Heat transfer and circular flow around a hydrodynamic turning point through a porous annular tube

Chinese Journal of Physics 61 (2019) 316–335

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Heat transfer and circular flow around a hydrodynamic turning point through a porous annular tube

T

⁎

Médard Marcus Nganbe IIa, Elisabeth Ngo Nyobeb, Jacques Honaa, , Elkana Pemhaa a

Applied Mechanics Laboratory, Faculty of Science, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon Department of Mathematics and Physical Science, National Advanced School of Engineering, University of Yaoundé I, P.O. Box 8390, Yaoundé, Cameroon

b

A R T IC LE I N F O

ABS TRA CT

Keywords: Hydrodynamic turning point Heat transfer Nonlinear two-point boundary-value problem Suction-driven circular flow Porous annulus

In this paper, the circular flow and heat transfer in a uniformly porous annular tube are investigated for the purpose of examining the influence of the fluid distribution and the effects due to different values of the control numbers on the variation of temperature. The attention is focused on the hydrodynamic turning point highlighted inside the annular conduct and formed by three solution branches that are the first branch Fr as well as two secondary branches S1 and S2. So, the flow structures and heat transfer patterns are analyzed through each branch of the turning point diagram. It is found that the opposite behaviors between the first branch Fr and the secondary branch of type S2 raised in the fluid distribution are also revealed in the heat transfer. The secondary branch of type S1 where a boundary layer type flow takes place by increasing the Reynolds number is also the seat of the thermal boundary layer as the Péclet number is close to the value of 10. In all the cases, the influence of the Péclet number is great compared to that of the Reynolds number on the variation of temperature.

1. Introduction The search for understanding the phenomenon of heat transfer is of great interest to scientists [1,2]. This transfer can occur by radiation, conduction and convection. There are even contexts where all these heat transfer modes are combined [3–5]. Heat transfer by radiation takes place when the matter is irradiated from a source of thermal energy. This mode of heat transfer is often associated with infrared radiation, because it is this part of spectrum that is preponderant in heat exchange. Whatever its temperature, a body emits radiation which is more or less intense depending on this temperature. The wavelength at which this radiation is emitted also depends on this temperature. Heat transfer by radiation is the only one which can occur without contact between the bodies that exchange heat, like the solar radiation arriving on the ground. In general, the distribution of energy by thermal radiation is often taken into account in the case of high temperature or simply in the absence of conduction and convection. When the matter is inert, the heat transfer is most often performed by conduction which therefore concerns solids and immobilized fluids. The conduction can be achieved within a single body or by contact between two bodies due to the gradual transmission of thermal vibration energy of the atoms around their equilibrium position. At the macroscopic level, the movement of matter or that of its components leads to the combination of conduction and convection in the energy distribution. In particular, the distribution of heat in a moving fluid is a very complex phenomenon, because its investigation requires a good mastery of the laws that govern the transport of mass and energy. There are many practical problems involving the coupling between the distribution of the mass of a fluid and the distribution of heat

⁎

Corresponding author. E-mail address: [email protected] (J. Hona).

https://doi.org/10.1016/j.cjph.2019.09.017 Received 11 January 2019; Received in revised form 6 June 2019; Accepted 18 September 2019 Available online 26 September 2019 0577-9073/ © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

