After double integration of two equations (15) in *α*_{2} direction we apply the low heat flux continuity (*q*_{p} = *q*_{c}) in the contact region between the elastic alloy layer and lubricant, and finally we obtain the external displacement of the sleeve surface in the following dimensional form:

$$\begin{array}{}{u}_{02}={u}_{2}\left({\alpha}_{1},{\alpha}_{2}={c}_{2},{\alpha}_{3}\right)=\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}={\epsilon}_{s}\frac{1+\vartheta}{1-\vartheta}\cdot \frac{1-2\vartheta}{E}p-\left[{f}_{p}-\frac{1}{2}\frac{{\epsilon}_{s}{q}_{p}}{{\kappa}^{\ast}}\right]{\epsilon}_{s}{\alpha}_{T}^{\ast}\frac{1+\vartheta}{1-\vartheta}+\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\frac{1}{2}\cdot \frac{1-2\vartheta}{E}\cdot \frac{1+\vartheta}{1-\vartheta}{\epsilon}_{s}^{2}\left[{\mathbf{N}}^{\ast}\mathrm{\nabla}{B}_{2}^{\ast}+{\left({\mathbf{J}}^{\ast}\times {\mathbf{B}}^{\ast}\right)}_{2}\right]\end{array}$$(25)

where *ϵ*_{s} = *c*_{3} – *c*_{2}, 0 < *α*_{1} < 2*π*, –*b*_{m} < *α*_{3} < +*b*_{m}. Solution (25) valid if and only if *E*_{Y} - Young modulus and solid body coefficients *α*_{T}, *κ*^{*}, *ϑ* - Poisson ratio, are independent of *α*_{2}, The hydrodynamic pressure p, the load carrying capacity and temperature T distributions with oil velocity components v_{1}, v_{2}, v_{3} in machine slide journal bearing are solved by the means of the quasi half numerical Frobenius small parameter method implemented by the finite difference method using Mathcad 15 Professional Program. A system of partial differential equations (12)-(14) defined in thin space (*α*_{1}, *α*_{2}, *α*_{3}, *t*) between two movable rotational surfaces and for continuous *m*(*n*), *η* == *m*(*n* = 1), has an analytical solution in the form of the following infinite uniform convergent power function series in following form:

$$\begin{array}{}{v}_{i}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})={v}_{i}^{(0)}+g{v}_{i}^{(1)}+\dots +{g}^{k}{v}_{i}^{(k)}+\dots \\ T({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})={T}^{(0)}+g{T}^{(1)}+\dots +{g}^{k}{T}^{(k)}+\dots \end{array}$$(26a)

$$\begin{array}{}p({\alpha}_{1},{\alpha}_{3})={p}^{(0)}+g{p}^{(1)}+\dots +{g}^{k}{p}^{(k)}+\dots \\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{q}_{c}={q}_{c}^{(0)}+g{q}_{c}^{(1)}+\dots +{g}^{k}{q}_{c}^{(k)}+\dots \end{array}$$(26b)

where
$\begin{array}{}{q}_{c}^{(0)}={q}_{c}^{(1)}=\dots ={q}_{c}^{(k)}\equiv q,\end{array}$ with *q*_{c} = *q*/(1 – *g*); for *i* = 1,2,3; *k* = 0,1,2,…, *g* = *g(n)*≡ *(n*-1)/2 where for 0 < *n* ≤ 3/2 the small parameter *g* is less than +1/4 and greater than –1/2. In order to achieve the linearization of apparent viscosity (16), we put functions (26a) into Eq. (16) and expand function in closed interval [1, *n*] or [*n*, 1] for 0 < *n* ≤ 3/2 using Taylor series in the neighborhood of point *n* = 1 with respect to the small parameter in the following form:

$$\begin{array}{}{\eta}_{p}\left({\alpha}_{1},{\alpha}_{2},{\alpha}_{3}\right)={\eta}_{0}\cdot \left[{\eta}_{1}\left(T,B,E\right)\right]\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\cdot \left(1+g{\eta}_{pr1}+\dots +{g}^{k}{\eta}_{prk}+\dots \right)\end{array}$$(27)

