Spectral Numerical Calculation of Non-isothermal Flow through a Rotating Curved Rectangular Duct with Moderate Curvature

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1 Available at Appl. Appl. Math. ISSN: Applications and Applied Mathematics: An International Journal (AAM) Special Issue No. (May 016), pp th International Mathematics Conference, March 0, 014, IUB Campus, Bashundhara Dhaka, Bangladesh Spectral Numerical Calculation of Non-isothermal Flo through a Rotating Curved Rectangular Duct ith Moderate Curvature Md. Zohurul Islam 1*, Md. Saidul Islam and Rabindra Nath Mondal 3 1 Department of Mathematics and Statistics Jessore University of Science and echnology Jessore-7408, Bangladesh Department of Mathematics Faculty of Science, Engineering and echnology Hamdard University Ne on, Narayangong, Bangladesh 3 Department of Mathematics Jagannath University Dhaka-1100, Bangladesh * zohurulmathku@gmail.com ABSRAC he present paper investigates non-isothermal flo characteristics through a rotating curved rectangular duct, here co-existence of the rotational forces and fluid temperature gradients leads to the emergence of rotation-induced buoyancy effects. A spectral-based numerical scheme is employed as the principal tool for the simulation hile Chebyshev polynomial and collocation method as the secondary tools. he outer all of the duct is heated hile the inner all cooled, the top and bottom alls being thermally insulated. he emerging parameters controlling the flo characteristics are the rotation parameter, i.e., the aylor number r ranging 0 to 000, the Grashof number Gr = 100, the Prandtl number Pr, the aspect ratio, and the pressure-driven parameter, i.e., the Dean number Dn beteen 100 and he flo structures are examined under combined action of the centrifugal, Coriolis and buoyancy forces. As a result, asymmetric -cell structures are computed for small values of r hile asymmetric 6-cell structures for large r. Unsteady flo characteristics sho that the flo undergoes in the scenario chaotic multi-periodic periodic steady-state, if r is increased in the positive direction. ypical contours of secondary flo patterns, temperature profiles and axial flo distribution are also obtained at several values of r, and it is found that there exist asymmetric to- to multi-vortex solutions. Heating the outer all is found to generate a significant temperature gradient at the outer concave all. KEYWORDS: Curved rectangular duct; Dean number; aylor number and time evolution AMS-MSC 010 No.: 76U05, 76E06 37

2 38 M. Zohurul Islam et al. 1. INRODUCION Rotating flo is an important branch of fluid dynamics and is full of complex physics. In practical applications, rotating thermal flos occur frequently in a variety of rotating machinery. For a long time this intriguing branch of thermal flos has attracted much attention of the researchers. In the past decades, there have appeared several books and revie articles on hydrodynamic and heat transfer characteristics of rotating flos. Since rotating machines ere introduced into engineering applications, such as rotating systems, gas turbines, electric generators, heat exchangers, cooling system and some separation processes, scientists have paid considerable attention to study rotating curved duct flos. he readers are referred to Nandakumar and Masliyah (1986), Ito (1987) and Yanase et al. (00) for some outstanding revies on curved duct flos. he fluid flo in a rotating curved duct is subjected to to forces: the Coriolis force due to rotation and the centrifugal force due to curvature of the duct. For isothermal flos of a constant property fluid, the Coriolis force tends to produce vortices hile centrifugal force is purely hydrostatic. In a curved passage, centrifugal forces are developed in the flo due to channel curvature causing a counter rotating vortex motion applied on the axial flo through the curved channel by Zohurul et al. (014). his creates characteristics spiraling fluid flo in the curved passage knon as secondary flo. At a certain critical flo condition and beyond, additional pairs of counter rotating vortices appear on the outer concave all of curved fluid passages. his flo condition is referred to as Dean s hydrodynamic instability and the additional vortices are knon as Dean Vortices, in recognition of the pioneering ork in this field by Dean (197). When a temperature field is applied to the fluid then the variation of fluid density occurs in non-isothermal flos, both Coriolis and centrifugal type buoyancy forces can contribute to the generation of vortices referred Mondal et al. (007). hese to effects of rotation either enhance or counteract each other in a non-linear manner depending on the direction of all heat flux and the flo domain. herefore, the effect of system rotation is more subtle and complicated and yields ne; richer features of flo and heat transfer in general, bifurcation and stability in particular, for non-isothermal flos. Hoever, analytical, numerical and experimental investigations, such as Baylis (1971), Mondal et al. (014) and Humphrey et al. (1977) concluded that Dean number as solely responsible for generating secondary flo and Dean instability in curved passages. Mondal et al. (014) applied an spectral-based numerical study to investigate combined effects of centrifugal and Coriolis instability of the flo through a rotating curved rectangular duct of small curvature, here they investigated unsteady flo characteristics for small values of the flo parameters ith positive rotation of the duct, and presented some preliminary results of the flo evolution. Selmi et al. (1994) examined combined effects of system rotation and curvature on the bifurcation structure of to-dimensional flos in a rotating curved duct ith square cross section. Wang and Cheng (1996), employing finite volume method, examined the flo characteristics and heat transfer in curved square ducts for positive rotation and found reverse secondary flo for the corotation cases. Selmi and Nandakumer (1999) and Yamamoto et al. (1999) performed studies on the flo in a rotating curved rectangular duct. Norouzi et al. (010) investigated on the inertial and creeping flo of a second-order fluid in a curved duct ith square cross-section by using finite difference method. he effect of centrifugal force due to the curvature of the duct and the opposing effects of the first and second normal stress difference on the flo field ere investigated in that study. Chandratilleke et al. (01) presented extensive 3D computational study using helicity function that describes the secondary vortex structure and thermal behavior in the fluid flo

