**3. Solution by Adomian Decomposition Method**

In this section, the Adomian solution is determined for the velocity field. According to the Adomian decomposition method, Equation (12) can be written in the operator form as

$$L\_r w = \frac{\mathrm{d}p}{\mathrm{d}z} + \mathrm{M}^2 w \tag{14}$$

where the differential operator *Lr* is defined in the form

$$L\_r = \frac{1}{r} \frac{\partial}{\partial r} \left( r\mu(r) \,\,\frac{\partial}{\partial r} \right) \tag{15}$$

and the inverse operator *L*−<sup>1</sup> *<sup>r</sup>* is defined by

$$L\_r^{-1}(.) = \int \left[ \frac{1}{r\mu(r)} \int r(.) \,\mathrm{d}r \right] \mathrm{d}r \tag{16}$$

Applying the inverse operator, Equation (12) takes the form

$$\begin{array}{l} w(r,z) = L\_r^{-1} \left[ \frac{\mathrm{d}p}{\mathrm{d}z} + M^2 w \right] + c\_1 r + c\_2\\ w(r,z) = \frac{\mathrm{d}p}{\mathrm{d}z} I(r) + L\_r^{-1} \left( M^2 w \right) + c\_1 r + c\_2 \end{array} \tag{17}$$

in which

$$L\_r^{-1} \left(\frac{\mathrm{d}p}{\mathrm{d}z}\right) = \int \left[\frac{1}{r\mu(r)} \int r \left(\frac{\mathrm{d}p}{\mathrm{d}z}\right) \mathrm{d}r\right] \mathrm{d}r = \frac{\mathrm{d}p}{\mathrm{d}z} I(r) \tag{17a}$$

and *I*(*r*) is given by

$$I(r) = \int \frac{r}{2\mu(r)} \mathrm{d}r \tag{18}$$

According to Adomian decomposition, it can be written as

$$w = \sum\_{n=0}^{\infty} w\_n \tag{19}$$

Using the Adomian decomposition method, the solution *w*(*r*, *z*) can be elegantly computed by the recurrence relation

$$\begin{array}{rcl} w\_0 &=& c\_1 r + c\_2\\ w\_1 &=& \frac{\mathrm{d}p}{\mathrm{d}z} I(r) + M^2 L\_r^{-1}(w\_0) \\ w\_{n+2} &=& M^2 L\_r^{-1}(w\_{n+1}), \; n \ge 0 \end{array} \tag{20}$$

The above equations give

$$w\_n = M^{2n-2} \left(\frac{\mathrm{d}p}{\mathrm{d}z} + M^2 c\_2\right) \left(L\_r^{-1}\right)^{n-1} I(r) + M^{2n} c\_1 \left(L\_r^{-1}\right)^{n-1} I\_1(r), \ n \ge 1\tag{21}$$

in which

$$I\_1(r) = \int \frac{r^2}{3\mu(r)} \mathrm{d}r \tag{22}$$

With the help of Equations (20) and (21), the closed form of *w* can be written as

$$\begin{aligned} w(r, z) &= w\_0 + \sum\_{n=1}^{\infty} w\_n\\ w(r, z) &= c\_1 \chi(r) + c\_2 \chi\_1(r) + \frac{\mathrm{d}p}{\mathrm{d}z} \chi\_2(r) \end{aligned} \tag{23}$$

Using the boundary conditions (13), the values of constants *c*<sup>1</sup> and *c*<sup>2</sup> can be written as

$$\begin{split} c\_{1} &= \frac{\chi\_{1}(r\_{1}) - \chi\_{1}(r\_{2})}{\chi(r\_{1})\ \chi\_{1}(r\_{2}) - \chi(r\_{2})\ \chi\_{1}(r\_{1})} - \frac{\mathrm{d}p}{\mathrm{d}z} \left[ \frac{\chi\_{2}(r\_{1})}{\chi(r\_{1})} \frac{\chi\_{1}(r\_{2}) - \chi\_{2}(r\_{2})}{\chi\_{1}(r\_{2}) - \chi(r\_{2})} \frac{\chi\_{1}(r\_{1})}{\chi\_{1}(r\_{1})} \right] \\ c\_{2} &= -\frac{1}{\chi\_{1}(r\_{2})} - \frac{\mathrm{d}p}{\mathrm{d}z} \frac{\chi\_{2}(r\_{2})}{\chi\_{1}(r\_{2})} - c\_{1} \frac{\chi(r\_{2})}{\chi\_{1}(r\_{2})} \end{split} \tag{24}$$

where these χ- *<sup>s</sup>* are defined in Appendix A.

The closed form solution (13) is represented in terms of integrals for any kind of general variable viscosity. These integrals can be computed for particular values of variable viscosity μ. Here, three cases of variable viscosity are taken into account, μ(*r*) = 1, *r*, and <sup>1</sup> *r* .
