Thermal analysis of conducting dusty fluid flow in a porous medium over a stretching cylinder in the presence of non-uniform source/sink
- Pattasale Thippeswamy Manjunatha^{1},
- Bijjanal Jayanna Gireesha^{2}Email author and
- Ballajja Chandrappa Prasannakumara^{3}
https://doi.org/10.1186/s40712-014-0013-8
© Manjunatha et al.; licensee Springer. 2014
Received: 14 March 2014
Accepted: 4 August 2014
Published: 24 September 2014
Abstract
Background
The principle aim of the present investigation is to study the heat transfer analysis of steady two dimensional flow of conducting dusty fluid over a stretching cylinder immersed in a porous media under the influence of non-uniform source/sink.
Methods
Governing partial differential equations are reduced into coupled non-linear ordinary differential equations using suitable similarity transformations. The resulting system of equations are then solved Numerically with efficient Runge Kutta Fehlberg-45 Method.
Results
Graphical display of the obtained numerical solution is performed to illustrate the influence of various flow controlling parameters like curvature parameter, magnetic parameter, porous parameter, Prandtl number, heat source/sink parameter, fluid-particle interaction parameter on velocity and temperature distributions of both fluid and dust phases. The numerical results for the skin-friction coefficient and Nusselt number are also presented. Finally, the obtained numerical solutions are compared and found to be in good agreement with previously published results under special cases.
Conclusion
The velocity within the boundary layer in the case of cylinder is larger than the flat surface and both the magnitude of the skin friction coefficient and heat transfer rate at the surface are higher for cylinder when compared to that of flat plate.
Keywords
Background
The study of hydrodynamic flow and heat transfer over a stretching cylinder has gained considerable attention due to its applications in industries and important bearings on several technological processes like thermal design of buildings, electronic cooling, solar collectors, drilling operations, commercial refrigeration, geothermal power generation, float glass production, heat-treated materials traveling between a feed roll and a wind-up roll, aerodynamic extrusion of plastic sheets, glass fiber and paper production, cooling of an infinite metallic plate in a cooling bath, manufacturing of polymeric sheets, etc. Due to these practical and industrial applications, the problem of boundary layer analysis over a stretching solid surfaces has become an area of attention for scientists, engineers, and mathematicians as well. Flow over a cylinder is generally considered as two dimensional, when radius of the cylinder is large enough compared to the boundary layer thickness. To study the viscous fluid flow and heat transfer outside a hollow stretching cylinder has importance in extrusion processes. Using local non-similarity transformations, Chen and Mucoglu (1975) investigated the effects of mixed convection flow over a vertical slender cylinder due to thermal diffusion with prescribed wall temperature. Wang (1988) obtained the exact solution of viscous flow and heat transfer due to uniformly stretching cylinder. Kumari and Nath (2004) analyzed the effects of localized cooling/heating and injection/suction on the mixed convection flow on a thin vertical cylinder. Chang (2008) numerically investigated the flow and heat transfer characteristics of natural convection in a micropolar fluid flow along a vertical slender hollow circular cylinder with conduction effects.
The hydrodynamic flow and heat transfer in porous medium have become hot topics of research for quite a long time, which is reflected in number of articles being published. Aydin and Kaya (2013) studied MHD mixed convection of a viscous dissipating fluid about a vertical slender cylinder. Mukhopadhyay and Ishak (2012) presented a result on the distribution of a solute undergoing a first-order chemical reaction in an axisymmetric laminar boundary layer flow along a stretching cylinder with velocity slip condition at the boundary instead of no-slip condition. A steady laminar flow caused by a stretching cylinder immersed in an incompressible viscous fluid with prescribed surface heat flux was investigated by Bachok and Ishak (2010). Mukhopadhyay and Ishak (2012) studied an axisymmetric laminar boundary layer flow and mixed convection of a viscous incompressible fluid and heat transfer over a stretching cylinder embedded in a porous medium. Chauhan et al. (1961) have investigated magnetohydrodynamic slip flow and heat transfer in a porous medium over a stretching cylinder by homotophy analysis method. Further, various aspects of heat transfer over a stretching cylinder in different geometrical models have been studied widely by many researchers (Ashorynejad et al. 2013; Gorla et al. 2012; Khalili et al. 2010; Mohammadiun et al. 2013; Mukhopadhyay 2011; Munawar et al. 2012; Rashad et al. 2013; Shateyi and Marewo 2013; Wang 2011; Weidman and Weidman 2010).
