Effects of Radiation on Unsteady Couette Flow between Two Vertical Parallel Plates with Ramped Wall Temperature

Size: px

Start display at page:

Download "Effects of Radiation on Unsteady Couette Flow between Two Vertical Parallel Plates with Ramped Wall Temperature"

Paula Lewis
5 years ago
Views:

1 Volume 39 No. February 01 Effects of Radiation on Unsteady Couette Flow between Two Vertical Parallel Plates wit Ramped Wall Temperature S. Das Department of Matematics University of Gour Banga Malda India C. Mandal Department of Applied Matematics Vidyasagar University Midnapore India R. N. Jana Department of Applied Matematics Vidyasagar University Midnapore India ABSTRACT Te effect of radiation on unsteady Couette flow between two vertical parallel plates wit ramped wall temperature ave been studied. Te fluid considered ere is a gray absorbing/emitting radiation but a non-scattering medium. An analytical solution of te governing equations as been obtained by employing Laplace transform tecnique. Te influence of te various parameters entering into te problem on te velocity field temperature field skin friction and rate of eat transfer is discussed wit te elp of graps and tables. It is found tat an increase in radiation parameter leads to decrease te fluid velocity and temperature. It is also found tat bot te velocity as well as te temperature of te fluid decrease wit an increase in Prandtl number. Furter it is observed tat te velocity increases wit an increase in asof number and an increase in time leads to increase te fluid velocity and temperature. Keywords: Couette flow Radiation parameter Prandtl number asof number and Ramped wall temperature. 1. INTRODUCTION An analysis of flow formation in Couette motion as predicted by classical fluid mecanics was presented by Sclicting [1]. Tis problem is of fundamental importance as it provides te exact solution of te Navier-Stokes equation and reveals ow te velocity profiles varies wit time approacing a linear distribution asymptotically and ow te boundary layer spreads trougout te flow field. Couette flow is one of te basic flow in fluid dynamics tat refers to te laminar flow of a viscous fluid in te space between two parallel plates one of wic is moving relative to te oter. Te flow is driven by virtue of viscous drag force acting on te fluid and te applied pressure gradient parallel to te plates. Couette flow is frequently used in pysics and engineering to illustrate sear-driven fluid motion. Te radiative eat transfer as an important fundamental penomena existing in practical engineering suc as tose found in solar radiation in buildings foundry engineering and solidification processes die forging cemical engineering composite structures applied in industry Korycki[]. Anoter important feature tat usually occurs in electronic devices over a period of continuous usage is te otness of te surface. Tis means tat a poor design could trap eat generated by te source of te power supply and could incapacitate te efficiency and durability of te systems. Terefore te efficiency in te functioning of tese systems is enanced wen tey are subjected to external cooling devices like air conditioners electric fans and some oters (e.g. laptop computers) inbuilt storage devices tat store electrical energy for tem to function for sometime even witout external source of power supply Gbaorun et al.[3]. Te IC components of tese electronic systems are termally coupled to te surrounding via convection and radiation. Radiation as a significant role in eat transfer in low-flow applications tere exists a larger temperature gradient between te components and te surrounding. Radiative free convection Couette flows are frequently encountered in many scientific and environmental processes suc as eating and cooling of cambers and solar power tecnology. Heat transfer by simultaneous radiation and convection as applications in numerous tecnological problems including combustion furnace design te design of ig temperature gas cooled nuclear reactors nuclear reactor safety fluidied bed eat excanger fire spreads solar fans solar collectors natural convection in cavities turbid water bodies poto cemical reactors and many oters. Furter te eat transfer effects under te conditions of free convection are now dominant in many engineering applications suc as rocket noels ig sinks in turbine blades ig speed aircrafts and teir atmosperic re-entry cemical devices and process equipment formation and dispersion of fog distribution of temperature and moisture over agricultural field and groves of fruit trees damage of crops due to freeing and pollution of te environment and so on. Sing [] as studied te natural convection in unsteady Couette motion. Ogulu and Motsa [5] ave investigated te radiative eat transfer to magnetoydrodynamic Couette flow wit variable wall temperature. Te radiation effects on MHD Couette flow wit eat transfer between two parallel plates ave been describbed by Mebine [6]. Effects of termal radiation and free convection currents on te unsteady Couette flow between two vertical parallel plates wit constant eat flux at one boundary ave been examined by Naraari [7]. Deka and Battacarya [8] ave analyed te unsteady free convective Couette flow of eat generating/absorbing fluid in porous medium. Te radiation effect on electroydrodynamic frot flow in vertical cannel ave been examined by Gbadeyan et al. [9]. Te present work concerns wit te effect of radiation on unsteady Coutte flow of a viscous incompressible fluid confined between two vertical parallel plates wit ramped wall temperature. Initially at time t 0 bot te fluid and plate are at rest and constant temperature T. An exact solution of te governing equations ave been obtained by using Laplace transformation tecnique. It is found tat te velocity decreases wit an increase in eiter radiation parameter R or Prandtl number Pr. An increase in asof number leads to rise te velocity u 1 of flow. It is seen tat te velocity u 1 increases wit an increase in time. It is also seen tat te temperature decreases wit an increase in eiter radiation parameter R 3

