Magnetohydrodynamic Stagnation Flow and Heat Transfer toward a Stretching Permeable Cylinder

Abstract

We perform an analysis of MHD flow and heat transfer on a stretching, permeable cylinder. We prove existence of solutions for all values of the relevant parameters and provide uniqueness results in the case of a monotonic solution. The nonlinear boundary value problem that is derived from a similarity transformation of the governing system of nonlinear partial differential equations is solved numerically and the results are presented with graphs and tables.

1. Introduction

The study of axisymmetric stagnation flow on a sheet or a cylinder is an important aspect in various industrial applications. Exact solutions were obtained by Homann [1] for axisymmetric stagnation point flow on a sheet and by Wang [2] for axisymmetric stagnation flow on a circular cylinder. The work of Wang was later extended by Gorla [3] who included an analysis of the heat transfer and by Crane [4] and Wang [5] who considered a stretching cylinder in their studies. A recent study of axisymmetric stagnation flow by Ishak et al. [6] included the effects of suction or injection of the fluid for flow toward a stretching cylinder. Wang and Ng [7] investigated the effects of a partial slip boundary condition.

Studies of magnetohydrodynamic flow and heat transfer due to a stretching cylinder were performed by Ishak et al. [8] who obtained numerical solutions using the Keller-box method, Joneidi et al. [9] who obtained solutions using the homotopy analysis method, and Butt and Ali [10] who included the effects of entropy generation. Chauhan et al. [11] generalized the results of Joneidi et al. by considering the cylinder to be embedded in a porous medium along with a partial slip boundary condition. Munawar et al. [12] examined unsteady flow and heat transfer due to a stretching cylinder in consideration of two general types of thermal boundary conditions.

In this paper, we extend the work of Ishak et al. [8] by including the effects of suction/injection and by obtaining existence and uniqueness results for the nonlinear boundary value problem governing the fluid flow. We provide figures of the velocity and temperature profiles for various values of the system parameters and tables of the relevant boundary derivatives. In addition, we compare our results with the aforementioned studies.

2. Mathematical Formulation

Consider the steady laminar flow and heat transfer of an incompressible electrically conducting fluid induced by an axially stretching cylinder of radius R as depicted in Figure 1. We assume that the flow is subject to a uniform magnetic field of intensity B0 that is perpendicular to the cylinder's axis of symmetry and that the temperature at the cylinder and the ambient fluid temperature are T _w and T_∞, respectively. The governing system of nonlinear partial differential equations for the fluid flow and heat transfer in terms of the cylindrical coordinate system (r,θ, z) is given by [8]

$\begin{array}{l}\frac{\partial }{\partial z}\left(rw\right)+\frac{\partial }{\partial r}\left(ru\right)=0,\\ w\frac{\partial w}{\partial z}+u\frac{\partial w}{\partial r}=\nu \left(\frac{{\partial }^{2}w}{\partial r^2}+\frac{1}{r}\frac{\partial w}{\partial r}\right)-\frac{\sigma B_0^2}{\rho }w,\\ w\frac{\partial u}{\partial z}+u\frac{\partial u}{\partial r}=-\frac{1}{\rho }\frac{\partial p}{\partial r}+\nu \left(\frac{{\partial }^{2}u}{\partial r^2}+\frac{1}{r}\frac{\partial u}{\partial r}-\frac{u}{r^2}\right),\\ w\frac{\partial T}{\partial z}+u\frac{\partial T}{\partial r}=\alpha \left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{1}{r}\frac{\partial T}{\partial r}\right),\end{array}$

(1)

subject to the boundary conditions

$\begin{array}{c}u=U_w,\hspace{1em}\hspace{1em}w=w_w,\hspace{1em}\hspace{1em}T=T_w,\hspace{1em}\text{at}\hspace{0.17em}\hspace{0.17em}r=a,\\ w⟶\mathrm{0},\hspace{1em}\hspace{1em}T⟶T_{\infty },\hspace{1em}\text{as}\hspace{0.17em}\hspace{0.17em}r⟶\infty ,\end{array}$

(2)

where u and w are the radial and axial velocity components, respectively, T is the temperature, ρ is the density, p is the pressure, ν is the kinematic viscosity, and α is the thermal diffusivity. To model the permeability and the stretching of the cylinder, we let U _w = caγ and w _w = 2cz, respectively, where a is the cylinder radius, c is the strain rate of the oncoming radial flow, and γ is the permeability parameter for which γ>0 (γ<0) corresponds to suction (injection).