The analysis of two-phase flows in which solid spherical particles are distributed in a fluid is important in areas like environmental pollution, smoke emission from vehicles, emission of effluents from industries, cooling effects of air conditioners, flying ash produced from thermal reactors and formation of raindrops, etc. On the basis of these applications, Saffman (1962) has formulated the basic equations of motion for fluid carrying small dust particles in which dust particles are uniformly distributed. Using the Saffman (1962) model, Soundalgekar and Gokhale (1984) studied the flow of a dusty gas past an impulsively started infinite vertical plate by employing an implicit finite difference technique. Datta and Mishra (1982) have analyzed the boundary layer flow of a dusty fluid over a semi-infinite flat plate. Later, Das et al. (1992) studied the flow of a dusty gas past a uniformly accelerated horizontal plate. Ganesan and Palani (2004) have obtained the numerical solution for an unsteady free convection flow of a dusty gas past a semi-infinite inclined plate with constant heat flux using an implicit finite difference method. Flow of a dusty gas past an impulsively started semi-infinite vertical plate was studied by Kulandaivel (2010). Vajravelu and Nayfeh (1992) examined hydromagnetic flow of a dusty fluid over a stretching sheet. Recently, Gireesha et al. (2011, 2012, 2013) have obtained the results for heat transfer analysis on dusty fluid flow due to linear stretching with various effects like nonuniform heat source or sink, radiation and viscous dissipation, etc. In this studies, they were analyzed by two types of heating processes namely, surface temperature and heat flux.
Different from our previous investigations, we extended the work to stretching cylinder. In the present paper, we try to investigate the flow and heat transfer of an electrically conducting dusty fluid over a stretching cylinder in porous media. The resulting nonlinear momentum and energy equations are simplified using similarity transformations. Numerical solutions have been developed for the velocity and temperature (PST and PHF). Graphical results for various values of the flow parameters are presented to gain thorough insight towards the physics of the problem. The results have possible technological applications in liquid-based systems involving stretchable materials.
Method
Flow analysis of the problem
where a prime denotes differentiation with respect to η.
Heat transfer analysis
where A^{∗} and B^{∗} are the coefficients of the space- and temperature-dependent internal heat generation/absorption. Here, we make a note that A^{∗}>0 and B^{∗}>0 corresponds to internal heat generation and that A^{∗}<0 and B^{∗}<0 corresponds to internal heat absorption. In this paper, we discussed two types of heating process namely prescribed surface temperature (PST) and prescribed heat flux (PHF case). Here, the prescribed surface temperature is defined as a quadratic function of z, while in the case of PHF, it is a the power law heat flux.
where T_{ w } and T_{ ∞ } denote the temperature at the wall and at large distance from the wall, respectively. A and D are positive constants.
where \(T-T_{\infty }=A\left (\frac {z}{l}\right)^{2}\theta (\eta)\) (PST case) and \(T_w-T_{\infty }=\frac {D}{k^{*}}\left (\frac {z}{l}\right)^{2}\sqrt {\frac {\nu }{c}}\) (PHF case).
