**2.1 Non-dimensionalization**

The following non-dimensional parameters are defined as:

$$t = \frac{t' \nu}{a^2}, \quad t\_0 = \frac{a^2}{v}, \quad R = \frac{r'}{a}, \quad \lambda = \frac{b}{a}, \quad M^2 = \frac{\sigma B\_0^2 a^2}{\rho \nu}, \quad \theta = \frac{(T' - T\_0)}{(T\_w - T\_0)}, \quad Pr = \frac{\rho c\_p}{k} \tag{5}$$

$$U = \frac{u'}{v\_0}, \quad K = \frac{V\_0}{U\_0}, \quad U\_0 = \frac{g\beta(T\_w - T\_0)a^2}{v}, \quad Gr = \frac{g\beta(T\_w - T\_0)a^3}{v^2}, \quad A = \frac{Q\_0 V t\_0}{k} \tag{6}$$

Where *θ* is the dimensionless temperature; *Pr* is the Prandtl number; *M* is the Hartmann number and *t* is the dimensionless time.

Using the non-dimensional parameters in Eq. (5) above, the governing equations of momentum (3) and energy (4) can be written in dimensionless form as:

$$\frac{\partial U}{\partial t} = \left[ \frac{\partial^2 U}{\partial R^2} + \frac{1}{R} \frac{\partial U}{\partial R} \right] - M^2 (U - K(f(t)) + Gr\theta \tag{6}$$

$$\frac{\partial \theta}{\partial t} = \frac{1}{Pr} \left[ \frac{\partial^2 \theta}{\partial R^2} + \frac{1}{R} \frac{\partial \theta}{\partial R} \right] - A \tag{7}$$

The initial conditions for velocity and temperature field in dimensionless form are:

$$t \le 0 \qquad U = 0, \quad \theta = 0 \text{ For } \ 1 \le R \le \lambda \tag{8}$$

While the boundary conditions in dimensionless form is given as:

$$t > 0 \qquad \begin{cases} U = f(t), & \theta = f(t) \quad \text{at } R = 1 \\ U = 0, & \theta = 0 \qquad \text{at } R = \lambda \end{cases} \tag{9}$$

Where *<sup>λ</sup>* <sup>¼</sup> *<sup>b</sup> <sup>a</sup>* >1

$$f(t) = \begin{cases} \frac{t}{t\_0} \text{ if } 0 \le t \le t\_0 \\\\ \mathbf{1} \quad \text{if } t \ge t\_0 \end{cases}$$

$$f(t) = H(t) \left(\frac{t}{t\_0}\right) - \left(\frac{1}{t\_0}\right) (t - t\_0) H(t - t\_0)$$

Where *H t*ð Þ is the Heaviside unit step function defined by *H t*ðÞ¼ 0, *<sup>t</sup>*<sup>&</sup>lt; <sup>0</sup> 1, *t*≥ 0 �

*Magneto-Hydrodynamic Natural Convection Flow in a Concentric Annulus with Ramped… DOI: http://dx.doi.org/10.5772/intechopen.100827*

### **2.2 Laplace transform**

Introducing the Laplace transform on the dimensionless velocity and temperature

$$\overline{U}(R,s) = \bigcap\_{t=0}^{\infty} U(R,t) \exp\left(-st\right)dt\tag{10}$$

$$\overline{\theta}(R,s) = \bigcap\_{t=0}^{\infty} \theta(R,t) \exp\left(-st\right)dt\tag{11}$$

(Where s is the Laplace parameter such that *s*>0) applying the properties of Laplace transform on Eqs. (6) and (7) subject to initial condition (8) gives

$$\left[\frac{d^2\overline{U}}{dR^2} + \frac{1}{R}\frac{d\overline{U}}{dR}\right] - \left(M^2 + s\right)\overline{U} = -Gr\overline{\theta} - M^2K\overline{f}(s) \tag{12}$$

$$\left[\frac{d^2\overline{\theta}}{dR^2} + \frac{1}{R}\frac{d\overline{\theta}}{dR}\right] - sPr\overline{\theta} - \frac{HPr}{s} = 0\tag{13}$$

