7.1.1. Problem definition

${\left(\frac{X}{a}\right)}^{2n}+{\left(\frac{Y}{b}\right)}^{2n}=1$

(7.1)

For the expression of the magnetic field strength, it can be considered that two magnetic sources at points $\left(\overline{a_1},\overline{b_1}\right)$ and $\left(\overline{a_2},\overline{b_2}\right)$. The components of the magnetic field intensity $\left(\overline{H_x},\overline{H_y}\right)$ and the magnetic field strength $(\overline{H})$ can be considered as [2]:

$\overline{H_x}=\frac{\gamma }{2\pi }\frac{1}{{\left(x-\overline{a_1}\right)}^2+{\left(y-\overline{b_1}\right)}^2}\left(y-\overline{b_1}\right)-\frac{\gamma }{2\pi }\frac{1}{{\left(x-\overline{a_2}\right)}^2+{\left(y-\overline{b_2}\right)}^2}\left(y-\overline{b_2}\right)$

(7.2)

$\overline{H_y}=-\frac{\gamma }{2\pi }\frac{1}{{\left(x-\overline{a_1}\right)}^2+{\left(y-\overline{b_1}\right)}^2}\left(x-\overline{a_1}\right)-\frac{\gamma }{2\pi }\frac{1}{{\left(x-\overline{a_2}\right)}^2+{\left(y-\overline{b_2}\right)}^2}\left(x-\overline{a_2}\right)$

(7.3)

$\overline{H}=\sqrt{{\overline{H}}_x^2+{\overline{H}}_y^2}$

(7.4)

where γ is the magnetic field strength at the source (of the wire). The contours of the magnetic field strength are shown in Fig. 7.2. In this chapter, magnetic source is located at $\left(\overline{a_1}=1.05\text{cols},\overline{b_1}=0.5\text{rows}\right)$ and $\left({\overline{a}}_2=-0.05\text{cols},\overline{b_2}=0.5\text{rows}\right)$.

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

(7.5)

${\rho }_{\text{nf}}\left(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}\right)=-\frac{\partial P}{\partial x}+{\mu }_{\text{nf}}\left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)+{\mu }_{0}M\frac{\partial \overline{H}}{\partial x}-{\sigma }_{\text{nf}}{B}_y^{2}u+{\sigma }_{\text{nf}}{B}_x{B}_{y}v$

(7.6)

$\begin{array}{l}{\rho }_{\text{nf}}\left(u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}\right)=-\frac{\partial P}{\partial y}+{\mu }_{\text{nf}}\left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)+{\mu }_{0}M\frac{\partial \overline{H}}{\partial y}-{\sigma }_{\text{nf}}{B}_x^{2}v+{\sigma }_{\text{nf}}{B}_x{B}_{y}u\\ +{\rho }_{\text{nf}}{\beta }_{\text{nf}}g\left(T-T_{\text{c}}\right)\end{array}$

(7.7)

$\begin{array}{l}{\left(\rho C_{\text{p}}\right)}_{\text{nf}}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}\right)=k_{\text{nf}}\left(\frac{{\partial }^{2}T}{\partial x^2}+\frac{{\partial }^{2}T}{\partial y^2}\right)+{\sigma }_{\text{nf}}{\left(u{B}_y-v{B}_x\right)}^2\\ -{\mu }_{0}T\frac{\partial M}{\partial T}\left(u\frac{\partial \overline{H}}{\partial x}+v\frac{\partial \overline{H}}{\partial y}\right)+{\mu }_{\text{nf}}\left\{2{\left(\frac{\partial u}{\partial x}\right)}^2+2{\left(\frac{\partial v}{\partial y}\right)}^2+{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2\right\}\end{array}$

(7.8)