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in a given geometrical domain. The subject is interesting because it leads to concurrently describe the mass and energy distributions within the geometrical domain. The boundaries of the geometrical domain may be impermeable or permeable. The case of a fluid flow with heat transfer in a permeable domain can serve as a model for the separation processes of binary mixtures by gaseous diffusion, transpiration cooling, respiration, filtration, hydraulic irrigation, biological transport processes, solar energy collectors, boundary layer separation, palm oil production and petroleum production. In most industrial settings, the geometrical domain of the problem is cylindrical or represents a channel of rectangular cross-section. In these two configurations, the scientists are interested in examining the influence of temperature on the momentum diffusion and vice versa [6–8]. If the physical properties of the fluid, such as the specific mass and the dynamic viscosity are sensitive to the variation of temperature, the equations for the momentum diffusion and energy distribution are fully coupled [9–11]. The other physical properties of the fluid, such as the thermal conductivity and the specific heat at constant pressure which do not intervene in the momentum diffusion, their respective influence is restricted to the energy equation [12,13] and their variations do not involve the full coupling of the equations governing the fluid flow and heat transfer. In particular, when the other physical properties are constant in a fluid flow, the dependence of the thermal conductivity on temperature gives rise to a partial coupling [12]. The partial coupling is also observed when the variation of temperature in a fluid flow does not affect any physical property of the fluid; this is the case where the physical properties of the fluid remain constant [13]. More precisely, the coupling is partial in the case where the Navier–Stokes equations which describe the fluid flow are solved independently of the parameters related to the energy equation, while the energy equation needs the solution of the Navier–Stokes equations to be solved. The Navier–Stokes equations are increasingly used to model the fluid flows in conducts. When a cylindrical circular conduct or a rectangular channel is modeled as a flow domain with porous walls, the problem becomes similar to the Berman flow [14] which has inspired many other studies [15–18]. The Navier–Stokes equations for viscous fluid flows in porous conducts are very complex, because they are highly nonlinear and sensitive to initial conditions and boundary conditions. They represent a system of differential equations used to solve several fluid flow problems giving rise to a variety of solutions. Most of these problems concern the control of the boundary layer. The boundary layer can be defined as the region of interface and of mutual influence between a moving fluid layer and a solid wall. Depending on the geometrical configuration of the problem and the type of solution sought, many methods for solving the Navier–Stokes equations modeling the fluid flows are recorded [11,19–21]. Regarding fluid flows between two porous walls, the pioneer work of Berman [14] provides an approach of seeking solutions called the similarity method that many authors have adopted [22–24]. In validating this similarity solution, investigators have often relied on numerical simulations. Extensive experimental verifications and numerous laboratory experiments have also been accomplished [25–27]. The similarity method is also used in the current study pertaining to a circular flow around a hydrodynamic turning point in an annular pipe composed of two coaxial porous cylinders fixed at different temperatures. It is more interesting and a novelty to study the heat transfer in a circular flow, because most flow problems relative to the cylindrical circular conduct concern the axial or longitudinal flow [13,17,28–30]. So, the present work is never performed elsewhere and represents a motivation for further exploration about the fluid flows and the energy transfer through circular cylindrical conducts. The present study especially deals with the investigation of the temperature behavior coupled to the flow structures related to the vorticity transport when a viscous fluid rotates between two coaxial porous cylinders kept at different temperatures. The suction-driven circular flow in the current context is the mass withdrawal process with respect to the core of the annular conduct as an incompressible viscous fluid rotates between the walls of two coaxial porous cylinders fixed at different temperatures. The main purpose is to investigate the behavior of temperature following the fluid distribution within the porous annulus. To reach this objective, Sections 2 and 3 deal with the mathematical model of the problem and the description of the similarity method, respectively. The analytical solution found for low values of the control numbers of the problem is presented in Section 4. The flow structures and temperature profiles are presented and discussed in Section 5. A final summary of the study is devoted to Section 6. 2. Mathematical model The circular flow of the viscous incompressible fluid and heat distribution develop within an annular area formed by two coaxial porous cylinders fixed at different temperatures. The geometry of the problem is described by means of a cylindrical polar coordinate system (r*, θ, z*), where r*, θ and z* denote the radial, the angular and the axial coordinates, respectively. The inner cylinder and the outer one have walls at r* = a and r* = a + 2 h, respectively, such that 2 h is the distance between the two. The length of the annular tube along the z*-axis which is the symmetry axis of both cylinders tends to infinity in order to neglect the influence at the ends in this direction. In this work, the axial motion is not considered, that is why Fig. 1 only presents a circular cross-section of the annulus. So, the velocity field has components as (u*, v*), where u* and v* denote the radial velocity and the linear velocity of rotation of fluid particles, respectively. The flow is driven by suction, such that V is the fluid suction speed at both uniformly permeable walls of the annulus. The variables describing the temperature and the pressure inside the flow domain are T* and p*, respectively. The walls of the annulus are fixed at different temperatures. Thus, the temperature of the cold wall which is the internal cylinder is T0 and the temperature of the hot wall representing the external cylinder is T1. The temperature difference between the two cylinders ΔT = T1− T0 does not influence the physical properties of the fluid that remain constant, notably the specific mass ρ, the dynamic viscosity µ, the thermal conductivity κ and the specific heat at constant pressure cp. The mathematical model of the problem is obtained from the general form of the Navier–Stokes equations and the energy equation [31–34]. The energy equation is derived by considering the absence of a heat source and neglecting the dissipation effects which could occur within the annulus, while the Navier–Stokes equations are obtained by neglecting gravity terms, due to the fact 317

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Fig. 1. Transversal circular section of the annular conduct having the z*-axis as the symmetry axis and showing the suction at r* = a as well as at r* = a + 2 h for the inner and outer cylinders kept at temperatures T0 and T1, respectively.

that the annular conduct is horizontal and the radii of the cylinders are great compared to the gap representing the flow domain. Taking into account these assumptions, the fluid distribution and heat transfer in the porous annular conduct are mathematically governed by the differential equations as in the following:

∂ ∂v * (r *u*) + =0 ∂r * ∂θ

(1)

∂p* ∂u* v * ∂u* v *2 ⎞ 1 ∂ ⎛ ∂u* ⎞ 1 ∂2u* u* 2 ∂v * ⎞ + μ⎛ + − − + − =− ρ ⎛u* r* 2 * * * *2 *2 *2 ∂θ ⎠ ∂r * ∂ ∂ ∂ r * ∂θ r* ⎠ r r r r θ r r ⎝ ⎠ ⎝ ∂r * ⎝ ⎜

⎟

(2)

∂v * v * ∂v * u*v * ⎞ 1 ∂p* 1 ∂ ⎛ ∂v * ⎞ 1 ∂ 2v * v* 2 ∂u* ⎞ + μ⎛ + − + + + =− ρ ⎛u* r* 2 * * * * * * * *2 *2 *2 ∂θ ⎠ ∂ ∂ ∂ ∂ ∂ ∂ r r θ r r θ r r r r θ r r ⎝ ⎠ ⎝ ⎠ ⎝

(3)

∂T * v * ∂T * ⎞ 1 ∂ ⎛ ∂T * ⎞ 1 ∂ 2T * ⎞ ρcp ⎛u* + = κ⎛ + r* * * * * * *2 ∂θ 2 ⎠ ∂ ∂ ∂ ∂ r r θ r r r r ⎝ ⎠ ⎝ ⎠ ⎝

(4)

⎜

⎟

⎜

⎟

where the incompressibility of the fluid is described by Eq. (1), the Navier–Stokes Eqs. (2) and (3) satisfy the momentum diffusion, while the energy conservation is verified by Eq. (4). The boundary conditions express the no-slip condition, equal permeability and the difference of temperatures at both walls of the annulus:

u* = −V , u* = +V ,

v * = 0, v * = 0,

T * = T0 T * = T1

for for

r* = a r * = a + 2h

(5)