where *η*_{1}(T, B, E) = *η*_{1}(*α*_{1}, *α*_{2}, *α*_{3}) and functions *η*_{prk} ≡ *η*_{prk}(*v*_{1}, *v*_{3}) for *k* = 1, 2, … – are the dimensionless expansion coefficients of dynamic viscosity dependent on *α*_{1}, *α*_{2}, *α*_{3}, whereas for *k* = 0 *η*_{pr0} = 1. Such functions after involved calculations are determined in analytical form. Putting the infinite series (26), (27) into the system (12)-(14), multiplying the series by Cauchy methods and equating the terms by the same powers *k* = 0,1,2,… of the small parameter g, and then using the L. Kronecker symbol *i*.*e*. *δ*_{k0} = 1 for *k* = 0, *δ*_{k0} = 0 for *k* = 1,2,…, we obtain the linear differential systems of equations for *k* = 0,1,2,… *i* = 1,3 in following form:

$$\begin{array}{}\frac{1}{{h}_{i}}\frac{\mathrm{\partial}{p}^{(k)}}{\mathrm{\partial}{\alpha}_{1}}=\frac{\mathrm{\partial}}{\mathrm{\partial}{\alpha}_{2}}\left[{\eta}_{0}{\eta}_{1}\left(B,E,{T}^{(k)}\right)\frac{\mathrm{\partial}{v}_{i}^{(k)}}{\mathrm{\partial}{\alpha}_{2}}\right]+\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}+\frac{\mathrm{\partial}}{\mathrm{\partial}{\alpha}_{2}}\left[{\eta}_{0}{\eta}_{1}\left(B,E,{T}^{(k)}\right){S}_{i}^{(k)}\right]+{M}_{i}{\delta}_{k0},\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\frac{\mathrm{\partial}{p}^{(k)}}{\mathrm{\partial}{\alpha}_{2}}=0,\end{array}$$(28)

$$\begin{array}{}& \frac{1}{{h}_{1}}\frac{\mathrm{\partial}}{\mathrm{\partial}{\alpha}_{1}}\left({v}_{1}^{(k)}\right)+\frac{\mathrm{\partial}}{\mathrm{\partial}{\alpha}_{2}}\left({v}_{2}^{(k)}\right)\\ & +\frac{1}{{h}_{1}{h}_{3}}\frac{\mathrm{\partial}}{\mathrm{\partial}{\alpha}_{3}}\left({h}_{1}{v}_{3}^{(k)}\right)=0,\end{array}$$(29)

$$\begin{array}{}\frac{\mathrm{\partial}}{\mathrm{\partial}\phantom{\rule{1pt}{0ex}}{\alpha}_{2}}\phantom{\rule{1pt}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(\kappa \frac{\mathrm{\partial}{T}^{(k)}}{\mathrm{\partial}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{1pt}{0ex}}{\alpha}_{2}}\right)=-{\eta}_{0}{\eta}_{1}\left(B,E,{T}^{(k)}\right)\phantom{\rule{thickmathspace}{0ex}}{V}_{T}^{(k)}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+{M}_{T}{\delta}_{k0}.\end{array}$$(30)

with

$$\begin{array}{}{\eta}_{1}\left(B,E,{T}^{(k)}\right)=\left[{\eta}_{1BE}\left(B,E\right)\right]{e}^{-{\delta}_{T}{T}^{(k)}\left({\alpha}_{1},{\alpha}_{2},{\alpha}_{3}\right)},\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\eta}_{1BE}\left(B,E\right)\equiv {\eta}_{1BE}\left({\alpha}_{1},{\alpha}_{3}\right),\phantom{\rule{1em}{0ex}}{S}_{i}^{(0)}=0,\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{V}_{T}^{(0)}\equiv V\left({v}_{1}^{(0)},{v}_{3}^{(0)}\right),\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{S}_{i}^{(1)}\equiv \frac{\mathrm{\partial}{v}_{i}^{(0)}}{\mathrm{\partial}{\alpha}_{2}}\left[{\left(\frac{{\epsilon}_{0}}{{v}_{0}}\right)}^{{}^{2}}{V}_{T}^{(0)}\right],\end{array}$$(31a)

$$\begin{array}{}{V}_{T}^{(1)}\equiv {V}_{T}^{(0)}\cdot \mathrm{ln}\left[{\left(\frac{{\epsilon}_{0}}{{v}_{0}}\right)}^{{}^{2}}{V}_{T}^{(0)}\right]\\ \phantom{\rule{2em}{0ex}}+2\left(\frac{\mathrm{\partial}{v}_{1}^{(0)}}{\mathrm{\partial}{\alpha}_{2}}\right)\left(\frac{\mathrm{\partial}{v}_{1}^{(1)}}{\mathrm{\partial}{\alpha}_{2}}\right)+2\left(\frac{\mathrm{\partial}{v}_{3}^{(0)}}{\mathrm{\partial}{\alpha}_{2}}\right)\left(\frac{\mathrm{\partial}{v}_{3}^{(1)}}{\mathrm{\partial}{\alpha}_{2}}\right),\end{array}$$(31b)

