows. In Sect.2, the magnitude of the radiation energy source is required to preserve theconservative integral thermal flux across the surface. An analytical result forthe steady thermocapillary droplet migration at large Re and Ma numbersis determined in Sect. 3. Finally, in Sect. 4, the conclusions and discussionsare given.
Consider the thermocapillary migration of a spherical droplet of radius R ,density γρ , dynamic viscosity αµ , thermal conductivity βk , and thermaldiffusivity λκ in a continuous phase fluid of infinite extent with density ρ ,dynamic viscosity µ , thermal conductivity k , and thermal diffusivity κ undera uniform temperature gradient G in the direction of movement and an in-homogeneous radiation energy source S in contrary to the direction of move-ment. It is assumed that the continuous phase fluid is transparent and thatthe radiation is absorbed totally on the droplet surface. The rate of change5f the interfacial tension between the droplet and the continuous phase fluidwith temperature is denoted by σ T . Unsteady energy equations for the con-tinuous phase and the fluid in the droplet in a laboratory coordinate systemdenoted by a bar are written as follows ∂ ¯ T∂t + ¯ v ¯ ∇ ¯ T = κ ¯∆ ¯ T , ∂ ¯ T ′ ∂t + ¯ v ′ ¯ ∇ ¯ T ′ = λκ ¯∆ ¯ T ′ , (1)where ¯ v and ¯ T are velocity and temperature, a prime denotes quantities inthe droplet. The solutions of Eqs. (1) have to satisfy the boundary conditionsat infinity ¯ T → T + G ¯ z, (2)where T is the undisturbed temperature of the continuous phase and theboundary conditions at the interface ¯ r b of the two-phase fluids¯ T (¯ r b , t ) = ¯ T ′ (¯ r b , t ) , ∂ ¯ T∂n (¯ r b , t ) + S = β ∂ ¯ T ′ ∂n (¯ r b , t ) . (3)Under the quasi-steady state assumptions, steady axisymmetric energyequations non-dimensionalized by taking the radius of the droplet R , theYGB model velocity v o = − σ T GR /µ and GR as reference quantities tomake coordinates, velocity and temperature dimensionless can be written inthe spherical coordinate system ( r, θ ) moving with the droplet velocity V ∞ as follows 1 + u ∂T∂r + vr ∂T∂θ = ǫ ∆ T, u ′ ∂T ′ ∂r + v ′ r ∂T ′ ∂θ = λǫ ∆ T ′ , (4)where the small parameter ǫ and Ma number are defined, respectively, as ǫ = 1 √ M aV ∞ (5)and M a = v R κ . (6)6he boundary conditions (2)(3) are rewritten, respectively, as T → r cos θ, as r → ∞ (7)and at the interface of two phase fluids T (1 , θ ) = T ′ (1 , θ ) , ∂T∂r (1 , θ ) + Ω sin θ cos θ = β ∂T ′ ∂r (1 , θ ) , ≤ θ ≤ π/ , ∂T∂r (1 , θ ) = β ∂T ′ ∂r (1 , θ ) , π/ < θ ≤ π. (8)The inhomogeneous radiation energy source non-dimensionalized by the ref-erence quantity kG is assumed as S = Ω sin θ . Its contribution to the inter-face thermal flux S cos θ is zero at θ = π/
