Natural convection heat transfer in an oscillating vertical cylinder

This paper studies the heat transfer analysis caused due to free convection in a vertically oscillating cylinder. Exact solutions are determined by applying the Laplace and finite Hankel transforms. Expressions for temperature distribution and velocity field corresponding to cosine and sine oscillations are obtained. The solutions that have been obtained for velocity are presented in the forms of transient and post-transient solutions. Moreover, these solutions satisfy both the governing differential equation and all imposed initial and boundary conditions. Numerical computations and graphical illustrations are used in order to study the effects of Prandtl and Grashof numbers on velocity and temperature for various times. The transient solutions for both cosine and sine oscillations are also computed in tables. It is found that, the transient solutions are of considerable interest up to the times t = 15 for cosine oscillations and t = 1.75 for sine oscillations. After these moments, the transient solutions can be neglected and, the fluid moves according with the post-transient solutions.


Introduction
Energy transfer due to convection is of great importance and arises in many physical situations [1]. Amongst the three different types of convections (free, forced, mixed), mixed convection is less investigated as compare to the other two types. When forced and free convections occur together, mixed convection induces. This phenomenon is usually seen in the channel flow due to heating or cooling of the channel walls. Energy transfer due to mixed convection is studied under different physical situations with various boundary constraints. For example, Fan, et al. [2] analyzed energy transfer because of mixed convection in a horizontal channel filled with nanofluids. Aaiza et al. [3,4] examined energy transfer due to mixed convection in channel flow for ferrofluid and nanofluid respectively. Aaiza et al. [4], further pointed out that in mixed convection energy transfer, the buoyancy force is responsible for free convection and at least one of the two, non-homogeneous boundary conditions on velocity or external pressure gradient results forced convection. Amongst the important studies on mixed convection energy transfer, we include here the attempts those made by Kumari et al. [5], Tiwari and Das a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 [6], Chamkha et al. [7], Sheikhzadeh et al. [8], Prasad et al. [9], Hasnain et al. [10] and Ganapathirao et al. [11]. However, most of these studies on energy transfer were focused in simple geometrical configurations.
In contrast, the energy transfer due to convection flow in stationary or moving cylinder has numerous applications in engineering and geophysics, such as nuclear reactor cooling system and underground energy transport and hence attracted the attention of many researchers. However, this area of research is not as much studied as flow over a flat plate, channel flow, flow over sheets etc. Most probably, it is due to complex nature of these problems. Most of these studies were investigated in the absence of heat or heat and mass transfer, see for example the work of Fetecau et al. [12][13][14], Jamil and Fetecau [15], Rubab et al. [16] and Abdulhameed et al. [17]. Such problems have also applications in biomagnetic fluid dynamics, see for example Sharma et al. [18], and Nehad et al. [19], where they used cylindrical coordinates and investigated the blood flow in cylindrical shaped arteries. Khan et al. [20][21][22], used cylindrical coordinates and investigated heat or heat and mass transfer in converging and diverging channels.
Free convection in cylindrical shape geometry is investigated in several earlier studies such as Goldstein and Briggs [23], in 1964 studied transient free convection over vertical plates and circular cylinders. Bottemanne [24] provided experimental results for pure and simultaneous heat and mass transfer by free convection over a vertical cylinder. Chen and Yuh [25] studied combined heat and mass transfer in free convection flow along a vertical cylinder. Some other related studies on free convection flow in a cylinder are given in [26][27][28][29][30]. In recent investigations, Deka et al. [31] analyzed transient free convection flow past an accelerated vertical cylinder in a rotating fluid whereas Deka and Paul [32] investigated unsteady one-dimensional free convection flow over an infinite moving vertical cylinder in the presence of thermal stratification. They used Laplace transform technique and obtained the exact solutions, expressed them in the forms of complicated integrals. Other interesting problems are studied in references [33][34][35][36][37][38][39].
The aim of this paper is to study the energy transfer in a vertically oscillating cylinder due to natural convection. Exact solutions are obtained by means of Laplace and Hankel transforms for velocity and temperature. The transient solutions for both cosine and sine oscillations of the cylinder are computed in tabular forms. Results of Prandtl and Grashof numbers for different times are shown in graphs and discussed.

