Effects of thermophoresis on mixed convection flow from a moving vertical plate in a parallel free stream in the presence of thermal radiation

Transcription

1 IOSR Journal of Matheatics (IOSR-JM) e-issn: p-ISSN: X Volue 7 Issue 4 (Jul. - Aug. 013) PP 8-9 Effects of therophoresis on ixed convection flow fro a oving vertical plate in a parallel free strea in the presence of theral radiation P. M. Patil Head of the Departent of Matheatics JSS s Banashankari Arts Coerce and S. K. Gubbi Science College Vidyagiri Dharwad India. Abstract: This article is focused on the nuerical investigation of a steady two-diensional ixed convection boundary layer flow along a oving sei-infinite vertical plate. The plate is assued to ove with a constant velocity in the direction of the flow. The influences of therophoresis and the theral radiation are also included in the analysis. The governing boundary layer equations for the flow theral and species concentration fields are transfored into a non-diensional for by a group of non-siilar transforations. The resulting syste of differential equations is solved by an iplicit finite difference schee in coupled highly nonlinear partial cobination with the quasi-linearization technique. The effects of various paraeters on the velocity and species concentration profiles and the wall therophoretic deposition velocity are presented in ters of graphs. The present results are copared with previously published work and are found to be in excellent agreeent. Keywords: Therophoresis; ixed convection; parallel free-strea; theral radiation; finite difference. I. Introduction Therophoresis is a phenoenon in which the otion of suspended particles in a fluid induced by a teperature gradient is of prie practical iportance in a variety of industrial and engineering applications including aerosol collection (theral precipitators) nuclear reactor safety gas cleaning corrosion of heat exchangers and icro-containation control. Further this phenoenon has any useful practical applications in reoving sall particles fro gas streas in deterining exhaust gas particle trajectories fro cobustion devices and in studying the particulate aterial deposition on turbine blades. It has also been shown that therophoresis is the doinant ass transfer echanis in the odified cheical vapor deposition process used in the fabrication of optical fiber perfor and is also iportant in view of its relevance to postulated accidents by radioactive particle deposition in nuclear reactors. In any industries the coposition of processing gases ay contain any of an unliited range of particle liquid or gaseous containants and ay be influenced by uncontrolled factors of teperature and huidity. hen such an ipure gas is bounded by a solid surface a boundary layer will develop and energy and oentu transfer gives rise to velocity and teperature gradients. Mass transfer caused by gravitation and olecular diffusion eddy diffusion and inertial ipact results in deposition of the suspended coponents onto the surface. In the application of pigents or cheical coating of etals or reoval of particles fro a gas strea by filtration there can be distinct advantages in exploring deposition echaniss to iprove efficiency. Goren [1] was the one of the pioneers to study the role of therophoresis in lainar flow of viscous and incopressible fluid. He used the classical proble of flow over a flat plate to calculate deposition rates and showed that substantial changes in surface depositions can be obtained by increasing the difference between the surface and free strea teperatures. This was later followed by siilarity solutions of two diensional lainar boundary layers and stagnation point flows by Gokoglu and Rosner [] and Park and Rosner [3]. Siilarity solutions were also obtained by Chiou [4] for the proble of a continuously oving surface in a stationary incopressible fluid including the cobined effects of convection diffusion wall velocity and therophoresis. The therophoretic deposition of sall particles in forced convection lainar flow over inclined plates was discussed by Garg and Jayaraj [5]. Ala et al. [6] studied the effects of variable suction and therophoresis on steady MHD ixed convection flow heat and ass transfer over a sei-infinite pereable inclined plate in the presence of theral radiation. Recently Bakier and Gorla [7] have investigated the effects of therophoresis and radiation on lainar flow along a sei-infinite vertical plate. The understanding of the boundary layer behavior over a oving vertical plate in a parallel free strea is very iportant because it has any iportant practical applications such as the aerodynaic extrusion of plastic sheets the cooling of an infinite etallic plate in a cooling bath the boundary layer along aterial handling conveyers the boundary layer along a liquid fil in condensation processes paper production etc. The analysis of boundary layer flow induced by a oving flat surface in an otherwise abient fluid was first initiated by 8 Page