2, which reveals that the upper andlower interface thermal boundary conditions in Eq. (8) are continuous. Aschematic diagram of the thermocapillary droplet migration in a coordinatesystem moving with the droplet speed V ∞ is shown in Fig. 1.For large Re numbers ( Re = v R ν ), the velocity fields of the continuousphase and the fluid within the droplet can be described by potential flowsand boundary layers flows, as shown in Fig. 2. The scaled potential flowfields around a fluid sphere u = − cos θ (1 − r ) ,v = sin θ (1 + r ) (9)and u ′ = cos θ (1 − r ) ,v ′ = − sin θ (1 − r ) (10)are taken as those in the continuous phase and within the droplet, respec-tively [21, 22]. It is noticed that the potential flow fields (9)(10) for large Renumbers may be obtained from the general solutions for small Re numbers7y setting D n = 0 , n ≥ T ≈ r cos θ − r cos θ + o (1) . (11)Integrating Eqs. (4) in the continuous phase domain ( r ∈ [1 , r ∞ ] , θ ∈ [0 , π ]) with the boundary condition (11) and within the droplet region ( r ∈ [0 , , θ ∈ [0 , π ]), respectively, we obtain Z π ∂T∂r (1 , θ ) sin θdθ + Z π/ Ω sin θ cos θdθ = − ǫ + Ω4 (12)and Z π ∂T ′ ∂r (1 , θ ) sin θdθ = 23 λǫ . (13)From Eq. (12) and Eq. (13), we have β R π ∂T ′ ∂r (1 , θ ) sin θdθ − R π ∂T∂r (1 , θ ) sin θdθ − R π/ Ω sin θ cos θdθ = ǫ (1 + βλ ) − Ω4 . (14)For large Ma numbers and finite V ∞ , Eqs. (12) and (13) should satisfy thethermal flux boundary condition (8), i.e., the right side of Eq. (14) will bezero. So, we have Ω = 43 ǫ (1 + 2 βλ ) = 43 (1 + 2 βλ ) V ∞ M a, (15)which preserves the conservative integral thermal flux across the surface. Infollowing, we will focus on the steady thermocapillary migration of a dropletin the uniform temperature gradient G combined with the external thermalradiation source S and determine the dependence of the migration speed onlarge Ma number. 8 Analysis and results
By using an outer expansion for the scaled temperature field in the continuousphase T = T + ǫT + o ( ǫ ) , (16)the energy equation for the outer temperature field in its leading order canbe obtained from Eqs.(4) as follows1 + u ∂T ∂r + vr ∂T ∂θ = 0 . (17)By using the coordinate transformation from ( r, θ ) to ( ψ, θ ) in the solvingEq. (17), its solution can be written as T ( r, θ ) = G ( ψ ) − Z r r + 1 dθ sin θ , (18)where G ( ψ ) is a function of ψ (the stream function in the continuous phase).Following [7, 13], the solution near r = 1 is simplified as T ( r, θ ) = (1 + π √ − ln 432) − ( π √ + ln 432)( r − r ) sin θ + ln( r − r )+ ln(1 + cos θ ) + ( r − r ) cos θ + ( r − r ) ln( r − r ) sin θ + ( r − r ) sin θ ln(1 + cos θ ) . (19)By using the boundary layer approximation x = r − ǫ , (20)the temperature field near interface can be expressed as t ( x, θ ) = 1 + π √ −
16 ln 48 + 23 ln( 1 + cos θ sin θ ) + 13 x sin θǫ ln ǫ + o ( ǫ ln ǫ ) . (21)9 .2 Outer temperature field within the drop By using the outer expansion for the scaled temperature field within thedroplet in Eqs. (4) T ′ = 1 ǫ T ′− + 1 ǫ T ′− + T ′ + o (1) , (22)the equation in its leading order can be written as u ′ ∂T ′− ∂r + v ′ r ∂T ′− ∂θ = 0 . (23)Its solution is T ′− = F ( ψ ′ ) , (24)where ψ ′ = sin θ ( r − r ) is the streamfunction within the droplet. Theunknown function F ( ψ ′ ) can be obtained from the following equation for thetemperature field T ′ in its second order1 + u ′ ∂T ′ ∂r + v ′ r ∂T ′ ∂θ = λ ∆ F . (25)Following [7], in the solving Eq. (25), the coordinate transformation from( r, θ ) to ( m, q ) is applied in the form of m = − ψ ′ ,q = r cos θ r − , (26)where m and q denote the streamlines and their orthogonal lines, respectively.The solution of Eq.