Mathematical formulation and solution of the problem
Let us consider transient free convection flow of an incompressible viscous fluid in an infinite vertical cylinder of radius r 0 . The z-axis is considered along the axis of cylinder in vertical upward direction and the radial coordinate r is taken normal to it. Initially at time t 0, it is assumed that the cylinder is at rest and the cylinder and fluid are at the same temperature T 1 . After time t = 0, the cylinder begins to oscillate along its axis and induces the motion in the fluid with velocity U 0 H(t)exp(iωt), where U 0 is the characteristic velocity, H(t) is the unit step function and ω is the frequency of oscillation. At the same time, the cylinder temperature raised to T w which is thereafter maintained constant (Fig 1). We assume that the velocity and temperature are the function of r and t only. For such a flow, the constraint of incompressibility is identically satisfied. It is also assumed that all the fluid properties are constant except for the density in the buoyancy term, which is given by the usual Boussinesq's approximation. In this paper, we have proposed to obtain analytical solutions for the temperature and velocity fields, in the negligible dissipation hypothesis. Under these assumptions, a well-defined problem is modeled in terms of the following partial differential equations: with appropriate initial and boundary conditions: Introducing the following dimensionless variables: the governing Eqs (1)-(4) reduce to (dropping out the star notation): where Gr ¼ Calculation for temperature Applying the Laplace transform to Eqs (7), (9) 2 and using the initial condition (8) 2 , we obtain the following transformed problem: where " yðr; qÞ is the Laplace transform of the function θ(r,t) and q is the transform variable. Applying the finite Hankel transform of order zero, to Eq (10), and using condition (11), we obtain: where " y H ðr n ; qÞ ¼ Taking inverse Laplace transform of Eq (12), we obtain: Taking inverse Hankel transform, we obtain In order to study the heat transfer from the cylinder surface to the fluid, we determine the Nusselt number. This dimensionless number is defined as ratio of the convective heat transfer to the conductive heat transfer and is given by Calculation for velocity Applying the Laplace transform to Eqs (6), (9) 1 , and using the initial condition (8) 1 , we obtain q" uðr; qÞ ¼ @ 2 " uðr; qÞ @r 2 þ 1 r @" uðr; qÞ @r þ Gr " yðr; qÞ; ð16Þ Natural convection in oscillating vertical cylinder Applying finite Hankel transform to Eq (16) and using Eqs (12), (17), we have where " u H ðr n ; qÞ ¼ Z 1 0 r" u H ðr; qÞJ 0 ðrr n Þdr is the finite Hankel transform of the function " uðr; qÞ: We consider where Applying the inverse Laplace transform to Eqs (19), (20), (21) and (22), we obtain  ðPr À 1Þ þ expðÀ r 2 n tÞ À Pr exp À r 2 n t Pr J 0 ðrr n Þ r 3 n J 1 ðr n Þ ! À À i 2ocosðotÞb 1 ðrÞ þ 2o 2 sinðotÞa 1 ðrÞ À 2o

Sine oscillation
For sine oscillations of cylinder, the velocity field is given as:

Numerical results and discussions
In order to obtain some information on the fluid flow parameters and heat transfer, we have made numerical simulations using Mathcad software. The obtained results are presented in the graphs from Figs 2-5. Geometry of the problem is given in Fig 1. We were interested, to analyze the influence of the Prandtl number on the temperature, Nusselt number and on fluid velocity. Also, the influence of the Grashof number on the fluid velocity was studied.  The decreasing of the transient solution u ct (r,t), given by Eq (31), is shown in the Table 1, for Gr = 5, Pr = 7 and ω = 0.449. It is observed from Table 1 that, for t = 15 the transient solution u ct (r,t), is of order 10 −6 , therefore, after this moment the transient solution can be neglected and, the fluid moves according with the post-transient solution.
Similarly, in Table 2 is presented the decreasing with time t of the transient solution corresponding to the sine oscillations of the cylinder, given by Eq (34). Comparing with the cosine oscillations, it is seen that, the critical time at which the transient solution is of order 10 −6 is lower for sine oscillations. For the same values of the system parameters, the transient solution for sine oscillations can be neglected after the value t = 1.75.

Conclusions
The problem of heat transfer due to free convection in an oscillating vertical cylinder is studied. Exact solutions for temperature and velocity are determined by applying the Laplace and