3 AAM: Intern. J., Special Issue, No. (May 016) 39 through curved rectangular ducts of aspect ratios ranging from 1 to 6. A curvilinear coordinate system as used in that study facilitating effective grid definition for capturing vortex generation and permitting efficient evaluation of local pressure gradient. Norouzi and Biglari (013) performed, for the first time, an analytical solution of Dean Flo inside a curved rectangular duct, here perturbation method as used to solve the governing equations. he main flo velocity (axial flo), vector plots of lateral velocity (secondary flos) and flo resistance ratios ere obtained in that study. heir study as limited to lo Reynolds numbers and obtained maximum four-vortex solutions. Wu et al. (013) performed numerical study of the secondary flo characteristics in a curved square duct by using the spectral method, here the alls of the duct except the outer all rotate around the Centre of curvature and an azimuthally pressure gradient as imposed. In that study, multiple solutions ith -vortex, 4-vortex, 8-vortex and even non-symmetric vortices ere obtained at the same flo condition. Recently, Kun et al. (014) performed an experimental investigation on laminar flos of pseudo-plastic fluids in a square duct of strong curvature using an ultrasonic Doppler velocimetry and microphones, here stream ise velocity in the cross-section of the duct and the fluctuating pressure on the alls ere measured for different flo rates. Guo et al. (011) used a laminar incompressible three- dimensional numerical model to explore the interactive effects of geometrical and flo characteristics on heat transfer and pressure drop. hey applied entropy generation as a hydro-thermal criterion and reported the influence of the Reynolds number and curvature ratio on the flo profile and the Nusselt number. Recently, Mondal et al. (01, 013a, 013b) performed numerical investigation on the non-isothermal flo through a rotating curved square and rectangular duct and obtained substantial results. o the best of the authors' knoledge, hoever, there has not yet been done detail investigation on the unsteady flo characteristics for the non-isothermal flo through a curved rectangular duct of moderate aspect ratio ith pressure gradient in the axial direction. But from the scientific as ell as engineering point of vie it is quite interesting to study the unsteady flo behavior in the presence of strong centrifugal and buoyancy forces for moderate aspect ratio of the duct, because this type of flo is often encountered in engineering applications such as in gas turbines, metallic industry and exhaustive pipes. he present paper investigates unsteady flo characteristics for the non-isothermal flo through a rotating curved rectangular duct by using the spectral method, and covering a ide range of the Dean number and the aylor number. Studying the effects of system rotation on the flo characteristics, caused by the centrifugal-coliolis-buoyancy forces ith pressure drop, is an important objective of the present study.. PHYSICAL MODEL Consider fully developed to-dimensional flo of viscous incompressible fluid through a rotating curved rectangular duct hose height and idth are h and d, respectively. he coordinate system ith the relevant notation is shon in Figure 1, here x and y axes are taken to be in the horizontal and vertical directions respectively, and z is the axial direction of the duct. he system rotates at a constant angular velocity and the representative v velocityu 0, here v is the kinematic viscosity. We introduce the non-dimensional variables d defined as:

4 40 M. Zohurul Islam et al. U u v x y z t t U0 U0 Uo d d d d d, P P, G P d, L U z U / u v x y z 0,,,,,,,, 0 0 here u, v, and are the non-dimensional velocity components in the x, y and z directions, respectively; t is the non-dimensional time, P the non-dimensional pressure, the nondimensional curvature, and temperature is non-dimensionalized by. Henceforth, all the variables are non-dimensionalized if not specified. Figure 1. Physical configuration of the system 3. MAHEMAICAL FORMULAION Since the flo field is uniform in the z -direction, the sectional stream function is introduced as, 1 1 u, v. 1x y 1x x (1) A ne coordinate variable y is introduced in the ydirection as y ay, here a h d is the aspect ratio of the duct cross section. From no on y denotes y for the sake of simplicity. hen the basic equations for the axial velocity, the stream function and the temperature are derived from the Navier-Stokes equations and the energy equation under the Boussinesq approximation as, 1 (, ) 1 (1 x) Dn (1 x) r, t a ( x, y) 1 x a (1 x) y x y () 1 1 (, ) x x t a (1 x) ( x, y) a 1 x y 1x x x

5 AAM: Intern. J., Special Issue, No. (May 016) x xy x x x x x x a y Gr(1 x) x 1 r, y (3) 1 (, ) 1 t (1 x) ( x, y) Pr 1 x x, (4) here 1 ( f, g) f g f g, (, ). x a y x y x y y x he non-dimensional parameters Dn, the Dean number; r, the aylor number; Gr, the Grashof number and Pr, the prandtl number, hich appear in equations () to (4) are defined as: Gd 3 d Dn, L d 3 r, gd 3 Gr, Pr, (5) here,, and g are the viscosity, the coefficient of thermal expansion, the co-efficient of thermal diffusivity and the gravitational acceleration respectively, is the viscosity of the fluid. In the present study, Dn and r are varied hile Gr, a, and Pr are fixed. he physical properties ith various parameters used in the present study, are shon in able 1.0 able 1. Parameters ith physical properties Parameters Physical properties Present Values Dean number (Dn) Pressure gradient 100 Dn 1000 aylor number (r) Non-dimensional rotational parameter 0 r 000 Grashof number (Gr) Dimensionless natural convection Gr = 100 parameter that governs the fluid flo Aspect ratio (a) - a =.0 Curvature ( ) Prandtl number (Pr) Characteristic of the fluid Pr = 7.0 (ater) 4. BOUNDARY CONDIIONS he boundary conditions for and of the present study are described as follos: ( 1, y) ( x, 1) ( 1, y) ( x, 1) ( 1, y) ( x, 1) 0. x y (6)

6 4 M. Zohurul Islam et al. Also, the temperature is constant on the alls as (1, y) 1, ( 1, y) 1, ( x, 1) x. (7) 5. Numerical methods 5.1. Method of numerical calculations In order to solve the system of non-linear partial differential equations () to (4) numerically, the spectral method is used. By this method the expansion functions n (x ) and n (x ) are expressed as,, n( x) (1 x ) Cn( x), n( x) (1 x ) Cn( x) (8) here Cn ( x) cos ncos 1 ( x) is the th n order Chebyshev polynomial. ( x, y, t), ( x, y, t) and ( x, y, t) are expanded in terms of the expansion functions n (x ) and n (x ) as M N ( x, y, t) mn ( t) m ( x) n ( y) m0 n0 M N ( x, y, t) mn ( t) m ( x) n ( y), m0 n0 M N mn m n m0 n0 ( x, y, t) ( x) ( y) x (9) here M and N are the truncation numbers along x and y directions respectively. We performed graphical representation of the time-evolution of the unsteady solutions, secondary flo behavior, temperature profiles and axial flo distribution. As a result, multiple solutions of steady-state, periodic, multi-periodic and chaotic solutions ith symmetric and asymmetric multi-vortex solutions are obtained. 5.. Grid sensitivity test he accuracy of the numerical calculations is investigated for the truncation numbers M and N used in this study. Five types of grid sizes ere used to check the dependence of grid size (i.e., M and N ). For good accuracy of the solutions, N is chosen equal to M. Different types of the grid sizes are taken in this study as14 8, 163,1836, 040, 44, and it is found that M = 16 and N = 3 gives sufficient accuracy of the numerical solutions, hich are shon in able.

