Approximate Analytic Solutions for Forced Convection Heat Transfer from a Shear-Flow Past a Rotating Cylinder

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1 Applied Mathematical Sciences, Vol. 2, 2008, no. 11, Approximate Analytic Solutions for Forced Convection Heat Transfer from a Shear-Flow Past a Rotating Cylinder K. Abdella 1 and P. S. Nalitolela Department of Mathematics, Trent University Peterborough, Ontario, K9J-7B8, Canada Abstract This paper presents analytic approximations to the thermal-fluid problem involving forced convective heat transfer from a rotating isothermal cylinder placed in a non-uniform stream shear flow. The approximations are obtained via series expansion of the scaled boundary layer equations in terms of an appropriate boundary layer variable. The resulting approximations are valid not only for small time but also for moderate and large times provided that the Reynolds number of the flow is sufficiently large. Mathematics Subject Classification: 76R05, 76D10 Keywords: Shear-flow, Forced convection, Rotating cylinder, Boundary layer, Viscous flow, Laminar flow 1 Introduction The steady and unsteady problems involving flow past a circular cylinder have been extensively studied numerically, and theoretically as well as experimentally see [1-2] for a comprehensive list of references). This is because, in addition to their direct applications, such flows exhibit the essential features commonly encountered in most industrial problems and therefore can serve as models which represent many fundamental fluid mechanics problems [3-11]. However, most of the focus has been on problems associated with uniform stream flows. These include the numerical and analytic studies involving natural convection heat transfer [12-16], free convection heat transfer [17-19], forced convection heat transfer [20-26], and mixed convection heat transfer 1 kabdella@trentu.ca

2 498 K. Abdella and P. S. Nalitolela [27-31]. Experimental investigations of heat transfer problems involving uniform stream flows include the first measurement reported in 1953 by Seban and Drake [32], and other studies later reported by Anderson and Saunders [33], Seban and [34], Farouk and Ball [35], and Ilgarubis, Ulinskas and Butkus [36]. In this paper, we consider the thermal-fluid flow problem involving forced convective heat transfer from a rotating circular cylinder placed in a nonuniform stream of shear flow. Previous studies on shear flow past a cylinder are numerous, including the theoretical investigations of [37-39], the numerical investigations of [40-47] and the experimental investigations of [48-52]. The main focus of these studies was on the effect of shear on the flow characteristics such as vortex shedding, boundary-layer separation and hydrodynamic forces. However, the focus of the present study is on the thermal-fluid problem of forced convective heat transfer from the cylinder which is assumed to have an isothermal surface. This problem has applications in a wide range of scientific and engineering fields including atmospheric flows, heat exchanger systems, and energy conservation [12,34,53,54]. The study is a direct extension of the recent work of Rohlf and D Alessio [55] which investigated the two-dimensional non-thermal problem of the unsteady shear flow of a viscous incompressible fluid past a rotating circular cylinder. Many theoretical and numerical studies have been carried out to investigate forced convective heat transfer involving cylinders. Investigations at low Reynolds numbers include the prediction of velocity and pressure variations, boundary layer studies, and isothermal field analysis [56-59]. Similar numerical and theoretical studies are also available for problems involving high Reynolds numbers [10,52-53,60-65]. In the present work an analytic approximation is obtained to describe the heat transfer process via a series expansion of the flow variable in 8t terms of a boundary layer variable λ = where t measures time and Re Re represents the Reynolds number. The analytic approximations are therefore valid not only for the initial stages of problems involving small and moderate Reynolds numbers but also for moderate and large times for sufficiently large Reynolds number problems. In the next section the governing equations and the associated boundary conditions are presented where a variable transformation is introduced in order to simplify the geometry of the problem. The flow variables are then scaled with respect to the boundary layer variable λ. In section 3 the boundary value problems corresponding to various orders of approximations are derived and solved. Results and discussions are presented in section 4 and concluding remarks are presented in section 5.