It is convenient at this stage to introduce the nondimensional formulation of the problem in order to provide the control numbers that govern the flow and heat transfer under study. Thus, nondimensional variables for length, velocity, temperature and pressure are measured in units of the half-width of the annulus (h), the absolute suction speed at walls (V), the temperature difference between the two walls (T1 – T0) and the reference pressure (ρV2), respectively. These nondimensional variables are defined as follows:

r=

r* , h

u=

u* , V

v=

v* , V

T=

T* , T1 − T0

p=

p* ρV 2

(6)

In terms of the above nondimensional variables, the governing equations of the problem become:

∂ ∂v (ru) + =0 ∂r ∂θ

(7)

u

∂p ∂u v ∂u v2 1 1 ∂ ⎛ ∂u ⎞ 1 ∂ 2u u 2 ∂v + ⎛ + 2 2 − 2 − 2 ⎞ + − =− r ∂r ∂r r ∂θ r R ⎝ r ∂r ⎝ ∂r ⎠ r ∂θ r r ∂θ ⎠

u

∂v v ∂v uv 1 ∂p 1 1 ∂ ⎛ ∂v ⎞ 1 ∂ 2v v 2 ∂u ⎞ + ⎛ + 2 2 − 2 + 2 + + =− r ∂r r ∂θ r r ∂θ R ⎝ r ∂r ⎝ ∂r ⎠ r ∂θ r r ∂θ ⎠

⎜

⎟

⎜

(8)

⎟

318

(9)

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u

∂T v ∂T 1 ⎛ 1 ∂ ⎛ ∂T ⎞ 1 ∂ 2T + = r + 2 2⎞ ∂r r ∂θ Pe´ ⎝ r ∂r ⎝ ∂r ⎠ r ∂θ ⎠ ⎜

⎟

(10)

where the Reynolds number R = ρVh/ μ and the Péclet number Pé = ρVhcp/κ have been introduced. The nondimensional boundary conditions are derived as follows:

u = −1,

v = 0,

u = +1,

v = 0,

T0 T1 − T0 T1 T= T1 − T0 T=

a h a + 2h for r = h for r =

(11)

The incompressibility of the fluid gives the existence of a stream function ψ prescribed in the governing equations. The stream function ψ describing the fluid distribution is associated with a new function C uniform to the nondimensional temperature. The functions ψ and C are defined by:

u=−

1 ∂ψ , r ∂θ

v=

T (r , θ) = C (r ) +

∂ψ ∂r

(12)

T0 T1 − T0

(13)

By taking the curl of the momentum equation and applying the transformation (12), the vorticity transport equation satisfied by the function ψ is obtained. In addition, considering the transformation (13), the heat transfer is coupled to the vorticity transport within the annulus in light of the following differential equations:

∂ψ ∂ 1 ⎛ ∂ψ ∂ 1 (D 2ψ) − (D 2ψ) ⎞ = D 4ψ R r ⎝ ∂r ∂θ ∂θ ∂r ⎠

−

(14)

1 ∂ψ dC 1 ⎛ 1 dC d 2C ⎞ = + r ∂θ dr Pe´ ⎝ r dr dr 2 ⎠ ⎜

where D 2ψ =

1 r 1 r

1 ∂2ψ r 2 ∂θ2

+

1 ∂ r ∂r

⎟

(15)

∂ψ

(r ∂r ) . The boundary conditions associated with the differential Eqs. (14) and (15) are derived as follows:

∂ψ ∂ψ a = 0, C = 0 for r = = 1, h ∂r ∂θ ∂ψ ∂ψ a + 2h = 0, C = 1 for r = = −1, ∂r h ∂θ

(16)

In fact, the vorticity characterizes vortex flows. In fluid dynamics, a vortex is a region in which the flow is primarily a rotational movement about a rectilinear or curved axis. The vortex movements can be observed almost at all scales, from the swirl of emptying a bathtub to the atmospheres of the planets, or the wakes observed in the vicinity of an obstacle placed inside the liquid and gas flows. Once formed, the vortex regions can move, stretch, twist and interact in complex ways. A simple way to visualize a vortex is to consider a fluid motion in which a small volume supposedly rigid is delimited. If this small volume rotates instead of being in translation, then it belongs to a vortex. 3. Similarity method According to the geometry of the annulus, the problem provides a similarity solution formulated as:

ψ (r , θ) = θ (δ−1 − 1) F (η) C (r ) = (1 − d ) f (h) 1 1 η = δr 2 − δ−1 (1 + δ 2) 2 2 d = h/(a + h)

(17)

where δ is the geometric parameter valid on the interval 0 ≤ δ < 1 and defined on the basis of the radii of the annulus. The new space variable η takes the values −1 and +1 for r = a/h and r = (a + 2 h)/h, respectively. Applying the transformations (17) in Eqs. (14) and (15), the resulting problem is:

(1 + δ 2 + 2δη)2F (4) + 8δ (1 + δ 2 + 2δη) F (3) + 8δ 2F (2) + 2Rδ (1 − δ )(2FF (2) − (F (1) )2) + R (1 − δ )(1 + δ 2 + 2δη)(FF (3) − F (1) F (2) ) = 0 (1 + δ 2 + 2δη) φ(2) + 2δφ(1) + Pe´ (1 − δ ) Fφ(1) = 0