(25) is thus written as T ′− ( r, θ ) = F = 1 λ [ B ′ − m + 3256 (3 ln 2 − m − m ln m ]+ o ( m ln m ) , (27)where B ′ is an unknown constant. By using the boundary layer approxima-tion x ′ = 1 − r √ λǫ , (28)10he temperature field near interface can be expressed as follows t ′ ( x ′ , θ ) = − √ λ x ′ sin θ ǫ − x ′ sin θ ln ǫ + o (ln ǫ ) . (29) By using inner expansions for the continuous phase and the fluid in thedroplet t ( x, θ ) = t − ǫ + t l ln ǫ + o (ln ǫ ) , (30) t ′ ( x ′ , θ ) = t ′− ǫ + t ′ l ln ǫ + o (ln ǫ ) (31)and the inner variables given in Eqs. (20) and (28), the scaled energy equa-tions for the inner temperature fields in the leading order can be written asfollows − x cos θ ∂t − ∂x + 32 sin θ ∂t − ∂θ = ∂ t − ∂x , (32) − x ′ cos θ ∂t ′− ∂x ′ + 32 sin θ ∂t ′− ∂θ = ∂ t ′− ∂x ′ . (33)The boundary conditions are t − (0 , θ ) = t ′− (0 , θ ) ,δ ∂t − ∂x (0 , θ ) + ωδ sin θ cos θ = − ∂t ′− ∂x ′ (0 , θ ) , ≤ θ ≤ π/ ,δ ∂t − ∂x (0 , θ ) = − ∂t ′− ∂x ′ (0 , θ ) , π/ < θ ≤ π,t − ( x → ∞ , θ ) → ,t ′− ( x ′ → ∞ , θ ) → B − √ λ x ′ sin θ, (34)where δ = √ λ/β and ω = Ω ǫ = (1 + βλ ). We transform the independentvariables from [( x, x ′ ) , θ ] to [( η, η ′ ) , ξ ] and the functions from ( t − , t ′− ) to( f , f ′ ) as ( η, η ′ ) = ( x sin θ, x ′ sin θ ) ,ξ = (2 − θ + cos θ ) = (2 + cos θ )(1 − cos θ ) (35)11nd f ( η, ξ ) = t − ( x, θ ) ,f ′ ( η ′ , ξ ) = t ′− ( x ′ , θ ) − B + √ λ x ′ sin θ. (36)The corresponding energy equations for f , f ′ and the boundary conditionscan be written as follows ∂f ∂ξ = ∂ f ∂η , ∂f ′ ∂ξ = ∂ f ′ ∂η ′ (37)and f (0 , ξ ) = f ′ (0 , ξ ) + B,δ ∂f ∂η (0 , ξ ) = − ∂f ′ ∂η ′ (0 , ξ ) + √ λ + Φ( ξ ) , ≤ ξ ≤ ,δ ∂f ∂η (0 , ξ ) = − ∂f ′ ∂η ′ (0 , ξ ) + √ λ , < ξ ≤ ,f ( η → ∞ , ξ ) = 0 ,f ′ ( η ′ → ∞ , ξ ) = 0 , (38)where Φ( ξ ) = − ωδ cos θ = − ωδ (cos φ − √ φ ) , φ = arccos(1 − ξ ) in theShengjin’s formula [19]. To solve Eqs. (37), initial conditions are providedbelow f ( η,
0) = 0 ,f ′ ( η ′ ,
0) = f ′ ( η ′ , ξ ( π )) = f ′ ( η ′ ,
2) = g ( η ′ ) ,g ( η ′ → ∞ ) → . (39)Following the methods given by Carslaw & Jaeger [20] and Harper & Moore[21], the solutions of Eqs.(37) for the continuous phase and the fluid in thedroplet can be respectively determined as f ( η, ξ ) = δ {− √ π R ξ Φ( ξ − τ ) exp( − η τ ) dττ / + ( B + η √ λ )erfc( η √ ξ )+ √ πξ R ∞ g ( η ∗ ) exp[ − ( η + η ∗ ) ξ ] dη ∗ } , (0 ≤ ξ ≤ ,f ( η, ξ ) = δ {− √ π R ξ Φ( ξ − τ ) exp( − η τ ) dττ / + √ π R ξ − Φ( ξ − − τ ) exp( − η τ ) dττ / +( B + η √ λ )erfc( η √ ξ ) + √ πξ R ∞ g ( η ∗ ) exp[ − ( η + η ∗ ) ξ ] dη ∗ } , (1 < ξ ≤ f ′ ( η ′ , ξ ) = δ δ {− √ πδ R ξ Φ( ξ − τ ) exp( − η ′ τ ) dττ / − ( B − η ′ δ √ λ )erfc( η ′ √ ξ ) } + √ πξ R ∞ g ( η ∗ ) { exp[ − ( η ′ − η ∗ ) ξ ] + − δ δ exp[ − ( η ′ + η ∗ ) ξ ] } dη ∗ , (0 ≤ ξ ≤ ,f ′ ( η ′ , ξ ) = δ δ {− √ πδ R ξ Φ( ξ − τ ) exp( − η ′ τ ) dττ / + √ πδ R ξ − Φ( ξ − − τ ) exp( − η ′ τ ) dττ / − ( B − η ′ δ √ λ )erfc( η ′ √ ξ ) } + √ πξ R ∞ g ( η ∗ ) { exp[ − ( η ′ − η ∗ ) ξ ] + − δ δ exp[ − ( η ′ + η ∗ ) ξ ] } dη ∗ , (1 < ξ ≤ . (41) Due to the zero net force acting on the droplet at the flow direction, themigration speed of the droplet can be obtained as V ∞ = − α ) Z π sin θ ∂t∂θ (0 , θ ) dθ = 12 + 3 α Z π sin θ cos θt (0 , θ ) dθ. (42)When the inner expansion in the temperature field (30) is truncated at the o (ln ǫ ) order, we rewrite Eq. (42) as V ∞ = 12 + 3 α Z π sin θ cos θ [ t − (0 , θ ) 1 ǫ + t l (0 , θ ) ln ǫ ] dθ. (43)Since ǫ = 1 / √ M aV ∞ , the migration speed of the droplet is evaluated as V ∞ ≈ a M a − a l ln M a + a , (44)where a = 12 + 3 α Z π sin θ cos θt − (0 , θ ) dθ (45)and a l = 12 + 3 α Z π sin θ cos θt l (0 , θ ) dθ. (46)13rom Eqs.(40), we obtain the inner temperature field in its leading order forthe continuous phase near the surface of the droplet t − (0 , θ ) = f (0 , ξ )= δ [ − √ π R ξ Φ( ξ − τ ) dττ / + B + √ πξ R ∞ g ( η ∗ ) exp( − η ∗ ξ ) dη ∗ ]= δ [ − √ π R √ ξ Φ( ξ − s ) ds + B + √ π R ∞ g (2 ξ / ζ ) exp( − ζ ) dζ ] , ≤ θ ≤ π/ ,t − (0 , θ ) = f (0 , ξ )= δ [ − √ π R ξ Φ( ξ − τ ) dττ / + √ π R ξ − Φ( ξ − − τ ) dττ / + B + √ πξ R ∞ g ( η ∗ ) exp( − η ∗ ξ ) dη ∗ ]= δ [ − √ π R √ ξ Φ( ξ − s ) ds + √ π R √ ξ − Φ( ξ − − s ) ds + B + √ π R ∞ g (2 ξ / ζ ) exp( − ζ ) dζ ] , π/ < θ ≤ π. (47)Substituting Eq. (47) into Eq. (45), we obtain a = − √ π (2+3 α )(1+ δ ) { R π sin θ cos θ [ R √ ξ Φ( ξ − s ) ds ] dθ − R ππ/ sin θ cos θ [ R √ ξ − Φ( ξ − − s ) ds ] dθ } + √ π (2+3 α )(1+ δ ) R π sin θ cos θ [ R ∞ g (2 ξ / ζ ) exp( − ζ ) dζ ] dθ. (48)To determine the function g in Eq. (48), we use the boundary conditionwithin the droplet at the front and rear stagnation points in Eq. (39) g ( η ′ ) = δ δ {− √ πδ R √ Φ(2 − s ) exp( − η ′ s ) ds + √ πδ R Φ(1 − s ) exp( − η ′ s ) ds − ( B − η ′ δ √ λ )erfc( η ′ √ ) } + √ π R ∞ g ( η ∗ ) { exp[ − ( η ′ − η ∗ ) ] + − δ δ exp[ − ( η ′ + η ∗ ) ] } dη ∗ . (49)The integral of the fourth term on the right-hand side of Eq. (49) is approx-imated as Z ∞ g ( η ∗ ) h ( η ′ , η ∗ ) dη ∗ = Z η ∗ l g ( η ∗ ) h ( η ′ , η ∗ ) dη ∗ + g ( η ∗ l ) Z ∞ η ∗ l h ( η ′ , η ∗ ) dη ∗ . (50)14hen, Eq. (49) is evaluated in a linear system of equations g ( η ′ ) − √ π g ( η ∗ ) { exp[ − ( η ′ − η ∗ ) ] + − δ δ exp[ − ( η ′ + η ∗ ) ] } ∆ η ∗ − √ π g ( η ∗ N +1 ) { exp[ − ( η ′ − η ∗ N +1 ) ] + − δ δ exp[ − ( η ′ + η ∗ N +1 ) ] } ∆ η ∗ − √ π P Nj =2 g ( η ∗ j ) { exp[ − ( η ′ − η ∗ j ) ] + − δ δ exp[ − ( η ′ + η ∗ j ) ] } ∆ η ∗ − g ( η ∗ N +1 )[erfc( η ∗ N +1 − η ′ √ ) + − δ δ erfc( η ∗ N +1 + η ′ √ )]= δ δ [ − √ πδ R √ Φ(2 − s ) exp( − η ′ s ) ds + √ πδ R Φ(1 − s ) exp( − η ′ s ) ds − ( B − η ′ δ √ λ )erfc( η ′ √ )] , (51)where η ∗ N +1 = η ∗ l and ∆ η ∗ = η ∗ l /N . The physical coefficients used in spaceexperiments [13] with the uniform temperature gradient G = 12 K/cm forthe continuous phase of Fluorinert FC-75 and the droplet of 5cst silicon oilat T = 333 K are adopted to yield α = 0 . β = 0 .
