4. Zeroth-order deformation problem

The equations of zeroth-order deformation problem are obtained as

$$(1 - p)L\_f[f(\eta; p) - f\_0(\eta)] = p h\_f N\_f[f(\eta; p)].\tag{29}$$

$$(1 - p)\mathcal{L}\_{\theta}[\theta(\eta; p) - \theta\_0(\eta)] = p h\_{\theta} \mathcal{N}\_{\theta}[f(\eta; p), \theta(\eta; p)],\tag{30}$$

$$(1 - p)L\_{\phi}[\phi(\eta; p) - \phi\_0(\eta)] = p\hbar\_{\phi}N\_{\phi}[f(\eta; p), \phi(\eta; p)].\tag{31}$$

Here hf , h<sup>θ</sup> and h<sup>ϕ</sup> are auxiliary nonzero parameters. The corresponding boundary conditions are written as

$$\left.f(\eta;p)\_{\eta=1} = 1, \frac{\partial f(\eta;p)}{\partial \eta}\right|\_{\eta=1} = 1, \left.\theta(\eta;p)\right|\_{\eta=1} = 1, \phi(\eta;p)\big|\_{\eta=1} = 1,\tag{32}$$

$$\left. \frac{\partial^2 f(\eta; p)}{\partial \eta} \right|\_{\eta = \beta} = 0, \left. \frac{\partial \theta(\eta; p)}{\partial \eta} \right|\_{\eta = \beta} = 0, \left. \frac{\partial \phi(\eta; p)}{\partial \eta} \right|\_{\eta = \beta} = 0. \tag{33}$$

Since

$$p = 0 \Rightarrow f(\eta; 0) = f\_0(\eta) = \eta \lrcorner \theta(\eta; 0) = \theta\_0(\eta) = 1, \phi(\eta; 0) = \phi\_0(\eta) = 1,\tag{34}$$

The Heat and Mass Transfer Analysis During Bunch Coating of a Stretching Cylinder by Casson Fluid 47 http://dx.doi.org/10.5772/intechopen.79772

$$p = 1 \Rightarrow f(\eta; 1) = f(\eta), \theta(\eta; 1) = \theta(\eta), \phi(\eta; 1) = \phi(\eta). \tag{35}$$

Using the Taylor's expansions of fðη; pÞ, θ ηð ; pÞ and ϕ ηð ; pÞ about p ¼ 0 in Eqs. (28)–(31), we obtained

$$f(\eta; p) = f\_0(\eta) + \sum\_{w=1}^{\infty} f\_w(\eta) p^w. \tag{36}$$

$$\Theta(\eta; p) = \Theta\_0(\eta) + \sum\_{w=1}^{\infty} \Theta\_w(\eta) p^w,\tag{37}$$

$$\phi(\eta; p) = \phi\_0(\eta) + \sum\_{w=1}^{\infty} \phi\_w(\eta) p^w,\tag{38}$$

where

$$f\_w(\eta) = \frac{1}{w!} \frac{\partial^w f}{\partial \eta^w} \Big|\_{\eta=0}, \theta\_w(\eta) = \frac{1}{w!} \frac{\partial^w \theta \,(\eta; p)}{\partial \eta^w} \Big|\_{p=0}, \phi\_w(\eta) = \frac{1}{w!} \frac{\partial^w \phi \,(\eta; p)}{\partial \eta^w} \Big|\_{p=0}.\tag{39}$$

The convergence of series depends on hf , hθ, and hφ. So let us suppose that series converges at p ¼ 1 for some values of hf , hθ, and hφ, then Eqs. (35)–(37) become

$$f(\eta) = f\_0(\eta) + \sum\_{w=1}^{\infty} f\_w(\eta),\tag{40}$$

$$\Theta(\eta) = \Theta\_0(\eta) + \sum\_{w=1}^{\infty} \Theta\_w(\eta), \tag{41}$$

$$
\phi(\eta) = \phi\_0(\eta) + \sum\_{w=1}^{\infty} \phi\_w(\eta). \tag{42}
$$