3 On forced convection Governing Equations Consider the problem of forced convection heat transfer from an unsteady flow past a circular cylinder of radius a centred at the origin and rotating at an angular velocity of Ω 0. The flow is assumed to be viscous and incompressible. It is also assumed that the flow remains laminar and two-dimensional for all times and for all parameter values considered in this paper. The surface of the cylinder is kept at a constant temperature T 0 while the approaching stream with linear velocity profile, Uy) = ky U 0 is kept at a constant temperature T as depicted in Figure 1 where x and y are the usual Cartesian coordinates. The temperature difference δt = T 0 T is assumed to be positive, giving rise to the buoyancy force and inducing fluid motion. Figure 1: Schematics of the physical problem Applying the Boussinesq approximation and neglecting the effects of viscous dissipation and radiation the governing equations are given by the equations of motion and the energy equation: ζ t + u ζ x + v ζ 2 ) ζ = ν y x + 2 ζ 1) 2 y 2 2 ) ψ ζ = x + 2 ψ 2) 2 y 2 T t + u T x + v T 2 ) T = κ y x + 2 T 3) 2 y 2

4 500 K. Abdella and P. S. Nalitolela where t is time, u = ψ ψ and v = are the velocity components in the x y x and y directions respectively, ζ = v x u is the vorticity, ψ is the stream y function, T is the temperature, ν is the kinematic viscosity and κ is the thermal diffusivity. Introducing the following non-dimensional quantities x = x a,y = y a,u = u U 0,v = v U 0,t = tu 0 a ψ = ψ,ζ = aζ au 0 equations 1-3 become,φ = T T 0 U 0 δt,v = v U 0,t= t U 0 a, ζ t ψ ζ y x + ψ ζ = 2 2 ) ζ x y Re x + 2 ζ 2 y 2 2 ) ψ ζ = x + 2 ψ 2 y 2 φ t ψ φ y x + ψ φ = 2 2 ) φ x y Pe x + 2 φ 2 y 2 4) 5) 6) where the primes have been omitted. Here the Reynolds and Peclet numbers are defined as Re = 2aU 0, Pe = RePr ν where Pr = ν is the Prandtl number. κ Because of the geometry of the problem and the existence of large velocity and temperature gradients near the cylinder s surface, it is convenient to use the polar coordinates r, θ) and the transformation ξ =lnr. Then in terms of the ξ,θ) coordinates, equations 4-6 become 2ξ ζ e = ψ ζ t θ ξ ψ ζ ξ θ + 2 Re e 2ξ 2 ) ψ ζ = ξ + 2 ψ 2 θ 2 2ξ φ e t = ψ φ θ ξ ψ φ ξ θ + 2 Pe 2 ) ζ ξ + 2 ζ 2 θ 2 2 ) φ ξ + 2 φ 2 θ 2 Note that the new coordinate system maps the surface of the cylinder to ξ =0 and the infinite region exterior to the cylinder to the semi-infinite rectangular strip ξ 0, 0 θ 2π. 7) 8) 9)

5 On forced convection 501 The boundary conditions on the surface of the cylinder for t>0 include the no-slip, the impermeability and isothermal conditions: ψ =0, ψ ξ =Ω, andφ =1, on ξ = 0 10) where Ω = aω 0 is a non-dimensional angular velocity. The free stream conditions far away from the cylinder surface are U 0 [55]: φ 0, ζ K, and ψ K 4 e2ξ + Ve ξ sinθ) K 4 e2ξ cos 2θ as ξ 11) where K = ak is a dimensionless shear parameter representing the transverse U 0 velocity gradient of the approaching stream and V is the dimensionless centreline velocity taking on the values 0, 1 or -1 with V = 1 representing a centre-line velocity moving from right to left. Since the flow variables must remain periodic with respect to θ, the following periodicity conditions are also invoked: ζξ,θ,t) =ζξ,θ+2π, t), ψξ,θ,t) =ψξ,θ+2π, t), and φξ,θ,t) =φξ,θ+2π, t). 12) Note that, at the cylinder s surface there are two boundary conditions for ψ while there are none for ζ. Hence ψ is over-determined while ζ is underdetermined. This can be resolved by applying Green s Second identity which is given by D g 2 h h 2 g ) dxdy = C g h n h g ) ds n where g and h are twice differentiable functions in the region D, C is the closed curve representing the boundary of D and represents the normal derivative. n Then, using ψ for g and the harmonic functions e mξ sinmθ) and e mξ cosmθ) for h in equation 11, we obtain the following global integral conditions: 1 2π e 2 m)ξ ζ sinmθ)dθdξ =2Vδ 1,m, m =1, 2,... 13) π π 2π 0 0 e 2 m)ξ ζ cosmθ)dθdξ = Kδ 2,m, m =1, 2,... 14) 1 ξ 2π e 2ξ ζdθdξ = Ke 2ξ 2Ω. 15) π 0 0