(18) (19)

where F (i) = diF / dηi and φ(i) = diφ / dηi . Following the transformations (17), the boundary conditions (16) become: 319

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F (1) = −(1 + δ )(1 − δ )−1

F (−1) = 1,

F (1) (−1) = 0, F (1) (1) = 0 ϕ (−1) = 0, ϕ (1) = (1 − δ )−1

(20)

It follows that the flow and heat transfer under study are reduced to a two-point boundary-value problem expressed as a secondorder nonlinear ordinary differential equation coupled to a fourth-order nonlinear ordinary differential equation with three boundary conditions at each wall of the annulus. Considering Eqs. (12) and (17), the similarity expressions of the velocity components are derived and given by:

u = −(1 − δ )(1 + δ 2 + 2δη)−1/2F (η) v = θ (δ−1 − 1)(1 + δ 2 + 2δη)1/2F (1) (η)

(21)

The similarity method is increasingly used in fluid dynamics and especially for the study of fluid flows between two porous walls [10,14,23–28]. The general objective in using this method is to transform the partial differential equations describing the fluid flows into ordinary differential equations modeling the same problems. This method is based essentially on the geometrical configuration of a given fluid flow. In the current study, Eq. (17) is adopted in order to transform the partial differential Eqs. (14) and (15) satisfied by the stream function ψ and the function C uniform to temperature into ordinary differential Eqs. (18) and (19). The geometry of the annulus which intervenes in formulas (17) is governed by the nondimensional parameter δ defined in terms of the half-width of the flow domain h and the radius of the internal cylinder a. As the stream function ψ is related to the velocity components, its expression as shown in Eqs. (17) is due to the cylindrical symmetry of the flow under study, such that the normal or the radial velocity u in light of Eqs. (21) is independent of the angular coordinate θ, because the radial movement is the same for the fluid particles belonging to the same arc inside the annulus, while the linear velocity of rotation v depends on the angular coordinate and the similarity normal variable η. The expression of η as shown in Eqs. (17) derives from the change of variables accompanying the similarity transformation, such that the quantity η varies between −1 and 1 as a centered variable, because the two porous walls of the annulus are opposite. On the other hand, if the dependence of the linear velocity of rotation on the angular variable is obvious because the function v is measured in this direction, its variation with η is due to the fact that the rotation of the fluid is sensitive to the proximity of the walls of the annular conduct. Relative to the temperature, the function ϕ only depends on η and does not vary with θ in light of Eq. (17) because of the cylindrical symmetry of the flow, since the fluid particles belonging to the same arc have the same temperature within the annular conduct. In other words, the temperature only changes with the radial or normal position of fluid particles inside the annulus. 4. Analytical approach When the Reynolds number and the Péclet number tend to zero, Eqs. (18) and (19) are linear and provide analytical solutions. In fact, these analytical solutions will help in Section 5 while testing the validity of the numerical scheme used to solve the nonlinear problem corresponding to the Reynolds and Péclet numbers different from zero. These analytical solutions are expressed by the formulas:

F = c0 + c1 η + (c2 + c3 η)ln(1 + δ 2 + 2δη)

(22)

φ = c4 + c5 ln(1 + δ 2 + 2δη)

(23)

Since the functions F and ϕ satisfy the boundary conditions (20), the constants of integration c0, c1, c2, c3, c4 and c5 are derived and given by: 2 1−δ

A−B+ c0 = 1 +

c1 =

c2 =

c3 =

− ) (ln ( ) + ) ln(1 − δ) + 2δ ( + (ln ( )+ ) ln ( ) − (1 − δ)(A − B) 2δ (1 + δ2) (1 − δ2)2

1+δ 1−δ

4δ2 ln(1 − δ2) (1 − δ2)2

ln(1 − δ ) (1 − δ )(1 + δ )2

2δ (1 + δ2) (1 − δ2)2

1+δ 1−δ

1+δ 1−δ

A−B 4δ2 ln(1 − δ2) (1 − δ2)2

( ( )+ ) ln ( ) − (1 − δ)(A − B) − (ln ( ) + ) + (ln ( )+ ) ln ( ) − (1 − δ)(A − B)

+ ln

2δ (1 + δ2) (1 − δ2)2

1+δ 1−δ

1 1−δ

4δ2 ln(1 − δ2) (1 − δ2)2

2δ (1 + δ2) (1 − δ2)2

1+δ 1−δ

c4 = −

( ( )+

+ ln

1+δ 1−δ

(1 − δ )−1 ln(1 − δ ) ln

( ) 1+δ 1−δ

,

2δ (1 + δ2) (1 − δ2)2

1+δ 1−δ

δ (1 − δ )(1 + δ )2 4δ2 ln(1 − δ2) (1 − δ2)2

1+δ 1−δ

−

2δ (1 + δ2) (1 − δ2)2

c5 =

1+δ 1−δ

δ (1 − δ )3

) ln ( ) − (1 − δ)(A − B) 1+δ 1−δ

(1 − δ )−1 2 ln

( ) 1+δ 1−δ

320

ln(1 − δ ) (1 − δ )3

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Fig. 2. Zeroth-order analytical solution for the stream function given by the formula (22) under different values of the geometric parameter δ. 2δ (ln(1 + δ ) +

δ (1 + δ )2

)

