Effect of Thermal Radiation on the Casson Thin Liquid Film Flow over a Stretching Sheet

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 3, Number 6 (207), pp Research India Publications Effect of Thermal Radiation on the Casson Thin Liquid Film Flow over a Stretching Sheet K. Kalyani, K. Sreelakshmi and G. Sarojamma * Department of Applied Mathematics, Sri Padmavati Mahila Visvavidyalayam, Tirupati, 57502, A.P, India * Corresponding Author Abstract The effect of thermal radiation and viscous dissipation on the characteristics of flow in a chemically reactive Casson liquid thin film subject to a transverse magnetic field is examined. By introducing appropriate similarity variables the non-linear partial differential equations governing the flow are reduced into a set of non-linear ordinary differential equations which are then solved using the shooting technique along with the Runge-Kutta-Fehlberg method. The velocity, temperature and species concentration, film thickness and free surface velocity are evaluated numerically. It is observed seen that thinner films are formed due to stronger magnetic field strengths. Increase in Casson parameter reduced the film thickness. Free surface velocity is found to enhance with unsteady parameter. Temperature distribution is found to be an increasing function of Casson parameter while it reduces with increasing Prandtl number. Species concentration is improved with Casson parameter while a reversal trend is noticed for an increasing variation in Schmidt s number and chemical reaction parameter. Rate of heat transfer is favorably enhanced due to thermal radiation and viscous dissipation. The Sherwood number is increased significantly for increasing values of Schmidt number. The present results are compared with the already published results and are found to agree favorably with them. Keywords: Unsteady flow, Casson thin liquid film, Thermal radiation, Chemical reaction.

2 576 K. Kalyani, K. Sreelakshmi and G. Sarojamma INTRODUCTION Several problems on flows in liquid thin films on stretching surfaces have been extensively investigated due to their abundant applications in the last two decades. In the melt spinning procedure when the extruded material is drawn through a die, the flow caused due to stretching surface is very close to the extruded material is an example. In all coating processes the objective is to obtain glossy smooth surface on the end product with best finish in terms of low friction, transparency and good strength. The properties of flow and heat transfer in a liquid thin film enable us to achieve the expected finish of the coating and also in the design and development of various heat exchangers as well as chemical processing equipments. Wang [] initiated the study of flow characteristics in a liquid thin film resting on an unsteady stretching sheet. Dandapat et al. [2] extended this study to include heat transfer analysis. It is reported that in a heavily viscous fluid, considerable heat can be generated even at low speeds of the fluid, for instance, in the case of extrusion of plastic sheets, and thus rate of heat transfer may change appreciably due to viscous dissipation. Sarma and Rao [3] obtained analytical solutions for the heat transfer in a steady laminar flow of a viscoelastic fluid over a stretching surface in the presence of viscous dissipation and internal heat generation. Sarojamma et al. [4] presented a mathematical model to study the effect of viscous dissipation on the time dependent flow of a Casson fluid due to a stretching sheet embedded in a rotating fluid subject to a uniform magnetic field with thermal radiation and chemical reaction of n th order. Abel et al. [5] examined the effect of viscous dissipation on the MHD flow and heat transfer in a liquid film due to a stretching surface. Vajravelu et al. [6] carried out a mathematical analysis of the effects of thermo physical properties on the thin film flow of an Ostwald-de Waele liquid over a stretching surface in the presence of viscous dissipation. Majority of the fluids used for protective coatings are usually non-newtonian. Hence, the study of the non-newtonian flow characteristics has significant relevance in industry, for example in polymer and plastic fabrication and in coating equipment. Chen [7] did a numerical investigation of heat transfer and flow characteristics in a thin liquid film of a power law fluid due to an unsteady stretching sheet. Wang and Pop [8] made a Homotopy analysis of the flow in a power-law fluid film on an unsteady stretching surface. Mahmoud and Megahed [9] studied the effect of variable viscosity and variable thermal conductivity on the flow and heat transfer of an electrically conducting non-newtonian power-law fluid within a thin liquid film over an unsteady stretching sheet in the presence of a transverse magnetic field. Effect of thermal radiation has significant applications in physics, space technology and processes operated at very high temperature. For example, in polymer processing