For *k* = 0 the abovementioned system of equations (28)-(31) determines functions:
$\begin{array}{}{v}_{i}^{(0)},{T}_{i}^{(0)},{p}^{(0)}\end{array}$ for Newtonian oil taking into account the boundary conditions in steady motion (18), (19). For *k* = 1,2,… and *i* = 1,2,3 the system of equations (28)-(31) determines the first, second,… corrections:
$\begin{array}{}{v}_{i}^{(k)},\end{array}$ *T*^{(k)}, *p*^{(k)} caused by the non-Newtonian oil. The boundary conditions for *k* = 0 are as follows:

$$\begin{array}{}{v}_{1}^{(0)}=\omega \phantom{\rule{thickmathspace}{0ex}}{h}_{1},\phantom{\rule{thinmathspace}{0ex}}{v}_{2}^{(0)}=0,{v}_{3}^{(0)}=0,\phantom{\rule{thickmathspace}{0ex}}{T}^{(0)}={T}_{0}+{f}_{c}\end{array}$$(32a)

for *α*_{2} = 0,

$$\begin{array}{}{v}_{1}^{(0)}=0,\phantom{\rule{thinmathspace}{0ex}}{v}_{2}^{(0)}=0,\phantom{\rule{thinmathspace}{0ex}}{v}_{3}^{(0)}=0,\phantom{\rule{thickmathspace}{0ex}}{T}^{(0)}={T}_{0}+{f}_{p}^{(0)}\end{array}$$

for *α*_{2} = *ε*_{T};

and for k = 1,2,… we have:

$$\begin{array}{}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{v}_{i}^{(k)}=0,\phantom{\rule{thickmathspace}{0ex}}{T}^{(k)}=s{f}_{p}^{k},\\ \text{for}\phantom{\rule{1em}{0ex}}s={\alpha}_{2}/{\epsilon}_{T},{\alpha}_{2}=0,{\alpha}_{2}={\epsilon}_{T};\end{array}$$(32b)

The total gap height *ϵ*_{T} includes surface displacement (25). Integrating twice (28) and (29) once with respect to the variable *α*_{2}, then the oil velocity components for *k* = 0,1,2,… by virtue of boundary conditions (32a, b) have the following form:

$$\begin{array}{}{\displaystyle {v}_{i}^{(k)}=\frac{1}{{\eta}_{0}{\eta}_{1BE\phantom{\rule{thickmathspace}{0ex}}}}\cdot \frac{1}{{h}_{i}}\frac{\mathrm{\partial}{p}^{(k)}}{\mathrm{\partial}{\alpha}_{i}}{K}_{i}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})+}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\left[{K}_{2}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})\right]\cdot \underset{0}{\overset{{\epsilon}_{T}}{\int}}{S}_{i}^{(k)}d{\alpha}_{2}-\underset{0}{\overset{{\alpha}_{2}}{\int}}{S}_{i}^{(k)}d{\alpha}_{2}+}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\phantom{\rule{thickmathspace}{0ex}}\omega {h}_{1}[1-{K}_{2}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})]{\delta}_{i1}{\delta}_{k0}+{\delta}_{k0}\mathit{\Delta}{M}_{i},}\end{array}$$(33)

$$\begin{array}{}{\displaystyle {v}_{2}^{(k)}=-\frac{1}{{h}_{1}}\underset{0}{\overset{{\alpha}_{2}}{\int}}\frac{\mathrm{\partial}{v}_{1}^{(k)}}{\mathrm{\partial}{\alpha}_{1}}d{\alpha}_{2}}\\ {\displaystyle \phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}-\frac{1}{{h}_{1}{h}_{3}}\underset{0}{\overset{{\alpha}_{2}}{\int}}\frac{\mathrm{\partial}}{\mathrm{\partial}{\alpha}_{3}}\left({h}_{1}{v}_{3}^{(k)}\right)\phantom{\rule{thickmathspace}{0ex}}d{\alpha}_{2},}\end{array}$$(34)

for *i* = 1, 3; 0 < *α*_{21} < *α*_{2} < *ϵ*_{T}, 0 < *α*_{1} < 2*π*, –*b*_{m} < *α*_{3} < +*b*_{m}, whereas we denote:

$$\begin{array}{}{\displaystyle \mathit{\Delta}{M}_{i}=-\frac{1}{{\eta}_{0}{\eta}_{1BE}}\underset{0}{\overset{{\alpha}_{2}}{\int}}\underset{0}{\overset{{\alpha}_{21}}{\int}}{M}_{i}d{\alpha}_{21}d{\alpha}_{2}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}+\frac{1}{{\eta}_{0}{\eta}_{1BE}}s\underset{0}{\overset{{\epsilon}_{T}}{\int}}\underset{0}{\overset{{\alpha}_{21}}{\int}}{M}_{i}d{\alpha}_{21}d{\alpha}_{2}=\frac{{\epsilon}_{T}^{2}}{2{\eta}_{0}{\eta}_{1BE}}{M}_{i}(s-{s}^{2}),}\end{array}$$(35a)

$$\begin{array}{}{\displaystyle {K}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})\equiv \frac{det\u2225{\mathit{\Omega}}^{(k)}\u2225}{{\mathit{\Omega}}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2}={\epsilon}_{T},{\alpha}_{3})},}\\ {\displaystyle {K}_{2}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})\equiv \frac{{\mathit{\Omega}}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})}{{\mathit{\Omega}}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2}={\epsilon}_{T},{\alpha}_{3})},}\end{array}$$(35b)

$$\begin{array}{}{\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\u2225{\mathit{\Omega}}^{(k)}\u2225\equiv}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\u2225\begin{array}{cc}{\mathit{\Omega}}_{2}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})& {\mathit{\Omega}}_{2}^{(k)}({\alpha}_{1},{\alpha}_{2}={\epsilon}_{T},{\alpha}_{3})\\ {\mathit{\Omega}}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})& {\mathit{\Omega}}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2}={\epsilon}_{T},{\alpha}_{3})\end{array}\u2225,}\\ {\displaystyle {\mathit{\Omega}}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})\equiv \underset{0}{\overset{{\alpha}_{2}}{\int}}{e}^{-{\delta}_{T}\cdot {T}^{(k)}}d{\alpha}_{2},\phantom{\rule{thickmathspace}{0ex}}{\mathit{\Omega}}_{2}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\equiv \underset{0}{\overset{{\alpha}_{2}}{\int}}{\alpha}_{2}\cdot {e}^{-{\delta}_{T}\cdot {T}^{(k)}}d{\alpha}_{2}.}\end{array}$$(35c)

It is easy to see, that:

$$\begin{array}{}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{K}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2}=0,{\alpha}_{3})=0,\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{K}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2}={\epsilon}_{T},{\alpha}_{3})=0,\\ \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{K}_{2}^{(k)}({\alpha}_{1},{\alpha}_{2}=0,{\alpha}_{3})=0,\phantom{\rule{1em}{0ex}}{K}_{2}^{(k)}({\alpha}_{1},{\alpha}_{2}={\epsilon}_{T},{\alpha}_{3})=1.\end{array}$$(35d)

Imposing the boundary condition (32ab) in point *α*_{2} = *ϵ*_{T} on the solution (34) and taking into account the law of integral differentiation with variable limits of integration, we obtain the following modified Reynolds equation, which determines the unknown function *p*^{(k)}(*α*_{1}, *α*_{3}):