The boundary conditions (8) becomes

$$\begin{aligned} \overline{U} &= \overline{f}(s), \quad \overline{\theta} = \overline{f}(s) \quad \quad \text{at } R = \mathbf{1} \\ \overline{U} &= \mathbf{0}, \quad \overline{\theta} = \mathbf{0} \quad \quad \text{at } R = \lambda \end{aligned} \tag{14}$$

#### **2.3 Solution**

The set of Bessel ordinary differential Eqs. (12) and (13) with the boundary condition (14) are solved for velocity and temperature in the Laplace domain as follows:

$$\overline{U}(R,s) = C\_3 I\_0 \left( R\sqrt{M^2 + s} \right) + C\_4 K\_0 \left( R\sqrt{M^2 + s} \right)$$

$$-\left[ \frac{Gr\left( C\_1 I\_0 \left( R\sqrt{sPr} \right) + C\_2 K\_0 \left( R\sqrt{sPr} \right) \right)}{sPr - \left( M^2 + s \right)} \right] + \frac{M^2 K\_0^7 (s)}{M^2 + s} - \frac{GrH}{(M^2 + s)s^2} \tag{15}$$

$$\overline{\theta}(R,s) = C\_1 I\_0 \left( R\sqrt{sPr} \right) + C\_2 K\_0 \left( R\sqrt{sPr} \right) - \frac{A}{s^2} \tag{16}$$

Where;

*<sup>C</sup>*<sup>1</sup> <sup>¼</sup> *<sup>s</sup>* <sup>2</sup>*f s*ð Þ*K*<sup>0</sup> *<sup>λ</sup>* ffiffiffiffiffiffi *sPr* � � <sup>p</sup> <sup>þ</sup> *A K*<sup>0</sup> *<sup>λ</sup>* ffiffiffiffiffiffi *sPr* � � <sup>p</sup> � *<sup>K</sup>*<sup>0</sup> ffiffiffiffiffiffi *sPr* � � � � <sup>p</sup> *s*<sup>2</sup> *I*<sup>0</sup> ffiffiffiffiffiffi *sPr* � � <sup>p</sup> *<sup>K</sup>*<sup>0</sup> *<sup>λ</sup>* ffiffiffiffiffiffi *sPr* � � <sup>p</sup> � *<sup>I</sup>*<sup>0</sup> *<sup>λ</sup>* ffiffiffiffiffiffi *sPr* � � <sup>p</sup> *<sup>K</sup>*<sup>0</sup> ffiffiffiffiffiffi *sPr* � � � � <sup>p</sup> *<sup>C</sup>*<sup>2</sup> <sup>¼</sup> *<sup>s</sup>* <sup>2</sup>*f s*ð Þ*I*<sup>0</sup> *<sup>λ</sup>* ffiffiffiffiffiffi *sPr* � � <sup>p</sup> <sup>þ</sup> *A I*<sup>0</sup> *<sup>λ</sup>* ffiffiffiffiffiffi *sPr* � � <sup>p</sup> � *<sup>I</sup>*<sup>0</sup> ffiffiffiffiffiffi *sPr* � � � � <sup>p</sup> *s*<sup>2</sup> *K*<sup>0</sup> ffiffiffiffiffiffi *sPr* � � <sup>p</sup> *<sup>I</sup>*<sup>0</sup> *<sup>λ</sup>* ffiffiffiffiffiffi *sPr* � � <sup>p</sup> � *<sup>K</sup>*<sup>0</sup> *<sup>λ</sup>* ffiffiffiffiffiffi *sPr* � � <sup>p</sup> *<sup>I</sup>*<sup>0</sup> ffiffiffiffiffiffi *sPr* � � � � <sup>p</sup> *<sup>C</sup>*<sup>3</sup> <sup>¼</sup> *f s*ð Þ*K*0ð Þ� *λδ <sup>A</sup>*1½*K*0ð Þ� *λδ <sup>K</sup>*0ð Þ*<sup>δ</sup>* � þ *<sup>A</sup>*4½*K*0ð Þ� *λδ <sup>K</sup>*0ð Þ*<sup>δ</sup>* � þ ½ � *<sup>A</sup>*2*K*0ð Þ� *λδ <sup>A</sup>*3*K*0ð Þ*<sup>δ</sup>* ½ � *I*0ð Þ*δ K*0ð Þ� *λδ I*0ð Þ *λδ K*0ð Þ*δ <sup>C</sup>*<sup>4</sup> <sup>¼</sup> *f s*ð Þ*I*0ð Þ� *λδ <sup>A</sup>*1½*I*0ð Þ� *λδ <sup>I</sup>*0ð Þ*<sup>δ</sup>* � þ *<sup>A</sup>*4½*I*0ð Þ� *λδ <sup>I</sup>*0ð Þ*<sup>δ</sup>* � þ ½ � *<sup>A</sup>*2*I*0ð Þ� *λδ <sup>A</sup>*3*I*0ð Þ*<sup>δ</sup>* ½ � *K*0ð Þ*δ I*0ð Þ� *λδ K*0ð Þ *λδ I*0ð Þ*δ*