3 Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 577 industry when the whole system containing the polymer extrusion processes is placed at high temperatures then radiation effect plays a vital role in controlling the heat transfer process. As the quality of the end product greatly depends on the rate of heat transfer, the knowledge of radiative heat transfer may be helpful to obtain the final product with best quality. Hossain et al. [0] examined the influence of thermal radiation on flow of viscous fluid over a heated vertical permeable plate with constant surface temperature. Khader and Megahed [] examined the flow and heat transfer in a thin liquid film over an unsteady stretching sheet in a saturated porous medium with thermal radiation. Prasad et al. [2] explored the effects of variable thermal conductivity, thermal radiation and viscous heating on the MHD flow and heat transfer of a non-newtonian power-law liquid film at a horizontal porous sheet. Khademinejad et al. [3] explored the effects of viscous dissipation, magnetic field and thermal radiation to analyze the heat transfer characteristics of a thin liquid film flow over an unsteady stretching sheet using HAM. Studies on flows in liquid thin films are very limited. Casson fluid is a non-newtonian fluid initially proposed by Casson during this study on the flow curves of printing inks. Subsequently this model was used to describe blood, varnishes, polymers etc. Megahed [4] examined the impact of variable heat flux, viscous heating and velocity slip flow on the heat transfer of a Casson fluid in thin film on a stretching sheet. Vijaya et al. [5] investigated the heat transfer on the flow of a Casson fluid film in the presence of viscous dissipation and temperature dependent heat source. In this paper an analysis to study the effect of thermal radiation and first order chemical reaction on the heat and mass transfer characteristics of flow in a Casson thin liquid film is carried out. MATHEMATICAL FORMULATION We consider a chemically reactive non-newtonian Casson liquid thin film with thickness h(t) over a heated stretching sheet that emerges from a narrow slit at the origin of the Cartesian coordinate system as shown schematically in figure. The motion of the fluid within the film is due to the stretching of the sheet. The continuous sheet is parallel to x-axis and moves in its own plane with a velocity U(x, t) = bx αt where α and b are positive constants with dimension per time. The stretching sheet s temperature and concentration T s and C s is assumed to vary with the distance x from the slit as T s (x, t) = T 0 T ref [ bx2 2v ] ( αt) 3/2 (2) ()

4 578 K. Kalyani, K. Sreelakshmi and G. Sarojamma C s (x, t) = C 0 C ref [ bx2 2v ] ( αt) 3/2 (3) where T 0 and C 0 are the temperature and concentration at the slit, v is the kinematic viscocity. A transverse magnetic field B = B 0 ( αt) /2 is applied to the thin liquid film. Effect of thermal radiation is taken into account. Figure.Physical model and coordinate system The constitutive equation of the Casson fluid can be written as [6] τ ij = { 2 (μ B + P y 2π ) e ij, π > π c 2 (μ B + P y ) e 2π ij, π < π c c where τ ij is the (i, j) th component of the stress tensor, μ B is the plastic dynamic viscosity of the non-newtonian fluid, P y is the yield stress of the fluid, π is the product of the component of deformation rate with itself, namely, π = e ij e ij, and e ij is the (i, j) th component of deformation rate, and π c is the critical value of π depends on non-newtonian model. Under these assumptions, equations of the flow in the liquid film are given by u + v = 0 (5) x y (4) u t u u + u + v = ν ( + ) 2 u x y β σb2 y 2 ρ u (6) T t T T + u + v = x y k 2 T + 6σ T T + μ ρc p y 2 3ρc p k y 2 ρc p ( + β ) ( u y )2 (7)