The terms ${\mu }_{0}M\frac{\partial \overline{H}}{\partial x}$ and ${\mu }_{0}M\frac{\partial \overline{H}}{\partial y}$ in (7.6) and (7.7), respectively, represent the components of magnetic force, per unit volume, and depend on the existence of the magnetic gradient on the corresponding x- and y-directions. These two terms are well known from ferrohydrodynamic (FHD) which is the so-called Kelvin force. The terms $-{\sigma }_{\text{nf}}{B}_y^{2}u+{\sigma }_{\text{nf}}{B}_x{B}_{y}v$ and $-{\sigma }_{\text{nf}}{B}_x^{2}v+{\sigma }_{\text{nf}}{B}_x{B}_{y}u$ appearing in (7.6) and (7.7), respectively, represent the Lorentz force per unit volume toward the x- and y-directions and arise due to the electrical conductivity of the fluid. These two terms are known in magnetohydrodynamic (MHD). The principles of MHD and FHD are combined in the mathematical model presented in [3] and the aforementioned terms arise together in the governing Eqs. (7.6) and (7.7). The term ${\mu }_{0}T\frac{\partial M}{\partial T}\left(u\frac{\partial \overline{H}}{\partial x}+v\frac{\partial \overline{H}}{\partial y}\right)$ in Eq. (7.8) represents the thermal power per unit volume due to the magnetocaloric effect. Also, the term ${\sigma }_{\text{nf}}{\left(u{B}_y-v{B}_x\right)}^2$ in (7.8) represents the Joule heating. For the variation of the magnetization M, with the magnetic field intensity $\overline{H}$ and temperature T, the following relation is used [4]:

$M=K^{\prime }\overline{H}\left({T^{\prime }}_{\text{c}}-T\right)$

(7.9)

where K′ is a constant and ${T^{\prime }}_{\text{c}}$ is the Curie temperature.

In the aforementioned equations, μ₀ is the magnetic permeability of vacuum (4π × 10⁻⁷ Tm/A), $\overline{H}$ is the magnetic field strength, $\overline{B}$ is the magnetic induction $\left(\overline{B}={\mu }_0\overline{H}\right)$, and the bar above the quantities denotes that they are dimensional. The effective density (ρ_nf) and heat capacitance (ρC_p)_nf of the nanofluid are defined as:

${\rho }_{\text{nf}}={\rho }_{\text{f}}(1-\phi )+{\rho }_{\text{s}}\phi$

(7.10)

${\left(\rho C_{\text{p}}\right)}_{\text{nf}}={\left(\rho C_{\text{p}}\right)}_{\text{f}}(1-\phi )+{\left(\rho C_{\text{p}}\right)}_{\text{s}}\phi$

(7.11)

${\alpha }_{\text{nf}}=\frac{k_{\text{nf}}}{{\left(\rho C_{\text{p}}\right)}_{\text{nf}}}$

(7.12)

${\beta }_{\text{nf}}={\beta }_{\text{f}}(1-\phi )+{\beta }_{\text{s}}\phi$

(7.13)

${\mu }_{\text{nf}}=\frac{{\mu }_{\text{f}}}{{(1-\phi )}^{2.5}}$

(7.14)

$\frac{k_{\text{n}\text{f}}}{k_{\text{f}}}=\frac{k_{\text{s}}+2k_{\text{f}}-2\phi (k_{\text{f}}-k_{\text{s}})}{k_{\text{s}}+2k_{\text{f}}+\phi (k_{\text{f}}-k_{\text{s}})}$

(7.15)

$\frac{{\sigma }_{\text{nf}}}{{\sigma }_{\text{f}}}=1+\frac{3\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}-1\right)\phi }{\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}+2\right)-\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}-1\right)\phi }$

(7.16)

$\begin{array}{l}a=\frac{\overline{a}}{L},b=\frac{\overline{b}}{L},X=\frac{x}{L},Y=\frac{y}{L},\Theta =\frac{T-T_{\text{c}}}{T_{\text{h}}-T_{\text{c}}},U=\frac{uL}{{\alpha }_{\text{f}}},V=\frac{vL}{{\alpha }_{\text{f}}},\\ H=\frac{\overline{H}}{\overline{H_0}},H_x=\frac{\overline{H_x}}{\overline{H_0}},H_y=\frac{\overline{H_y}}{\overline{H_0}},P=\frac{p}{{\rho }_{\text{f}}{\left({\alpha }_{\text{f}}/L\right)}^2}\end{array}$