$$\text{Where:} = \sqrt{M^2 + s}, A\_1 = \frac{M^2 \tilde{K}^{(s)}}{M^2 + s}, A\_4 = \frac{G r A}{(M^2 + s)^{2^s}}, A\_2 = \frac{G r \left[ C\_1 l\_0 \left( \sqrt{s Pr} \right) + C\_2 K\_0 \left( \sqrt{s Pr} \right) \right]}{s Pr - \left( M^2 s \right)}$$

$$A\_3 = \frac{G r \left[ C\_1 l\_0 \left( \lambda \sqrt{s Pr} \right) + C\_2 K\_0 \left( \lambda \sqrt{s Pr} \right) \right]}{s Pr - \left( M^2 + s \right)}$$

#### **2.4 Skin friction**

The skin – friction is the measure of the frictional force between the fluid and the surface of the cylinder. ð Þ *τ*<sup>1</sup> is the skin-friction at the outer surface of the inner cylinder and ð Þ *τλ* is the skin- friction at the inner surface of the outer cylinder. These are obtained by taking the first derivative of the velocity *U R*ð Þ , *s* given in Eq. (15) with respect to *R* as follows:

$$\begin{aligned} \left. \overline{\tau\_{1}} = \left. \frac{d\overline{U}}{dR} \right|\_{R=1} &= \sqrt{M^2 + s} \left( \mathbf{C}\_3 I\_1 \left( \sqrt{M^2 + s} \right) - \mathbf{C}\_4 K\_1 \left( \sqrt{M^2 + s} \right) \right) \\ &- \frac{Gr \sqrt{sPr}}{\left[ sPr - \left( M^2 + s \right) \right]} \left( \mathbf{C}\_1 I\_1 \left( \sqrt{sPr} \right) - \mathbf{C}\_2 K\_1 \left( \sqrt{sPr} \right) \right) \\ \left. \overline{\tau\_{\lambda}} = \left. \frac{d\overline{U}}{dR} \right|\_{R=\lambda} &= \sqrt{M^2 + s} \left( \mathbf{C}\_3 I\_1 \left( \lambda \sqrt{M^2 + s} \right) - \mathbf{C}\_4 K\_1 \left( \lambda \sqrt{M^2 + s} \right) \right) \\ &- \frac{Gr \sqrt{sPr}}{\left[ sPr - \left( M^2 + s \right) \right]} \left( \mathbf{C}\_1 I\_1 \left( \lambda \sqrt{sPr} \right) - \mathbf{C}\_2 K\_1 \left( \lambda \sqrt{sPr} \right) \right) \end{aligned} \tag{18}$$

