# Approximate solutions for MHD squeezing fluid flow by a novel method

## Abstract

In this paper, a steady axisymmetric MHD flow of two-dimensional incompressible fluids has been investigated. The reproducing kernel Hilbert space method (RKHSM) has been implemented to obtain a solution of the reduced fourth-order nonlinear boundary value problem. Numerical results have been compared with the results obtained by the Runge-Kutta method (RK-4) and optimal homotopy asymptotic method (OHAM).

MSC: 46E22, 35A24.

## 1 Introduction

Squeezing flows have many applications in food industry, principally in chemical engineering [14]. Some practical examples of squeezing flow include polymer processing, compression and injection molding. Grimm [5] studied numerically the thin Newtonian liquids films being squeezed between two plates. Squeezing flow coupled with magnetic field is widely applied to bearing with liquid-metal lubrication [2, 68].

In this paper, we use RKHSM to study the squeezing MHD fluid flow between two infinite planar plates. This problem has been solved by RKHSM and for comparison it has been compared with the OHAM and numerically with the RK-4 by using Maple 16.

The RKHSM, which accurately computes the series solution, is of great interest to applied sciences. The method provides the solution in a rapidly convergent series with components that can be elegantly computed. The efficiency of the method was used by many authors to investigate several scientific applications. Geng and Cui [9] and Zhou et al. [10] applied the RKHSM to handle the second-order boundary value problems. Yao and Cui [11] and Wang et al. [12] investigated a class of singular boundary value problems by this method and the obtained results were good. Wang and Chao [13], Li and Cui [14], Zhou and Cui [15] independently employed the RKSHSM to variable-coefficient partial differential equations. Du and Cui [16] investigated the approximate solution of the forced Duffing equation with integral boundary conditions by combining the homotopy perturbation method and the RKM. Lv and Cui [17] presented a new algorithm to solve linear fifth-order boundary value problems. Cui and Du [18] obtained the representation of the exact solution for the nonlinear Volterra-Fredholm integral equations by using the RKHSM. Wu and Li [19] applied iterative RKHSM to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. For more details about RKHSM and the modified forms and its effectiveness, see [937] and the references therein.

The paper is organized as follows. We give the problem formulation in Section 2. Section 3 introduces several reproducing kernel spaces. A bounded linear operator is presented in Section 4. In Section 5, we provide the main results, the exact and approximate solutions. An iterative method is developed for the kind of problems in the reproducing kernel space. We prove that the approximate solution converges to the exact solution uniformly. Some numerical experiments are illustrated in Section 6. There are some conclusions in the last section.

## 2 Problem formulation

Consider a squeezing flow of an incompressible Newtonian fluid in the presence of a magnetic field of a constant density ρ and viscosity μ squeezed between two large planar parallel plates separated by a small distance 2H and the plates approaching each other with a low constant velocity V, as illustrated in Figure 1, and the flow can be assumed to quasi-steady [13, 39]. The Navier-Stokes equations [3, 4] governing such flow in the presence of magnetic field, when inertial terms are retained in the flow, are given as [38]

$\mathrm{\nabla }V\cdot u=0$
(2.1)

and

$\rho \left[\frac{\partial u}{\partial t}+\left(u\cdot \mathrm{\nabla }\right)u\right]=\mathrm{\nabla }\cdot T+J×B+\rho f,$
(2.2)

where u is the velocity vector, denotes the material time derivative, T is the Cauchy stress tensor,

$T=-pI+\mu {A}_{1}$

and

${A}_{1}=\mathrm{\nabla }u+{u}^{T},$

J is the electric current density, B is the total magnetic field and

$B={B}_{0}+b,$

${B}_{0}$ represents the imposed magnetic field and b denotes the induced magnetic field. In the absence of displacement currents, the modified Ohm law and Maxwell’s equations (see [40] and the references therein) are given by [38]

$J=\sigma \left[E+u×B\right]$
(2.3)

and

$divB=0,\phantom{\rule{2em}{0ex}}\mathrm{\nabla }×B={\mu }_{m}J,\phantom{\rule{2em}{0ex}}curlE=\frac{\partial B}{\partial t},$
(2.4)

in which σ is the electrical conductivity, E is the electric field and ${\mu }_{m}$ is the magnetic permeability.

The following assumptions are needed [38].

1. (a)

The density ρ, magnetic permeability ${\mu }_{m}$ and electric field conductivity σ are assumed to be constant throughout the flow field region.

2. (b)

The electrical conductivity σ of the fluid is considered to be finite.

3. (c)

Total magnetic field B is perpendicular to the velocity field V and the induced magnetic field b is negligible compared with the applied magnetic field ${B}_{0}$ so that the magnetic Reynolds number is small (see [40] and the references therein).

4. (d)

We assume a situation where no energy is added or extracted from the fluid by the electric field, which implies that there is no electric field present in the fluid flow region.