571 and λ = 0 . η ∗ l is chosen as 3. Using the trial and error method to satisfythe above approximation, we determine the unknown constant B = 1 .
419 andobtain the dependence of g on η ′ as shown in Fig. 3. From Eq. (48), we candetermine the root-mean-square of the leading order term of the migrationspeed as a = 4 . × − . (52)Although equations and boundary conditions describing the second orderterm of the migration speed can be obtained, we are unable to find an ana-lytical result for t l in Eq. (46). Under the truncation after the leading orderterm in Eq.(44), we obtain the migration speed of the droplet V ∞ ≈ . × − M a, (53)which indicates that the thermocapillary droplet migration speed increasesas Ma number increases. Using the migration speed V ∞ , Eq. (15) is rewrittenas Ω = (1 + βλ ) V ∞ M a ≈ . × − M a . (54)15herefore, to reach steady thermocapillary droplet migration in the spaceexperiment at large Ma numbers [13], the external radiation energy source S = Ω sin θ ≈ . × − M a sin θ (55)in contrary to the direction of movement should be provided. In this paper, the steady thermocapillary droplet migration in a uniform tem-perature gradient combined with a radiation energy source at large Re andMa numbers is studied. The magnitude of the radiation energy source withthe sine square dependence is determined to preserve the conservative inte-gral thermal flux across the surface. Under the assumption of quasi-steadystate, we have determined an analytical result for the steady thermocapillarymigration of droplet at large Re and Ma numbers. The result shows that thethermocapillary droplet migration speed increases with the increasing of Manumber.In general, when the droplet in a uniform temperature gradient movesupward, the thermal energy is not only transferred into the droplet from thetop surface but also out the droplet from the bottom surface. Meanwhile,the thermal flux across the surface is balanced. For large Ma numbers, oncethermocapillary droplet migration reaches a quasi-steady state, the relationof the nonconservative integral thermal flux across the surface will be required[11, 12]. To satisfy the challenge, a thermal source at the surface throughthe absorption from an external radiation energy source is provided for thesystem to make a balance of the integral thermal flux across the surface.The thermal source at the surface can bring more heat to the droplet, whilethe heat transfer in the system due to the thermal conduction across/around16he droplet is weaker than that due to the thermal convection around thedroplet at large Ma numbers. The thermocapillary migration of a droplet inthe uniform temperature gradient combined with the radiation energy sourceat large Ma numbers can thus arrive at a quasi-steady state process.To perform a real space experiment to confirm the above theoretical anal-ysis of the steady thermocapillary migration of a droplet, the laser beamheating technology may be one of the possible physical means to providethe external radiation energy source in contrary to the droplet movementdirection. 17 cknowledgments
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J. Coll.Interface Sci. , (1981) 11.[24] R. S. Subramanian, Thermocapillary migration of bubbles and droplets, Adv. Space Res. , (1983) 145. 21 igure caption Fig. 1. A schematic diagram of the thermocapillary droplet migrationunder the combined actions of a temperature gradient G and a radiationenergy source S = Ω sin θ in an axisymmetric spherical coordinate systemmoving with the droplet velocity V ∞ .Fig. 2. A schematic diagram of potential flows and boundary layer flowsof the thermocapillary droplet migration at large Re numbers. Solid line:the interface of the droplet; Dashed/Longdash lines: the interface betweenpotential flow (white/green zone) and boundary layer flow (blue/yellow zone)in the continuous flow/within the droplet; DashDot lines: streamlines of thepotential flows inside and outside the droplet.Fig. 3. Function g versus η ′ determined from Eq. (51).22 S= sin² GV ’ g ( η ’ ))