One can observe that, if γ=0(a⇒∞), then, the present problem under consideration (S=0 and Q−0) reduces to boundary layer flow along a stretching flat plate considered by Gireesha et al. (2011) and if γ=0 and S=0, then, it reduces to the problem of Gireesha et al. (2012) with λ=0, i.e., in the absence of the ratio of free stream velocity parameter to stretching sheet parameter and that of Gireesha et al. (2013) with \(a=\frac {\pi }{2}\) (horizontal stretching sheet) in that paper.
Method of solution
In this method, the edge of the boundary layer η_{ ∞ } has been chosen as η=5, which is sufficient to achieve the far field boundary conditions asymptotically for all values of the parameters considered. A comprehensive numerical parametric computations have been carried out for various values of Curvature parameter, fluid particle interaction parameter, Prandtl number, heat source/sink parameter, fluid-particle interaction parameter on velocity, and temperature in both PST and PHF cases, and then, the results are reported in terms of graphs.
Results and discussion
The numerical solutions are presented through graphs for physical interpretation of the proposed study. For the validation of numerical results, we compared our results with previously published works with the absence of the curvature parameter and dust particles.
The values of f ^{ ′ ′ } (0), − θ ^{ ′ } (0), and θ (0) for various values A ^{ ∗ } , B ^{ ∗ } , S , N , γ , E c , and β
A ^{∗} | B ^{∗} | P r | Q | β | γ | S | −f"(0) | −θ^{ ′ }(0) | θ(0) |
---|---|---|---|---|---|---|---|---|---|
0.2 | 0.2 | 3.0 | 1.0 | 0.1 | 0.5 | 0.5 | 1.82157 | 1.81112 | 0.764171 |
0.4 | 1.82157 | 1.74642 | 0.7829819 | ||||||
0.6 | 1.82157 | 1.68173 | 0.801791 | ||||||
0.2 | 0.2 | 3.0 | 1.0 | 0.1 | 0.5 | 0.5 | 1.82157 | 1.81112 | 0.764171 |
0.4 | 1.82157 | 1.75226 | 0.778672 | ||||||
0.6 | 1.82157 | 1.69122 | 0.794108 | ||||||
0.2 | 0.2 | 3.0 | 1.0 | 0.1 | 0.5 | 0.5 | 1.82157 | 1.81112 | 0.764171 |
5.0 | 1.82157 | 2.29439 | 0.720947 | ||||||
7.0 | 1.82157 | 2.65646 | 0.705487 | ||||||
0.2 | 0.2 | 3.0 | 1.0 | 0.1 | 0.5 | 0.5 | 1.82157 | 1.81112 | 0.764171 |
1.5 | 1.97352 | 1.81919 | 0.757088 | ||||||
2.0 | 2.11317 | 1.82094 | 0.752149 | ||||||
0.2 | 0.2 | 3.0 | 1.0 | 0.0 | 0.5 | 0.5 | 1.80043 | 0.8622 | 1.041606 |
0.05 | 1.81322 | 1.40414 | 0.880658 | ||||||
0.1 | 1.82157 | 1.81112 | 0.764171 | ||||||
0.2 | 0.2 | 3.0 | 1.0 | 0.1 | 0.0 | 0.5 | 1.60946 | 1.61563 | 0.823394 |
0.2 | 1.69472 | 1.66004 | 0.806224 | ||||||
0.4 | 1.77971 | 1.7598 | 0.777936 | ||||||
0.6 | -1.863 | 1.86194 | 0.750953 | ||||||
0.2 | 0.2 | 3.0 | 1.0 | 0.1 | 0.5 | 0.0 | 1.65356 | 1.79255 | 0.774557 |
0.5 | 1.65356 | 1.79255 | 0.774557 | ||||||
1.0 | 1.97352 | 1.81919 | 0.757088 | ||||||
1.5 | 2.11317 | 1.82094 | 0.752149 |
Conclusions
In this study, a two-dimensional MHD flow and heat transfer of dusty fluid generated by stretching cylinder immersed in a porous medium is investigated. The governing equations are reduced to a set of non-linear ordinary differential equations by means of similarity transformations. Due to non-linearity, a numerical approach called Runge-Kutta-Felhberg 45 technique has been used to compute the values of velocity function and temperature field at different points of dynamic region. Comparison of obtained numerical results is made with previously published results for some special cases and found to be in a good agreement. The study is mainly focused on the effect of curvature of stretching cylinder, which is a very vital parameter affecting both flow and