Figure 1:

Schematic diagram of axisymmetric stagnation flow toward a circular cylinder of radius a with uniform transpiration U _w subject to a uniform magnetic field B0.

Following [5], we define the similarity transformations:

$\begin{array}{c}\eta ={\left(\frac{r}{a}\right)}^{\mathrm{2}},\hspace{1em}\hspace{1em}u=-ca\left[\frac{f\left(\eta \right)}{\sqrt{\eta }}\right],\\ w=\mathrm{2}c{f}^{\prime }\left(\eta \right)z,\hspace{1em}\hspace{1em}\theta =\frac{(T-T_{\infty })}{(T_w-T_{\infty })}.\end{array}$

(3)

Substituting (3) into (1) yields the following nonlinear boundary value problem:

$\eta f^{\mathrm{\prime \prime \prime }}+f^{\mathrm{\prime \prime }}+\mathrm{Re}\left[f{f}^{\mathrm{\prime \prime }}-{\left(f^{\prime }\right)}^{\mathrm{2}}\right]-M{f}^{\prime }=\mathrm{0},\hspace{1em}\mathrm{1}<\eta <\infty ,$

(4)

subject to

$\eta {\theta }^{\mathrm{\prime \prime }}+\left(\mathrm{1}+\text{Re\hspace{0.17em}Pr}f\right){\theta }^{\prime }=\mathrm{0},\hspace{1em}\mathrm{1}<\eta <\infty ,$

(5)

where the Reynolds number Re, magnetic parameter M, and Prandtl number Pr are defined by

$f\left(\mathrm{1}\right)=\gamma ,\hspace{1em}\hspace{1em}{f}^{\prime }\left(\mathrm{1}\right)=\mathrm{1},\hspace{1em}\hspace{1em}{f}^{\prime }\left(\infty \right)=\mathrm{0},$

(6)

The physical quantities of interest are the skin friction coefficient and the Nusselt number, which are defined as

$\theta \left(\mathrm{1}\right)=\mathrm{1},\hspace{1em}\hspace{1em}\theta \left(\infty \right)=\mathrm{0},$

(7)

where τ _w and q_w are the shearing stress and heat transfer at the surface cylinder, respectively, which are given by

$\mathrm{Re}=\frac{c{a}^{\mathrm{2}}}{\mathrm{2}\nu },\hspace{1em}\hspace{1em}M=\frac{\sigma B_{\mathrm{0}}^{\mathrm{2}}{a}^{\mathrm{2}}}{\mathrm{4}\nu \rho },\hspace{1em}\hspace{1em}\text{Pr}=\frac{\nu }{\alpha }.$

(8)

Substituting (10) into (9) yields

$C_f=\frac{{\tau }_w}{(\mathrm{1}/\mathrm{2})\rho w_w^{\mathrm{2}}},\hspace{1em}\hspace{1em}Nu=\frac{a{q}_w}{k(T_w-T_{\infty })},$

(9)

3. Existence and Uniqueness Results

Using a topological shooting argument, the existence of a solution to (4) and (6) can be determined by formulating a related initial value problem (IVP) in which the boundary conditions in (6) are replaced by

${\tau }_w=\mu {\left(\frac{\partial w}{\partial r}\right)}_{r=a},\hspace{1em}\hspace{1em}{q}_w=-k{\left(\frac{\partial T}{\partial r}\right)}_{r=a}.$

(10)

In this formulation, α is a free parameter corresponding to the skin friction coefficient. We follow Mastroberardino and Paullet [13] and denote the solution of (4) and (12) by f(η;α). We will show that α can be chosen so that f′(η;α) exists for all η>1 and also satisfies f′(∞) = 0. To this end, we define the sets as follows:

$C_f\left(\mathrm{Re}\frac{z}{a}\right)=f^{\mathrm{\prime \prime }}\left(\mathrm{1}\right),\hspace{1em}\hspace{1em}Nu=-\mathrm{2}{\theta }^{\prime }\left(\mathrm{1}\right).$

(11)

$f\left(\mathrm{1}\right)=\gamma ,\hspace{1em}\hspace{1em}{f}^{\prime }\left(\mathrm{1}\right)=\mathrm{1},\hspace{1em}\hspace{1em}{f}^{\mathrm{\prime \prime }}\left(\mathrm{1}\right)=\alpha .$

(12)