6 502 K. Abdella and P. S. Nalitolela where δ i,j is the Kronecker delta function which is zero when i j and 1 when i = j. In essence these integral conditions convert the surface and the free stream boundary conditions into conditions that are valid throughout the entire domain of the problem. Finally, since the fluid is initially at rest the initial conditions for ζ and ψ are: ζξ,θ,t =0)=0, ζξ,θ,t =0)=0. 16) However, the temperature on the surface of the cylinder is assumed to increase suddenly to T 0 at t = 0 so that the initial temperature distribution is given by: { 1 if ξ =0 φξ,θ,t =0)= 0 if ξ 0. 17) It should be noted that this initial distribution is singular and results in a thin boundary-layer region close to the surface of the cylinder. 3 Approximate analytic solutions Due to the nonlinearity of the governing equations, it is not possible to obtain analytical solutions valid for all time and all parameter values. However, approximations for the early development of the flow and the heat transfer process may be obtained via series expansion in terms of an appropriate boundary layer parameter. As mentioned above, the structure of the flow field and heat transfer process in the initial stages of the flow is characterized by a thin boundary layer region near the cylinder surface. By examining the dominant terms of the initial solutions, it can be shown that the boundary layer thickness is given by 8t λ = Pe. 18) Since the variable λ measures the diffusive growth of the boundary-layer structure, it can be used to rescale the space coordinate ζ and the flow variables via the following changes of variables: ξ = λz, ψ = λψ,,ζ = ω λ,,φ = Φ λ 19) This change of variables stretches the thin boundary-layer and rescales the flow variables and removes the initial singularity. The scaling of the vorticity and the stream functions has successfully been used in other studies including Dennis and Staniforth [66], D Alessio et al. [67] and Badr and Dennis [68].

7 On forced convection 503 In terms of the boundary-layer variables, equations 7-9 become: 2 ω z Pr e2λz z ω ) z + ω Ψ ω θ z Ψ z e 2λz 2 ) Ψ ω = z + 2 Ψ 2 λ2 θ 2 2 Φ z 2 +2e2λz z Φ ) z +Φ = 2 ω λe2λz Pr λ 2 ω λ2 θ Reλ λz Φ =2λe λ 2 Φ λ2 θ Peλ2 2 2 Ψ θ Φ z Ψ z ) ω θ 20) 21) ) Φ. 22) θ Note that λ is a small parameter if Re is large or if t is small. Therefore, analytic approximations of these equation may be obtained via a series expansion of the variables in terms of λ. However, D Alessio et al. [26], Rohlf and D Alessio [55] and others have used double series expansion in both λ and t instead of a single expansion in λ. As pointed out by D Alessio et al. [26], the double series expansion is necessary in order to avoid the analytical complications arising from the single λ expansions. However, they also point out that, while the double series expansion simplifies the analytic calculations, it is more advantageous to have the single expansion in λ than the double expansion since the validity of the single expansion in λ would only require that λ be small which can be achieved for moderate and large times provided that Re is sufficiently large. However, the double series expansion is valid only for small times. Therefore, in this paper we use the following single series expansions in λ which are valid not only for small times but also for large times for problems with large Re. Ψ=Ψ 0 + λψ 1 + λ 2 Ψ ) ω = ω 0 + λω 1 + λ 2 ω ) Φ=Φ 0 + λφ 1 + λ 2 Φ ) Substituting these expansions into the boundary layer equations and equating like powers of λ results in a hierarchy of boundary value problems for the expansion coefficients. 3.1 Linear approximation for all values of Pr In this section we obtain the O1) and Oλ) approximations valid for all values of Pr.