#### **2.5 Nusselt number**

The expression for Nusselt number which is the measure of heat transfer rate on the cylinder is presented in the following form. *Nu<sup>α</sup>* <sup>¼</sup> *<sup>d</sup><sup>θ</sup> dR* � � � *R*¼*α*

$$\left.Nu\_1 = \frac{d\overline{\theta}}{dR}\right|\_{R=1} = \sqrt{sPr}\left(\mathbf{C}\_1 I\_1\left(\sqrt{sPr}\right) - \mathbf{C}\_2 K\_1\left(\sqrt{sPr}\right)\right) \tag{19}$$

$$\left. \mathrm{Nu}\_{\lambda} = \frac{d\overline{\theta}}{d\mathsf{R}} \right|\_{\mathsf{R}=\lambda} = \sqrt{sPr} \left( \mathbf{C}\_{1}I\_{1} \left( \lambda \sqrt{sPr} \right) - \mathbf{C}\_{2}K\_{1} \left( \lambda \sqrt{sPr} \right) \right) \tag{20}$$

#### **2.6 Mass flow rate**

Mass flow rate evaluates the rate of fluid flow through the annulus. It is achieved by taking a definite integral of Eq. (15) with respect to *R* as shown below:

$$\begin{split} \overline{Q} &= 2\pi \int\_{1}^{\hat{l}} R \overline{U}(R,s) dR = 2\pi \frac{C\_{3}}{\sqrt{M^{2}+s}} \left( \lambda I\_{1} \Big( \lambda \sqrt{M^{2}+s} \Big) - I\_{1} \Big( \sqrt{M^{2}+s} \Big) \right) \\ &- \frac{C\_{4}}{\sqrt{M^{2}+s}} \Big( \lambda K\_{1} \Big( \lambda \sqrt{M^{2}+s} \Big) - K\_{1} \Big( \sqrt{M^{2}+s} \Big) \Big) - \frac{GrC\_{1}}{\sqrt{s} Pr \left[ \operatorname{SPr} - \left( M^{2}+s \right) \right]} \left( \lambda I\_{1} \Big( \lambda \sqrt{s} Pr \Big) - I\_{1} \Big( \sqrt{s} Pr \Big) \right) \\ &+ \frac{GrC\_{2}}{\sqrt{s} Pr \left[ \operatorname{sPr} - \left( M^{2} \right) \right]} \Big) \Big( \lambda K\_{1} \Big( \lambda \sqrt{s} Pr \Big) - K\_{1} \Big( \sqrt{s} Pr \Big) \Big) \\ &+ \left( \frac{\lambda^{2}-1}{2} \right) \left( \frac{M^{2}K\_{1}^{2}(s)}{M^{2}+s} \right) - \left( \frac{\lambda^{2}-1}{2} \right) \Big( \frac{GrA}{(M^{2}+s)^{2}} \Big) \end{split} \tag{21}$$

*Magneto-Hydrodynamic Natural Convection Flow in a Concentric Annulus with Ramped… DOI: http://dx.doi.org/10.5772/intechopen.100827*

### **2.7 Riemann sum approximation**

Eqs. (15) and (16) are to be inverted in order to determine the velocity and temperature in time domain. Since these equations are difficult to invert in closed form. We use a numerical procedure used in Jha and Apere [14] which is based on the Riemann-sum approximation. In this method, any function in the Laplace domain can be inverted to the time domain as follows.

$$U(R,t) = \frac{\varepsilon^{ct}}{t} \left[ \frac{1}{2} \overline{U}(R,\varepsilon) + \operatorname{Re} \sum\_{n=1}^{M} \overline{U} \left( R, \varepsilon + \frac{in\pi}{t} \right) (-1)^{n} \right], \mathbf{1} \le R \le \lambda \tag{22}$$

where Re refers to the real part of *<sup>i</sup>* <sup>¼</sup> ffiffiffiffiffiffi �<sup>1</sup> <sup>p</sup> the imaginary number. M is the number of terms used in the Riemann-sum approximation and *ε* is the real part of the Bromwich contour that is used in inverting Laplace transforms. The Riemann-sum approximation for the Laplace inversion involves a single summation for the numerical process, its accuracy depends on the value of *ε* and the truncation error dictated by M. According to Tzou [29], the value of *εt* that best satisfied the result is 4.7.