$\begin{array}{c}\mathcal{A}=\{\phantom{f^{\prime }\left(\eta \right)\left[\mathrm{1},{\eta }_{\mathcal{A}}\right]}\alpha <\mathrm{0}\mid \text{There\hspace{0.17em}\hspace{0.17em}is\hspace{0.17em}\hspace{0.17em}a\hspace{0.17em}\hspace{0.17em}first\hspace{0.17em}\hspace{0.17em}point}\hspace{0.17em}{\eta }_{\mathcal{A}}>\mathrm{1}\\ \text{such\hspace{0.17em}\hspace{0.17em}that}\hspace{0.17em}{f}^{\mathrm{\prime \prime }}\left({\eta }_{\mathcal{A}}\right)=\mathrm{0},\\ f^{\prime }\left(\eta \right)>\mathrm{0}\hspace{0.17em}\mathrm{on}\hspace{0.17em}\left[\mathrm{1},{\eta }_{\mathcal{A}}\right]\},\\ \mathcal{B}=\{\phantom{f^{\mathrm{\prime \prime }}}\phantom{\left[\mathrm{1},{\eta }_{\mathcal{B}}\right]f\prime }\alpha <\mathrm{0}\mid \text{There\hspace{0.17em}\hspace{0.17em}is\hspace{0.17em}\hspace{0.17em}a\hspace{0.17em}\hspace{0.17em}first\hspace{0.17em}\hspace{0.17em}point\hspace{0.17em}\hspace{0.17em}}{\eta }_{\mathcal{B}}>\mathrm{1}\\ \text{such\hspace{0.17em}\hspace{0.17em}that\hspace{0.17em}\hspace{0.17em}}{f}^{\prime }\left({\eta }_{\mathcal{B}}\right)=\mathrm{0},\\ f^{\mathrm{\prime \prime }}\left(\eta \right)<\mathrm{0}\text{\hspace{0.17em}\hspace{0.17em}on\hspace{0.17em}\hspace{0.17em}}\left[\mathrm{1},{\eta }_{\mathcal{B}}\right]\}.\end{array}$

(13)

With appropriate modifications to the proofs in [13], we present the following two lemmas.

Lemma 1. The set $\mathcal{A}$ is nonempty and open.

Lemma 2. The set $\mathcal{B}$ is nonempty and open.

We now state a theorem regarding existence of solutions.

Theorem 3. For Re>0, M>0, and -∞<γ<∞, there exists a solution to (4) and (6) satisfying f′(η)>0 and f″(η)<0 for all η>1.

Proof. By Lemmas 1 and 2, the sets $\mathcal{A}$ and $\mathcal{B}$ are nonempty and open. Since they are also disjoint, there exists an α^* such that ${\alpha }^{*}\notin \mathcal{A}$ and ${\alpha }^{*}\notin \mathcal{B}$ . This follows from the fact that the interval (–∞,0) is connected and thus $\mathcal{A}\cup \mathcal{B}\ne (-\infty ,\mathrm{0})$ . Now suppose f′ = 0 and f″ = 0 simultaneously. Substituting these conditions into (4) yields f″ = 0 implying that f′(η)≡0 for all η, contradicting (6). From this we conclude that f(η;α^*) satisfies f′(η)>0 and f″(η)<0 for all η>1.

Turning our attention to the behavior of f′(η;α^*) as η approaches infinity, we note that since f′ is bounded below and decreasing, f′(∞;α^*) = C exists, where 0 ≤ C<1. The fact that C = 0 follows from a similar proof by contradiction given in [13].

The uniqueness of a monotone solution can be established with appropriate modifications to the proof in [13].

Theorem 4. If Re>0, M>0, and -∞<γ<∞, then there exists exactly one solution to (4) and (6) with the property f′(η)>0.

4. Results and Discussion

Equations (4) and (6) have been solved numerically using a shooting method, and our solutions have been used to generate tables of the relevant boundary derivatives and to plot the velocity and temperature profiles for various values of the system parameters. We have compared our results with those in [6, 8] for special cases of our study and we note that there is good agreement between the different numerical solutions. It is worth mentioning that the recent studies in [9, 11] do not provide tables of the relevant boundary derivatives and this can be considered a shortcoming.

Figures 2 and 3 show the variation in velocity profiles for varying values of the Reynolds number Re and magnetic parameter M, respectively, for both the suction γ>0 and injection γ<0 cases. It is evident that as either Re or M is increased, the velocity profiles decrease, thus resulting in an increase in the velocity gradient. These observations are supported by the magnitudes of the skin friction coefficient reported in Table 1.