$$\begin{array}{}{\displaystyle \frac{1}{{h}_{1}}\frac{\mathrm{\partial}}{\mathrm{\partial}{\alpha}_{1}}\left\{\frac{1}{{\eta}_{0}{\eta}_{1BE}({\alpha}_{1},{\alpha}_{3})}\frac{\mathrm{\partial}{p}^{(k)}({\alpha}_{1},{\alpha}_{3})}{\mathrm{\partial}{\alpha}_{1}}\right.}\\ {\displaystyle \left.\underset{0}{\overset{{\epsilon}_{T}}{\int}}{K}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})d{\alpha}_{2}\right\}+\frac{1}{{h}_{3}}\frac{\mathrm{\partial}}{\mathrm{\partial}{\alpha}_{3}}\left\{\frac{1}{{\eta}_{0}{\eta}_{1BE}({\alpha}_{1},{\alpha}_{3})}\frac{{h}_{1}}{{h}_{3}}\right.}\\ {\displaystyle \left.\frac{\mathrm{\partial}{p}^{(k)}({\alpha}_{1},{\alpha}_{3})}{\mathrm{\partial}{\alpha}_{3}}\underset{0}{\overset{{\epsilon}_{T}}{\int}}{K}_{1}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})d{\alpha}_{2}\right\}+}\\ {\displaystyle +\frac{\mathrm{\partial}}{\mathrm{\partial}{\alpha}_{1}}\left[{\mathit{\Pi}}_{NNi}^{(k)}-\frac{{\delta}_{k0}{\epsilon}_{T}^{3}{M}_{1}}{{\eta}_{0}{\eta}_{1BE}({\alpha}_{1},{\alpha}_{3})}\right]}\\ {\displaystyle +\frac{\mathrm{\partial}{h}_{1}}{{h}_{3}\mathrm{\partial}{\alpha}_{3}}\left[{\mathit{\Pi}}_{NN3}^{(k)}-\frac{{\delta}_{k0}{\epsilon}_{T}^{3}{M}_{3}}{{\eta}_{0}{\eta}_{1BE}({\alpha}_{1},{\alpha}_{3})}\right]=}\\ {\displaystyle =\omega {h}_{1}\frac{\mathrm{\partial}}{\mathrm{\partial}{\alpha}_{1}}\left[\underset{0}{\overset{{\epsilon}_{T}}{\int}}{K}_{2}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})d{\alpha}_{2}-{\epsilon}_{T}\right],}\end{array}$$(36)

with

$$\begin{array}{}{\displaystyle {\mathit{\Pi}}_{NNi}^{(k)}({\alpha}_{1},{\alpha}_{3})\equiv \left[\underset{0}{\overset{{\epsilon}_{T}}{\int}}{S}_{i}^{(k)}d{\alpha}_{2}\right]}\\ {\displaystyle \left[\underset{0}{\overset{{\epsilon}_{T}}{\int}}{K}_{2}^{(k)}({\alpha}_{1},{\alpha}_{2},{\alpha}_{3})d{\alpha}_{2}\right]-\left[\underset{0}{\overset{{\epsilon}_{T}}{\int}}\left(\underset{0}{\overset{{\alpha}_{2}}{\int}}{S}_{i}^{(k)}d{\alpha}_{2}\right)d{\alpha}_{2}\right],}\end{array}$$(37)

for i = 1,3; k = 1,2,3.

To obtain the temperature functions *T*^{(k)}(*α*_{1}*α*_{2}, *α*_{3}) for k = 0,1,2,…, we put solutions of oil velocity components (33), (34) for i = 1,3; into the temperature equation (30). After double integration with respect to the variable *α*_{2} *i*.*e*. in film thickness direction, using boundary conditions (19ab) in the form: T^{(k)} = *f*_{c}*δ*_{k0} for *α*_{2} = 0, T^{(k)} = *f*_{p}*δ*_{k0} for (*α*_{2} = *ϵ*_{T}, and after the term ordering, we finally obtain:

$$\begin{array}{}{\displaystyle {T}^{(k)}={\eta}_{0}{\eta}_{1BE}\left\{s\underset{0}{\overset{{\epsilon}_{T}}{\int}}\left[\frac{1}{\kappa}\underset{0}{\overset{{\alpha}_{2}}{\int}}{V}_{T}^{(k)}{e}^{-{\delta}_{T}{T}^{(k)}}d{\alpha}_{2}\right]d{\alpha}_{2}\right.}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\left.-\underset{0}{\overset{{\alpha}_{2}}{\int}}\left[\frac{1}{\kappa}\underset{0}{\overset{{\alpha}_{2}}{\int}}{V}_{T}^{(k)}{e}^{-{\delta}_{T}{T}^{(k)}}d{\alpha}_{2}\right]d{\alpha}_{2}\right\}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}+\left\{\underset{0}{\overset{{\alpha}_{2}}{\int}}\left[\frac{1}{\kappa}\underset{0}{\overset{{\alpha}_{2}}{\int}}{M}_{T}\phantom{\rule{thickmathspace}{0ex}}d{\alpha}_{2}\right]d{\alpha}_{2}\right.}\\ {\displaystyle \phantom{\rule{2em}{0ex}}\left.-s\underset{0}{\overset{{\epsilon}_{T}}{\int}}\left[\frac{1}{\kappa}\underset{0}{\overset{{\alpha}_{2}}{\int}}{M}_{T}\phantom{\rule{thickmathspace}{0ex}}d{\alpha}_{2}d{\alpha}_{2}\right]d{\alpha}_{2}\right\}{\delta}_{k0}}\\ {\displaystyle \phantom{\rule{2em}{0ex}}+{\delta}_{k0}\left[{f}_{c}\left(1-s\right)+{f}_{p}s\right],}\end{array}$$(38)

where *s* ≡ *α*_{2}/*ϵ*_{T}, 0 ≤ *s* ≤ 1. Non linear differential equation (38) determines temperature functions T^{(k)} for k = 0,1,2,… Imposing the condition (19c) on the temperature T^{(k)} determined by the (38), we obtain the unknown temperature *f*_{p} (*α*_{1}, *α*_{3}) on the sleeve.