8 504 K. Abdella and P. S. Nalitolela The O1) approximation Substituting the expansions into equations and into the boundary and integral conditions and collecting the leading order terms yields the following boundary value problem for the coefficients Ψ 0, ω 0 and Φ 0 : 2 Φ 0 z +2 z Φ ) 0 2 z +Φ 0 = 0 26) 2 ω 0 z Pr ) z ω 0 z + ω 0 = 0 27) 2 Ψ 0 z 2 = ω 0 28) subject to Ψ 0 Ψ 0 =0, z =Ω, and Φ 0 =0 on z = 0 29) Φ 0 0 and ω 0 0 as z 30) 1 2π ω 0 sinmθ)dθdz =2Vδ 1,m, m =1, 2,... 31) π π ω 0 cosmθ)dθdz = Kδ 2,m, m =1, 2,... 32) π ξ 2π ω 0 dθdz = K 2Ω. 33) π 0 0 Note that the free stream condition on Ψ 0 will not be applied here. Instead, the integral conditions are applied to appropriately capture the dynamic behaviour of the evolving boundary layer structure near the cylinder surface where vorticity is generated. A similar approach was used by D Allesio et al. [26] and others. It is clear that Φ 0 = 0 and ω 0 = A 0 θ)fe fz)2 34) where f = 1 satisfies the boundary value problem in which A 0 θ) would Pr be determined from the integral conditions. Applying the integral conditions then yields A 0 θ) = 2 π 2V sin θ K cos 2θ + K 2Ω 2 ) ). 35) Then integrating equation 28 gives: ) π 1 Ψ 0 =Ωz + A 0 θ) zerffz)+ e fz)2 2 f π 36)

9 On forced convection The Oλ) approximation The first order approximation obtained by collecting Oλ) terms leads to the following boundary value problems: 2 Φ 1 z +2z Φ 1 2 z 2 ω 1 z +2f 2 z ω 1 2 z = 4z 2 Φ 0 z 4zΦ 0 37) = 4f 2 z 2 ω 0 z 4zf2 ω 0 38) 2 Ψ 1 z 2 = 2zω 0 + ω 1 39) subject to Ψ 1 Ψ 1 =0, z =0, and Φ 1 =1 on z = 0 40) Φ 1 0 and ω 1 K as z 41) 1 2π ω 1 +2 m)zω 0 ) sinmθ)dθdz =0, m =1, 2,... 42) π π ω 1 +2 m)zω 0 ) cosmθ)dθdz =0, m =1, 2,... 43) π ξ 2π ω 1 +2zω 0 ) dθdz =2Kz. 44) π 0 0 Solving equation 37 with respect to the Φ 1 boundary conditions yields Φ 1 = erfcz). 45) Similarly, solving equation 38 subject to the free-stream condition on ω 1 gives: π ω 1 = K + A 1 θ)erfcfz) A 0 θ) 4 erfcfz) 1 2 A 0θ)e fz)2 ) 2f 3 z 3 + fz ). 46) Applying the integral condition given by equations then gives the function A 1 θ): A 1 θ) =2V sin θ 2K cos 2θ. 47) Finally, integrating equation 39) subject to the surface boundary condition for Ψ 1 yields: Ψ 1 = Kz f F mθ) erffz) 2f 2 z 2 erfcfz) ) ze fz)2 2 8f π F pθ)+ za 1θ) f π, 48) where F p θ) = πa 0 θ)+4a 1 θ) and F m θ) = πa 0 θ) 4A 1 θ). 49)