2 Volume 39 No. February 01 or Prandtl number Pr but it increases wit an increase in time. Te sear stress x decreases for bot te cases of ramped and isotermal wall temperature wit an increase in eiter radiation parameter R or Prandtl number Pr for fixed values of asof number. Furter te rate of eat transfer (0) decreases wit an increase in eiter radiation parameter R or Prandtl number Pr for bot ramped as well as isotermal wall temperature at te wall 0.. FORMULATION OF THE PROBLEM AND ITS SOLUTIONS Consider te unsteady Couette flow of a viscous incompressible radiative eat generating fluid between two infinite vertical parallel walls separated by a distance. Te flow is set up by te buoyancy force arising from te temperature gradient occurring as a result of asymmetric eating of te parallel plates as well as constant motion of one of te plates. Coose a cartesian co-ordinates system wit te x - axis along one of te plates in te vertically upward direction and te y - axis normal to te plates [See Figure 1]. Initially at time t 0 te two plates and te fluid are assumed to be at te same temperature T and stationary. At time t >0 te plate at y 0 starts moving in its own plane wit a velocity Ut () and te t temperature of te plate is raised or lowered to T ( T0 T) t 0 wen 0<t t0 and te constant temperature T is maintained at t > t 0 as te plate at y is stationary and maintained at a constant temperature T is a constant. Our aim is to analye te radiation effects on te unsteady Couette flow resulting from te ramped temperature profile of te moving plate. We assume tat te flow is laminar and is suc tat te effects of te convective and pressure gradient terms in te momentum and energy equations can be neglected. It is assumed tat te effect of viscous dissipation is negligible. It is also assumed tat te radiative eat flux in te x - direction is negligible as compared to tat in te y - direction. As te plates are infinite long te velocity and temperature fields are functions of y and t only. Figure 1. Geometry of te problem Te free convection flow of a radiating fluid under usual Boussinesq approximation to be governed by te following system of equations u u g( T T ) t y c p T T q k r t y y u is te velocity in te x -direction T te temperature of te fluid g te acceleration due to gravity te kinematic coefficient of viscosity te fluid density k te termal conductivity (1) () c p te specific eat at constant pressure and te radiative eat flux. Te eating due to viscous dissipation is neglected for small velocities in te energy equation (). Te initial and te boundary conditions are u = 0 T = T for 0 y and t 0 u = U( t) at y = 0 t T ( T0 T) for 0 < t t0 T = t0 at y = 0 T for t > t0 u 0 T T at y for t > 0 Te radiative eat flux can be found from Rosseland approximation and its formula is derived from te diffusion concept of radiative eat transfer in te following way T qr () 3k y is te Stefan-Boltman constant and k te spectral mean absorption coefficient of te medium. It sould be noted tat by using te Rosseland approximation we limit our analysis to optically tick fluids. If te temperature differences witin te flow are sufficiently small ten te equation () can be linearied by expanding T into te Taylor series about T and neglecting iger order terms to give: 3 T TT 3 T. (5) It is empasied ere tat equation (5) is widely used in computational fluid dynamics involving radiation absorption problems [10] in expressing te term T as a linear function. In view of () and (5) equation () reduces 3 16 T T T T c p k. t y 3k y Introducing non-dimensionless variables y t t0 u1= u t0 U0 T T U ( t ) U 0 f ( ) T0 T equations (1) and (6) become u 1 u 1 3RPr and 3R 3 T c p Pr te Prandtl number and k asof number. q r (3) (6) (7) (8) (9) kk R is te radiation parameter g ( T0 T ) te U0 35