Under these assumptions, the magnetohydrodynamic force involved in Eq. (2.2) can be put into the form

$J×B=-\sigma {B}_{0}^{2}u.$
(2.5)

An axisymmetric flow in cylindrical coordinates r, θ, z with z-axis perpendicular to plates and $z=±H$ at the plates. Since we have axial symmetry, u is represented by

$u=\left({u}_{r}\left(r,z\right),0,{u}_{z}\left(r,z\right)\right),$

when body forces are negligible, Navier-Stokes Eqs. (2.1)-(2.2) in cylindrical coordinates, where there is no tangential velocity (${u}_{\theta }=0$), are given as [38]

$\rho \left({u}_{r}\frac{\partial {u}_{r}}{\partial r}+{u}_{z}\frac{\partial {u}_{r}}{\partial z}\right)=-\frac{\partial p}{\partial r}+\left(\frac{{\partial }^{2}{u}_{r}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial {u}_{r}}{\partial r}-\frac{{u}_{r}}{{r}^{2}}+\frac{{\partial }^{2}{u}_{r}}{\partial {z}^{2}}\right)+\sigma {B}_{0}^{2}u$
(2.6)

and

$\rho \left({u}_{z}\frac{\partial {u}_{z}}{\partial r}+{u}_{z}\frac{\partial {u}_{z}}{\partial z}\right)=-\frac{\partial p}{\partial r}+\left(\frac{{\partial }^{2}{u}_{z}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial {u}_{z}}{\partial r}+\frac{{\partial }^{2}{u}_{z}}{\partial {z}^{2}}\right),$
(2.7)

where p is the pressure, and the equation of continuity is given by [38]

$\frac{1}{r}\frac{\partial }{\partial r}\left(r{u}_{r}\right)+\frac{\partial {u}_{z}}{\partial z}=0.$
(2.8)

The boundary conditions require

(2.9)

Let us introduce the axisymmetric Stokes stream function Ψ as

${u}_{r}=\frac{1}{r}\frac{\partial \mathrm{\Psi }}{\partial z},\phantom{\rule{2em}{0ex}}{u}_{z}=-\frac{1}{r}\frac{\partial \mathrm{\Psi }}{\partial r}.$
(2.10)

The continuity equation is satisfied using Eq. (2.10). Substituting Eqs. (2.3)-(2.5) and Eq. (2.10) into Eqs. (2.7)-(2.8), we obtain

$-\frac{\rho }{{r}^{2}}\frac{\partial \mathrm{\Psi }}{\partial r}{E}^{2}\mathrm{\Psi }=-\frac{\partial p}{\partial r}+\frac{\mu }{r}\frac{\partial {E}^{2}\mathrm{\Psi }}{\partial z}-\frac{\sigma {B}_{0}^{2}}{r}\frac{\partial \mathrm{\Psi }}{\partial z}$
(2.11)

and

$-\frac{\rho }{{r}^{2}}\frac{\partial \mathrm{\Psi }}{\partial z}{E}^{2}\mathrm{\Psi }=-\frac{\partial p}{\partial z}+\frac{\mu }{r}\frac{\partial {E}^{2}\mathrm{\Psi }}{\partial r}.$
(2.12)

Eliminating the pressure from Eqs. (2.11) and (2.12) by the integrability condition, we get the compatibility equation as [38]

$-\rho \left[\frac{\partial \left(\mathrm{\Psi },\frac{{E}^{2}\mathrm{\Psi }}{{r}^{2}}\right)}{\partial \left(r,z\right)}\right]=\frac{\mu }{r}{E}^{2}\mathrm{\Psi }-\frac{\sigma {B}_{0}^{2}}{r}\frac{{\partial }^{2\mathrm{\Psi }}}{\partial {z}^{2}},$
(2.13)

where

${E}^{2}=\frac{{\partial }^{2}}{\partial {r}^{2}}-\frac{1}{r}\frac{\partial }{\partial r}+\frac{{\partial }^{2}}{\partial {z}^{2}}.$

The stream function can be expressed as [1, 3]

$\mathrm{\Psi }\left(r,z\right)={r}^{2}F\left(z\right).$
(2.14)

In view of Eq. (2.14), the compatibility equation (2.13) and the boundary conditions (2.9) take the form

${F}^{\left(iv\right)}\left(z\right)-\frac{\sigma {B}_{0}^{2}}{r}{F}^{″}\left(z\right)+2\frac{\rho }{\mu }F\left(z\right){F}^{‴}\left(z\right)=0,$
(2.15)

subject to

$\begin{array}{r}F\left(0\right)=0,\phantom{\rule{2em}{0ex}}{F}^{″}\left(0\right)=0,\\ F\left(H\right)=\frac{V}{2},\phantom{\rule{2em}{0ex}}{F}^{\prime }\left(H\right)=0.\end{array}$
(2.16)

Non-dimensional parameters are given as [38]

${F}^{\ast }=2\frac{F}{V},\phantom{\rule{2em}{0ex}}{z}^{\ast }=\frac{z}{H},\phantom{\rule{2em}{0ex}}Re=\frac{\rho HV}{\mu },\phantom{\rule{2em}{0ex}}m={B}_{0}H\sqrt{\frac{\sigma }{\mu }}.$

For simplicity omitting the , the boundary value problem (2.15)-(2.16) becomes [38]

${F}^{\left(iv\right)}\left(z\right)-{m}^{2}{F}^{″}\left(z\right)+ReF\left(z\right){F}^{‴}\left(z\right)=0,$
(2.17)

with the boundary conditions

$\begin{array}{r}F\left(0\right)=0,\phantom{\rule{2em}{0ex}}{F}^{″}\left(0\right)=0,\\ F\left(1\right)=1,\phantom{\rule{2em}{0ex}}{F}^{\prime }\left(1\right)=0,\end{array}$
(2.18)

where Re is the Reynolds number and m is the Hartmann number.

## 3 Reproducing kernel spaces

In this section, we define some useful reproducing kernel spaces.

