**2. Mathematical Formulation**

Consider the magnetohydrodynamic flow of an electrically conducting viscous fluid through the gap between two coaxial tubes. It is assumed that a uniform magnetic field *B*<sup>0</sup> is applied transversely to the flow. Further, considering that the magnetic Reynolds number is small, the induced magnetic field is negligible. A schematic diagram of the geometry of the problem under consideration is shown in Figure 1.

**Figure 1.** Effects of an endoscope on peristaltic motion.

The geometry of the wall surface is described as

$$
\overline{R}\_1 = a\_1 \tag{1}
$$

$$
\overline{R}\_2 = a\_2 + b \cos \frac{2\pi}{\lambda} (\overline{Z} - c\overline{t}) \tag{2}
$$

where *a*1, and *a*<sup>2</sup> are the radii of the inner and the outer tubes, respectively; *b* is the amplitude of the wave; λ is the wavelength; *c* is the propagation velocity; and *t* is the time.

In the laboratory frame (*R*, *Z*), the flow is unsteady, but, by introducing a wave frame (*r* , *z*) moving with velocity *c* away from the fixed frame, the flow can be treated as steady [10]. The coordinate frames are related by the transformations.

$$
\overline{z} = \overline{Z} - c\overline{t}, \; \overline{r} = \overline{R}, \; \overline{w} = \overline{\mathcal{W}} - c, \; \overline{u} = \overline{\mathcal{U}} \tag{3}
$$

where (*u*, *<sup>w</sup>*) and -*<sup>U</sup>*, *<sup>W</sup>* are the velocity components in radial and axial directions in moving and fixed coordinates, respectively. Using the transformations (3), the equations that govern the flow are

$$
\frac{\partial \overline{u}}{\partial \overline{r}} + \frac{\partial \overline{w}}{\partial \overline{z}} + \frac{\overline{u}}{\overline{r}} = 0 \tag{4}
$$

$$\begin{split} \rho \left[ \overline{u} \frac{\partial \overline{u}}{\partial r} + \overline{w} \frac{\partial \overline{u}}{\partial \overline{z}} \right] &= \quad -\frac{\partial \overline{p}}{\partial \overline{r}} + \frac{\partial}{\partial \overline{r}} \Big( 2\mu (\overline{r}) \left. \frac{\partial \overline{u}}{\partial \overline{r}} \right) \\ &+ \frac{2\mu (\overline{r})}{\overline{r}} \Big[ \frac{\partial \overline{u}}{\partial \overline{r}} - \frac{\overline{u}}{\overline{r}} \Big] \\ &+ \frac{\partial}{\partial \overline{z}} \Big( \mu (\overline{r}) \left( \frac{\partial \overline{u}}{\partial \overline{z}} + \frac{\partial \overline{w}}{\partial \overline{r}} \right) \Big) \end{split} \tag{5}$$

$$\begin{split} \partial \left[ \overline{u} \frac{\partial \overline{u}}{\partial \overline{\tau}} + \overline{w} \frac{\partial \overline{w}}{\partial \overline{z}} \right] &= \begin{split} -\frac{\partial \overline{p}}{\partial \overline{z}} + \frac{\partial}{\partial \overline{z}} \Big( 2\mu (\overline{r}) \frac{\partial \overline{w}}{\partial \overline{z}} \Big) \\ + \frac{1}{\overline{r}} \frac{\partial}{\partial \overline{r}} \Big( \overline{r} \mu (\overline{r}) \left( \frac{\partial \overline{u}}{\partial \overline{z}} + \frac{\partial \overline{w}}{\partial \overline{r}} \right) \Big) \\ - \sigma B\_0^2 (\overline{w} + c) \end{split} \tag{6}$$

where *u* and *w* are the velocity components in the *r* and *z* directions, respectively; ρ is the density; σ is the electrically conductivity of the fluid; and μ is the variable viscosity. The governing equations can be dimensionalized by the following non-dimensional parameters.

$$\begin{aligned} r &= \frac{\overline{r}}{a\_2}, z = \frac{\overline{z}}{\overline{\lambda}}, w = \frac{\overline{w}}{c}, u = \frac{\lambda \overline{u}}{a\_2 c}, p = \frac{a\_2^2 \overline{p}}{\lambda c \mu}, r\_1 = \frac{\overline{r}\_1}{a\_2}, \ \delta = \frac{a\_2}{\lambda} \\\ r\_2 &= \frac{\overline{r}\_2}{a\_2} = 1 + \varphi \cos(2\pi z), \ Re = \frac{\rho c a\_2}{\mu}, M = \sqrt{\frac{\underline{a}}{\mu}} B\_0 a\_2 \end{aligned} \tag{7}$$

where φ is the amplitude ratio, Re is the Reynolds number, δ is the dimensionless wave number, and *M* is the magnetic parameter.

Using the above non-dimensional parameters in Equations (4)–(6), the following system of equations is obtained.

$$\frac{1}{r}\frac{\partial (ru)}{\partial r} + \frac{\partial w}{\partial z} = 0 \tag{8}$$

$$\begin{split} \text{Re}\delta^3 \Big[ u \frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{w} \frac{\partial \boldsymbol{u}}{\partial z} \Big] &= \quad -\frac{\partial p}{\partial r} + \boldsymbol{\delta^2} \frac{\partial}{\partial r} \Big( 2\mu(r) \left. \frac{\partial \boldsymbol{u}}{\partial r} \right|\_{} + \boldsymbol{\delta^2} \Big( \frac{2\mu(r)}{r} \left( \frac{\partial \boldsymbol{u}}{\partial r} - \frac{\boldsymbol{u}}{r} \right) \Big) \\ &\quad + \boldsymbol{\delta^2} \Big( \frac{\partial}{\partial z} \Big( \mu(r) \left( \frac{\partial \boldsymbol{u}}{\partial z} \boldsymbol{\delta^2} + \frac{\partial \boldsymbol{w}}{\partial r} \right) \Big) \Big) \end{split} \tag{9}$$

$$\begin{split} \delta \text{Re} \Big[ \mu \frac{\partial w}{\partial r} + w \frac{\partial w}{\partial z} \Big] &= -\frac{\partial p}{\partial z} + \delta^2 \frac{\partial}{\partial z} \Big( 2\mu(r) \left. \frac{\partial w}{\partial z} \right| \\ &+ \frac{1}{r} \frac{\partial}{\partial r} \Big( r\mu(r) \left( \frac{\partial \mu}{\partial z} \delta^2 + \frac{\partial w}{\partial r} \right) \Big) - M^2 w \end{split} \tag{10}$$

Using the long wavelength approximation and dropping terms of order δ and higher, the above equations reduce to

$$\frac{\partial p}{\partial r} = 0\tag{11}$$

$$\frac{1}{r}\frac{\partial}{\partial r}\Big(r\mu(r)\Big.\frac{\partial w}{\partial r}\Big)=\frac{\partial p}{\partial z}+M^2w\tag{12}$$

The relevant boundary conditions in new parameters are

$$\begin{cases} w = -1 & at \quad r = r\_1 \\ w = -1 & at \quad r = r\_2 \end{cases} \tag{13}$$