5 Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 579 C C C + u + v = D 2 C k t x y y (C C 2 ) (8) where u and v are the velocity components of fluid in x- and y- directions, T is temperature, C is the fluid concentration, μ is dynamic viscosity, σ is electrical conductivity, β = μ B 2π c /P y is the Casson parameter, ρ is density, c p is specific heat at constant pressure, k is the thermal conductivity, σ is the Stefen-Boltzman constant, k is the absorption coefficient, D is the mass diffusivity and k (t) = k 0 /( αt) is the time dependent reaction rate. The boundary conditions on the stretching sheet are no slip, no penetration and imposed sheet temperature and concentration distributions and are represented respectively as u = U, v = 0, T = T s, C = C s at y = 0, (9) u = 0, T C dh = 0, = 0, v = y y y dt The following similarity transformations are introduced: η = [ b ] ν( αt) 2 y, ψ = x [ νb bx 2 αt ] at y = h(t) (0) 2 f(η), () T = T 0 T ref [ 2ν( αt) 3/2] θ(η), θ(η) = T T 0 (2) T S T 0 bx 2 C = C 0 C ref [ 2ν( αt) 3/2] φ(η), φ(η) = C C 0 (3) C S C 0 Also, ψ(x, y) is the stream function which automatically fulfils mass conservation equation (5) and the velocity components are can be obtained as u = ψ y = bx αt f (η) where prime denotes differentiation with respect to η. ψ v = = ( νb x αt )/2 f(η) (4) METHOD OF SOLUTION The mathematical problem defined through equations (5) (8) are transformed to the following non-linear boundary value problem on the finite range of 0 γ: ( + β ) f + [ff S(f + η 2 f ) f 2 Mf ] = 0 (5) ( Nr) θ + Pr (fθ 2f θ S 2 (ηθ + 3θ) + Ec ( + β ) f 2 ) = 0 (6) φ + Sc (fφ 2f φ S 2 (ηφ + 3φ) δφ) = 0 (7)

6 580 K. Kalyani, K. Sreelakshmi and G. Sarojamma subject to the boundary conditions f(0) = 0, f (0) =, θ(0) =, φ(0) = (8) f(γ) = 2 Sγ, f (γ) = 0, θ (γ) = 0, φ (γ) = 0 (9) Where, S = α bis the unsteadiness parameter, M = σb 2 0 ρb is the magnetic field parameter, Pr = ρc p ν k is the Prandtl number, Nr = 4σ T 3 kk is the thermal radiation parameter, Ec = U 2 c p (T s T 0 ) is the Eckert number, Sc = ν D is the Schmidt number and δ = k 0 b is the chemical reaction parameter. Further, γ denotes the value of the similarity variable η at the free surface so that the first term of equation () gives b γ = ( ν( αt) )/2 h(t) (20) Since γ is an unknown constant, which should be determined, as a whole, from the set of the present boundary-value problem, the rate of change of the film thickness can be obtained as follows: dh = αγ ( ν dt 2 b( αt) )/2 (2) Thus, the kinematic constraint at y = h(t) given by equation (0) transforms to the free surface condition (20). The surface drag coefficient C fx, Nusselt number Nu x and Sherwood number Sh x which play a significant role in estimating the surface drag force, rate of heat and mass transfer are defined respectively, as C fx Re x /2 = 2 ( + β ) f (0), Nu x Re x /2 = θ (0), Sh x Re x /2 = φ (0) (22) where Re x = Ux/ν is the local Reynolds number. The coupled ordinary differential equations (5) (7) are non-linear and exact analytical solutions are not possible. Equations (5) (7) with the appropriate boundary conditions (8) and (9) are solved numerically by the efficient fourth order Runge-Kutta-Fehlberg algorithm along with numerical shooting technique. These equations are converted into a set of first order equations as follows: df 0 dη = f, df dη = f 2,

7 Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 58 ( + β ) df 2 dη = S (f + η 2 f 2) + f 2 f 0 f 2 + Mf, (23) dθ 0 dη = θ, ( Nr) dθ dη = Pr (S 2 (3θ 0 + ηθ ) + 2θ 0 f θ f 0 Ec ( + β ) f 2 2 ). (24) dφ 0 dη = φ, dφ dη = Sc (S 2 (3φ 0 + ηφ ) + 2φ 0 f φ f 0 + δφ 0 ) (25) The associated boundary conditions take the form, f 0 (0) = 0, f (0) =, θ 0 =, φ 0 = (26) f 0 (γ) = 2 Sγ, f 2(γ) = 0, θ (γ) = 0, φ (γ) = 0. (27) Here f 0 (η) = f(η) and θ 0 (η) = θ(η) and φ 0 (η) = φ(η). This requires the initial values f 2 (0), θ (0) and φ (0) and hence suitable guess values are chosen and later integration is performed. A step size of η = 0.0 is chosen. The value of γ is obtained in such a way that the boundary condition f 0 (γ) = Sγ 2 error of tolerance of 0 6. is satisfied with an Table. Comparison of γ and f (0) with published values when M = 0 and β for various values of S S Wang [] Abel et al. [5] Megahed [4] Present study γ f (0)/γ γ f (0) γ f (0) γ f (0) Accuracy of the present scheme is ensured by comparing the present results, viz., non dimensional thickness of the film γ, surface drag coefficient f (0) with the corresponding values evaluated by Wang [], Abel et al. [5] and Megahed [4] in the