temperature fields. The effects of applied magnetic field, flow-dependent heat source/sink parameter, temperature-dependent heat source/sink parameter, Prandtl number, and fluid-particle interaction parameter are also taken into account. As expected, an increase of the curvature parameter leads to the increase in the velocity and temperature profiles. Also, an increase in the value of magnetic parameter leads to the decrease in the velocity boundary layer thickness. However, quite the opposite is true with the thermal boundary layer thickness in both PST and PHF cases. Comparing the results in Figures 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17, we see that the temperature of fluid phase is parallel to that of dust phase and also, the temperature of the fluid is always more than that of the dust phase. It is worth to mention that PHF boundary conditions are best suitable for cooling and PST for heating of the stretching cylinder. Finally, we conclude that both the magnitude of the skin friction coefficient and heat transfer rate at the surface are higher for cylinder when compared to that of flat plate.
Notation
(w,u) and (w_{ p },u_{ p }) are the velocity components of the fluid and dust particle phase. ν is the kinematic viscosity of fluid, ρ is the density of the fluid, N is the number density of the dust particles, K=6πμd is the Stoke’s constant, μ is the dynamic viscosity of the fluid, d is the radius of dust particles, B_{0} is the magnetic field, kp is the permeability of the porous medium, ρ_{ p } is is the density of dust phase, m is the mass of the dust particle, σ is the electric conductivity of the fluid, a is the radius of the cylinder, \(U_{w}(z) = b \frac {z}{l}\) is the stretching velocity, b>0 is the stretching rate, l is the reference length, ω is the density ratio, \(\gamma =\sqrt {\frac {l \nu }{a^{2}}}\) is the curvature parameter, \(A=\frac {\sigma B_{0}^{2} l}{\rho b}\) is the magnetic parameter, \(S=\frac {\nu l}{b k p}\) is the porous parameter, \(\tau =\frac {m}{K}\) is the relaxation time of the particle phase, \(\beta =\frac {l}{\tau b}\) is the fluid-particle interaction parameter, \(Pr=\frac {N}{\rho }\) is the relative density and \(l^{*}=\frac {mN}{\rho }\) is the mass concentration of dust particles, T and T_{ p } are the temperatures of fluid and dust particles, respectively, c_{ p } and c_{ m } are respectively specific heat of fluid and dust particles, τ_{ T } is the thermal equilibrium time i.e., the time required by a dust cloud to adjust its temperature to the fluid, τ_{ v } is the relaxation time of the dust particle, k^{∗} is the thermal conductivity, q^{′′′} is the space- and temperature-dependent internal heat generation/absorption, A^{∗} and B^{∗} are the coefficients of the space- and temperature-dependent internal heat generation/absorption, T_{ w } and T_{ ∞ } are the temperature at the wall and at large distance from the wall, respectively, A and D are positive constants, \(Pr=\frac {\mu c_{p}}{k^{*}}\) is the Prandtl number, \({Re}_x=\frac {U_{w} x}{\nu }\) is the local Reynold’s number, and \(Ec=\frac {bl^{2}}{A c_{p}}\) (PST case) and \(Ec=\frac {k^{*} l^{2} b^{\frac {3}{2}} }{A c_{p}}\) (PHF case) is the Eckert number.
Declarations
Acknowledgments
One of the co-authors (B.C.Prasannakumara) is thankful to the Vision Group of Science and Technology, Government of Karnataka, India, for supporting financially under Seed Money to Young Scientist for Research.
Authors’ Affiliations
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