2 Sakiadis [8] by both analytical and nuerical schees. Rearkable results were found between this analysis and the analysis of the boundary layer on a stationary plate in a oving fluid presented by Blasius [9]. In fact the skin friction coefficient is approxiately 30% higher for the Sakiadis [8] case copared to the Blasius case [9]. Tsou et al. [10] have showed experientally that the flow and heat transfer proble fro a continuously oving surface is a physically realizable one and perfored investigations on its basic characteristics. Abdelhafez [11] exained the boundary layer flow on a oving flat plate in a parallel strea. He showed that Blasius and Sakiadis probles are two special cases of his proble. Chappidi and Gunnerson [1] have studied the lainar boundary layer in two cases Uw U and U U w separately and forulated two sets of boundary value probles. Afzal et al. [13] forulated a single set of boundary conditions by eploying the coposite reference velocity U Uw U where U w is the oving plate velocity and U is the free strea velocity instead of considering U and U separately as done by Abdelhafez [11] irrespective of whether U w U or U Uw w. Lin and Haung [14] have analyzed for horizontal isotheral plate oving in parallel or reversibly to a free strea where siilarity and nonsiilarity equations are used to obtain the flow and theral fields. The detailed analysis of the flow field considering the cobined otions of a bounding surface and free strea in the sae direction have been discussed by Abraha and Sparrow [15] and Sparrow and Abraha [16]. Karwe and Jaluria [17 18] have presented ixed convection flow over a continuous plate oving at a unifor speed in aterial processes such as hot rolling extrusion and drawing etc. Ingha [19] investigated the existence of the singular and non-unique solutions for the free convection boundary layer flow near a continuously oving vertical plate with teperature inversely proportional to the distance along the plate. Ali and Al-Yousef [0] have discussed the proble of lainar ixed convection flow adjacent to a uniforly oving vertical plate with suction or injection. The steady lainar flow and heat transfer characteristics of a continuously oving vertical sheet of extruded aterial are studied close to and far downstrea fro the extrusion slot by Al-sanea [1]. Soundalgekar and Murty [] have discussed the effects of power law surface teperature variation on the heat transfer fro a continuous oving surface with constant surface velocity. The effects of variable surface teperature and linear surface stretching were investigated by Grubka and Bobba [3]. Siilarity solutions were reported by Ali [4] for the case of power-law surface velocity and three different theral boundary conditions. Ali [5] has extended his work for the stretching surface subject to suction or injection. Moutsoglou and Chen [6] have considered buoyancy effects on flow and heat transfer fro an inclined continuous sheet with either unifor wall teperature or unifor surface heat flux. More recently Cortell [7] extended the work of Afzal et al. [13] by taking onto account viscous dissipation effects in the energy balance. The effects of transpiration on the flow and heat transfer over a oving pereable surface in a parallel strea are analyzed by Ishak et al [8]. Ishak et al [9] exained the boundary layer flow over a continuously oving thin needle in a parallel strea. Patil [30] discussed the effects of free convection on the oscillatory flow of a polar fluid through a porous ediu in presence of variable wall heat flux. Patil and Hireath [31] have studied the effects of couple stresses on the flow through a porous ediu. The developent of the boundary layer on a fixed or oving surface parallel to a unifor free strea in presence of surface