(7.17)

$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$

(7.18)

$\begin{array}{l}U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\mathrm{Pr}\left[\frac{{\mu }_{\text{nf}}/{\mu }_{\text{f}}}{{\rho }_{\text{nf}}/{\rho }_{\text{f}}}\right]\left(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}\right)\\ +M{n}_{\text{F}}\mathrm{Pr}\left(\frac{{\rho }_{\text{f}}}{{\rho }_{\text{nf}}}\right)\left({\varepsilon }_2-{\varepsilon }_1-\Theta \right)H\frac{\partial H}{\partial X}-H{a}^2\mathrm{Pr}\left[\frac{{\sigma }_{\text{nf}}/{\sigma }_{\text{f}}}{{\rho }_{\text{nf}}/{\rho }_{\text{f}}}\right]\left(H_y^{2}U-H_x{H}_{y}V\right)0\end{array}$

(7.19)

$\begin{array}{l}U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial X}+\mathrm{Pr}\left[\frac{{\mu }_{\text{nf}}\text{/}{\mu }_{\text{f}}}{{\rho }_{\text{nf}}\text{/}{\rho }_{\text{f}}}\right]\left(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}\right)\\ +M{n}_{\text{F}}\mathrm{Pr}\left(\frac{{\rho }_{\text{f}}}{{\rho }_{\text{nf}}}\right)({\epsilon }_2-{\epsilon }_1-\Theta )H\frac{\partial H}{\partial Y}-H{a}^2\mathrm{Pr}\left[\frac{{\sigma }_{\text{nf}}\text{/}{\sigma }_{\text{f}}}{{\rho }_{\text{nf}}\text{/}{\rho }_{\text{f}}}\right](H_x^{2}V-H_x{H}_{y}U)\\ +Ra\mathrm{Pr}\left[\frac{{\beta }_{\text{nf}}}{{\beta }_{\text{f}}}\right]\Theta \end{array}$

(7.20)

$\begin{array}{l}U\frac{\partial \Theta }{\partial X}+V\frac{\partial \Theta }{\partial Y}=\left[\frac{\frac{k_{\text{nf}}}{k_{\text{f}}}}{\frac{{\left(\rho C_{\text{P}}\right)}_{\text{nf}}}{{\left(\rho C_{\text{P}}\right)}_{\text{f}}}}\right]\left(\frac{{\partial }^2\Theta }{\partial X^2}+\frac{{\partial }^2\Theta }{\partial Y^2}\right)\\ +H{a}^{2}Ec\left[\frac{\frac{{\sigma }_{\text{nf}}}{{\sigma }_{\text{f}}}}{\frac{{\left(\rho C_{\text{P}}\right)}_{\text{nf}}}{{\left(\rho C_{\text{P}}\right)}_{\text{f}}}}\right]{\left\{U{H}_y-V{H}_x\right\}}^2\\ +M{n}_{\text{F}}Ec\frac{{\left(\rho C_{\text{P}}\right)}_{\text{f}}}{{\left(\rho C_{\text{P}}\right)}_{\text{nf}}}\left\{U\frac{\partial H}{\partial X}+V\frac{\partial H}{\partial Y}\right\}H\left({\varepsilon }_1+\Theta \right)\\ +Ec\left[\frac{\frac{{\mu }_{\text{nf}}}{{\mu }_{\text{f}}}}{\frac{{\left(\rho C_{\text{P}}\right)}_{\text{nf}}}{{\left(\rho C_{\text{P}}\right)}_{\text{f}}}}\right]\left\{2{\left(\frac{\partial U}{\partial X}\right)}^2+2{\left(\frac{\partial V}{\partial Y}\right)}^2+{\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)}^2\right\}\end{array}$