8 582 K. Kalyani, K. Sreelakshmi and G. Sarojamma absence of magnetic field parameter (M = 0) for a Newtonian fluid (β ) for various values of unsteady parameter. Since Wang [] used different similarity variables, the values of f (0) γ evaluated by Wang [], shall be same as f (0) of the present analysis. These values are presented in Table and it is observed that they are in excellent agreement. RESULTS AND DISCUSSION To obtain a flow in the thin film, numerical computations of flow velocity, temperature and concentration for various sets of governing parameters have been obtained and graphically illustrated. Figures 2-4 depict the influence of non-newtonian rheology of the fluid through the Casson parameter (β) on velocity, temperature and concentration. The velocity in the vicinity of the boundary is seen to be a constant function of the Casson parameter. However, a considerable reduction in the velocity of the fluid within the film away from the boundary is observed for higher values of β. Reduction in the velocity shall be due to the non-newtonian nature of the fluid as increase in Casson parameter amounts to an increase in the plastic dynamic viscosity of the fluid. As a consequence film thickness also decreases for higher values of β. However, for the same variation of β, the temperature and concentration are found to increase as shown in Figures 3 and 4. From figure 5 it is observed that in the absence of magnetic field (M), velocity steadily decreases in the film. Presence of magnetic field leads to a rapid reduction of velocity in the vicinity of the boundary due to the action of Lorentz force which opposes the fluid motion. Figure 6 reveals that velocity distribution in the film decreases monotonically for small values of unsteadiness parameter (S). As the unsteadiness parameter assumes higher values fluid gets accelerated and hence higher velocities occur. For increasing values of unsteadiness parameter the films become thinner. When unsteadiness parameter S =.4, film thickness is found to be decreased by two and half times than that of the film corresponding to S =. Figure 7 illustrates the variation of thermal radiation parameter (Nr) on temperature. Increasing values of Nr enhances the temperature prominently as the presence of thermal radiation releases higher thermal energy. Figure 8 presents the temperature profiles for a variation in Prandtl number (Pr). It is revealed that temperature falls from its higher value on the wall to its minimum value on the surface. For higher values of Pr, temperature decreases rapidly near the boundary. As higher values of Pr indicate that the thermal conductivity of the fluid is smaller and hence lower temperatures are resulted. Figure 9 illustrates the variation of Eckert number (Ec) on temperature.

9 Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 583 Profiles of temperature reveal that increasing values of Ec heat up the fluid in the film resulting in higher temperatures. This enhancement is due to internal heating in the fluid layers. In particular near the boundary when Ec = 3.0 an over shoot of the temperature occurs. Figure 0 highlights the variation of Schmidt s number (Sc) on species concentration. It is noticed that concentration decreases considerably for increasing values of Schmidt s number which is in conformity with fact that higher values of Sc corresponds to smaller mass diffusivity. From Figure it is observed that the effect of chemical reaction parameter (γ) on concentration is similar o that of Schmidt s number. From Figure 2 it is clear that the free surface velocity remains almost steady for all values of the magnetic field parameter (M). Increasing values of unsteadiness parameter (S) is found to reduce the free surface velocity. As S various its value from to.2 there is a twofold reduction in the free surface velocity is seen. Figure 3 presents the variation of film thickness versus magnetic field parameter for different values of the unsteadiness parameter. It can be seen that film thickness reduces for increasing values of S. Film thickness decreases rapidly for smaller values of magnetic field and a further reduction is noticed for stronger magnetic field strength. 0.9 =.0 = 2.0 = 3.0 = 4.0 f ' () M = ; S = ; Pr =.0; Nr = ; Ec = 0.; Sc = ; = 0. = = = = Figure 2.Velocity profiles for different values of β