heat flux has been investigated by Ishak et al [3]. Effects of radiation on the flow and heat transfer over a oving plate in a parallel strea exained by Anuar Ishak [33]. Very recently Patil et al. [34] analyzed the role of internal heat generation or absorption on flow and heat transfer over a oving vertical plate in a parallel free strea. The coprehensive analysis on steady ixed convection flow due to a oving seiinfinite vertical plate in a parallel free strea with theral diffusion is also iportant to be investigated and such studies are yet to appear in the literature. Furtherore steady ixed convection flows do not necessarily posses siilarity solutions in any flow patterns. The nonsiilarity in such flows ay be due to the free strea velocity or due to the curvature of the body or due to the surface ass transfer or even possibly due to all these effects. A solution is called self-siilar if a syste of partial differential equations can be reduced to a syste of ordinary differential equations. If the siilarity transforations are only able to reduce the nuber of independent variables then the transfored equations are called as sei-siilar and the corresponding solutions are the sei-siilar solutions. Therefore an attept ade towards the developent of investigation of ixed convection flows it is of prie iportance as well as useful to study the steady ixed convection boundary layer flow along a sei-infinite vertical oving plate in a parallel free strea to obtain non-siilar solutions. In this paper our ain goal is to analyze the cobined effects of buoyancy and therophoresis on a steady ixed convection flow fro a sei-infinite vertical plate in presence of theral radiation. The plate is supposed to ove parallel to the free strea velocity. The coupled non-linear partial differential equations governing the flow have been solved nuerically using an iplicit finite difference schee in cobination with 83 Page

3 the quasi-linearization technique [35-37]. In order to verify the correctness of the present investigation nuerical results are copared with previous results available in the open literature. II. Matheatical forulation e consider a steady lainar viscous and incopressible ixed convection boundary layer flow along a sei-infinite vertical plate oving with velocity U in the x- direction. The x-axis is taken along the plate in the vertically upward direction and the y- axis is taken noral to it. It is assued that the viscous dissipation and heat generation during therophoresis be neglected. The buoyancy force arises due to the teperature difference in the fluid. The fluid is considered to be gray; absorbing-eitting radiation but non-scattering ediu and the Rosseland approxiation [38] is used to describe the radiative heat flux in the x- direction is considered negligible in coparison to the y- direction. All thero-physical properties of the fluid in the flow odel are assued constant except the density variations causing a body force in the oentu equation. The Boussinesq approxiation is invoked for the fluid properties to relate density changes and to couple in this way the teperature field to the flow field [39]. Under these assuptions the diensional equations of conservation of ass oentu energy and concentration governing the steady ixed convection boundary layer flow over a oving vertical plate are given by: u x v y 0 (1) u u u u v g T T C C x y y * () T T T 1 qr u v x y y C p y (3) C C C u v D VT C x y y y (4) The physical boundary conditions are given by y 0 : u Uw v 0 T T C C y : u U T T T T. (5) here u and v are the fluid velocity coponents along the x- and y- axes; g the acceleration due to gravity; U the oving plate velocity; U the free strea velocity; T the teperature of the fluid in the boundary layer; T the teperature of the wall; T the abient teperature; C the species concentration in the boundary layer; C the species concentration at the wall;c the abient species concentration; D the density; the kineatic viscosity; theral diffusivity and the Brownian diffusion coefficient respectively; and * the coefficients of theral expansions of teperature and concentration respectively; and CP the specific heat at constant pressure. The effect of therophoresis is usually prescribed by eans of an average velocity which a particle acquire when exposed to teperature gradient. However in the boundary flow the teperature gradient in the y- direction is very uch larger than in the x- direction and therefore only the therophoretic velocity in y- direction is considered. The radiation heat flux q under Rosseland approxiation [38] we take q r 4 3a SB R 4 T y r where SB is the Stefan-Boltzann constant and a R is the Rosseland ean spectral absorption coefficient. 4 e assue that the teperature difference within the flow are sufficiently sall such that T can be expressed as a linear function of teperature. e now expand T T 4T T T 6T T T 4 T in a Taylor series about T as follows: Neglecting higher order ters in the above equation beyond the first degree in (T - T ) we get (6) (7) 84 Page