10 506 K. Abdella and P. S. Nalitolela 3.2 Higher order approximations for Pr=1 The equations describing the second and higher order approximations turn out to be analytically intractable. Therefore, we assume that Pr=1 for second and higher order approximations The Oλ 2 ) approximation The second order approximation is obtained by collecting the Oλ 2 ) terms resulting in the following boundary value problems: 2 Φ 2 z +2z Φ 2 2 z 2Φ 2 = 4z 2 z Φ 0 z +Φ 0 + Φ ) 1 z ) Ψ0 Pe 2 θ 2 ω 2 z 2 +2f 2 z ω 2 z 2f 2 ω 2 = 4f 2 z 2 Re 2 Ψ0 θ Φ 0 z Ψ 0 z Φ 0 θ z ω 0 z + ω 0 + ω 1 z ) ω 0 z Ψ 0 z ω 0 θ 2 Φ 0 θ 2 50) ) 2 ω 0 θ 2 51) 2 Ψ 2 z 2 = 2z 2 ω 0 +2zω 1 + ω 2 2 Ψ 0 θ 2 52) subject to Ψ 2 =0, Ψ 2 z =0, and Φ 2 =0 on z = 0 53) 1 π 1 π 2π 0 0 2π 0 0 ω 2 +2 m)zω 1 + ω 2 +2 m)zω ξ 2π π 0 0 Φ 2 0 and ω 2 0 as z 54) ) 2 m)2 z 2 ω 0 sinmθ)dθdz =0, m =1, 2, ) ) 2 m)2 z 2 ω 0 cosmθ)dθdz =0, m =1, 2, ) ω2 +2zω 1 +2z 2 ω 0 ) dθdz =2Kz 2. 57) Solving equation 50 with f = 1 and with respect to the Φ 2 boundary conditions yields Φ 2 = 1 2 zerfcz) z2 e z2 π. 58)

11 On forced convection 507 Similarly, solving equation 51 with f = 1 subject to the free-stream condition on ω 2, gives: ω 2 = A 2 θ)z + A 3 θ)e z2 + πzerfz)) 1 2 π A 1θ)2z 2 e z2 πzerfz)) 59) A 0e z2 9z 2 2z 4 +12z 6) πzerfz) 2ReΩA 0 θ) 3A 0 θ) 4A 0θ) ) ReA 0θ)A 0 θ) 6πzerf 2 z)+8ze z2 + πerfz)3 6z 2 )e z2 6ze 2z2) where A 3 θ) = 1 2 π 2A 2θ)+A 1 θ))+ 1 4A 0 16 θ)+3a 0θ) ReA 0 θ)2ω + πa 0 θ)) ) 60) Applying the integral condition given by equations then gives the function A 2 θ): 4 A 2 θ) =a 0 + a i siniθ)+b i cosiθ)) 61) where i=1 a 0 =0, a 1 = V, a 2 = Re 6πV 2 +3πK 2 4KΩ+4V 2 +2K 2 ), a 3 =0, 6π 62) 3πK +2K 2Ω, b 2 =3K, 3π 3π +2 b 3 =ReVK 2π, b 4 =0. Finally, integrating equation 52) subject to the surface boundary condition for Ψ 2 yields: a 4 = ReK2 3π +2), b 1 = V Re 6π Ψ 2 = A 4 θ)z + A 5 θ)+k z3 3 + A 3θ) 192 π z3 A 2 θ)+ A 1θ) ) πz 2 +1)e z2 + πzerfz)32z ) πz πzerfz) 48z 2 e z2 + ) πzerfz)) π π 192 ReA 0θ)erfz) 2 πz 3 erfz) 3 ) πzerfz) 11e z2 +4z 2 e z2 4 π 96 ReA 0θ) ze 2z2 + 32πerf ) 2z) + 63) π 192 za 0θ) erfz)10z 2 9) 16 πz 2) z2 A 0 θ)e z2 4z 2 +3 ) + 1 πerfz)2z 96 ΩReA 0 θ)z 2 3) + 2e z2 z 2 2) 1 16 za 0 θ) πerfz)f1 ) z)+2ze z2 2) where the functions A 4 θ) and A 5 θ) determined by applying the surface boundary conditions are given by A 4 θ) = ReA 0 θ)a 0θ), A 5 θ) = A 3θ) ReΩA 0 64) )

