International Journal of Engineering Mathematics

Volume 2017 (2017), Article ID 8410691, 13 pages

https://doi.org/10.1155/2017/8410691

## Unsteady Natural Convection Flow past an Infinite Cylinder with Thermal and Mass Stratification

^{1}Department of Mathematics, Cotton College State University, Guwahati 781001, India^{2}Department of Mathematics, Gauhati University, Guwahati 781014, India

Correspondence should be addressed to Ashish Paul

Received 29 June 2016; Revised 8 November 2016; Accepted 6 December 2016; Published 11 January 2017

Academic Editor: Alberto Cardona

Copyright © 2017 Ashish Paul and Rudra Kanta Deka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents an analytical solution of unsteady one-dimensional free convection flow past an infinite vertical circular cylinder in a stratified fluid medium. The dimensionless coupled linear governing partial differential equations are solved by Laplace transform technique for unit Prandtl number and Schmidt number. Effects of various physical parameters are presented with graphs. Numerical values of boundary layer thickness for different parameters are presented in table. Due to the effects of thermal and mass stratifications, the velocity, temperature, and skin friction, Nusselt number shows oscillatory behaviour at smaller times and then reaches steady state at larger times.

#### 1. Introduction

Natural convection flows with heat and mass stratification are frequently encountered in nature. These types of problem over vertical cylinder have wide range of applications in the field of science and technology such as startup of chemical reactors and emergency cooling of nuclear fuel elements. In glass and polymer industries, hot filaments are considered as vertical cylinder and cooled as they pass through the surrounding environment. Free convective flows driven by temperature and concentration difference have been studied extensively. When both the temperature and concentration differences occur simultaneously, the free convective flow can become quite complex.

Gebhart and Pera [1] analysed the steady combined buoyancy effects on vertical natural convection flows. Bottemanne [2] studied the combined effect of heat and mass transfer in the steady laminar boundary layer of a vertical cylinder placed in still air for Pr = 0.71 and Sc = 0.63. Chen and Yuh [3] presented a numerical study of steady heat and mass transfer processes near cylinder with uniform wall heat and mass fluxes and wall temperature. Their study covered a wide range of radii and Prandtl numbers. Velusamy and Garg [4] studied natural convection adjacent to a heat generating vertical cylinder. Ganesan and Rani [5] presented a numerical solution for the transient natural convection flow over a vertical cylinder under the combined buoyancy effect of heat and mass transfer. Numerical analysis of two-dimensional unsteady natural convective flow past semi-infinite vertical cylinder with heat and mass transfer under different physical situations was studied by Ganesan and Loganathan [6–8]. Rani [9] presented a numerical study on transient natural convection along vertical cylinder with variable surface temperature and mass diffusion. Devakar et al. [10] presented closed form solution for Poiseuille flow, Couette flow, and generalized Couette flows of an incompressible couple stress fluid between two concentric circular cylinders with slip boundary condition. Recently, Deka et al. [11] presented the analytical investigation of one-dimensional unsteady natural convection flow past an infinite vertical cylinder with heat and mass transfer under the effect of constant heat flux at the surface of the cylinder. They have shown that the velocity and temperature increase unboundedly with time, while the concentration approaches steady state at larger times.

In recent times many researchers have shown interests in the study of stratification effects on transient natural convective flows along vertical bodies under various physical situations. Takhar et al. [12] presented a numerical study of natural convection boundary layer flow over a continuously moving vertical surface immersed in a thermally stratified medium by an implicit finite difference scheme. Again, Takhar et al. [13] investigated the natural convection flow past a vertical cylinder embedded in a thermally stratified high-porosity medium. They solved the coupled nonlinear partial differential equations by finite difference as well as perturbation technique and found that separation of flow occurs for some values of stratification parameter. Loganathan and Ganesan [14] presented a numerical study of free convective flow of a viscous incompressible fluid past a moving, semi-infinite vertical cylinder with constant temperature and mass diffusion in a thermally stratified medium by employing a finite difference scheme of Crank-Nicolson type.

Shapiro and Fedorovish [15] presented analytical solution of one-dimensional laminar natural convection along an infinite vertical plate by introducing the pressure work term and the ambient thermal stratification in the thermodynamic energy equation for the case of unit Prandtl number. They have shown that thermal stratification provides a negative feedback mechanism: warm fluid rises, expands, and cools relative to the environment, whereas cool fluid subsides, compresses, and warms relative to the environments. Later on, Shapiro and Fedorovich [16] carried out study on natural convection in a stably stratified fluid along vertical plates and circular cylinders seeking solutions in the form of harmonic oscillators.

In recent times, the effect of double stratifications, namely, thermal stratification and mass stratification, has been considered by different researchers. For example, Cheng [17] studied the coupled heat and mass transfer by natural convection near a vertical wavy surface in a non-Newtonian fluid saturated porous medium with thermal and mass stratification and obtained solutions by collocation method. Recently, Srinivasacharya and Reddy [18], Srinivasacharya and RamReddy [19], Rathish Kumar and Krishna Murthy [20], and Neagu [21] have investigated numerically on flow past plates and wavy surfaces taking double stratifications into account.

Deka and Paul [22, 23] presented the analytical investigation of transient free convection flow past an infinite moving vertical cylinder in a stably stratified fluid including thermal stratification by employing Laplace transform technique. Very recently Deka and Paul [24] presented analytical investigation to study the effects of thermal stratification and mass stratification on natural convection heat and mass transfer over moving vertical cylinder. This motivates undertaking this study. This paper presents an analytical investigation of one-dimensional free convective flow past a stationary infinite vertical cylinder with combined effects of thermal and mass stratification. The unsteady nondimensional governing linear equations are solved by Laplace transform technique for the case of unit Prandtl number and unit Schmidt number. Solutions are presented in closed form and this is always necessary for validating numerical models. Also solutions of unsteady state for larger time are compared with the solutions of steady state.

#### 2. Mathematical Analysis

Consider an unsteady, laminar, and incompressible viscous flow past an infinite vertical cylinder of radius with constant temperature and concentration in presence of thermal and mass stratification. The -axis of the cylinder is taken vertically upward along the axis of the cylinder and the radial coordinate is taken normal to the cylinder as shown in Figure 1. The physical model and coordinate system of the flow problem is shown in Figure 1. Upon commencement of the transient, we consider the fluid moving up from the leading edge () parallel to the cylinder as a wave, in front of which the velocity, temperature, and concentration are only functions of the time and the radial distance from the cylinder. Behind the wave there must be a dependence on the vertical coordinate, . The basic premise in this work is that convective effects will begin at a position, , as soon as fluid which was initially located at the leading edge rises to this position, regardless of the distance, , away from the cylinder at which it first arrives. Since the surface temperature and concentration above the leading edge are uniform with , the temperature of the fluid and concentration may be assumed to be independent of . In addition, the vertical velocity, , must be independent of and from the continuity equation, the velocity normal to the plate is seen to be zero, except that the temperature and concentration of the ambient fluid are function of the vertical distance only. At time , the uniform temperature () and concentration () are specified at the surface of the cylinder. Viscous dissipation terms have been neglected. All derivatives in the direction parallel to the cylinder are zero, except and termed as thermal stratification and mass stratification, respectively. Here, and are the temperature and concentration of the undisturbed fluid. It is to be noted that initially the fluid may not be stratified, but upon commencement of the transient the fluid gets self-stratifications. Then following Boussinesq’s approximation, the one-dimensional equations for momentum, energy, and concentration are as follows:with initial and boundary conditions as