4 4 4 3 T 3T 4 T T. (8) In view of Eqs. (6) and (8) the Eq. (3) becoes 3 T T 16 SB T T u v 1. (9) x y 3k ar y As a consequence the therophoretic velocity V T which appears in Eq. (4) can be expressed in the following for as: V T Q T T y (10) where Q is the therophoretic coefficient. Eq. (4) becoes in view of Eq. (10) u v D C x y y T y y y T y Applying the following transforations: C C C Q C T T C T. Ux 1/ U x 1/ y x y U x 1/ f 1/ u y v x xu u U f U F U U Uw v f f f x T T T T G C C C C H ; (1) to Eqs. (1) () (9) and (11) we find that Eq. (1) is satisfied identically. Then Eqs. () (9) and (11) becoe f F F G N H F F f F (13) Pr f Pr G G F G f G 1 R 1R (14) H HG H H G H G G G Sc F H f H where f F d fw; fw0. Sc f ScQ 0 The corresponding boundary conditions are: F 1 G 1 H 1 at 0 F G 0 H 0 as. (16) here f the diensionless strea function; F the diensionless velocity; G the diensionless teperature; H the diensionless species concentration; Pr / is the Prandtl nuber Sc / D is the Schidt nuber T / ( T T ) is the therophoresis paraeter is the ixed convection paraeter and N is the diensionless paraeter representing the ratio between the theral and the concentration buoyancy forces which are defined as (11) (15) * Grx Grx N. (17) Re Gr x 3 Here the paraeter Gr g ( T T ) x / is the local theral Grashof nuber * * 3 x w x w Gr g ( C C ) x / is the local solutal Grashof nuber and Re Ux/ is the local Reynolds nuber. It should be entioned that 0 ( ) corresponds to an assisting flow while 0 ( T w T ) corresponds to an opposing flow and 0 ( T w T ) corresponds to the forced x T w T x 85 Page

5 convection flow case. Also N 0 for the cobined buoyancy forces are driving the flow N 0 for the buoyancy forces opposing each other N 0 for no buoyancy effect due to ass diffusion N 1 for the theral and ass buoyancy forces of the sae strength and N for no buoyancy effect due to theral diffusion and U UU corresponds to the ratio of free strea velocity to the coposite reference velocity. It ay be rearked that a single constant Pr/(1+R) known as Pr effective [40] can display the effects of radiation (R) and Prandtl nuber (Pr) when the sole interest is to study the effects of radiation (R) and Prandtl nuber (Pr) for a wide range of paraeter values. In such situation a single paraeter Pr/(1+R) known as Pr effective [40] could serve the purpose for the whole range of paraeter values. Since the present study concerned about the effect of Newtonian heating only for air (Pr = 0.7) and water (P r = 7.0) nuerical results are presented by indicating known Pr values of air (P r = 0.7) and water (P r = 7.0). F and species concentration Of interest in this investigation are the non-diensional velocity H profiles as well as the wall therophoretic deposition velocity V Q Sc G 0. 1 V which is defined by III. Solution Procedure The set of diensionless non-linear coupled partial differential equations (13) - (15) under the boundary conditions (16) have been solved nuerically using an iplicit finite-difference schee in cobination with the quasi-linearization technique (Bhattacharyya and Layek [35] Patil [36] and Patil et al. [37]). The quasilinearization technique is viewed as a generalized of the Newton-Raphson technique for the functional equations. The advantage of this technique is its quadratic convergence property. An iterative sequence of linear equations is carefully constructed to approxiate the nonlinear equations (13) - (15) under the boundary conditions (16) to achieve quadratic convergence. Applying the quasi-linearization technique the nonlinear coupled partial differential equations (13) - (15) under the boundary conditions (16) are replaced by the following sequence of linear partial differential equations: i1 i i1 i