3 Volume 39 No. February 01 Te corresponding initial and te boundary conditions are u1 0 0 for 0 1 and 0 for 0 < 1 u1 f( ) at 0 1 for > 1 u1 0 0 at 1 for > 0 is te accelerating parameter. U 0 (10) On te use of Laplace transformation equations (8) and (9) become d u 1 1 su (11) d d s d (1) s s u1( s) u1( s) e d ( s) ( s) e d 0 0 (13) Te corresponding boundary conditions for u 1 and are u1(0 s) = f ( s) u1(1 s) = 0 1 s (0 s) (1 e ) (1 s) 0. (1) s Te solution of te equations (11) and (1) subject to te boundary conditions (1) are easily obtained and are given by for 1 s 1 e a s b s e e s ( s) = s 1 e a s b s e e for 1 s a s b s f () s e e s (1 e ) a s b s a s b s 3 e e e e for 1 s ( 1) u1( s) = s a s b s (1 e ) a s b s f () s e e 5/ e e s b s d s ( n 1){ e e } for 1. Te inverse transforms of (15) and (16) give te solution for te temperature distribution and velocity field as ( ) F( a ) F( b ) H ( 1){ F( a 1) F( b 1)} for 1 F ( a ) F ( b ) H ( 1){ F ( a 1) F ( b 1)} for 1 (15) (16) (17) 36

4 Volume 39 No. February 01 F( a ) F( b ) F1( a ) F1( b ) 1 { F1( a ) F1( b )} H ( 1){ F1( a 1) F1( b 1) [ F1( a 1) F1( b 1)]} for 1 u1( ) F( a ) F( b ) F3( a ) F3( b ) H 1 { F3( a 1) F3( b 1)} ( n 1) F3( b ) F3( d ) H ( 1){ F3( b 1) F3( d 1)} for 1 (18) a n b n d n F ( ) erfc 1 1 F1 ( ) erfc e 6 ( ) = erfc F e (19) 1 F3 ( ) = e erfc 3 6 and is a dummy variable and F1 F and F 3 are dummy functions defined in equation (19) erfc ( x ) being complementary error function and and H( 1) is te unit step function. It is noticed tat in te absence of radiation parameter ( R ) te solutions (17) and (18) are identical wit te equations (8) and (9)..1 SOLUTION FOR ISOTHERMAL PLATES For isotermal plates te boundary conditions for te temperature and velocity fields are u 0 T T for 0 y and t 0 u U( t) T T0 at y 0 for t > 0 (0) u 0 T T at y for t > 0. On te use of (7) te boundary conditions for u 1 and become u1 0 0 for 0 1 and 0 u1 f( ) 1 at 1 for > 0 (1) u1 0 0 at 1 for > 0. Te corresponding boundary conditions for and u 1 are u1(0 s) f ( s) u1(1 s) 0 1 (0 s) (1 s) 0. () s Te solution of te equations (11) and (1) subject to te boundary conditions () are given by 1 a s b s e e for 1 s ( s) 1 a s b s e e for = 1 s Te inverse transforms of (3) and () give te solution for te temperature distribution and velocity field as a s b s f () s e e a s b s a s b s e e e e for 1 s ( 1) u1( s) a s b s a s b s f () s e e 3/ e e s b s d s ( n 1){ e e } for 1. () (3) 37