Definition 3.1 (Reproducing kernel)

Let E be a nonempty abstract set. A function $K:E×E⟶C$ is a reproducing kernel of the Hilbert space H if and only if

$\left\{\begin{array}{l}\mathrm{\forall }t\in E,\phantom{\rule{1em}{0ex}}K\left(\cdot ,t\right)\in H,\\ \mathrm{\forall }t\in E,\mathrm{\forall }\phi \in H,\phantom{\rule{1em}{0ex}}〈\phi \left(\cdot \right),\phantom{\rule{0.25em}{0ex}}K\left(\cdot ,t\right)〉=\phi \left(t\right).\end{array}$
(3.1)

The last condition is called ‘the reproducing property’: the value of the function φ at the point t is reproduced by the inner product of φ with $K\left(\cdot ,t\right)$.

Definition 3.2 We define the space ${W}_{2}^{5}\left[0,1\right]$ by

The fifth derivative of u exists almost everywhere since ${u}^{\left(4\right)}$ is absolutely continuous. The inner product and the norm in ${W}_{2}^{5}\left[0,1\right]$ are defined respectively by

${〈u,v〉}_{{W}_{2}^{5}}=\sum _{i=0}^{4}{u}^{\left(i\right)}\left(0\right){v}^{\left(i\right)}\left(0\right)+{\int }_{0}^{1}{u}^{\left(5\right)}\left(x\right){v}^{\left(5\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}u,v\in {W}_{2}^{5}\left[0,1\right]$

and

${\parallel u\parallel }_{{W}_{2}^{5}}=\sqrt{{〈u,u〉}_{{}_{{W}_{2}^{5}}}},\phantom{\rule{1em}{0ex}}u\in {W}_{2}^{5}\left[0,1\right].$

The space ${W}_{2}^{5}\left[0,1\right]$ is a reproducing kernel space, i.e., for each fixed $y\in \left[0,1\right]$ and any $u\in {W}_{2}^{5}\left[0,1\right]$, there exists a function ${R}_{y}$ such that

$u={〈u,{R}_{y}〉}_{{W}_{2}^{5}}.$

Definition 3.3 We define the space ${W}_{2}^{4}\left[0,1\right]$ by

The fourth derivative of u exists almost everywhere since ${u}^{\left(3\right)}$ is absolutely continuous. The inner product and the norm in ${W}_{2}^{4}\left[0,1\right]$ are defined respectively by

${〈u,v〉}_{{W}_{2}^{4}}=\sum _{i=0}^{3}{u}^{\left(i\right)}\left(0\right){v}^{\left(i\right)}\left(0\right)+{\int }_{0}^{1}{u}^{\left(4\right)}\left(x\right){v}^{\left(4\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}u,v\in {W}_{2}^{4}\left[0,1\right]$

and

${\parallel u\parallel }_{{W}_{2}^{4}}=\sqrt{{〈u,u〉}_{{}_{{W}_{2}^{4}}}},\phantom{\rule{1em}{0ex}}u\in {W}_{2}^{4}\left[0,1\right].$

The space ${W}_{2}^{4}\left[0,1\right]$ is a reproducing kernel space, i.e., for each fixed $y\in \left[0,1\right]$ and any $u\in {W}_{2}^{4}\left[0,1\right]$, there exists a function ${r}_{y}$ such that

$u={〈u,{r}_{y}〉}_{{W}_{2}^{4}}.$

Theorem 3.1 The space ${W}_{2}^{5}\left[0,1\right]$ is a reproducing kernel Hilbert space whose reproducing kernel function is given by

${R}_{y}\left(x\right)=\left\{\begin{array}{ll}{\sum }_{i=1}^{10}{c}_{i}\left(y\right){x}^{i-1},& x\le y,\\ {\sum }_{i=1}^{10}{d}_{i}\left(y\right){x}^{i-1},& x>y,\end{array}$

where ${c}_{i}\left(y\right)$ and ${d}_{i}\left(y\right)$ can be obtained easily by using Maple 16 and the proof of Theorem  3.1 is given in Appendix.

Remark 3.1 The reproducing kernel function ${r}_{y}$ of ${W}_{2}^{4}\left[0,1\right]$ is given as

${r}_{y}\left(x\right)=\left\{\begin{array}{ll}1+xy+\frac{1}{4}{y}^{2}{x}^{2}+\frac{1}{36}{y}^{3}{x}^{3}+\frac{1}{144}{y}^{3}{x}^{4}-\frac{1}{240}{y}^{2}{x}^{5}+\frac{1}{720}y{x}^{6}-\frac{1}{5\text{,}040}{x}^{7},& x\le y,\\ 1+yx+\frac{1}{4}{y}^{2}{x}^{2}+\frac{1}{36}{y}^{3}{x}^{3}+\frac{1}{144}{x}^{3}{y}^{4}-\frac{1}{240}{x}^{2}{y}^{5}+\frac{1}{720}x{y}^{6}-\frac{1}{5\text{,}040}{y}^{7},& x>y.\end{array}$

This can be proved easily as the proof of Theorem 3.1.

## 4 Bounded linear operator in ${W}_{2}^{5}\left[0,1\right]$

In this section, the solution of Eq. (2.17) is given in the reproducing kernel space ${W}_{2}^{5}\left[0,1\right]$.