i1 i1 i1 i i1 i i1 i F X F X F X F X G X H X (19) G Y G Y G Y F Y (0) i1 i i1 i i1 i i1 i H Z H Z H Z H Z F Z G Z G Z G Z (1) i1 i i1 i i1 i i1 i i1 i i1 i i1 i i1 i The coefficient functions with iterative index i are known and the functions with iterative index (i+1) are to be deterined. The corresponding boundary conditions are given by i1 i1 i1 F ( 0) 1 G ( 0) 1 H ( 0) 1 () i1 i1 i1 F ( ) G ( ) 0 H ( ) 0 at where represents the edge of the boundary layer. Since the ethod is presented in detail for partial differential equations in a recent study by Bhattacharyya and Layek [35] and Patil [36] and Patil et al. [37] its detailed description is not provided here. At each iteration step the sequence of linear partial differential equations (19) - (1) were expressed in finite difference for using central difference schee in the -direction and backward difference schee in the direction. Thus in each step the resulting equations were then reduced to a syste of linear algebraic equations with a block tri-diagonal atrix which is solved by Varga s algorith [41]. To ensure the convergence of the nuerical solution to the exact solution the step sizes and are optiized and taken to be and respectively. The presented results are independent of the step sizes up to the fourth significant figure. A convergence criterion based on the relative difference between the current and the previous iteration values is eployed. The solution is assued to have converged and the iteration process terinated once the following convergence criterion is et: Max F F G G H H k k k k k k w w w w w w (18) (3) IV. Results and Discussion 86 Page

6 N Effects of therophoresis on ixed convection flow fro a oving vertical plate in a parallel free The coputations have been carried out for various values of N 1.0 Sc0. Sc Q1 Q 50 0R1 and0 R. The edge of the boundary layer has been taken between 5.0 and 1.0 depending on the values of the paraeters. Afzal et al. [13] have discussed only self-siilar solutions where all solutions along streawise direction were ade congruent using siilarity transforations. Governing equations were finally reduced to a set of ordinary differential equations. In such cases single asyptotic solution will not be able to represent all physical solutions. Further it is reported by any other investigators also that there are dual solutions with different asyptotic nature. In contrast authors have captured non-siilar solutions at each streawise location in different tie-step by solving coupled set of partial differential equations. In this investigation authors have observed a single non-siilar solution at each streawise location. Accuracy of the presented approach is verified by direct coparison with the results previously reported by Tsou et al. [10] Soundalgekar and Murty [] Ali [4] Moutsoglou and Chen [6] and Patil et al. [3]. The results of this coparison are presented in Table 1 and are found to be in excellent agreeent. Figures 1 and display the effects of varying the ixed convection paraeter and the Schidt nuber Sc on the velocity F( ) and species concentration H( ) profiles with Pr N = Q = 1 and R = 1. In the case of buoyancy-aiding flow 0 the buoyancy force shows the overshoot in the velocity profiles near the surface for both fluids of saller Sc 0.66 and higher Sc.57 values of Sc but for higher valuessc.57 the overshoot of the velocity profiles is reduced copared to saller Sc 0.66 values of Sc as shown in Fig. 1. The physical reason is that the lower Sc values ay ean the lower viscosity of the fluid which increases the local velocity everywhere within the oving boundary layer. Thus the velocity overshoot occurs. For the higher values of Sc the overshoot is reduced because fluids with higher values of the Sc iply ore viscous fluids which has less ipact on the buoyancy paraeter. However the influence of the paraeter is ore proinent for the lower value of Sc ( 0.66). Fig. also indicates that the agnitude of the species concentration decreases rearkably for 87 Page