(7.21)

where $R{a}_{\text{f}}=g{\beta }_{\text{f}}{L}^3\left(T_{\text{h}}-T_{\text{c}}\right)/\left({\alpha }_{\text{f}}{\upsilon }_{\text{f}}\right),P{r}_{\text{f}}={\upsilon }_{\text{f}}/{\alpha }_{\text{f}},Ha=L{\mu }_0{H}_0\sqrt{{\sigma }_{\text{f}}/{\mu }_{\text{f}}},{\varepsilon }_1=T_1/∆T$, ${\epsilon }_2={T^{\prime }}_{\text{c}}\text{/}∆T,\mathrm{Ec}=({\mu }_{\text{f}}{\alpha }_{\text{f}})\text{/}[{(\rho C_{\text{P}})}_{\text{f}}∆T\hspace{0.17em}{L}^2]$, and $M{n}_{\text{F}}={\mu }_0{H}_0^2{K}^{\prime }\left(T_{\text{h}}-T_{\text{c}}\right)L^2/\left({\mu }_{\text{f}}{\alpha }_{\text{f}}\right)$ are the Rayleigh number, Prandtl number, Hartmann number arising from MHD, temperature number, curie temperature number, Eckert number, and Magnetic number arising from FHD the for the base fluid, respectively. The thermophysical properties of the nanofluid are given in Table 7.1 [3].

$\begin{array}{l}u=\frac{\partial \psi }{\partial y},v=-\frac{\partial \psi }{\partial x},\omega =\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\\ \Omega =\frac{\omega L^2}{{\alpha }_{\text{f}}},\Psi =\frac{\psi }{{\alpha }_{\text{f}}}\end{array}$

(7.22)

$\begin{array}{ll}\Theta =1.0\hfill & \text{on}\text{the}\text{inner}\text{circular}\text{boundary}\hfill \\ \Theta =0.0\hfill & \text{on}\text{the}\text{outer}\text{wall}\text{boundary}\hfill \\ \Psi =0.0\hfill & \text{on}\text{all}\text{solid}\text{boundaries}\hfill \end{array}$

(7.23)

$N{u}_{\text{loc}}=\left(\frac{k_{\text{nf}}}{k_{\text{f}}}\right)\frac{\partial \Theta }{\partial r}$

(7.24)

$N{u}_{\text{ave}}=\frac{1}{\gamma }\underset{0}{\overset{\gamma }{\int }}N{u}_{\text{loc}}\left(\zeta \right)d\zeta$

(7.25)

$En=\frac{Nu\left(\phi =0\text{.04}\right)-Nu\left(\text{base fluid}\right)}{Nu\left(\text{base fluid}\right)}\times 100$

(7.26)

7.1.2. Effects of active parameters

$\begin{array}{l}N{u}_{\text{ave}}=a_{15}+a_{25}{Y}_3+a_{35}{Y}_4+a_{45}{Y}_3^2+a_{55}{Y}_4^2+a_{65}{Y}_4{Y}_3\\ Y_1=a_{11}+a_{21}R{a}^{*}+a_{31}M{n}_{\text{F}}^{*}+a_{41}R{a}^{{*}^2}+a_{51}M{n}_{\text{F}}^{{*}^2}+a_{61}R{a}^{*}M{n}_{\text{F}}^{*}\\ Y_2=a_{12}+a_{22}\phi +a_{32}R{a}^{*}+a_{42}{\phi }^2+a_{52}R{a}^{{*}^2}+a_{62}\phi R{a}^{*}\\ Y_3=a_{13}+a_{23}{Y}_1+a_{33}H{a}^{*}+a_{43}{Y}_1^2+a_{53}H{a}^{{*}^2}+a_{63}{Y}_{1}H{a}^{*}\\ Y_4=a_{14}+a_{24}H{a}^{*}+a_{34}{Y}_2+a_{44}H{a}^{{*}^2}+a_{54}{Y}_2^2+a_{64}H{a}^{*}{Y}_2\end{array}$

(7.27)