On defining the linear operator $L:{W}_{2}^{5}\left[0,1\right]\to {W}_{2}^{4}\left[0,1\right]$ as

$Lu={u}^{\left(4\right)}\left(x\right)+Re\frac{{e}^{x}}{e}\left({x}^{3}-4{x}^{2}+4x\right){u}^{\left(3\right)}\left(x\right)-{m}^{2}{u}^{″}\left(x\right)+Re\frac{{e}^{x}}{e}\left({x}^{3}+5{x}^{2}-2x-6\right)u\left(x\right).$

Model problem (2.17)-(2.18) changes the following problem:

$\left\{\begin{array}{l}Lu=M\left(x,u,{u}^{\left(3\right)}\right),\phantom{\rule{1em}{0ex}}x\in \left[0,1\right],\\ u\left(0\right)=0,\phantom{\rule{2em}{0ex}}u\left(1\right)=0,\phantom{\rule{2em}{0ex}}{u}^{\prime }\left(1\right)=0,\phantom{\rule{2em}{0ex}}{u}^{″}\left(0\right)=0,\end{array}$
(4.1)

where

$F\left(x\right)=u\left(x\right)+\frac{{e}^{x}}{e}\left({x}^{3}-4{x}^{2}+4x\right)$

and

$\begin{array}{rcl}M\left(x,u,{u}^{\left(3\right)}\right)& =& -Re{u}^{\left(3\right)}\left(x\right)u\left(x\right)-Re{\left(\frac{{e}^{x}}{e}\right)}^{2}\left({x}^{3}-4{x}^{2}+4x\right)\left({x}^{3}+5{x}^{2}-2x-6\right)\\ -\frac{{e}^{x}}{e}\left({x}^{3}+8{x}^{2}+8x-2\right)+{m}^{2}\frac{{e}^{x}}{e}\left({x}^{3}+2{x}^{2}-6x\right).\end{array}$

Theorem 4.1 The operator L defined by (4.1) is a bounded linear operator.

Proof We only need to prove

${\parallel Lu\parallel }_{{W}_{2}^{4}}^{2}\le P{\parallel Lu\parallel }_{{W}_{2}^{5}}^{2},$

where P is a positive constant. By Definition 3.3, we have

${\parallel u\parallel }_{{W}_{2}^{4}}^{2}={〈u,u〉}_{{W}_{2}^{4}}=\sum _{i=0}^{3}{\left[{u}^{\left(i\right)}\left(0\right)\right]}^{2}+{\int }_{0}^{1}{\left[{u}^{\left(4\right)}\left(x\right)\right]}^{2}\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}u\in {W}_{2}^{4}\left[0,1\right],$

and

$\begin{array}{rcl}{\parallel Lu\parallel }_{{W}_{2}^{4}}^{2}={〈Lu,Lu〉}_{{W}_{2}^{4}}& =& {\left[\left(Lu\right)\left(0\right)\right]}^{2}+{\left[{\left(Lu\right)}^{\prime }\left(0\right)\right]}^{2}+{\left[{\left(Lu\right)}^{″}\left(0\right)\right]}^{2}\\ +{\left[{\left(Lu\right)}^{\left(3\right)}\left(0\right)\right]}^{2}+{\int }_{0}^{1}{\left[{\left(Lu\right)}^{\left(4\right)}\left(x\right)\right]}^{2}\phantom{\rule{0.2em}{0ex}}dx.\end{array}$

By the reproducing property, we have

$u\left(x\right)={〈u,{R}_{x}〉}_{{W}_{2}^{5}},$

and

$\begin{array}{c}\left(Lu\right)\left(x\right)={〈u,\left(L{R}_{x}\right)〉}_{{W}_{2}^{5}},\phantom{\rule{2em}{0ex}}{\left(Lu\right)}^{\prime }\left(x\right)={〈u,{\left(L{R}_{x}\right)}^{\prime }〉}_{{W}_{2}^{5}},\hfill \\ {\left(Lu\right)}^{″}\left(x\right)={〈u,{\left(L{R}_{x}\right)}^{″}〉}_{{W}_{2}^{5}},\phantom{\rule{2em}{0ex}}{\left(Lu\right)}^{\left(3\right)}\left(x\right)={〈u,{\left(L{R}_{x}\right)}^{\left(3\right)}〉}_{{W}_{2}^{5}},\hfill \\ {\left(Lu\right)}^{\left(4\right)}\left(x\right)={〈u,{\left(L{R}_{x}\right)}^{\left(4\right)}〉}_{{W}_{2}^{5}}.\hfill \end{array}$

Therefore, by the Cauchy-Schwarz inequality, we get

Thus

${\left[\left(Lu\right)\left(0\right)\right]}^{2}+{\left[{\left(Lu\right)}^{\prime }\left(0\right)\right]}^{2}+{\left[{\left(Lu\right)}^{″}\left(0\right)\right]}^{2}+{\left[{\left(Lu\right)}^{\left(3\right)}\left(0\right)\right]}^{2}\le \left({a}_{1}^{2}+{a}_{2}^{2}+{a}_{3}^{2}+{a}_{4}^{2}\right){\parallel u\parallel }_{{W}_{2}^{5}}^{2}.$