12 508 K. Abdella and P. S. Nalitolela 3.3 The Oλ 3 ) approximation The boundary value problems associated with the third order approximation turn out to be highly complicated. In particular, the ω 3 and therefore the Ψ 3 equations are too complex to solve analytically. Therefore, we only consider the Φ 3 problem which is given by: 2 Φ 3 z 2 +2z Φ 3 z 4Φ 3 = 8 3 z3 subject to 2 Φ 1 θ 2 z Φ 0 Re 2 z +Φ 0 Ψ1 θ ) 4z 3 Φ 1 4z z Φ ) 2 z +Φ 2 Φ 0 z + Ψ 0 Φ 1 θ z Ψ 1 Φ 0 z θ Ψ 0 z 65) ) Φ 1 θ Φ 3 =0, on z =0, and Φ 3 0 as z. 66) Then, solving equation 63), we obtain the following expression for Φ 3 : φ 3 = A 6 θ)f 1 z)+a 7 θ)f 2 z) 3z erfz) π ze z ReA 0θ)zerfz)e z z2 erfz)+ 1 where and + πf 1 z) I 1 z) = z It can be shown that 0 I 1 z)erfcz)+i 3 z) erfs)e s2 ds, I 2 z) = π z3 e z 2 π z5 e z π ReA 0θ)e z π 2 erfcz)erfiz) ReA 0θ)erf 2 z) z 0 erfcs)e s2 ds, I 3 z) = z ) ) +2zI 2 e z2 erfcs) 2 e s2 ds 68) F 1 z) =1+2z 2, F 2 z) =2ze z2 + πf 1 z)erfz). 69) I 1 z) = 1 π z 2 2F 2 1, 1; 3 2, 2; z2 ), I 2 z) = where 2 F 2 is the generalized hypergeometric function. π 2 erfiz) I 1z) 70) Then applying the boundary conditions of equation 68, we obtain the following expressions for the functions A 6 θ) and A 7 θ): A 6 θ) = ReA 0θ) 12 π, A 7θ) = 51 π +8 3π)ReA 0 θ) + C 71) 96π where C = 0 erfc 2 s)e s2 ds = )

13 On forced convection Results and Discussion In this section, we test the validity of the analytic solution at the initial stages of a moderate Re flow and at the fully developed stage of a high Re flow. The test is carried out by comparing the results of the analytic solution with those of a high-resolution numerical solution obtained using a spectral finite difference scheme [55]. In this numerical approach, the flow variables as well as the temperature function are approximated in terms of truncated Fourier series expansion of N terms in the angular direction. The resulting 6N+3 two dimensional partial differential equations in time and in the radial variable are then integrated using a finite difference procedure. In order to gain insight into the patterns of the heat transfer rate, we compute the local Nusselt number and the average Nusselt number variations with respect to time and radial component. The local Nusselt number Nu and average Nusselt number Nu are respectively defined as: Nuθ, t) = 2 ) Φ 73) λ 2 z z=0 Nut) = 1 2π Nuθ, t)dθ. 74) 2π 0 Using the analytic approximation of the temperature function Φ and taking the derivative with respect to z and then finding its value at z = 0 yields: Nuθ, t) = 2 2 λ 1 ) ) 51 π +8 3π)ReA λ 2 π 2 λ θ + C) λ 3. 96π 75) Similarly, the average Nusselt number is given by: Nut) = 4 λ π +1 λ ) 17 π +16πC. 76) 2π 2 We begin with Figure 2 where the time evolution of the numerically simulated and the analytically determined local Nusselt s number are depicted for Re=1000, K =0.0, Ω=0.25 and t =0.01, 0.05, 0.1, 0.3, 0.5 and 1.0. As we can see from the figures, there is excellent agreement between the analytic approximations and the numerical simulations for all times presented. Note that, since Re is moderately large, the analytic solution is in good agreement even at t =1. This is because our expansion is valid in this limit as well. Similar results are obtained for Re=500 as depicted in Figure 3. However,