5 Volume 39 No. February 01 F( a ) F( b ) for 1 ( ) F( a ) F( b ) for 1 F( a ) F( b ) F ( a ) F ( b ) { F ( a ) F ( b )} for 1 1 u1( ) F( a ) F( b ) F ( a ) F ( b ) ( n 1) F ( b ) F ( d ) for 1 (5) (6) a b d and F ( ) are given by (19) and ( ) erfc F e. (7) 3. RESULTS AND DISCUSSION We ave presented te non-dimensional velocity u 1 and temperature for several values of Radiation parameter R Prandtl number Pr asof number and time in Figures -8. It is found from Figures and 3 tat te velocity u decreases wit an increase in eiter radiation parameter R 1 or Prandtl number Pr. Tis is possible pysically because fluids wit ig Prandtl number ave greater viscosity wic makes te fluid tick and ence move slowly. It is also found tat te velocity u 1 is maximum near te plate and decreases away from te plate and finally takes asymptotic value for all values of R and Pr. It is observed from Figures and 5 tat an increase in eiter asof number or time leads to rise in te velocity u 1. Pysically tis is due to fact tat as te asof number or time increases te contribution from te buoyancy force near te plate becomes significant and ence a rise in te velocity near te plate as observed. It is observed from Figures 6 and 7 tat te temperature decreases wit an increase in eiter radiation parameter R or Prandtl number Pr. In te presence of radiation te termal boundary layer always found to ticken wic implies tat te radiation provides an additional means to diffuse energy. Tis means tat te termal boundary layer decreases and more uniform temperature distribution across te boundary layer. An increase in te Prandtl number means an increase of fluid viscosity wic causes a decrease in te flow velocity and te temperature decreases. Tis is consistent wit te fact tat te termal boundary layer tickness decreases wit te increasing Prandtl number. Te temperature increases wit an increase in time sown in Figure 8. Figure. Velocity profiles for R wen Pr = 10 = 10 and = 1.1 Figure 3. Velocity profiles for Pr wen R = 1.5 = 10 and =

6 Volume 39 No. February 01 Figure. Velocity profiles for wen Pr =7 R = 0.8 and = 1.1 Figure 7. Temperature profiles for Pr wen R = and = 1. Figure 5. Velocity profiles for time wen Pr = 10 = 10 and R = 1.5 Figure 6. Temperature profiles for R wen Pr =7 and = 1. Figure 8. Temperature profiles for time wen Pr = 10 and R = For ramped wall temperature te sear stress at te plate 0 is given by H1( a ) H1( b ) H( a ) H( b ) 1 { H( a ) H( b )} H 1 { H( a 1) H( b 1) [ H( a 1) H( b 1)]}] for 1 x H 1( a ) H1( b ) F3( a ) F3( b ) ( n 1){ H3( b ) H3( d )} H 1 { F3( a 1) F3( b 1) ( n 1){ H3( b 1) H3( d 1)}}] for 1 (8) 39

7 Volume 39 No. February 01 a n b n (9) 1 H 1( ) e (30) 1 3 H( ) erfc 6 1 e (31) 3 3 ( ) erfc H e. (3) For isotermal plates te sear stress at te plate 0 is obtained as: H1( a ) H1( b ) H5( a ) H5( b ) 1 { H5( a ) H5( b )}] for 1 x H1( a ) H1( b ) F( a ) F( b ) ( n 1){ H( b ) H( b )}] for = 1 ( ) erfc H 5 ( ) erfc H e. and aand b and H1 ( ) are given by (9) and (30). (33) (3) (35) Figure 9. Sear stress x for time wen Pr = 10 and R = 1.5 Figure 10. Sear stress x for R wen Pr = 10 and = 1.1. Numerical values of te non-dimensional sear stress at te moving plate 0 are presented in Figures 9-11 against asof number for various values of radiation parameter R Prandtl number Pr and time. Figure 9 sows tat te sear stress x at te moving plate 0 increases for bot ramped as well as isotermal wall temperature wit an increase in eiter asof number or time. It is seen from Figures 10 and 11 tat for fixed values of asof number te sear stress x decreases for bot te cases of ramped and isotermal wall temperature wit an increase in eiter radiation parameter R or time. Figure 11. Sear stress x for Pr wen = 0.5 and R = 1.5 For ramped wall temperature te rate of eat transfer at te moving plate 0 is obtained as: 0