7 AAM: Intern. J., Special Issue, No. (May 016) 43 able. he values of and (0,0) for various M and N at Dn 700 and r 830 M N (0,0) In order to calculate the unsteady solutions, Crank-Nicolson and Adams-Bashforth methods together ith the function expansion (9) and the collocation methods are applied. Details of these methods are discussed in Mondal (006). By applying the Crank- Nicolson and the Adams- Bashforth methods to the non-dimensional basic equations ()-(4), and rearranging, e get 1 1 ( t) ( tt) tt t x P( x, y). t t t (10) t t t t t t Q( x, y). 1 x t (11) 1 1 ( ) ( ) t t x x (, ). Pr t t t t Pr t R x y (1) In the above formulations, P, Q and R are the non-linear terms. hen applying the Adams- Bashforth method for the second term of R. H. S of Equations (10), (11) and (1) and simplifying e calculate t t, t t and t t by numerical computation Resistance coefficient We use the resistance coefficient as one of the representative quantities of the flo state. It is also called the hydraulic resistance coefficient and is generally used in fluids engineering defined as, P * 1 P z * * dh 1 * *, here quantities ith an asterisk denote the dimensional ones, stands for the mean over * the cross section of the rectangular duct, and d h 4d 4dh/ 4d 8dh. Since * * * P P / z is related to the mean non-dimensional axial velocity as, 1 G Where 16 Dn, (14) 3 (13) * d / v. In this paper, is used to calculate the unsteady solutions by numerical computations.

8 44 M. Zohurul Islam et al. 6. Results and Discussion In the present model, our objective is to study the effects of rotation on the flo characteristics. he expressions for velocity profile, flo rate, all heat transfer (temperature profile) are obtained by solving the governing equation of flo using the spectral method. In this paper, time evolution calculations of the resistant coefficient are performed for the nonisothermal flos over a ide range of the Dean Numbers (Dn) and the aylor Number (r) for the to cases of the duct rotation, Case I: Dn = 700 and Case II: Dn = Case I: Dean Number, Dn = 700 We performed time evolution of for 0 r 000 and 100 Dn Figure (a) shos time evolution of for r = 100 and Dn = 700 at Gr = 100. It is found that the unsteady flo at r = 100 is a chaotic solution hich is ell justified by draing the phase spaces as shon in Figure (b). Figure (c) shos typical contours of secondary flo patterns, temperature profiles and axial flo distribution for r = 100 and Dn = 700, here e find that the unsteady flo is a four-vortex solution. his is caused by the combined action of the Coriolis force and centrifugal force by Wang and Cheng (1996). hen e performed time evolution for r = 800 as shon in Figure 3(a). It is found that the unsteady flo at r =800 is a eak chaotic solution hich is transformed into the multi-periodic solution, and it is ell justified by draing the phase spaces as shon in Figure 3(b). Contours of secondary flo patterns, temperature profiles and axial flo distribution for the corresponding flo patterns are shon in Figure 3(c), here it is found that the multiperiodic oscillation at r = 800 is a four-vortex solution. he time evolution for r = 830 is shon in Figure 4(a). It is found that the unsteady flo at r =830 is a fully multi-periodic solution hose orbit is different and it is ell justified by draing the phase space as shon in Figure 4(b). Contours of secondary flo patterns, temperature profiles and axial flo distribution for the corresponding flo patterns are shon in Figure 4(c), here it is found that the multi-periodic oscillation at r = 830 is a four-vortex solution. In this paper, the buoyancy effects caused by the differentially heated vertical sidealls are clearly shon. he buoyancy effects are due to the centrifugal and Coriolis forces in the presence of a density gradient. When a duct is rotated, heating or cooling of the fluid hich is contained in or flo through it, can give rise to significant buoyant force effects. he present ork, therefore, confines itself to the thermal buoyancy effects in rotating non-isothermal flos, in hich the coexistence of the rotational forces and fluid temperature gradients leads to the emergence of rotation-induced buoyancy effects. Since the rotational forces are spatially varying ith the local flo information, the buoyancy effects induced are obviously more sophisticated than the gravitational buoyancy resulted from constant gravity in a conventional natural/mixed convection flo. he coupling nature of the thermal flo phenomena ith the rotational buoyancy is very influential to transport phenomena in the rotating system.