2δ (ln(1 − δ ) −

δ (1 − δ )2

)

with A = and B = . (1 − δ )3 (1 − δ )(1 + δ )2 The zeroth-order analytical solutions found pertain to the first step in seeking the results of the problem and help while analyzing the changes that happen with respect to other values of the Reynolds and Péclet numbers not close to zero. It appears that the analytical solution (22) plotted in Fig. 2 shows that the function F decreases within the flow domain for each value of δ and descends as the geometric parameter δ ascends. On the other hand, the zeroth-order analytical solution for the temperature (23) presented in Fig. 3 increases inside the flow field for a given value of δ and ascends with the growth of the geometric parameter. It follows that, as the Reynolds and Péclet numbers are close to zero, the stream function and the temperature exhibit the opposite behaviors in the flow domain and with respect to the growth of the parameter δ. When the Reynolds and Péclet numbers are not close to zero, the problem is nonlinear and a numerical integration is performed in order to derive further solutions. 5. Numerical results and discussion The numerical results are obtained by applying the rapidly converging shooting method associated with the fourth-order RungeKutta algorithm [35]. In fact, the procedure is as follows; the two-point boundary-value problem that consists of the nonlinear ordinary differential Eqs. (18) and (19) with boundary conditions (20), is transformed into an initial value problem. Since three of the six auxiliary conditions (20) are of the boundary-value type, the numerical solution becomes dependent upon three initial guesses. The numerical scheme ensures the determination of the values of the successful initial guesses in an iterative fashion, and then the solution of the problem is obtained. The validity of the numerical scheme based on the shooting method associated with a fourth-order Runge-Kutta algorithm is tested for small values of R and Pé. In fact, this test is performed in order to verify if the numerical solutions obtained agree with the analytical solutions found in Section 4. For this, a comparison between the two sets of solutions is made in Figs. 4 and 5 for δ = 0.01. These figures show that the numerical curves of the function F obtained for small values of R (R = 0.05, 0.1, 0.5, 1, and 1.5) tend to the zeroth-order analytical solution plotted for R = 0 in light of Fig. 4. On the other hand, the numerical curves of the function ϕ for R = 1, under small values of Pé (Pé = 0.07, 0.2, 0.6, 0.8, and 1) approach the analytical solution plotted for Pé = 0 as shown in Fig. 5. Since the numerical results approach the analytical solutions under different low Reynolds and Péclet numbers, this leads to the conclusion that the numerical scheme applied to solve the nonlinear problem is efficient and satisfactory. 5.1. Hydrodynamic structures The problem under study is highly nonlinear and is very sensitive to the control parameters. In particular, the geometric 321

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Fig. 3. Zeroth-order analytical solution for the temperature given by the formula (23) under different values of the geometric parameter δ.

Fig. 4. Comparison between the numerical results of Eq. (18) obtained by the shooting method associated with the fourth-order Runge–Kutta algorithm and the analytical solution given by the formula (22) for δ = 0.01 under different low Reynolds numbers.

322

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Fig. 5. Comparison between the numerical results of Eq. (19) obtained by the shooting method associated with the fourth-order Runge–Kutta algorithm and the analytical solution given by the formula (23) for δ = 0.01 and R = 1 under different low Péclet numbers.

parameter δ is very important as it describes the differential equations and boundary conditions. When δ is equal to zero, the problem coincides with a two-dimensional porous rectangular channel flow [12,14,18,24,36]. The cited works examine the behavior of the flow field components and many of them analyze the stability of similarity solutions. In addition, in these works, while varying the Reynolds number R, multiple solutions are found. Some of these solutions have been shown to become unstable to time-dependent perturbations, others lead to pitchfork bifurcations, saddle-node bifurcations, cusp type bifurcations, Hopf bifurcations. This set of previous solutions also includes stable asymmetric steady solutions, stable time-periodic solutions, stable quasi-periodic solutions, phase locking and chaos. Most of the above porous channel flow structures are also found in the case of the axial fluid flows through porous circular tubes [13,17,28–30]. Since the porous rectangular channel flows and axial fluid flows through porous circular tubes were covered in a series of studies [7–9,13,17], the current configuration is a new exploration. Indeed, this work especially concerns the investigation of heat transfer coupled to a circular flow through a porous annular pipe. In order to treat an annular problem, the value of the geometric parameter must be such that 0 < δ < 1. Thus, in accordance with the narrow gap approximation, the value of the geometric parameter is taken as δ = 0.01. One of the main purposes of the present work is the analysis of the circular flow characteristics around a turning point revealed within the annulus. The diagrams of this turning point are presented in Fig. 6(a) and (b) which show respectively the values of the functions F(2)(+1) and F(2)(−1) proportional to the wall friction coefficient in the close vicinity of the outer wall and in the close neighborhood of the inner wall as the Reynolds number varies. It is important to note that the flow process or the fluid motion manifests itself at the macroscopic level as a phenomenon of mass distribution. The momentum diffusion develops due to the rubs of fluid layers from each other and the diffusion coefficient associated is the dynamic viscosity or the shear viscosity. This shear viscosity is at the origin of friction effects when the fluid is in contact with a solid wall. A turning point occurs when the wall friction coefficient undergoes a qualitative change at a critical value of the Reynolds number. Through these turning point diagrams, a first branch denoted Fr exists for low and great values of the Reynolds number as shown in Fig. 6. This first branch is the unique set of solutions that exists for R < RTP = 6.659, where RTP is the value of the Reynolds number corresponding to the turning point which gives birth to two new secondary branches S1 and S2. Fig. 6 presenting the values of the wall friction coefficients F(2)(+1) and F(2)(−1) for δ = 0.01 in the close neighborhoods of the outer and the inner cylinders as the fluid rotates within the annular tube shows different shapes of the turning point. Indeed, the shapes of the turning point are different due to the fact that the viscous stress as the moving fluid is in contact with the solid cylinders is not identical at both walls. This is clearly shown in Fig. 7 where the comparison between the wall friction coefficients at the outer and inner walls is performed. Fig. 7 shows that the functions F(2)(+1) and F(2)(−1) plotted versus R are very different, because with the exception of a few point coincidences, there is no interval of Reynolds number values where their respective solution branches meet or approach each other asymptotically. A point coincidence is transient and obvious and does not attract much attention. It is the meeting of the functions 323

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Fig. 6. Diagram of the turning point representing the values of the wall friction coefficient, (a) in the close neighborhood of the outer cylinder, (b) in the close neighborhood of the inner cylinder.

