*3.1. Case 1 (When* μ = *1)*

With the help of Equations (16), (18) and (22), the following are obtained:

$$I(r) = \frac{r}{2} \text{d}r = \frac{r^2}{4} \tag{25}$$

$$\left(L\_r^{-1}\right)^n I(r) = \frac{\left(\frac{r}{2}\right)^{2n+2}}{\left[\left(n+1\right)!\right]^2}, \ n \ge 1, \ 2, \ 3, \dots \tag{26}$$

$$\left(L\_r^{-1}\right)^n I(r) = \frac{\left(\frac{r}{2}\right)^{2n+2}}{\left[(n+1)!\right]^2}, \ n \ge 1, \ 2, \ 3, \dots \tag{27}$$

$$I\_1(r) = \int \frac{r^2}{3} \mathrm{d}r = \frac{r^3}{3^2},$$

$$\left(L\_r^{-1}\right)^n I\_1(r) = \sum\_{n=0}^\infty \frac{r^{2n+3}}{3^2 \cdot 5^2 \cdot 7^2 \dots \left(2n+3\right)^2} \tag{28}$$

The closed form of *w*(*r*, *z*) can be written as

$$w(r,z) = c\_1 \chi\_3(r) + c\_2 I\_0(Mr) + \frac{1}{M^2} \frac{\mathrm{d}p}{\mathrm{d}z} (-1 + I\_0(Mr))\tag{29}$$

Using boundary conditions (13), the solution of (29) can be written as

$$w(r,z) = a\_{14}\chi\_3(r) + a\_{15}I\_0(Mr) + \frac{1}{M^2}\frac{d\mu}{dz}(-1 - a\_{15}I\_0(Mr) - a\_{14}\chi\_3(r))\tag{30}$$

The constants appearing in the above equations are defined in the equations and *I*<sup>0</sup> are the modified Bessel functions, with the first kind of order 0.

#### 3.1.1. Volume Flow Rate and Pressure Rise

The instantaneous volume flow rate *Q*(*z*) is given by

$$\overline{Q}(z) = \int\_{r\_1}^{r\_2} r w(r, z) \,\mathrm{d}r = \frac{\mathrm{d}p}{\mathrm{d}z} a\_{22} + a\_{20} + a\_{21} \tag{31}$$

From Equation (31), the following is obtained:

$$\frac{d\underline{p}}{dz} = \frac{1}{a\_{22}} (\overline{Q}(z) - a\_{20} - a\_{21}) \tag{32}$$

The volume flow *Q* over a period is obtained as

$$Q = \overline{Q}(z) + \left(1 + \frac{\varphi^2}{2}\right) - r\_1^2 \tag{33}$$

and

$$\frac{\mathrm{d}p}{\mathrm{d}z} = \frac{1}{a\_{22}} \Big( \mathbb{Q} - \left( 1 + \frac{\mathrm{q}^2}{2} \right) + r\_1^2 - a\_{20} - a\_{21} \Big) \tag{34}$$

The pressure rise Δ*p* and the friction force (at the wall) on the outer and inner tubes are *F*(0) and *F*(1), respectively, are

$$
\Delta p = \int\_0^1 \frac{\mathrm{d}p}{\mathrm{d}z} \mathrm{d}z \tag{35}
$$

$$\begin{aligned} F^{(0)} &= \int\_0^1 r\_2^2 \left( -\frac{\mathrm{d}p}{\mathrm{d}z} \right) \mathrm{d}z \\ F^{(1)} &= \int\_0^1 r\_1^2 \left( -\frac{\mathrm{d}p}{\mathrm{d}z} \right) \mathrm{d}z \end{aligned} \tag{36}$$

3.1.2. Stream Function

The corresponding stream function - *<sup>u</sup>* = <sup>−</sup><sup>1</sup> *r* ∂Ψ <sup>∂</sup>*<sup>z</sup>* and *<sup>w</sup>* <sup>=</sup> <sup>1</sup> *r* ∂Ψ ∂*r* can be written as

$$\Psi = a\_{14}g(r) - \frac{1}{M^2} \frac{\mathrm{d}p}{\mathrm{d}z} \Big( a\_{14}g(r) + \frac{r^2}{2} + a\_{15} \frac{r}{M} I\_1(Mr) \Big) + a\_{15} \frac{r}{M} I\_1(Mr) \tag{37}$$

where the constants appears in the above equations are defined in Appendix A; *I*<sup>1</sup> is a modified Bessel functions of the first order; and *a*14, *a*<sup>15</sup> are defined in Appendix A.