8 Volume 39 No. February 01 [ { H6( a ) H6( b )} H 1 { H6( a 1) H6( b 1)}] for 1 (0) =0 [{ H6( a ) H6( b )} H 1 { H6( a 1) H6( b 1)}] for = 1 (36) 1 1 H6( ) erfc e and aand b are given by (9). (37) For isotermal plates te rate of eat transfer at te moving plate 0 is given by (0) 0 H1( a ) for 1 H1( b ) H1( a ) H1( b ) for = 1 (38) Numerical results of te rate of eat transfer (0) at te moving plate ( 0) against te radiation parameter R are presented in Table 1 and for various values of Prandtl number Pr and time. Table 1 sows tat te rate of eat transfer (0) increases wit an increase in eiter radiation parameter R or Prandtl number Pr for bot ramped as well as isotermal wall temperature at te moving plate 0. It is observed from Table tat te rate of eat transfer (0) first increases and reaces maximum and ten decreases wit an increase in time wile it increases wit an increase in radiation parameter R for ramped wall temperature. On te oter and te rate of eat transfer (0) increases wit an increase in radiation parameter R as it decreases wit an increase in time for isotermal wall temperature. Te negative value of (0) pysically explains tat tere is eat flow from te plate. aand b and H1 ( ) are given by (9) and (30). Table 1. Rate of eat transfer (0) at te moving plate ( 0) wen = 1.1 Ramped Isotermal R \ Pr Table. Rate of eat transfer (0) at te moving plate ( 0) wen Pr = 10. Ramped Isotermal R \

9 Volume 39 No. February 01. CONCLUSION Effect of radiation on unsteady Coutte flow between two vertical parallel plates wit a ramped wall temperature as been investigated. It is found tat te velocity decreases wit increase in eiter radiation parameter R or Prandtl number Pr. An increase in asof number leads to rise in te velocity u 1 of flow field. It is seen tat te velocity u 1 increases wit an increase in time. It is also seen tat te temperature decreases wit increase in eiter radiation parameter R or Prandtl number Pr. Te temperature increases wit an increase in time. Furter it is observer tat for fixed value of asof number te sear stress x decreases for bot te cases of ramped and isotermal wall temperature wit an increase in eiter radiation parameter R or time. Te rate of eat transfer (0) increases wit an increase in eiter radiation parameter R or Prandtl number Pr for bot ramped as well as isotermal wall temperature at te moving plate REFERENCES [1] Sclicting H. (1979). Boundary Layer Teory. Mcaw- Hill New York. [] Korycki R.(006). Sensitivity analysis and sape optimiation for transient eat conduction wit radiation. Int. J. Heat Mass Transfer. 9: [3] Gbaorun F. K. Ikyo B. Iyoror B. and Okanigbun R. A.(008). Heat model for temperature distribution in a laptop computer. J. of NAMP. 1: [] Sing A. K. (1988). Natural convection in unsteady Couette motion. Def. Sci. J. 38(1): [5] Ogulu A. and Motsa S. (005). Radiative eat transfer to magnetoydrodynamic Couette flow wit variable wall temperature. Pysica Scripta. 71(): [6] Mebine P. (007). Radiation effects on MHD Couette flow wit eat transfer between two parallel plates. Global J. Pure and Appl.Mat. 3(): 1-1. [7] Naraari M. (010). Effects of termal radiation and free convection currents on te unsteady Couette flow between two vertical parallel plates wit constant eat flux at one boundary. WSEAS Transactions on Heat and Mass Transfer. 5(1): [8] Deka R. K. and Battacarya A. (011). Unsteady free convective Couette flow of eat generating/absorbing fluid in porous medium. Int. J. Mat. Arc. (6): [9] Gbadeyan J. A. Daniel S. and Kefas E. G. (005). Te radiation effect on electroydrodynamic frot flow in vertical cannel. J. Mat. Associ. Nigeria. 3(B): [10] Cung T. J.(00). Computational Fluid Dynamics. Cambridge University Press.

Radiation Effects on Free Convection MHD Couette Flow Started Exponentially with Variable Wall Temperature in Presence of Heat Generation

Open Journal of Fluid Dynamics,,, 4-7 http://dx.doi.org/.436/ojfd.. Published Online March (http://www.scirp.org/journal/ojfd) Radiation Effects on Free Convection MHD Couette Flow Started Exponentially