9 [ AAM: Intern. J., Special Issue, No. (May 016) 45 t Figure. (a) ime evolution of for Dn = 700 and r = 100, (b) Phase space for r =100, (c) Contours of secondary flo patterns (top), temperature profiles (middle) and axial flo (bottom) for r = 100 at 3.0 t t Figure 3. (a) ime evolution of for Dn = 700 and r = 800. (b) Phase space for r = 800, (c) Contours of secondary flo patterns (top), temperature profiles (middle) and axial flo distribution (bottom) for r = 800 at t 31.3 t Figure 4. (a) ime evolution of for Dn = 700 and r = 830. (b) Phase space for r = 830, (c) Contours of secondary flo patterns (top), temperature profiles (middle) and axial flo distribution (bottom) for r = 830 at 36.0 t 36.5

10 46 M. Zohurul Islam et al. t Figure 5. (a) ime evolution of for Dn = 700 and r = 870. (b) Phase space for r = 870, (c) Contours of secondary flo patterns (top), temperature profiles (middle) and axial flo (bottom) for r = 870 at t (a) t = 36.0 (b) Figure 6. (a) ime evolution of for Dn = 700 and r = 880, (b) Contours of secondary flo patterns (left), temperature profiles (middle) and axial flo distribution (right) for r = 880 at t ime evolution result for r = 870 is shon in Figure 5(a). It is found that the unsteady flo at r = 870 is a periodic solution hose orbit is single and it is ell justified by draing the phase space as shon in Figure 5(b). Contours of secondary flo patterns temperature profiles and axial flo distribution for the corresponding flo patterns are shon in Figure 5(c), here it is found that the periodic oscillation at r = 870 is a four-vortex solution. If the rotational speed is increased more in the positive direction, for example r = 880 up to 000, it is found that the flo becomes steady state hich is shon in Figure 6(a) for r = 880. Since the unsteady flo is a steady-state solution, single contours of secondary flo pattern, temperature profile and axial flo distribution is shon in Figure 6(b), and it is found that steady-state solution is a to-vortex flo.

11 AAM: Intern. J., Special Issue, No. (May 016) Case II: Dean Number, Dn = 1000 Here, e performed time evolution of for 0 r 000 and Dn = Figure 7(a) shos time evolution of for r = 500. It is found that the unsteady flo at r = 500 is a strongly chaotic solution, hich is ell justified by draing the phase spaces as shon in Figure 7(b). Figure 7(c) shos typical contours of secondary flo patterns, axial flo distribution and temperature profiles for r = 500, here e find that the unsteady flo is a six- and eightvortex solution. hen e performed time evolution for r = 1470 as shon in Figure 8(a). It is found that the unsteady flo at r =1470 is a multi-periodic solution hich is ell justified by depicting the phase space as shon in Figure 8(b). Contours of secondary flo patterns, temperature profiles and axial flo for the corresponding flo patterns has been shon in Figure 8(c) and here it is found that the multi-periodic oscillation at r =1470 is a fourvortex solution. Continuing this process the time evolution for r = 1485 is displayed in Figure 9(a). It is observed that the unsteady flo at r =1485 is a multi-periodic solution hose orbit is different and it is justified by draing the phase space as shon in Figure 9(b). Contours of secondary flo patterns, temperature profiles and axial flo for the corresponding flo patterns are expressed in Figure 9(c), here the four-vortex solution has been observed. Next, time evolution for r = 1490 is shon in Figure 10(a). It is found that the unsteady flo at r =1490 is a periodic solution hose orbit is single and it is ell justified by draing the phase space as shon in Figure 10(b). Contours of secondary flo patterns temperature profiles and axial flo for the corresponding flo patterns are shon in Figure 10(c), here it is found that the periodic oscillation at r =1485 is a four-vortex solution. hen, the rotational speed is increased more in the positive direction, for example r = 1600 up to 000, and it is found that the flo becomes steady-state hich is shon in Figure 11(a) for r = Since the unsteady flo is a steady-state solution, a single contour of secondary flo pattern, temperature profile and axial flo distribution is shon in Figure 11(b) and it is found that steady-state solution is an asymmetric to-vortex solution. t Figure 7. (a) ime evolution of for Dn = 1000 and r = 500. (b) Phase space for r =500, (c) Contours of secondary flo patterns (top), temperature profiles (middle) and axial flo distribution (bottom) for r = 500 at 46.3 t 46.7