14 510 K. Abdella and P. S. Nalitolela we notice that the analytic accuracy is not as high in Figure 2 for t =1. A more pronounced difference is observed in Figure 4 where the Re=50 case is depicted. In this case, the analytic solution is not valid except for small time t since Re is also small. This is clearly evident as shown in Figures 4e) and 4f) where the t =0.5 and t =1.0 comparisons are presented. This is further quantified in Figure 5 where the percentage relative differences between the analytic and the numerical solutions are presented for various values of Re. As we can see from Figure 5a), for small times the percentage relative difference between the analytic and the numerical method is within 1% for all Re values considered. However, for t = 1.0 the percentage relative difference for smaller Re values is close to 10% resulting from the fact that, in these cases, λ is not small enough for the analytic assumptions to be valid. The effect of shear on the local Nusselt number can be seen in Figure 6 where the Re=1000 case with K =0.5 is presented. The integrated average Nusselt numbers are also compared in Figure 7 for various values of Re. Again we see that there is excellent agreement between the two solutions for small values of t. However, note that as t increases the two solutions tend to deviate from each other as expected. Another useful comparison is the surface vorticity distribution which is given by the following analytic expression ζ0,θ,t)= 1 λ A 0θ)+ K + A 1 θ) ) π 4 A 0θ) λa 3 θ). 77) Figures 8-10 depict the comparisons of the analytic with that of the numerical surface vorticity for K = 0.2, Ω = 0.4 and Re values of 100, 500 and 50 respectively. Again we notice that the analytic solution becomes less accurate as time increases and as Re decreases. It is also worth noting that the results are more accurate for the bottom half of the cylinder 180 θ 360) which is a result of slower flow at the bottom half. The dependency on the shear parameter K is demonstrated in Figure 11 for Re=1000 and Ω = 0.5. We note that the vorticity distribution becomes less symmetric with increasing shear. The figure also shows that shear enhances the surface vorticity in the upper half of the cylinder where there are faster moving fluid particles with the maximum occurring at the top tip of the cylinder. Again the analytic approximations are in excellent agreement with the numerically predicted solutions. The effect of the rotation parameter Ω is depicted in Figure 12 where a non-rotating and a rotating cylinder in a sheared flow is presented for Re=1000. As we can see, the effect of the rotation is to shift the vorticity distribution curve down, resulting in a highly enhanced rotation for the bottom half of the cylinder with

15 On forced convection 511 Figure 2: Local Nusselt Number comparison for Re=1000, K =0.0, Ω = 0.25, V = 1 at times a) t =0.01, b) t =0.05, c) t =0.1, d) t =0.3, e) 0.5, and f) t =1.0

16 512 K. Abdella and P. S. Nalitolela Figure 3: Local Nusselt Number comparison for Re=500, K =0.0, Ω = 0.25, V = 1 at times a) t =0.01, b) t =0.05, c) t =0.1, d) t =0.3, e) 0.5, and f) t =1.0

17 On forced convection 513 Figure 4: Local Nusselt Number comparison for Re=50, K = 0.0, Ω = 0.25, V = 1 at times a) t =0.01, b) t =0.05, c) t =0.1, d) t =0.3, e) 0.5, and f) t =1.0

18 514 K. Abdella and P. S. Nalitolela Figure 5: The percentage relative differences between numerical and analytical local Nusselt numbers at Re=10,1000, Re=1000, Re=500, Re=100 and Re=50, K =0.0, Ω = 0.25, V = 1 at times: a) t =0.01, b) t =0.2