10 584 K. Kalyani, K. Sreelakshmi and G. Sarojamma 0.9 =.0 = 2.0 = 3.0 = 4.0 () M = ; S = ; Pr =.0; Nr = ; Ec = 0.; Sc = ; = 0. = = = = Figure 3.Temperature profiles for different values of β 0.9 =.0 = 2.0 = 3.0 = 4.0 () M = ; S = ; Pr =.0; Nr = ; Ec = 0.; Sc = ; = 0. = = = = Figure 4.Concentration profiles for different values of β

11 Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet M = 0.0 M =.0 M = 2.0 M = 3.0 = ; S = ; Pr =.0; Nr = ; Ec = 0.; Sc = ; = 0. f ' () = = = Figure 5.Velocity profiles for different values of M = M = ; = ; Pr =.0; Nr = ; Ec = 0.; Sc = ; = 0. S = S =.0 S =.2 S =.4 f ' () = = = = Figure 6.Velocity profiles for different values of S

12 586 K. Kalyani, K. Sreelakshmi and G. Sarojamma 0.9 Nr = Nr =.0 Nr =.5 Nr = 2.0 M = ; = ; S = ; Pr =.0; Ec = 0.; Sc = ; = 0. () Figure 7.Temperature profiles for different values of Nr 0.9 Pr = Pr =.0 Pr = 2.0 Pr = 3.0 () M = ; = ; S = ; Nr = ; Ec = 0.; Sc = ; = Figure 8.Temperature profiles for different values of Pr

13 Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 587 Ec = 0.0 Ec =.0 Ec = 2.0 Ec = 3.0 () M = ; = ; S = ; Pr =.0; Nr = ; Sc = ; = Figure 9.Temperature profiles for different values of Ec 0.9 Sc = Sc =.0 Sc =.5 Sc = 2.0 M = ; = ; S = ; Pr =.0; Nr = ; Ec = 0.; = 0. () Figure 0.Concentration profiles for different values of Sc

14 588 K. Kalyani, K. Sreelakshmi and G. Sarojamma 0.9 = 0. = = =.0 M = ; = ; S = ; Pr =.0; Nr = ; Ec = 0.; Sc = () Figure.Concentration profiles for different values of δ 5 S = S =.2 f ' () M Figure 2.Variation of free surface velocity f (γ) with M

15 Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet S = S = M Figure 3.Variation of film thickness γ with M Local skin friction coefficient, Nusselt number and Sherwood number on the stretching surface for different variations of the governing parameters are presented in Table 2. Surface drag coefficient is found to increase with elapse of time. Rate of heat transfer is observed to reduce for an increase in the unsteadiness parameter which is in conformity with the variation observed in temperature with unsteadiness parameter. Sherwood number increases with higher values S. Larger values of Casson parameter decrease skin friction coefficient due to smaller velocities. Rate of heat and mass transfer is found to be smaller for a variation in Casson parameter. Surface drag coefficient is significantly reduced due to stronger magnetic field strengths. Lorentz force favors the rate of heat and mass transfer. Table 3 illustrates the Nusselt number for different values of Pr, Ec and Nr. Increase in Prandtl number leads to an enhancement in the temperature gradient. Eckert number and thermal radiation parameter decrease the Nusselt number. Table 4 shows that Schmidt s number increases the mass concentration gradient significantly while the chemical reaction parameter increases the mass concentration gradient moderately.