7 .57 higher values of the Sc as copared with the lower value of Sc ( 0.66). Further it is very interesting to note fro Figs. 1 and that the buoyancy opposing force 0 reduces the agnitude of the velocity profile considerably within the oving boundary layer while the opposite behavior is found in the case of species concentration profile. The effects of the ratio of buoyancy forces paraeter N and the Schidt nuber Sc on velocity F( ) and species concentration H( ) profiles are displayed in Figs. 3 and 4 when Pr Q =1and R = 1. It is observed that the agnitude of velocity profile increases with the increase of the ratio of buoyancy forces paraeter N in the both cases for Sc( 0.66) and Sc(.57) (See Fig. 3). The physical reason is that the assisting ratio of buoyancy forces N 0 iplies a favorable pressure gradient for which the fluid gets accelerated which results in thicker oentu boundary layer while the opposing buoyancy forces N 0 iply that the fluid gets decelerated. However the agnitude of species concentration profile decreases with the increase of the ratio of buoyancy forces paraeter N in the both cases for Sc( 0.66) and Sc(.57) (see Fig. 4). The physical reason is that the assisting ratio of buoyancy forces N 0 acts as a non-favorable pressure gradient due to therophoresis effect for which the fluid gets decelerated which results in the thinner species concentration boundary layer while the opposing ratio of buoyancy forces N 0 acts as a favorable pressure gradient and for which the fluid gets accelerated. Thus there is an increase in the concentration boundary layer. Figure 5 displays the effects of (the ratio of free strea velocity to the coposite reference velocity) and the therophoretic paraeter on the velocity profiles F( ) when = 1.0 R = 1 N = 1 = 0.5 Q = 1 Sc =.57 and Pr = 0.7. The velocity profile is strongly depending on because it occurs in the oentu 88 Page

8 equation as well as in the boundary condition for F. It has been observed that the agnitude of the velocity profile within the oving boundary layer increases with the increase of. The physical reason is that the assisting buoyancy force due to theral and solutal gradients as well as increase in acts like a favorable pressure gradient which accelerates the fluid for unifor otion causing the velocity overshoots near the surface within the oving boundary layer. However the agnitude of the velocity profile increases with increasing therophoretic paraeter when 0.5 while it decreases when 0.5. It reveals that the therophoretic effects are crucial for the velocity profile. However it clearly indicates that the effect of therophoresis on oving boundary with free strea affects the velocity profile. Figure 6 illustrate the role of theral radiation paraeter R and Schidt nuber Sc on the species concentration profile H( ) with Pr N = Q = 0.5and = 1. It is seen that the species concentration profile is substantially influenced by the paraeters R and Sc. The values of the Schidt nuber (Sc) are chosen to be ore realistic representing diffusing cheical species of ost coon interest like water Aonia Propyl Benzene hydrogen water vapor and Propyl Benzene etc. at 5 degree Celsius at one atospheric pressure. It is observed that the concentration and velocity boundary layers are to decrease as the Schidt nuber Sc is increased. The physical reason is that the Schidt nuber Sc leads to a thinning of the concentration boundary layer. As a result the concentration of the fluid decreases and thus leads to a decrease in the fluid velocity. This is siilar to the effect of increasing the Prandtl nuber Pr on the thickness of a theral boundary layer. Furtherore species concentration profile H( ) leads to a fall in the agnitude due to the effect of theral radiation applied at the wall which works as heat source and thus species concentration boundary layer decreases. 89 Page

9 Figure 7 presents the variations of the species concentration profiles H( ) with the non-siilar variable and Schidt nuber Sc for with Pr N= 0.5 R =1 0.5 Q = 1and = 1.The profiles drawn at 0 represent the siilarity solutions when all solutions along the x-direction are ade congruent using siilarity transforations. This figure shows that the concentration profile H( ) decreases with the increase of fro 0 to 1. This clearly indicates that in the therophoretic flow an increase in acts as a decelerating force and hence fluid flow gets slower. Thus there is a fall in the agnitude of the species concentration profiles H( ). Also an increase in Sc decreases the ass diffusion and thus in turn the agnitude of the species concentration profile H( ) decreases. The effects of theral radiation paraeter R on variations of the wall therophoretic deposition velocity V against non-siilar variable are