#### **2.8 Validation of results**

In order to validate the results obtained from the Riemann sum approximation methods we use the partial differential equation parabolic and elliptic (PDEPE) method and compared the result. In General, it is given in the form:

$$\mathcal{L}\left(\mathbf{x},t,u,\frac{\partial u}{\partial t}\right)\frac{\partial u}{\partial t} = \mathbf{x}^{-m}\frac{\partial}{\partial \mathbf{x}}\left(\mathbf{x}^{m}f\left(\mathbf{x},t,u,\frac{\partial u}{\partial \mathbf{x}}\right)\right) + s\left(\mathbf{x},t,u,\frac{\partial u}{\partial \mathbf{x}}\right) \tag{23}$$

Initial condition *U x*ð Þ¼ , *t*<sup>0</sup> *U*0*x* Boundary conditions- one at each boundary ð Þþ *<sup>x</sup>*, *<sup>t</sup>*, *<sup>u</sup> q x*ð Þ , *<sup>t</sup> f x*, *<sup>t</sup>*, *<sup>u</sup>*, *<sup>∂</sup><sup>u</sup> ∂x* � � <sup>¼</sup> 0 . These comparisons are analyzed on the tables.

#### **2.9 Result and discussion**

A MATLAB program is written in order to depict the effect of the flow parameters such as the Hartman number ð Þ *M* , Prandlt number ð Þ *Pr* , Grashoff number (Gr), Heat source/sink parameter (A) and ratio of radii (*λ*) on Velocity ð Þ *U* , Temperature ð Þ *T* , Nusselt number at the outer surface of the inner cylinder ð Þ *Nu*<sup>1</sup> , Nusselt number at the inner surface of the outer cylinder ð Þ *Nu<sup>λ</sup>* , Skin friction (*λ*) at the outer surface of the inner cylinder (*τ*1), Skin friction at the inner surface of the outer cylinder (*τλ*) and mass flow rate ð Þ *Q* .

**Figure 2** illustrated the temperature profile for different values of time. It is seen from this graph that the temperature increases with increase in time. Increase in radii ratio lead to an increase in temperature as show in **Figure 3**. As the heat generating or absorbing parameter increase, a decrease in temperature is noticed as depicted in **Figure 4**.

**Figure 5** illustrated the effect of time on fluid velocity for cases (*K* ¼ 0*:*0 if the magnetic field is fixed relative to the fluid), (*K* ¼ 0*:*5 if the velocity of the magnetic field is less than the velocity of the moving cylinder) and (*K* ¼ 1*:*0 if the magnetic field is fixed relative to the moving cylinder) and for Prandtl number ð Þ *Pr* ¼ 0*:*71 *Air and Pr* ¼ 7*:*0 *Water* . These graphs show that the fluid velocity