F(2)(+1) and F(2)(−1) in an interval of values of R which has greater hydrodynamic significance and is the most interesting case to characterize. Two types of coincidences in an interval of Reynolds number values can be envisaged. Indeed, two branches of the same type or different type can meet in a given interval of values of R. Each of the two cases raised represents a hydrodynamic behavior that deserves a significant interpretation. Relative to the present work, as the cylindrical symmetry seems natural and considering the fact that the two walls of the inner and the outer cylinders are uniformly permeable, Fig. 7 is achieved in order to verify if the functions F(2)(+1) and F(2)(−1) are also symmetric, but the noticeable asymmetric behavior is due to the nonlinearity of the problem 324

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Fig. 7. Comparison between the values of the wall friction coefficient in the close neighborhoods of the inner and the outer cylinders.

under study. Since the highlighted turning point inside the annular permeable conduct is formed by three solution branches Fr, S1 and S2, another motivation of this work deals with the distribution of the fluid through each branch by examining the behavior of the velocity components. Thus, the suction-driven circular flow within the porous annulus is characterized by a radial velocity u and a linear velocity of rotation per unit angle v/θ. The fluid permanently rotates inside the annular conduct, that is why it is convenient and even

Fig. 8. Radial velocity through the first branch Fr. 325

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Fig. 9. Linear velocity of rotation through the first branch Fr.

realistic to examine the linear velocity of rotation per unit angle. Through the first branch, a region near the inner cylinder where the magnitude of the radial velocity component u exceeds its fixed values at walls is shown in Fig. 8, especially for the great Reynolds numbers. In fact, the magnitude of the radial velocity exceeding its values at walls highlights flow reversal which manifests itself as positive values of the linear velocity of rotation v /θ near the inner wall according to Fig. 9. For great values of the Reynolds number R > RTP, while flow reversal occurs near the inner cylinder, the motion of the fluid is not reversed in the rest of the annulus through the first branch Fr. More precisely, when the Reynolds number increases above the turning point RTP, a clockwise and a counterclockwise circular flows occur concurrently inside the annulus relative to the first branch Fr. In fact, in this study, since the flow primarily develops in the clockwise direction and obeys v < 0, the reversal occurs in the counterclockwise direction and corresponds to v > 0. For R < RTP, the linear velocity of rotation presents a parabolic behavior and negative values in light of Fig. 9, while the radial velocity plotted in Fig. 8 becomes indistinguishable from the law u = sin (πη/2) in order to satisfy the well-known Taylor profile [37]. In addition, relative to the first branch, Fig. 8 shows that the function u descends as R ascends. With respect to the secondary branch of type S2, the reverse flow moves from the region near the internal cylinder to the region close to the external cylinder by referring to Figs. 10 and 11 in which the magnitudes of the normal velocity and the linear velocity of rotation exceeding their respective values at the boundaries are shown in the neighborhood of the outer wall. Solutions of type S2 show an increase of u with the growth of R. It appears that an exchange of behaviors of the flow exists between the two neighborhoods of the porous walls with respect to solution branches of types Fr and S2 for R > RTP. In addition, the radial velocity plotted in Fig. 8 for the first branch Fr and in Fig. 10 for the secondary branch of type S2 shows the opposite behaviors as the Reynolds number increases above RTP. These opposite behaviors are due to the fact that, for R > RTP solutions of types Fr and S2 tend to be mirror images of each other as illustrated their comparison in terms of radial velocities in Fig. 12 and linear velocities of rotation in Fig. 13. In all the cases, the suction corresponding to the secondary branch S2 is dominant on the fluid withdrawal process relative to the first branch Fr in light of Fig. 12, since the magnitude of u for the solution branch S2 is great compared to that of the solution branch Fr under different values of the Reynolds number. The growth of the Reynolds number causes the radial velocity through the secondary branch of type S1 to undergo a linear law of the form u = η which is its expected profile for an inviscid suction flow [38,39] as presented in Fig. 14. In this figure, the function u tends to a same constant curve for different great values of the Reynolds number and seems to be not sensitive to the variations of R. More precisely, the curves plotted for different large values of R are indistinguishable. The described behavior of the radial velocity tending to satisfy a linear law is due to the presence of a boundary layer type flow that develops across the secondary branch of type S1. For a better understanding of the phenomenon of the boundary layer, an extended analysis is needed at this stage. In fact, the boundary layer takes place when the effects of viscosity are significant despite the importance of inertia effects in the case where a fluid layer moves in the immediate vicinity of a solid surface. There are many types of boundary layers including the external and internal boundary layers. The external boundary layer occurs when a fluid layer possesses an interface with a solid wall and another free boundary. For example, the atmospheric boundary layer is the air layer near the ground affected by diurnal heat, moisture or 326

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Fig. 10. Radial velocity relative to the secondary branch of type S2.