Since

${\left(Lu\right)}^{\left(4\right)}={〈u,{\left(L{R}_{x}\right)}^{\left(4\right)}〉}_{{W}_{2}^{5}},$

then

Therefore, we have

${\left[{\left(Lu\right)}^{\left(4\right)}\right]}^{2}\le {a}_{5}^{2}{\parallel u\parallel }_{{W}_{2}^{5}}^{2}$

and

${\int }_{0}^{1}{\left[{\left(Lu\right)}^{\left(4\right)}\left(x\right)\right]}^{2}\phantom{\rule{0.2em}{0ex}}dx\le {a}_{5}^{2}{\parallel u\parallel }_{{W}_{2}^{5}}^{2},$

that is,

$\begin{array}{rcl}{\parallel Lu\parallel }_{{W}_{2}^{4}}^{2}& =& {\left[\left(Lu\right)\left(0\right)\right]}^{2}+{\left[{\left(Lu\right)}^{\prime }\left(0\right)\right]}^{2}+{\left[{\left(Lu\right)}^{″}\left(0\right)\right]}^{2}+{\left[{\left(Lu\right)}^{\left(3\right)}\left(0\right)\right]}^{2}+{\int }_{0}^{1}{\left[{\left(Lu\right)}^{\left(4\right)}\left(x\right)\right]}^{2}\phantom{\rule{0.2em}{0ex}}dx\\ \le & \left({a}_{1}^{2}+{a}_{2}^{2}+{a}_{3}^{2}+{a}_{4}^{2}+{a}_{5}^{2}\right){\parallel u\parallel }_{{W}_{2}^{5}}^{2}=P{\parallel u\parallel }_{{W}_{2}^{4}}^{2},\end{array}$

where $P=\left({a}_{1}^{2}+{a}_{2}^{2}+{a}_{3}^{2}+{a}_{4}^{2}+{a}_{5}^{2}\right)>0$ is a positive constant. This completes the proof. □

## 5 Analysis of the solution of (2.17)-(2.18)

Let ${\left\{{x}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ be any dense set in $\left[0,1\right]$ and ${\mathrm{\Psi }}_{x}\left(y\right)={L}^{\ast }{r}_{x}\left(y\right)$, where ${L}^{\ast }$ is the adjoint operator of L and ${r}_{x}$ is given by Remark 3.1. Furthermore

${\mathrm{\Psi }}_{i}\left(x\right)\stackrel{\text{def}}{=}{\mathrm{\Psi }}_{{x}_{i}}\left(x\right)={L}^{\ast }{r}_{{x}_{i}}\left(x\right).$

Lemma 5.1 ${\left\{{\mathrm{\Psi }}_{i}\left(x\right)\right\}}_{i=1}^{\mathrm{\infty }}$ is a complete system of ${W}_{2}^{5}\left[0,1\right]$.

Proof For $u\in {W}_{2}^{5}\left[0,1\right]$, let

$〈u,{\mathrm{\Psi }}_{i}〉=0\phantom{\rule{1em}{0ex}}\left(i=1,2,\dots \right),$

that is,

$〈u,{L}^{\ast }{r}_{{x}_{i}}〉=\left(Lu\right)\left({x}_{i}\right)=0.$

Note that ${\left\{{x}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ is the dense set in $\left[0,1\right]$. Therefore $\left(Lu\right)\left(x\right)=0$. Assume that (4.1) has a unique solution. Then L is one-to-one on ${W}_{2}^{5}\left[0,1\right]$ and thus $u\left(x\right)=0$. This completes the proof. □

Lemma 5.2 The following formula holds:

${\mathrm{\Psi }}_{i}\left(x\right)=\left({L}_{\eta }{R}_{x}\left(\eta \right)\right)\left({x}_{i}\right),$

where the subscript η of the operator ${L}_{\eta }$ indicates that the operator L applies to a function of η.

Proof

$\begin{array}{rcl}{\mathrm{\Psi }}_{i}\left(x\right)& =& {〈{\mathrm{\Psi }}_{i}\left(\xi \right),{R}_{x}\left(\xi \right)〉}_{{W}_{2}^{5}\left[0,1\right]}\\ =& {〈{L}^{\ast }{r}_{{x}_{i}}\left(\xi \right),{R}_{x}\left(\xi \right)〉}_{{W}_{2}^{5}\left[0,1\right]}\\ =& {〈\left({r}_{{x}_{i}}\right)\left(\xi \right),\left({L}_{\eta }{R}_{x}\left(\eta \right)\right)\left(\xi \right)〉}_{{W}_{2}^{4}\left[0,1\right]}\\ =& \left({L}_{\eta }{R}_{x}\left(\eta \right)\right)\left({x}_{i}\right).\end{array}$

This completes the proof. □

Remark 5.1 The orthonormal system ${\left\{{\overline{\mathrm{\Psi }}}_{i}\left(x\right)\right\}}_{i=1}^{\mathrm{\infty }}$ of ${W}_{2}^{5}\left[0,1\right]$ can be derived from the Gram-Schmidt orthogonalization process of ${\left\{{\mathrm{\Psi }}_{i}\left(x\right)\right\}}_{i=1}^{\mathrm{\infty }}$ as

${\overline{\mathrm{\Psi }}}_{i}\left(x\right)=\sum _{k=1}^{i}{\beta }_{ik}{\mathrm{\Psi }}_{k}\left(x\right)\phantom{\rule{1em}{0ex}}\left({\beta }_{ii}>0,i=1,2,\dots \right),$
(5.1)

where ${\beta }_{ik}$ are orthogonal coefficients.

In the following, we give the representation of the exact solution of Eq. (2.17) in the reproducing kernel space ${W}_{2}^{5}\left[0,1\right]$.

Theorem 5.1 If u is the exact solution of (4.1), then

$u=\sum _{i=1}^{\mathrm{\infty }}\sum _{k=1}^{i}{\beta }_{ik}M\left({x}_{k},u\left({x}_{k}\right),{u}^{\left(3\right)}\left({x}_{k}\right)\right){\overline{\mathrm{\Psi }}}_{i}\left(x\right),$

where ${\left\{{x}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ is a dense set in $\left[0,1\right]$.

