3.2.2. Stream Function

Stream function, in this case, is defined as

$$\Psi = b\_{14}\mathbf{g}\_0(r) + b\_{18}\mathbf{g}\_1(r) + \frac{\mathbf{d}p}{\mathbf{d}z}(\mathbf{g}\_2(r) + b\_{19}\mathbf{g}\_1(r) - b\_{15}\mathbf{g}\_0(r))\tag{46}$$

*3.3. Case 3 (When* μ = <sup>1</sup> *r )*

Using the similar procedure as discussed in previous sections, it can be written as

$$w(r,z) = d\_{16}\chi\_{\gamma}(r) + d\_{18}\chi\_{8}(r) + \frac{\mathrm{d}p}{\mathrm{d}z}(\chi\_{\theta}(r) + d\_{19}\chi\_{8}(r) - d\_{17}\chi\_{\gamma}(r))\tag{47}$$

The constants appearing in the above equations are defined in Appendix A.

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

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

$$\overline{Q}(z) = \frac{\mathrm{d}p}{\mathrm{d}z}d\sharp + d\sharp\_{26} + d\sharp\_{25} \tag{48}$$

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

$$Q = \overline{Q}(z) + \left(1 + \frac{\varphi^2}{2}\right) - r\_1^2 \tag{49}$$

The pressure rise Δ*p* and the friction force *F*(0) and *F*(1) can be computed using (35) and (36).

$$\frac{\mathrm{d}p}{\mathrm{d}z} = \frac{1}{d\_{27}} \Big( \mathbb{Q} - \left( 1 + \frac{\mathrm{q}^2}{2} \right) + r\_1^2 - d\_{25} - d\_{26} \Big) \tag{50}$$

3.3.2. Stream Function

Stream function for this case is

$$\Psi = d\_{16}h(r) + d\_{18}h\_1(r) + \frac{\mathrm{d}p}{\mathrm{d}z}(h\_2(r) + h\_1(r)d\_{19} - h(r)\,d\_{17})\tag{51}$$

The constants appearing in the above equations are defined in Appendix A.
