**3. Conclusion**

The study of MHD natural convection flow of constant heat source/sink in an annulus due to ramped motion and ramped temperature of the boundaries have been carried out. The Laplace transform techniques have been used and the time domain solution was obtained using the Riemann sum approximation. The effect of the governing parameters such as the Hartmann number (*M*), radii ratio (*λ*), time (*t*), Grashof number ð Þ *Gr* Heat generating/absorbing parameter ð Þ *A* and

**Figure 14.**

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

**Figure 15.**

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

Prandtl number water (*Pr* ¼ 7*:*0) on the dimensionless fluid velocity ð Þ *U* , temperature ð Þ*θ* , mass flow rate ð Þ *Q* and skin – friction ð Þ *τ*<sup>1</sup> *and τλ* at both surfaces of the cylinder considering three cases of the velocity of the magnetic field ð*K* ¼ 0*:*0 when the magnetic field is fixed relative to the fluid, *K* ¼ 0*:*5 when the velocity of the magnetic field is less than the velocity of the moving cylinder and *K* ¼ 1*:*0 when the magnetic field is fixed relative to the moving cylinder) have been analyzed with the help of line graphs.

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

**Figure 16.**

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

**Figure 17.**

*Variation of skin friction* ð Þ *τλ for different values of Grashof number Gr and K* ð Þð*K* ¼ 0*:*0, *M* ¼ 2*:*0, *λ* ¼ 2*:*0 *A* ¼ �2*:*0Þ*.*

The noteworthy conclusions are summarized as follows:

• Hartmann number has a retarding effect on the skin friction ð Þ *τ*<sup>1</sup> and mass flow rate. It also decreases fluid velocity for cases ð Þ *K* ¼ 0*:*0 *and K* ¼ 0*:*5 the reverse

**Figure 18.**

*Variation of mass flow rate Q*ð Þ *for different values of Hartmann number M and K* ð Þð*K* ¼ 0*:*0, *λ* ¼ 2*:*0, *A* ¼ �2*:*0, *Gr* ¼ 5*:*0Þ*.*

#### **Figure 19.**

*Variation of mass flow rate Q*ð Þ *for different values of radii ratio* ð Þ *λ and K* ð*K* ¼ 0*:*0, *M* ¼ 2*:*0, *A* ¼ �2*:*0, *Gr* ¼ 5*:*0Þ*.*

effect is noticed for case ð Þ *K* ¼ 1*:*0 . increase in Hartmann number lead to an increase in skin friction ð Þ *τλ* for case ð Þ *K* ¼ 0*:*0 but decreases it for cases ð Þ *K* ¼ 0*:*0 *and K* ¼ 0*:*5

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

**Figure 20.**

*Variation of mass flow rate Q*ð Þ *for different values of heat source/sink A and K* ð Þð*K* ¼ 0*:*0, *M* ¼ 2*:*0 *λ* ¼ 2*:*0, *Gr* ¼ 5*:*0Þ*.*

#### **Figure 21.**

*Variation of mass flow rate Q*ð Þ *for different values of Grashof number Gr and K* ð Þð*K* ¼ 0*:*0, *M* ¼ 2*:*0, *λ* ¼ 2*:*0, *A* ¼ �2*:*0Þ*.*