Proof From (5.1) and the uniqueness of solution of (4.1), we have

$\begin{array}{rcl}u& =& \sum _{i=1}^{\mathrm{\infty }}{〈u,{\overline{\mathrm{\Psi }}}_{i}〉}_{{W}_{2}^{5}}{\overline{\mathrm{\Psi }}}_{i}=\sum _{i=1}^{\mathrm{\infty }}\sum _{k=1}^{i}{\beta }_{ik}{〈u,{L}^{\ast }{r}_{{x}_{k}}〉}_{{W}_{2}^{5}}{\overline{\mathrm{\Psi }}}_{i}\\ =& \sum _{i=1}^{\mathrm{\infty }}\sum _{k=1}^{i}{\beta }_{ik}{〈Lu,{r}_{{x}_{k}}〉}_{{W}_{2}^{4}}{\overline{\mathrm{\Psi }}}_{i}=\sum _{i=1}^{\mathrm{\infty }}\sum _{k=1}^{i}{\beta }_{ik}{〈M\left(x,u,{u}^{\left(3\right)}\right),{r}_{{x}_{k}}〉}_{{W}_{2}^{4}}{\overline{\mathrm{\Psi }}}_{i}\\ =& \sum _{i=1}^{\mathrm{\infty }}\sum _{k=1}^{i}{\beta }_{ik}M\left({x}_{k},u\left({x}_{k}\right),{u}^{\left(3\right)}\left({x}_{k}\right)\right){\overline{\mathrm{\Psi }}}_{i}\left(x\right).\end{array}$

This completes the proof. □

Now the approximate solution ${u}_{n}$ can be obtained by truncating the n-term of the exact solution u as

${u}_{n}=\sum _{i=1}^{n}\sum _{k=1}^{i}{\beta }_{ik}M\left({x}_{k},u\left({x}_{k}\right),{u}^{\left(3\right)}\left({x}_{k}\right)\right){\overline{\mathrm{\Psi }}}_{i}\left(x\right).$

Lemma 5.3 ([30])

Assume that u is the solution of (4.1) and ${r}_{n}$ is the error between the approximate solution ${u}_{n}$ and the exact solution u. Then the error sequence ${r}_{n}$ is monotone decreasing in the sense of ${\parallel \cdot \parallel }_{{W}_{2}^{5}}$ and ${\parallel {r}_{n}\left(x\right)\parallel }_{{W}_{2}^{5}}\to 0$.

## 6 Numerical results

In this section, comparisons of results are made through different Reynolds numbers Re and magnetic field effect m. All computations are performed by Maple 16. Figure 5.7 shows comparisons of $F\left(z\right)$ for a fixed Reynolds number with increasing magnetic field effect $m=1,3,8,20$. From this figure, the velocity decreases due to an increase in m. Figure 5.8 shows comparisons of $F\left(z\right)$ for a fixed magnetic field $m=1$ with increasing Reynolds numbers $Re=1,4,10$. It is observed that much increase in Reynolds numbers affects the results. The RKHSM does not require discretization of the variables, i.e., time and space, it is not affected by computation round of errors and one is not faced with necessity of large computer memory and time. The accuracy of the RKHSM for the MHD squeezing fluid flow is controllable and absolute errors are small with present choice of x (see Tables 1-6 and Figures 2-7). The numerical results we obtained justify the advantage of this methodology. Generally it is not possible to find the exact solution of these problems.

## 7 Conclusion

In this paper, we introduced an algorithm for solving the MHD squeezing fluid flow. We applied a new powerful method RKHSM to the reduced nonlinear boundary value problem. The approximate solution obtained by the present method is uniformly convergent. Clearly, the series solution methodology can be applied to much more complicated nonlinear differential equations and boundary value problems. However, if the problem becomes nonlinear, then the RKHSM does not require discretization or perturbation and it does not make closure approximation. Results of numerical examples show that the present method is an accurate and reliable analytical method for this problem.

## Appendix

Proof of Theorem 3.1 Let $u\in {W}_{2}^{5}\left[0,1\right]$. By Definition 3.2 we have

${〈u,{R}_{y}〉}_{{}_{{W}_{2}^{5}}}=\sum _{i=0}^{4}{u}^{\left(i\right)}\left(0\right){R}_{y}^{\left(i\right)}\left(0\right)+{\int }_{0}^{1}{u}^{\left(5\right)}\left(x\right){R}_{y}^{\left(5\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.$
(A.1)

Through several integrations by parts for (A.1), we have

$\begin{array}{rcl}{〈u,{R}_{y}〉}_{{}_{{W}_{2}^{5}}}& =& \sum _{i=0}^{4}{u}^{\left(i\right)}\left(0\right)\left[{R}_{y}^{\left(i\right)}\left(0\right)-{\left(-1\right)}^{\left(4-i\right)}{R}_{y}^{\left(9-i\right)}\left(0\right)\right]\\ +\sum _{i=0}^{4}{\left(-1\right)}^{\left(4-i\right)}{u}^{\left(i\right)}\left(1\right){R}_{y}^{\left(9-i\right)}\left(1\right)-{\int }_{0}^{1}u\left(x\right){R}_{y}^{\left(10\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
(A.2)