To obtain the oil velocity components (33), hydrodynamic pressure (36) and temperature (38) for Newtonian oil, and taking into account oil viscosity variations crosswise the gap height, we put k = 0 and hence:
$\begin{array}{}{S}_{i}^{(0)}=0,{\mathit{\Pi}}_{NNi}^{(0)}=0\end{array}$ for i = 1.2.3. In this case we obtain oil velocity components
$\begin{array}{}{v}_{i}^{(0)},\end{array}$ pressure *p*^{(0)}, temperature *T*^{(0)}. Assuming cylindrical coordinates: *α*_{1} = *ϕ*, *α*_{2} = *r*, *α*_{3} = *z*, *h*_{1} = *R*, *h*_{2} = *h*_{3} = 1, where *R* is the radius of the cylindrical journal, we put the temperature *T*^{(0)}, determined from the reduced Eq. (38), into the reduced pressure Equation (36). Finally, the Eq. (36) determines the pressure *p*(*ϕ*, *z*), 0 ≤ *ϕ* ≤ 2*π*, –*b*≤ *z* ≤ *b*, and tends to the following form:

$$\begin{array}{}& \frac{\mathrm{\partial}}{R\mathrm{\partial}\varphi}\left(\frac{\mathrm{\partial}{p}^{(0)}}{R\mathrm{\partial}\varphi}{\mathbf{K}}_{J1}\right)+\frac{\mathrm{\partial}}{\mathrm{\partial}z}\left(\frac{\mathrm{\partial}{p}^{(0)}}{\mathrm{\partial}z}{\mathbf{K}}_{J1}\right)=\omega \frac{\mathrm{\partial}{\mathbf{K}}_{J2}}{\mathrm{\partial}\varphi},\\ & {\mathbf{K}}_{J1}\equiv \frac{{\epsilon}_{T}^{3}}{{\eta}_{o}{\eta}_{1BE}}+{\delta}_{T}\frac{{\epsilon}_{T}^{3}}{{\eta}_{o}{\eta}_{1BE}}\{{f}_{c}-\frac{q}{2\kappa}{\epsilon}_{T}-\frac{1}{5}\frac{{\left(\omega R\right)}^{2}{\eta}_{o}{\eta}_{1BE}}{\kappa}\\ & -\frac{1}{60{\eta}_{o}{\eta}_{1BE}\kappa}{\epsilon}_{T}^{4}\left[{\left(\frac{\mathrm{\partial}{p}^{(0)}}{R\mathrm{\partial}\varphi}\right)}^{2}+{\left(\frac{\mathrm{\partial}\phantom{\rule{thickmathspace}{0ex}}{p}^{(0)}}{\mathrm{\partial}\phantom{\rule{thickmathspace}{0ex}}z}\right)}^{2}\right]\},\\ & {\mathbf{K}}_{J2}=6{\epsilon}_{T}-6{\delta}_{T}{\epsilon}_{T}\{\frac{1}{6}\frac{q}{\kappa}{\epsilon}_{T}+\frac{1}{24}\frac{{\left(\omega R\right)}^{2}{\eta}_{o}{\eta}_{1BE}}{\kappa}\\ & +\frac{1}{144{\eta}_{o}{\eta}_{1BE}\kappa}{\epsilon}_{T}^{4}\left[{\left(\frac{\mathrm{\partial}{p}^{(0)}}{R\mathrm{\partial}\varphi}\right)}^{2}+{\left(\frac{\mathrm{\partial}{p}^{(0)}}{\mathrm{\partial}z}\right)}^{2}\right]\},\end{array}$$(39)

0≤ *ϕ* ≤ 2*π*, –*b* ≤ *z* ≤ *b*, *p* = *p*(*ϕ*, *z*), *η*_{1BE} = exp(*δ*_{B}B + *δ*_{E}E).

For *δ*_{T} = 0, viscosity is constant in gap height direction and (39) tends to the well known classical form of Reynolds equation.

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