16 590 K. Kalyani, K. Sreelakshmi and G. Sarojamma Table 2. Variation of ( + β ) f (0) and θ (0) for various values of S, β and M S β M ( + β ) f (0) θ (0) φ (0) Table 3. Variation of θ (0) for various values of Pr, Ec and Nr Pr Ec Nr θ (0)

17 Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 59 Table 4. Variation of φ (0) for various values of Sc and δ Sc δ φ (0) CONCLUSIONS Some of the significant conclusions of the study are: Surface velocity is found to decrease with increasing values of magnetic field parameter. Higher values of unsteady parameter decreases film thickness. A qualitatively similar trend is found for increasing values of magnetic and Casson parameters. Temperature is an increasing function of Eckert number. Species concentration is found to be a decreasing function of Schmidt s number and chemical reaction parameter. Viscous heating and thermal radiation enhance the rate of heat transfer. REFERENCES []. Wang, C. Y., 990, Liquid film on an unsteady stretching surface, Quarterly of Applied Mathematics, 48(3), pp [2]. Dandapat, B. S., Andersson, H. I., and Aarseth, J. B., 2000, Heat transfer in a liquid film on an unsteady stretching surface, Int. J. Heat and Mass Transfer, 43(), pp [3]. Sarma, M. S., Rao, B. N., 998, Heat transfer in a viscoelastic fluid over a stretching sheet, J. Math. Anal. Appl., (), pp [4]. Sarojamma, G., Sreelakshmi, K., and Vasundhara, B., 206, Mathematical model of MHD unsteady flow induced by a stretching surface embedded in a rotating Casson fluid with thermal radiation, IEEE, /6/$3.00_c 206, pp [5]. Abel, M. S., Tawade, J., and Nandeppanavar, M. M., 2009, Effect of nonuniform heat source on MHD heat transfer in a liquid film over an unsteady stretching sheet, Int. J. Non-Linear Mechanics, 44(9), pp

18 592 K. Kalyani, K. Sreelakshmi and G. Sarojamma [6]. Vajravelu, K., Prasad K. V., and Raju, B. T., 203, Effects of variable fluid properties on the thin film flow of Ostwald de Waele fluid over a stretching surface, Journal of Hydrodynamics, 25(), pp [7]. Chen, C. H., 2003, Heat transfer in a power-law film over an unsteady stretching sheet, Heat and Mass Transfer, 39(8), pp [8]. Wang, C., and Pop, I., 2006, Analysis of the flow of a power-law fluid film on an unsteady stretching surface by means of homotopy analysis method, J. Non- Newtonian Fluid Mechanics, 38(2), pp [9]. Mahmoud, M. A. A., and Megahed, A. M., 2009, MHD flow and heat transfer in a non-newtonian liquid film over an unsteady stretching sheet with variable fluid properties, Canadian Journal of Physics, 87(0), pp [0]. Hossain, M. A., Khanafer, K., and Vafai, K., 200, The effect of radiation on free convection flow of fluid with variable viscosity from a porous vertical plate, Int. J. Therm. Sci., 40(2), pp []. Khader, M. M. and Megahed, A. M., 203, Numerical simulation using the finite difference method for the flow and heat transfer in a thin liquid film over an unsteady stretching sheet in a saturated porous medium in the presence of thermal radiation, Journal of King Saud University Engineering Sciences, 25(), pp [2]. Prasad, K. V., Vajravelu, K., Datti, P. S., and Raju, B. T., 203, MHD flow and heat transfer in a Power-law liquid film at a porous surface in the presence of Thermal radiation, Journal of Applied Fluid Mechanics, 6(3), pp [3]. Khademinejad, T., Khanarmuei, M. R., Talebizadeh, P., and Hamidi, A., 205, On the use of the homotopy analysis method for solving the problem of the flow and heat transfer in a liquid film over an unsteady stretching sheet, Journal of Applied Mechanics and Technical Physics, 56(4), pp [4]. Megahed, A. M., 205, Effect of slip velocity on Casson thin film flow and heat transfer due to unsteady stretching sheet in presence of variable heat flux and viscous dissipation, Appl. Math. Mech. Engl. Ed., 36, pp [5]. Vijaya, N., Sreelakshmi, K., and Sarojamma, G., 206, Effect of magnetic field on the flow and heat transfer in a Casson thin film on an unsteady stretching surface in the presence of viscous and internal heating, Open Journal of Fluid Dynamics, 6(4), pp [6]. Eldabe, N. T. M., and Salwa, M. G. E., 995, Heat transfer of MHD non- Newtonian Casson fluid flow between two rotating cylinders, J. Phys. Soc. Japan, 64, pp. 4.

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