**Figure 2.** *Temperature distribution for different values of time t*ð Þð Þ *λ* ¼ 2*:*0, *A* ¼ �2*:*0 *.*

**Figure 3.** *Temperature distribution for different values of radii ratio* ð Þ*λ* ð Þ *t* ¼ 0*:*4, *A* ¼ �2*:*0 *.*

increase as time increases for all cases. It is interesting to note that, *Pr* ¼ 0*:*71 converges faster than *Pr* ¼ 7*:*0. Also, the value of velocity becomes higher as the value of *K* increases. **Figure 6** depicted the influence of Hartmann number on the fluid velocity for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 *and Pr* ¼ 7*:*0 . the fluid velocity decreases for cases *K* ¼ 0*:*0 *and K* ¼ 0*:*5 This is physically true because application of magnetic field to an electrically conducting fluid give rise to resistivity force which is known as Lorentz force, this force has the tendency to decelerate fluid flow in the boundary layer region. Hartman number increase the fluid velocity for case *K* ¼ 1*:*0. This implies that the magnetic field is supporting the fluid motion. Although a flow reversal is noticed for ¼ 7*:*0 *atR*>1*:*4 . **Figure 7** show the outcome of radii ratio on fluid velocity for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 respectively and for *Pr* ¼ 7*:*0. It is evident from the graph that, increase in radii ratio result to an increase in fluid velocity for all cases of *K* considered. The value of velocity becomes higher as the value of *K* increases. Heat generating/absorbing parameter ð Þ *A* has a decreasing influence on the fluid velocity for all cases considered

*Magneto-Hydrodynamic Natural Convection Flow in a Concentric Annulus with Ramped… DOI: http://dx.doi.org/10.5772/intechopen.100827*

#### **Figure 4.**

*Temperature profile for different values of heat source/sink (A) (t* ¼ 0*:*4, *λ* ¼ 2*:*0*).*

**Figure 5.** *Velocity distribution for different values of time t and K* ð Þð Þ *Pr* ¼ 7*:*0, *λ* ¼ 2*:*0, *M* ¼ 2*:*0 *Gr* ¼ 5*:*0, *A* ¼ �2*:*0 *.*

ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 as illustrated in **Figure 8**. **Figure 9** presents the effect of Grashof number on fluid velocity for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for *Pr* ¼ 7*:*0. It is evident from the graph that increase in thermal buoyancy force lead to an increase in the fluid velocity for all cases of *K*.

**Figure 10** presents the effect of Hartmann number on skin friction at the outer surface of the inner cylinderð Þ *τ*<sup>1</sup> for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and forð Þ *Pr* ¼ 7*:*0 . It is evident from the graph that, the Hartmann number decrease the skin friction ð Þ *τ*<sup>1</sup> for all cases of *K*. **Figure 11** describe the impact of radii ratio on the skin friction ð Þ *τ*<sup>1</sup> for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and forð Þ *Pr* ¼ 7*:*0 . The graph show that, increase in radii ratio result to an increase in skin friction ð Þ *τ*<sup>1</sup> . **Figure 12** depicts the influence of heat generating/absorbing parameter ð Þ *A* for for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for ð Þ *Pr* ¼ 7*:*0 . It is noticed from these

**Figure 6.**

*Velocity distribution for different values of Hartmann number M and K* ð Þ ð*Pr* ¼ 7*:*0, *λ* ¼ 2*:*0, *t* ¼ 0*:*4 *Gr* ¼ 5*:*0, *A* ¼ �2*:*0Þ*.*

**Figure 7.** *Velocity distribution for different values of radii ratio* ð Þ *λ and K* ð Þ *Pr* ¼ 7*:*0, *A* ¼ �2*:*0, *t* ¼ 0*:*4 *Gr* ¼ 5*:*0, *M* ¼ 2*:*0 *.*

graphs that, Heat absorption has a retarding effect on the skin friction ð Þ *τ*<sup>1</sup> for all cases of *K:* It is essential to note that, a reverse flow occur on *Pr* ¼ 7*:*0 at *t* ¼ 0*:*8 for all cases of K considered. **Figure 13** illustrate the effect of Grashof number on skin friction ð Þ *τ*<sup>1</sup> for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for ð Þ *Pr* ¼ 7*:*0 . The thermal buoyance force is seen to increase the skin friction ð Þ *τ*<sup>1</sup> from the graph. **Figure 14** depict the effect of Hartmann number on skin friction at the inner surface of the outer cylinder ð Þ *τλ* for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for ð Þ *Pr* ¼ 7*:*0 . It is seen from the graph that, Hartmann number increase the skin frictionð Þ *τλ* for case *K* ¼ 0*:*0 while the opposite effect is noticed for both case

*Magneto-Hydrodynamic Natural Convection Flow in a Concentric Annulus with Ramped… DOI: http://dx.doi.org/10.5772/intechopen.100827*