12 48 M. Zohurul Islam et al. t Figure 8. (a) ime evolution of for Dn = 1000 and r = (b) Phase space for r =1470, (c) Contours of secondary flo patterns (top), temperature profiles (middle) and axial flo distribution (bottom) for r = 1470 at 35.4 t 36.5 t Figure 9. (a) ime evolution of for Dn = 1000 and r = (b) Phase space for r =1490, (c) Contours of secondary flo patterns (top), temperature profiles (middle) and axial flo (bottom) for r = 1490 at 35.4 t 36.5 t Figure 10. (a) ime evolution of for Dn = 1000 and r = (b) Phase space for r =1490, (c) Contours of secondary flo patterns (top), temperature profiles (middle) and axial flo (bottom) for r = 1490 at 40.8 t 43.5.

13 AAM: Intern. J., Special Issue, No. (May 016) 49 (a) t = 15.0 (b) Figure 11. (a) ime evolution of for Dn = 1000 and r = (b) Phase space for r =1600, (c) Contours of secondary flo patterns (left), temperature profiles (middle) and axial flo (right) for r = 1600 at t CONCLUSION A numerical study is presented for the flo characteristics through a rotating curved rectangular duct of aspect ratio.0 and curvature 0.1. Numerical calculations are carried out by using a spectral method and covering a ide range of the aylor number 0 r 000 for the Dean numbers 100 Dn 1000 and for the Grashof number Gr =100. We have investigated unsteady solutions for the positive rotation of the duct by time evolution calculations, and it is found that the unsteady flo undergoes in the scenario Chaotic multi-periodic periodic steady-state, if r is increased in the positive direction. In order to investigate the transition from multi-periodic oscillations to chaotic states more explicitly, the orbit of the solution is dran in phase space. Draing the phase spaces is found to be very fruitful to justify the transition of the unsteady flo characteristics as ell as chaotic flo behavior. In this regard, it should be orth mentioning that irregular oscillation of the flo through a curved duct has been observed experimentally by Ligrani and Niver (1988) for a large aspect ratio and by Wang and Yang (005) for the square duct. ypical contours of secondary flo patterns, temperature profiles and axial flo distribution are also obtained at several values of r, and it is found that there exist to-, four-, six- and eight-vortex solutions. he reason is that due to the combined action of the centrifugal-coriolis-buoyancy force, e obtained to- to multi-vortex solution. It is found that the temperature distribution is consistent ith the secondary vortices and axial flo distributions, and convective heat transfer is significantly enhanced as the secondary vortices become stronger. he present study shos, there exists an asymmetric to-vortex solution for the steady-state solution, hile asymmetric to-, three-, and four-vortex solutions for the periodic and multi-periodic solutions. For chaotic solution, on the other hand, e obtain asymmetric four-vortex solutions only. he present study also shos that chaotic flo enhances heat transfer more effectively than the steady-state or periodic solutions as the secondary flo become stronger for this case. REFERENCES Baylis, J. A., (1971). Experiments on Laminar Flo in Curved Channels of Square Section, Journal of Fluid Mechanics, Vol. 48(3), pp