Fig. 11. Linear velocity of rotation relative to the secondary branch of type S2.

momentum transport to or from the earth surface. In addition, on an aircraft wing, the boundary layer is the part of the flow close to the wing, where viscous forces distort the surrounding inviscid flow. This external boundary layer involves disastrous effects in certain circumstances, because when the friction between the air stream and the wings of the aircraft becomes irregular, it can cause a plane crash. On the other hand, the contact between the wind and the metal sheets can imply the removal of the roof of a house. Due to the importance of the movement of a fluid in contact with a solid wall, the boundary layer flows are widely investigated. In 327

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Fig. 12. Comparison of the radial velocities relative to branches Fr and S2.

Fig. 13. Comparison of the linear velocities of rotation relative to branches Fr and S2.

particular, the laminar boundary layers can be classified according to their structures and the circumstances under which they are developed. The thin shear layer which develops on a body is an example of a Stokes boundary layer [40], while the Blasius boundary layer [41–43] refers to the well-known similarity solution near a flat plate held in an oncoming unidirectional flow. In addition, the Falkner–Skan boundary layer [44] is a generalized Blasius flow where the flat plate is not parallel to the direction of the fluid motion. All that has just been mentioned concerns the external boundary layer which can occur in the case where the flow of a fluid layer 328

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Fig. 14. Radial velocity relative to the secondary branch of type S1.

has a border in contact with a solid wall and another free boundary. Regarding the internal boundary layer, it can occur in the case of a confined fluid flow, in other words, when the movement of the fluid develops inside a conduct. Some disadvantages of the outer boundary layer have been raised. There are also the disadvantages related to the inner boundary layer. The presence of the boundary layer within a conduct that serves as a model for the functioning of a motor can reduce the performance of this motor. For example, the functioning of a slab rocket motor is usually modeled as a fluid flow between two solid walls. The manifestation of the boundary layer in such a flow type considerably reduces the performance of the slab rocket motor. Since the boundary layer can be a serious problem for an outer flow as well as an inner flow, the scientists have incorporated the porosity and/or the motion of the solid walls [23,30,45,46] that are in the immediate vicinity of the fluid flow in order to attenuate or to separate the boundary layer through the devices where this boundary layer is envisaged. The porosity of the walls that contain the currently studied fluid flow is therefore in order to reduce the manifestation of the boundary layer, but it is detected through the branch S1, while it is attenuated through the branches Fr and S2. The ambiguity encountered in the control of the boundary layer in accordance with the observation made in this present work as well as in some previous works [12,47] is its appearance especially when it seems to be avoided. Indeed, the increase in the Reynolds number tends to reduce the effects of the viscosity, so to avoid the development of the boundary layer, but the boundary layer appears in this same condition inside the annular pipe, because in some circumstances, the growth in R makes the flow more sensitive to the proximity of the walls which can give rise to existence of the boundary layer. This boundary layer is characterized by the profiles approaching asymptotically a same constant curve of the linear velocity of rotation with the increase of the Reynolds number as shown in Fig. 15. Through the branch S1, the movement of the fluid begins from a creeping type flow for low Reynolds numbers to a boundary layer type flow for large Reynolds numbers. The velocity profiles presented enable to assume that solutions pertaining to the first branch Fr for R < RTP and solutions of type S1 present the same behaviors. On the other hand, the secondary branch of type S1 characterizes the intermediate phenomena between the branches Fr and S2 for R > RTP. The control of the boundary layer is also performed in the heat distribution where it is denoted thermal boundary layer. 5.2. Heat transfer patterns The temperature profiles are plotted through each branch of the turning point diagram presented in Fig. 6. For a given value of the Reynolds number through the first branch Fr, the temperature presented in Fig. 16 shows an asymptotic curve as the Péclet number approaches the value of 10. By decreasing the Péclet number, the temperature tends to exhibit a linear behavior. In light of Fig. 16, the temperature increases inside the flow domain and ascends with the Péclet number while it descends through the secondary branch of type S2 as shown in Fig. 17. It follows that, the first branch Fr and the secondary branch of type S2 are characterized by the opposite thermal behaviors with the growth of the Péclet number. In other words, for a fixed Reynolds number, the progressive increase of the Péclet number augments the temperature inside the flow domain towards the upper limit which is the upper 329

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Fig. 15. Linear velocity of rotation relative to the secondary branch of type S1.

Fig. 16. Temperature through the first branch Fr for R = 11.5, under different Péclet numbers.

horizontal asymptote through the first branch Fr. However, this same growth in the Péclet number decreases the temperature within the annulus towards the lower limit that represents the lower horizontal asymptote through the secondary branch of type S2. Hence, the antagonistic behaviors of the branches Fr and S2 described in the fluid distribution is generalized in the heat transfer. On the other 330

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Fig. 17. Temperature through the secondary branch of type S2 for R = 11.5, under different Péclet numbers.

Fig. 18. Temperature through the secondary branch of type S1 for R = 11.5, under different Péclet numbers.