$\begin{array}{l}En=b_{13}+b_{23}{Y}_1+b_{33}{Y}_2+b_{43}{Y}_1^2+b_{53}{Y}_2^2+b_{63}{Y}_1{Y}_2\\ Y_1=b_{11}+b_{21}R{a}^{*}+b_{31}H{a}^{*}+b_{41}R{a}^{{*}^2}+b_{51}H{a}^{{*}^2}+b_{61}R{a}^{*}H{a}^{*}\\ Y_2=b_{12}+b_{22}R{a}^{*}+b_{32}M{n}_{\text{F}}^{*}+b_{42}R{a}^{{*}^2}+b_{52}M{n}_{\text{F}}^{{*}^2}+b_{62}\mathrm{Re}M{n}_{\text{F}}^{*}\end{array}$

(7.28)

where Ra^* = log(Ra), Ha^* = Ha/10, $M{n}_{\text{F}}^{*}=M{n}_{\text{F}}/100$. Also a_ij, b_ij can be found in Tables  7.2 and  7.3.

The steady, two-dimensional, incompressible, laminar flow is considered taking place in a channel. The width of channel is D and the length of the channel is L, such that L/D = 25. The flow at the entrance is assumed to be fully developed $\left(\frac{u}{u_{\text{in}}}=4\frac{y}{D}\left(1-\frac{y}{D}\right)\right)$ (Fig. 7.7A) [5]. The flow is subjected to a magnetic source, which is placed very close to the lower plate and below it (a = 0.25L, b = −0.1D) (Fig. 7.8). The fluid in vessel is considered a homogenous mixture of Fe₃O₄–blood plasma. Thermophysical properties of the nanoscale ferromagnetic particle and blood plasma are assumed to be constant (Table 7.4).

$H_x=\frac{\left|b\right|}{{\left(x-a\right)}^2+{\left(y-b\right)}^2}\left(y-b\right)$

(7.29)

$H_y=-\frac{\left|b\right|}{{\left(x-a\right)}^2+{\left(y-b\right)}^2}\left(x-a\right)$

(7.30)

$H=\sqrt{H_x^2+H_y^2}=\frac{\left|b\right|}{\sqrt{{\left(x-a\right)}^2+{\left(y-b\right)}^2}}$

(7.31)

One of the novel computational fluid dynamics (CFD) methods, which solves the Boltzmann equation to simulate the flow instead of solving the Navier–Stokes equations, is called lattice Boltzmann methods (LBM). LBM has several advantages such as simple calculation procedure and efficient implementation for parallel computation, over other conventional CFD methods, because of its particulate nature and local dynamics. The LB model uses one distribution function f for the flow. It uses modeling of movement of fluid particles to capture macroscopic fluid quantities such as velocity and pressure. In this approach, the fluid domain is discretized to uniform Cartesian cells. Each cell holds a fixed number of distribution functions, which represent the number of fluid particles moving in these discrete directions. The D₂Q₉ model [6] was used and values of $w_0=4/9$ for |c₀| = 0 (for the static particle), $w_{1-4}=1/9$ for |c_1–4| = 1, and $w_{5-9}=1/36$ for $\left|c_{5-9}\right|=\sqrt{2}$ are assigned in this model (Fig. 7.7B). The density and distribution functions are calculated by solving the lattice Boltzmann equation, which is a special discretization of the kinetic Boltzmann equation. After introducing BGK approximation, the general form of lattice Boltzmann equation with external force is as follows.

$f_i(x+c_i∆t,t+∆t)=f_i(x,t)+\frac{∆t}{{\tau }_v}[f_i^{eq}(x,t)-f_i(x,t)]+∆t{c}_i{F}_k$

(7.32)

where ∆t denotes lattice time step, c_i is the discrete lattice velocity in direction i, F_k is the external force in direction of lattice velocity, and ${\tau }_v$ denotes the lattice relaxation time for the flow. The kinetic viscosity υ is defined in terms of its respective relaxation times, that is, $\upsilon =c_s^2({\tau }_v-1/2)$. Note that the limitation 0.5 < τ should be satisfied for both relaxation times to ensure that viscosity and thermal diffusivity are positive. Furthermore, the local equilibrium distribution function determines the type of problem that needs to be solved. It also models the equilibrium distribution functions, which are calculated with Eq. (7.32) for flow field, respectively.