14 50 M. Zohurul Islam et al. Chandratilleke,.., Nadim, N. and Narayanasamy, R., (01). Vortex Structure-based Analysis of Laminar Flo Behavior and hermal Characteristics in Curved Ducts, Int. J. hermal Sciences, Vol. 59, pp Dean, W. R., (197). Note on the Motion of Fluid in a Curved Pipe, Philos. Mag., Vol. 4, pp Guo, J., Xu, M., Cheng, L., (011). Second La Analysis of Curved Rectangular Channels, Int. J. hermal Sciences, Vol. 50, pp Humphrey, J. A. C., aylor, A. M. K. and Whitela, J. H., (1977). Laminar flo in a Square Duct of Strong Curvature, Journal of Fluid Mechanics, Vol. 83, pp Islam, M. Z., Islam, M. S., & Islam, M. M, (014). Pressure-Driven Flo Instability ith Convective Heat ransfer through a Rotating Curved Channel ith Rectangular Cross- Section: he Case of Negative Rotation, Journal of Mechanics of Continua and Mathematical Sciences, Vol. 9(1), pp Ito, H. (1987). Flo in curved pipes. JSME International Journal, Vol. 30, pp Kun, M. A, Yuan S., Chang, H. and Lai, H., (014). Experimental Study of Pseudo plastic Fluid Flos in a Square Duct of Strong Curvature, Journal of hermal Science, Vol. 3(4), pp Ligrani, P. M. and Niver, R. D., (1988). Flo Visualization of Dean Vortices in a Curved Channel ith 40 to I Aspect Ratio, Phys. Fluids, Vol. 31, pp Mondal, R. N., Alam M. M. and Yanase, S. (007). Numerical prediction of non- isothermal flos through a rotating curved duct ith square cross section, hommasat Int. J. Sci and ech., Vol. 1, No. 3, pp Mondal, R. N., Islam, M. Z., & Perven, R. (014). Combined effects of centrifugal and coriolis instability of the flo through a rotating curved duct of small curvature. Procedia Engineering, Vol. 90, pp Mondal, R. N., Datta, A. K. and Uddin, M. K. (01). A Bifurcation Study of Laminar hermal Flo through a Rotating Curved Duct ith Square Cross-section, Int. J. Appl. Mech. and Engg., Vol. 17 (). Mondal, R. N., Islam, M. S., Uddin, M. K. and Hossain, M. A. (013a). Effects of Aspect Ratio on Unsteady Solutions through a Curved Duct Flo, Appl. Math. & Mech., Vol. 34(9), pp.1-16 Mondal, R. N., Islam, M. Z. and Islam, M. S., (013b). ransient Heat and Fluid Flo through a Rotating Curved Rectangular Duct: he Case of Positive and Negative Rotation, Procedia Engineering, Vol. 56, pp Mondal, R. N., (006). Isothermal and Non-isothermal Flos through Curved Duct ith Square and Rectangular Cross-section, Ph.D. hesis, Department of Mechanical Engineering, Okayama University, Japan. Nandakumar, K. and Masliyah, J. H. (1986). Sirling Flo and Heat ransfer in Coiled and isted Pipes, Adv. ransport Process., Vol. 4, pp Norouzi, M., Kayhani, M. H., Shu, C., and Nobari, M. R. H., (010). Flo of Second-order Fluid in a Curved Duct ith Square Cross-section, Journal of Non-Netonian Fluid Mechanics, Vol. 165, pp Norouzi, M. and Biglari, N., (013). An Analytical Solution for Dean Flo in Curved Ducts ith Rectangular Cross Section, Physics of Fluids, Vol. 5, 05360, pp Selmi, M. and Namdakumar, K. (1999). Bifurcation Study of the Flo through Rotating Curved Ducts, Physics of Fluids, Vol. 11, pp Selmi, M. and Namdakumar, K. and Finlay W. H., (1994). A bifurcation study of viscous flo through a rotating curved duct, Journal of Fluid Mechanics, Vol. 6, pp Wang, L. and Yang,., (005). Periodic Oscillation in Curved Duct Flos, Physica D, Vol. 00, pp

15 AAM: Intern. J., Special Issue, No. (May 016) 51 Wang, L. Q. and Cheng, K.C., (1996). Flo ransitions and combined Free and Forced Convective Heat ransfer in Rotating Curved Channels: the Case of Positive Rotation, Physics of Fluids, Vol. 8, pp Wu, X. Y., Lai, S. D., Yamamoto, K., and Yanase, S. (013). Numerical Analysis of the Flo in a Curved Duct, Advanced Materials Research, Vol , pp Yamamoto, K., Yanase, S. and Alam, M. M. (1999). Flo through a Rotating Curved Duct ith Square Cross-section, J. Phys. Soc. Japan, Vol. 68, pp Yanase, S., Kaga, Y. and Daikai, R. (00). Laminar flo through a curved rectangular duct over a ide range of the aspect ratio, Fluid Dynamics Research, Vol. 31, pp