Note the property of the reproducing kernel

${〈u,{R}_{y}〉}_{{W}_{2}^{5}}=u\left(y\right).$

Now, if

$\left\{\begin{array}{l}{R}_{y}^{{}^{\prime }}\left(0\right)+{R}_{y}^{\left(8\right)}\left(0\right)=0,\\ {R}_{y}^{\left(3\right)}\left(0\right)+{R}_{y}^{\left(6\right)}\left(0\right)=0,\\ {R}_{y}^{\left(4\right)}\left(0\right)-{R}_{y}^{\left(5\right)}\left(0\right)=0,\\ {R}_{y}^{\left(5\right)}\left(1\right)=0,\\ {R}_{y}^{\left(6\right)}\left(1\right)=0,\\ {R}_{y}^{\left(7\right)}\left(1\right)=0,\end{array}$
(A.3)

then (A.2) implies that

${R}_{y}^{\left(10\right)}\left(x\right)=-\delta \left(x-y\right),$

when $x\ne y$

${R}_{y}^{\left(10\right)}\left(x\right)=0,$

and therefore

${R}_{y}\left(x\right)=\left\{\begin{array}{ll}{\sum }_{i=1}^{10}{c}_{i}\left(y\right){x}^{i-1},& x\le y,\\ {\sum }_{i=1}^{10}{d}_{i}\left(y\right){x}^{i-1},& x>y.\end{array}$

Since

${R}_{y}^{\left(10\right)}\left(x\right)=\delta \left(x-y\right),$

we have

${R}_{{y}^{+}}^{\left(k\right)}\left(y\right)={R}_{{y}^{-}}^{\left(k\right)}\left(y\right),\phantom{\rule{1em}{0ex}}k=0,1,2,3,4,5,6,7,8,$
(A.4)

and

${R}_{{y}^{+}}^{\left(9\right)}\left(y\right)-{R}_{{y}^{-}}^{\left(9\right)}\left(y\right)=-1.$
(A.5)

Since ${R}_{y}\left(x\right)\in {W}_{2}^{5}\left[0,1\right]$, it follows that

${R}_{y}\left(0\right)=0,\phantom{\rule{2em}{0ex}}{R}_{y}\left(1\right)=0,\phantom{\rule{2em}{0ex}}{R}_{y}^{\prime }\left(1\right)=0,\phantom{\rule{2em}{0ex}}{R}_{y}^{\prime \prime }\left(0\right)=0.$
(A.6)

From (A.3)-(A.6), the unknown coefficients ${c}_{i}\left(y\right)$ and ${d}_{i}\left(y\right)$ ($i=1,2,\dots ,12$) can be obtained. This completes the proof. □

## References

1. Papanastasiou TC, Georgiou GC, Alexandrou AN: Viscous Fluid Flow. CRC Press, Boca Raton; 1994.

2. Stefa Hughes WF, Elco RA: Magnetohydrodynamic lubrication flow between parallel rotating disks. J. Fluid Mech. 1962, 13: 21-32. 10.1017/S0022112062000464

3. Ghori QK, Ahmed M, Siddiqui AM: Application of homotopy perturbation method to squeezing flow of a Newtonian fluid. Int. J. Nonlinear Sci. Numer. Simul. 2007, 8: 179-184.

4. Ran XJ, Zhu QY, Li Y: An explicit series solution of the squeezing flow between two infinite plates by means of the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 119-132. 10.1016/j.cnsns.2007.07.012

5. Grimm RG: Squeezing flows of Newtonian liquid films an analysis include the fluid inertia. Appl. Sci. Res. 1976, 32: 149-166. 10.1007/BF00383711

6. Kamiyama S: Inertia effects in MHD hydrostatic thrust bearing. J. Lubr. Technol. 1969, 91: 589-596. 10.1115/1.3555005

7. Hamza EA: The magnetohydrodynamic squeeze film. J. Tribol. 1988, 110: 375-377. 10.1115/1.3261636

8. Bhattacharya S, Pal A: Unsteady MHD squeezing flow between two parallel rotating discs. Mech. Res. Commun. 1997, 24: 615-623. 10.1016/S0093-6413(97)00079-7

9. Geng F, Cui M: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 2007, 327: 1167-1181. 10.1016/j.jmaa.2006.05.011

10. Zhou Y, Lin Y, Cui M: An efficient computational method for second order boundary value problems of nonlinear differential equations. Appl. Math. Comput. 2007, 194: 357-365.

11. Yao H, Cui M: A new algorithm for a class of singular boundary value problems. Appl. Math. Comput. 2007, 186: 1183-1191. 10.1016/j.amc.2006.07.157

12. Wang W, Cui M, Han B: A new method for solving a class of singular two-point boundary value problems. Appl. Math. Comput. 2008, 206: 721-727. 10.1016/j.amc.2008.09.019

13. Wang YL, Chao L: Using reproducing kernel for solving a class of partial differential equation with variable-coefficients. Appl. Math. Mech. 2008, 29: 129-137. 10.1007/s10483-008-0115-y

14. Li F, Cui M: A best approximation for the solution of one-dimensional variable-coefficient Burgers’ equation. Numer. Methods Partial Differ. Equ. 2009, 25: 1353-1365. 10.1002/num.20428

15. Zhou S, Cui M: Approximate solution for a variable-coefficient semilinear heat equation with nonlocal boundary conditions. Int. J. Comput. Math. 2009, 86: 2248-2258. 10.1080/00207160903229881

16. Du J, Cui M: Solving the forced Duffing equations with integral boundary conditions in the reproducing kernel space. Int. J. Comput. Math. 2010, 87: 2088-2100. 10.1080/00207160802610843

17. Lv X, Cui M: An efficient computational method for linear fifth-order two-point boundary value problems. J. Comput. Appl. Math. 2010, 234: 1551-1558. 10.1016/j.cam.2010.02.036

18. Du J, Cui M: Constructive proof of existence for a class of fourth-order nonlinear BVPs. Comput. Math. Appl. 2010, 59: 903-911. 10.1016/j.camwa.2009.10.003

19. Wu BY, Li XY: Iterative reproducing kernel method for nonlinear oscillator with discontinuity. Appl. Math. Lett. 2010, 23: 1301-1304. 10.1016/j.aml.2010.06.018

20. Cui M, Lin Y: Nonlinear Numerical Analysis in the Reproducing Kernel Spaces. Nova Science Publishers, New York; 2009.