depicted for various values of therophoretic coefficient Q in Fig. 8 with Pr Sc = 0.94 N=1 0.5 and = 1. It is observed that the values of V increase with the therophoretic coefficient Q and non-siilar variable steadily. Further the values ofv is to decrease with the theral radiation paraeter when it increases fro 0 to 1. This indicates that theral radiation has a pronounced effect on therophoresis. In particular for exaple the percentage of increase of wall therophoretic deposition velocity V by about 97 % and 100 % as therophoretic coefficient increases fro Q=10 to Q=0 when R 0 and R 1 respectively at 1.0. V. Conclusions A nuerical investigation has been perfored to study the ixed convection boundary layer flow fro a oving vertical plate in a parallel free strea in the presence of theral radiation and therophoresis. The governing boundary layer equations are transfored into a non-diensional for by a group of non-siilar transforations. The resulting syste of coupled non-linear partial differential equations is solved by an iplicit 90 Page

10 finite difference schee in cobination with the quasi-linearization technique. The following conclusions have been drawn in the present nuerical investigation: The velocity profile exhibits significant enhanceent for low Schidt nuber fluids as copared to the agnitude of the velocity for higher Schidt nuber. Results reveal that velocity and species concentration boundary layers are to increase with the increase of ixed convection paraeter and the ratio of buoyancy forces N. An increase in the strea wise co ordinate acts as a nonfavorable pressure gradient and hence fluid flow gets slower. The agnitude of the velocity within the oving boundary layer increases with the increase of (the ratio of free strea velocity to the coposite reference velocity). The influence of theral radiation paraeter R on the velocity and species concentration profiles is significant and which leads to reduction in the profiles. The effects of therophoretic coefficient leads to an increase in the value of wall therophoretic deposition velocityv. Acknowledgeent: Author acknowledge with thanks to the University Grants Coission South estern Regional Office Bangalore for financial assistance with grant No. MRP(S)-77/1-13/KAKA060/UGC-SRO Dated 9 th March 013. Author also expresses sincere thanks to Dr. Ajith Prasad Finance Officer Janata Shikshana Saiti Dharwad for encouraging the research activities. Table 1. Coparison of - 0 G for 0 0 R = 0 0 and selected values of Pr the plate being isotheral ( T = constant) to previously published work. References: [1] S.L. Goren Therophoresis of aerosol particles in the lainar boundary layer on a flat surface Journal of Colloid and Interface Science 61 (1977) [] S.A. Gokoglu D.E. Rosner Therophoretically augented ass transfer rate to solid walls across lainar boundary layers AIAA Journal 4 (1986) 17. [3] H.M. Park D.E. Rosner Cobined inertial and therophoretic effects on particle deposition rates in highly loaded systes Cheical Engineering Science 44 (1989) [4] M.C. Chiou Effect of therophoresis on subicron particle deposition fro a forced lainar boundary layer flow onto an isotheral oving plate Acta Mechanica 19 (1998) [5] V. K. Garg S. Jayaraj Therophoresis of aerosol particles in lainar flow over inclined plates International Journal of Heat and Mass Transfer 31 (1988) [6] M. S. Ala M. M. Rahan M. A. Sattar Effects of variable suction and therophoresis on steady MHD cobined free-forced convective heat and ass transfer flow over a sei-infinite pereable inclined plate in the presence of theral radiation International Journal Theral Sciences 47 (008) [7] A. Y. Bakier R. S. R. Gorla Effects of therophoresis and radiation on lainar flow along a sei-infinite vertical plate 47 (011) [8] H. Blasius Grenzchiechten in flussigkeiten rnit kleiner reibung Z. Math. Phys. 56 (1908) [9] B. C. Sakiadis Boundary-layer behavior on continuous solid surfaces II. The boundary layer behavior on continuous flat surface AIChE J 7 (1967) 1 5. [10] F.K. Tsou E.M. Sparrow R.J. Goldstein (1967) Flow and heat transfer in the boundary layer on a continuous oving surface Int. J. Heat Mass Transfer 10 (1967) [11] T. A. Abdelhafez Skin friction and heat transfer on a continuous flat surface oving in a parallel free strea Int. J. Heat Mass Transfer 8 (1985) Page

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