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Fig. 19. Temperature across the first branch Fr for Pé = 1 and Pé = 10, under different Reynolds numbers.

hand, the secondary branch of type S1 according to Fig. 18 reveals the presence of the thermal boundary layer which manifests itself as a large inflection area that appears within the annulus with the growth of the Péclet number. This inflection area presents two concavities. The first concavity located near the cold wall is turned towards the stocking, and the second concavity situated in the neighborhood of the hot wall is turned towards the top. In fact, the increase in the Péclet number tends to decrease the effects of the thermal conductivity of the fluid, so that the fluid behaves as a thermal insulator, but the presence of the temperature difference between the two borders of the annular conduct imposes the energy transfer from the hot wall to cold wall. The thermal boundary layer appears when the effects due to the temperature difference between the walls compensate the growth of the Péclet number. Once again, a behavior similar to the distribution of the fluid is found in the heat transfer. Indeed, the branch S1 that has shown the presence of the boundary layer type flow also reveals the existence of the thermal boundary layer. In all the cases, Figs. 16–18 enable to observe that, through the three branches Fr, S1 and S2, the decrease of the Péclet number at a fixed Reynolds number implies the linear law in the variation of temperature. It is convenient at this stage to examine the influence of the Reynolds number in the temperature distribution. In fact, Figs. 19–21 show that the shapes of the profiles obtained depend mainly on the value of the Péclet number. More precisely, the influence of the Péclet number Pé is great compared to that of the Reynolds number R on the heat transfer through each branch of the turning point diagram. Indeed, as the linear law is due to low values of the Péclet number, in Figs. 19–21, this behavior does not change under different Reynolds numbers. In addition, in Figs. 19 and 21, as the Péclet number is close to the value of 10, the respective upper and lower horizontal asymptotes presented through the branches Fr and S2 remain unchanged despite the variation in the Reynolds number. Moreover, the variation of the Reynolds number R does not destroy the thermal boundary layer revealed across the secondary branch of type S1 for Pé = 10 as shown in Fig. 20. The change due to the increase of the Reynolds number is the very weak growth of the temperature through the branches Fr and S1 in light of Figs. 19 and 20 as well as the very weak decrease of the temperature through the branch S2 according to Fig. 21.

6. Conclusion A cylindrical polar coordinate system, the Navier–Stokes equations and the energy equation are used to investigate the flow structures and heat transfer in an annular tube that consists of two coaxial cylinders possessing uniformly porous walls kept at different temperatures. Due to the incompressibility of the working fluid and by using the similarity method, the problem satisfied by two velocity components, the temperature and the pressure is transformed in order to be described by two coupled nonlinear ordinary differential equations. This problem is very sensitive to the geometric parameter δ defined on the basis of the radii of the internal and external cylinders, the Reynolds number R as well as the Péclet number Pé. The analytical solution found for low values of the Reynolds and Péclet numbers involving linear differential equations, enables to 332

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Fig. 20. Temperature across the secondary branch of type S1 for Pé = 1 and Pé = 10, under different Reynolds numbers.

Fig. 21. Temperature across the secondary branch of type S2 for Pé = 1 and Pé = 10, under different Reynolds numbers.

test the validity of the rapidly converging shooting method associated with a fourth-order Runge–Kutta algorithm applied to obtain the numerical results corresponding to nonlinear differential equations due to large values of R and Pé. The attention is focused on the analysis of the behaviors of the velocity components and temperature through three solution branches, known as the first branch Fr as 333

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well as two secondary branches S1 and S2 that form the turning point diagram revealed in the fluid distribution. The first branch Fr and the secondary branch of type S2 seem to be mirror images of each other. For the great values of R, flow reversal is identified near the inner cylinder relative to the first branch Fr, but it takes place near the outer cylinder through the secondary branch of type S2. The results corresponding to the secondary branch of type S1 show the presence of a boundary layer type flow as the Reynolds number increases. This boundary layer causes the normal velocity to approach a linear profile as a same constant curve for different large values of the Reynolds number. Concerning the distribution of temperature at a fixed Reynolds number, the secondary branch of type S1 is the seat of the thermal boundary layer which appears as the Péclet number is close to the value of 10. For this value of the Péclet number Pé = 10, the temperature distribution inside the annular conduct presents the upper and lower horizontal asymptotes across the solution branches Fr and S2, respectively. On the other hand, the opposite behaviors between the branches Fr and S2 recorded in the fluid distribution are also revealed in the heat transfer. It is also found that the Péclet number plays a more important role than the Reynolds number in the variation of temperature. In fact, when the boundary layer type flow or the thermal boundary layer occurs through a given solution branch and from a critical value of the control number, no further flow structure or heat transfer pattern can appear above this critical value. This is illustrated by an almost identical and constant profile of the flow field component whose behavior is examined. More precisely, this profile does not change for all the values above this critical value of the control number. In other words, when the boundary layer develops, it prevents the existence of another behavior for higher than the critical value of the control number. That is why in this study, whenever the boundary layer appears across a given solution branch at a critical value of the control number, the investigation of the flow field behaviors stopped around this critical value through this solution branch. The validation of the results obtained in this work is performed by the comparison with theoretical data of other authors. Indeed, when the geometric parameter δ is equal to zero, the mathematical model of the problem treated in this work coincides with that of a previously studied fluid flow between two parallel rectangular porous walls [14,24,36,48] in the absence of the energy equation. In addition, the turning point highlighted in the current annular flow for δ different from zero gives rise to a pitchfork bifurcation in some previous works [24,36,48]. A typical turning point is also found in the case of an axial movement of a viscous incompressible fluid inside a porous annular pipe [13,17]. Moreover, the appearance of the boundary layer type flow through the secondary solution branch of type S1 with the increase of the Reynolds number gives rise to a linear profile of the normal velocity component as found in the previous studies [38,39], while this normal or radial velocity behaves as a Taylor profile [37] through the first solution branch Fr for R < RTP. On the other hand, the thermal boundary layer found through the solution branch of type S1 is also revealed in a previous study of heat transfer coupled to a fluid flow within a symmetric porous channel [12]. 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