21. Lü X, Cui M: Analytic solutions to a class of nonlinear infinite-delay-differential equations. J. Math. Anal. Appl. 2008, 343: 724-732. 10.1016/j.jmaa.2008.01.101

22. Jiang W, Cui M: Constructive proof for existence of nonlinear two-point boundary value problems. Appl. Math. Comput. 2009, 215: 1937-1948. 10.1016/j.amc.2009.07.044

23. Cui M, Du H: Representation of exact solution for the nonlinear Volterra-Fredholm integral equations. Appl. Math. Comput. 2006, 182: 1795-1802. 10.1016/j.amc.2006.06.016

24. Jiang W, Lin Y: Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space. Commun. Nonlinear Sci. Numer. Simul. 2011, 16: 3639-3645. 10.1016/j.cnsns.2010.12.019

25. Lin Y, Cui M: A numerical solution to nonlinear multi-point boundary-value problems in the reproducing kernel space. Math. Methods Appl. Sci. 2011, 34: 44-47. 10.1002/mma.1327

26. Mohammadi M, Mokhtari R: Solving the generalized regularized long wave equation on the basis of a reproducing kernel space. J. Comput. Appl. Math. 2011, 235: 4003-4014. 10.1016/j.cam.2011.02.012

27. Wu BY, Li XY: A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method. Appl. Math. Lett. 2011, 24: 156-159. 10.1016/j.aml.2010.08.036

28. Yao H, Lin Y: New algorithm for solving a nonlinear hyperbolic telegraph equation with an integral condition. Int. J. Numer. Methods Biomed. Eng. 2011, 27: 1558-1568. 10.1002/cnm.1376

29. Inc M, Akgül A: The reproducing kernel Hilbert space method for solving Troesch’s problem. J. Assoc. Arab Univ. Basic. Appl. Sci. 2013, 14: 19-27.

30. Inc, M, Akgül, A, Geng, F: Reproducing kernel Hilbert space method for solving Bratu’s problem. Bul. Malays. Math. Sci. Soc. (in press)

31. Inc M, Akgül A, Kilicman A: Explicit solution of telegraph equation based on reproducing kernel method. J. Funct. Spaces Appl. 2012., 2012: Article ID 984682

32. Inc M, Akgül A, Kilicman A: A new application of the reproducing kernel Hilbert space method to solve MHD Jeffery-Hamel flows problem in non-parallel walls. Abstr. Appl. Anal. 2013., 2013: Article ID 239454

33. Inc M, Akgül A, Kilicman A: On solving KdV equation using reproducing kernel Hilbert space method. Abstr. Appl. Anal. 2013., 2013: Article ID 578942

34. Inc M, Akgül A, Kilicman A: Numerical solutions of the second-order one-dimensional telegraph equation based on reproducing kernel Hilbert space method. Abstr. Appl. Anal. 2013., 2013: Article ID 768963

35. Akram G, Rehman HU: Numerical solution of eighth order boundary value problems in reproducing Kernel space. Numer. Algorithms 2013, 62(3):527-540. 10.1007/s11075-012-9608-4

36. Wenyan W, Bo H, Masahiro Y: Inverse heat problem of determining time-dependent source parameter in reproducing kernel space. Nonlinear Anal., Real World Appl. 2013, 14(1):875-887. 10.1016/j.nonrwa.2012.08.009

37. Mokhtari R, İsfahani FT, Mohammadi M: Reproducing kernel method for solving nonlinear differential-difference equations. Abstr. Appl. Anal. 2012., 2012: Article ID 514103

38. Islam S, Ullah M, Zaman G, Idrees M: Approximate solutions to MHD squeezing fluid flow. J. Appl. Math. Inform. 2011, 29(5-6):1081-1096.

39. Idrees M, Islam S, Haq S, Islam S: Application of the optimal homotopy asymptotic method to squeezing flow. Comput. Math. Appl. 2010, 59: 3858-3866. 10.1016/j.camwa.2010.04.023

40. Mohyuddin MR, Gotz T: Resonance behavior of viscoelastic fluid in Poiseuille flow in the presence of a transversal magnetic field. Int. J. Numer. Methods Fluids 2005, 49: 837-847. 10.1002/fld.1026

## Acknowledgements

We presented this paper in the International Symposium on Biomathematics and Ecology Education Research in 2013. We would like to thank the organizers of this conference and the reviewers for their kind and helpful comments on this paper. Ali Akgül gratefully acknowledge that this paper was partially supported by the Dicle University and the Firat University. This paper is a part of PhD thesis of Ali Akgül.

## Author information

Authors

### Corresponding author

Correspondence to Ali Akgül.

### Competing interests

The authors declare that they do not have any competing or conflict of interests.

### Authors’ contributions

Both authors contributed equally to this paper.

## Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

## Rights and permissions

Reprints and Permissions

Inc, M., Akgül, A. Approximate solutions for MHD squeezing fluid flow by a novel method. Bound Value Probl 2014, 18 (2014). https://doi.org/10.1186/1687-2770-2014-18