19 On forced convection 515 Figure 6: Local Nusselt Number comparison for Re=1000, K =0.5, Ω = 0.25, V = 1 at times a) t =0.01, b) t =0.05, c) t =0.1, d) t =0.3, e) 0.5, and f) t =1.0

20 516 K. Abdella and P. S. Nalitolela Figure 7: Average Nusselt number comparison up to time t =3.0, K =0.1, Ω=0.25, V = 1 and a) Re=1000, b) Re=500, and c) Re=100.

21 On forced convection 517 Figure 8: Surface vorticity comparison for K = 0.2, Ω = 0.4, Re=1000 at times a) t =0.01, b) t =0.1, c) t =0.5, and d) t =1.0. its maximum at the bottom tip of the cylinder. Moreover, the surface vorticity in the vicinity of the front stagnation point is no longer zero. Finally, we present comparisons between the numerical and analytical drug coefficient C D t)) and the lift coefficient C L. The drag and lift coefficients can then be derived by computing the resultant surface forces in the x and y directions respectively which are obtained by integrating the pressure and frictional stresses on the surface. In terms of the θ, ξ) coordinate system the drag coefficient therefore becomes: C D t) = 2π 0 P 0 cosθ)dθ + 2 Re 2π 0 ζ 0 sin θdθ,

22 518 K. Abdella and P. S. Nalitolela Figure 9: Surface vorticity comparison for K =0.2, Ω = 0.4, Re=500 at times a) t =0.01, b) t =0.1, c) t =0.5, and d) t =1.0.

23 On forced convection 519 Figure 10: Surface vorticity comparison for K =0.2, Ω = 0.4, Re=50 at times a) t =0.01, b) t =0.1, c) t =0.5, and d) t =1.0.

24 520 K. Abdella and P. S. Nalitolela Figure 11: Surface vorticity comparison at Re=1000, t =0.1, Ω = 0.5, V =1, a) K =0.1 and b) K =1.0. Figure 12: Surface vorticity comparison at Re=1000, t =0.1, K =0.1, V =1, a) Ω = 0.0 and b) Ω = 1.0.

25 On forced convection 521 where P 0 surface pressure which satisfies P = P 0 θ ξ=0 θ = 2 Re ζ. θ ξ=0 Therefore, after integration, we obtain the following analytic expression for the drag coefficient: C D t) = 2V ) π 8+2πλ λ2. 78) λre 4 Similarly the following expression can be derived for the analytic approximation of the lift coefficient: C L t) = 2V π1 + λ) 3πK +2K 2Ω 3π ) π 2 λ 4V Ω K). 79) 3Re The comparison of the time evolution of the analytically approximated and the numerically simulated drug and lift coefficients for the Re value of 1000 and Ω = 0.25 is depicted in Figure 13. The figure shows that, while the lift coefficient decreases with shear and even becomes negative in time, its magnitude is greatly enhanced by the presence of shear. The magnitude of the drag coefficient is also slightly enhanced by the shear. This finding is consistent with the observations reported by Chew, Luo and Chen[43]. 5 Conclusion In this paper analytic approximations to the thermal-fluid problem involving forced convective heat transfer from a rotating isothermal cylinder placed in a non-uniform stream shear flow are presented. A convenient coordinate system is first introduced in order to simplify the geometry of the problem. The flow variables are then scaled with respect to the boundary layer parameter λ resulting in a set of boundary layer equations subject to appropriate initial and boundary conditions. The analytic approximations for the boundary value problem are obtained via a Fourier series expansion in terms of the boundary layer variable. The resulting approximations are valid not only for small time but also for moderate and large times provided that the Reynolds number of the flow is sufficiently large. ACKNOWLEDGEMENTS. K. Abdella s general research in Applied Mathematics and Mathematical Modelling is supported by the Natural Sciences and Engineering Research Council of Canada NSERC).

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