*Velocity distribution for different values of heat source/sink A and K* ð Þ ð*Pr* ¼ 7*:*0, *λ* ¼ 2*:*0, *t* ¼ 0*:*4 *Gr* ¼ 5*:*0, *M* ¼ 2*:*0Þ*.*

#### **Figure 9.**

*Velocity distribution for different values of Grashof number Gr and K* ð Þ ð*Pr* ¼ 7*:*0, *λ* ¼ 2*:*0, *t* ¼ 0*:*4, *A* ¼ �2*:*0, *M* ¼ 2*:*0Þ*.*

*K* ¼ 0*:*5 *and K* ¼ 1*:*0. **Figure 15** show the impact of radii ratio on the skin friction ð Þ *τλ* for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for ð Þ *Pr* ¼ 7*:*0 . The graph show that, increase in radii ratio lead to a decrease in skin friction ð Þ *τλ* . **Figure 16** present the influence of heat generating/absorbing parameter ð Þ *A* for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for ð Þ *Pr* ¼ 7*:*0 . It is evident from the graph that, there is an enhancement in skin frictionð Þ *τλ* as the heat generating/absorbing parameter increase for all cases of *K*. **Figure 17** demonstrate the effect of Grashof number on skin friction ð Þ *τλ* for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for ð Þ *Pr* ¼ 7*:*0 . The thermal buoyance force is seen to decrease the skin friction ð Þ *τλ* from the graph.

**Figure 18** illustrate the effect of Hartmann number on mass flow rate ð Þ *Q* for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for ð Þ *Pr* ¼ 7*:*0 . The Hartman number decreases the volume flow rate. **Figure 19** show the impact of radii ratio on mass flow rate ð Þ *Q* for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for ð Þ *Pr* ¼ 7*:*0 . It is seen

#### **Figure 10.**

*Variation of skin friction* ð Þ *τ*<sup>1</sup> *for different values of Hartmann number M and K* ð Þð*Pr* ¼ 7*:*0, *A* ¼ �2*:*0, *λ* ¼ 2*:*0 *Gr* ¼ 5*:*0Þ*.*

#### **Figure 11.**

*Variation of skin friction* ð Þ *τ*<sup>1</sup> *for different values of radii ratio* ð Þ *λ and K* ð*Pr* ¼ 7*:*0, *A* ¼ �2*:*0, *M* ¼ 2*:*0 *Gr* ¼ 5*:*0Þ*.*

from the graph that, the radii ratio increase the mass flow rate for all cases of *K*. **Figure 20** present the influence of heat generating/absorbing parameter on mass flow rate ð Þ *Q* for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for ð Þ *Pr* ¼ 7*:*0 . It is noticed from the graph that, the heat generating/absorbing parameter decrease the mass flow rate for all cases of *K*. **Figure 21** depict the effect of Grashof number on mass flow rate ð Þ *Q* for cases ð Þ *K* ¼ 0*:*0,*K* ¼ 0*:*5 *and K* ¼ 1*:*0 and for ð Þ *Pr* ¼ 7*:*0 . It is evident from the graph that, the thermal buoyancy force increase the mass flow rate for all cases of *K*.

*Magneto-Hydrodynamic Natural Convection Flow in a Concentric Annulus with Ramped… DOI: http://dx.doi.org/10.5772/intechopen.100827*

**Figure 12.**

*Variation of skin friction* ð Þ *τ*<sup>1</sup> *for different values of heat source/sink A and K* ð Þ ð*Pr* ¼ 7*:*0, *λ* ¼ 2*:*0, *M* ¼ 2*:*0 *Gr* ¼ 5*:*0Þ*.*

**Figure 13.**

*Variation of skin friction* ð Þ *τ*<sup>1</sup> *for different values of Grashof number Gr and K* ð Þ ð*Pr* ¼ 7*:*0, *λ* ¼ 2*:*0, *M* ¼ 2*:*0 *Gr* ¼ 5*:*0, *A* ¼ �2*:*0Þ*.*