Table 1:

Values of the skin friction coefficient f″(1). Values in parentheses are from [6].

Re	M	γ = 0.5	γ = −0.5
		f″(1)	f″(1)
1	2	−2.21659	−1.72075
5		−4.33228	−1.86364
10		−6.88470	−1.92938
10	0	−6.62227 (−6.6222)	−1.67757 (−1.6776)
	2	−6.88470	−1.92938
	5	−7.24505	−2.27933

Figure 2:

Velocity profiles f′(η) for various values of Re for suction γ>0 and injection γ<0 cases.

Figure 3:

Velocity profiles f′(η) for various values of M for suction γ>0 and injection γ<0 cases.

It is worth mentioning that Re represents the relative significance of inertial effects versus viscous effects and an increase in Re results in a resistance to fluid transport. In regard to the magnetic parameter M, a Lorentz force is induced by the transverse magnetic field, which provides a resistance to fluid transport. It is also evident from Figures 2 and 3 that suction increases the velocity gradient whereas injection results in the opposite effect as one would expect from intuition, and, indeed, our conclusions are supported by the magnitudes of the skin friction coefficient reported in Table 1.

Figures 4 and 5 show the variation in temperature profiles for varying values of the Prandtl number Pr and Reynolds number Re, respectively, for both the suction γ>0 and injection γ<0 cases. For the suction case, it is clear that as either Re or M is increased, the temperature profiles decrease, thus resulting in an increase in the temperature gradient at the surface of the cylinder. These observations are supported by the magnitudes of the Nusselt number reported in Table 2. For the injection case, the temperature gradients are near zero at the surface of the cylinder, but further away, the effects of Re and Pr are in opposition; that is, as Re increases, the thermal boundary increases, whereas the thermal boundary layer decreases as Pr increases.

Table 2:

Values of the Nusselt number -θ′(1). Values in parentheses are from [8] and [6], respectively.

γ	Re	M	Pr	−θ′(1)
0.5	10	2	7	36.60283
0				6.08375 (6.0864)
−0.5				0.00002
0.5	1	2	7	4.57611
	5			18.99556
	10			36.60283
0.5	10	0	7	36.60105 (36.6120)
		2		36.60283
		5		36.60551
0.5	10	2	0.7	4.18133
			2	11.13801
			7	36.60283

Figure 4:

Temperature profiles θ(η) for various values of Pr for suction γ>0 and injection γ<0 cases.

Figure 5:

Temperature profiles θ(η) for various values of Re for suction γ>0 and injection γ<0 cases.

In Figures 6 and 7, we plot skin friction coefficient and rate of heat transfer, respectively, as functions of the permeability parameter γ for various values of the system parameters. It is clear that the magnitude of the skin friction coefficient increases as the magnetic parameter increases and as the permeability varies from injection to suction. It is worth noting that the skin friction coefficient is more sensitive to changes in permeability for the case of suction. The rate of heat transfer shows little variability in the case of moderate to high level of suction; on the other hand, it is markedly affected in the case of injection; that is, as injection increases the magnitude of the rate of heat transfer increases significantly. Regarding the effect of Prandtl number, it clearly plays a much more critical role in altering the rate of heat transfer for the case of suction, and, in particular, the magnitude of the rate of heat transfer increases considerably as the Prandtl number increases.

Figure 6:

Skin friction coefficient as a function of permeability for various values of M and Re = 10.

Figure 7:

Rate of heat transfer as a function of permeability for various values of Pr, M = 2, and Re = 10.

5. Conclusions

We have investigated MHD flow and heat transfer on a stretching, permeable cylinder. Our model extends the work of Ishak et al. [8] by including wall transpiration and by providing existence and uniqueness results in regard to the nonlinear boundary value problem that governs the fluid flow. We have provided figures of the velocity and temperature profiles for various values of the system parameters and tables of the relevant boundary derivatives. Below we present a summary of our results.

1. It is observed that injection decreases the skin friction coefficient whereas suction has the opposite effect.

2. Increasing either the Reynolds number or the Prandtl number results in an increase in the skin friction coefficient.

3. For both suction and injection cases, the rate of heat transfer—characterized by the magnitude of the Nusselt number—increases as either the Reynolds number or the Prandtl number is increased.

4. For the injection case, the thermal boundary layer increases as the Reynolds number is increased whereas an increase in the Prandtl number has the opposite effect.

We hope that this investigation could be used as a basis for modeling various engineering applications and as motivation for further experimental as well as theoretical studies.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.