$f_i^{eq}=w_i\rho \left[1+\frac{c_i.u}{c_s^2}+\frac{1}{2}\frac{{(c_i.u)}^2}{c_s^4}-\frac{1}{2}\frac{u^2}{c_s^2}\right]$

(7.33)

where $w_i$ is a weighting factor and ρ is the lattice fluid density.

$\begin{array}{l}F=F_x+F_y\\ F_x=3w_i\rho \left[A\left(-H_y^{2}u+H_x{H}_{y}v\right)+B\left(H\frac{\partial H}{\partial x}\right)\right],\\ F_y=3w_i\rho \left[A\left(-H_x^{2}v+H_x{H}_{y}u\right)+B\left(H\frac{\partial H}{\partial y}\right)\right]\end{array}$

(7.34)

where A and B are $A=\frac{H{a}^2\upsilon }{L^2}$ and $B=M{n}_{\text{F}}{u}_{\text{in}}^2$, respectively. $\mathrm{Re}=\frac{D{u}_{\text{in}}}{\upsilon },Ha={\mu }_0{H}_{0}D\sqrt{\frac{\sigma }{\mu }}$, and $M{n}_{\text{F}}=\frac{{\mu }_0\chi H_0^2}{\rho u_{\text{in}}^2}$ are Reynolds number, Hartmann number, and Magnetic number arising from FHD [3].

$\begin{array}{l}\text{Flow}\text{density}:\rho =\sum _i{f}_i,\\ \text{Momentum}:\rho u=\sum _i{c}_i{f}_i,\end{array}$

(7.35)

${\rho }_{\text{n}\text{f}}={\rho }_{\text{f}}(1-\phi )+{\rho }_{\text{s}}\phi \text{,}$

(7.36)

${\mu }_{\text{nf}}=\frac{{\mu }_f}{{(1-\phi )}^{2.5}}$

(7.37)

$\frac{{\sigma }_{\text{nf}}}{{\sigma }_{\text{f}}}=1+\frac{3\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}-1\right)\phi }{\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}+2\right)-\left(\frac{{\sigma }_{\text{s}}}{{\sigma }_{\text{f}}}-1\right)\phi }$

(7.38)

$C_{\text{f}}=\frac{2}{\mathrm{Re}}\frac{{\rho }_{\text{f}}}{{\rho }_{\text{nf}}}\frac{1}{{\left(1-\phi \right)}^{2.5}}{\left.\frac{\partial U}{\partial Y}\right|}_{X=0}\text{and}\overline{C_{\text{f}}}={\int }_0^L{C}_{\text{f}}dX$

(7.39)

7.2.2. Effects of active parameters

Fig. 7.13 shows the effects of Magnetic number and Reynolds number on velocity profile at x = 0.25L. At x = 0.25L flow is separated from the bottom wall and then reattaches with the lower plate. Finally, the flow at the outlet is again reverted to be fully developed. This phenomenon is more pronounced for higher values of Reynolds number. The effects of Magnetic number and Reynolds number on local skin friction coefficient along the lower and upper walls are shown in Fig. 7.14. The value of local skin friction coefficient at the lower wall increases rapidly near x = 0.2L, where it reaches its maximum value. Very close to the region where the source is located x = 0.25L, a corresponding decrement takes place and at x = 0.25L this parameter takes its minimum negative value. ${\left.C_{\text{f}}\right|}_{Y=1}$ increases near the area of the magnetic source in a smoother way, and its sign does not change as happens with the lower plate where reverse of the flow occurs. The wall shear is more influenced on the lower plate, below of which the magnetic source is located. It is an interesting observation that there are two points with zero values of skin friction for lower plate. In these points there is no skin friction, and this result may be exciting in the case of creation of fibrinoid. Fig. 7.15 shows the effects of Magnetic number and Reynolds number on average skin friction coefficient. The drag acting on the upper plate is greater than that of the lower. Average skin friction coefficient decreases with the increase of Reynolds number and Magnetic number.