1. Introduction

Considerable attention has been paid in recent years to the problems heat transfer analysis rectangular domain with regular boundary conditions yielding a well-posed problem. The governing couple equations are very complicated and highly nonlinear, with mostly numerical solutions.

The theoretical studies of mixed convective flow in a lid-driven cavity finds applications in flow and heat transfer in solar ponds and solar collectors, dynamics of lakes, reservoirs and cooling ponds, cooling of electronic systems, thermal-hydraulics of nuclear reactors, thermal convection in micropolar fluids, chemical processing equipment, lubricating grooves, crystal growing, materials processing such as float glass production, galvanizing, metal coating and casting, food processing, and industrial processes where a solid ribbon or a solid material is heated as it moves through a furnace, among others [1-10].

The above literature shows that the convective heat transfer and flow in a rectangular enclosure driven by a horizontal wall while being cooled from one horizontal and vertical wall, with the other vertical wall thermally isolated have not been investigated. This configuration finds practical applications in the cooling of an extruded plate in a hot rolling process. The fluid flow and the heat transfer patterns within the enclosure dictate the degree of cooling and hence the quality of the final product. This fact motivates the present study. The purpose of this work is therefore to present a parametric investigation of the Richardson and Prandtl numbers, and the aspect ratio on the flow patterns, energy distribution and heat transfer behaviour for this configuration.

The aim of the present work is to solve the problem of mixed convection flow in rectangular enclosures driven by a continuously moving horizontal plate using an analytical technique, OHAM. In view of this we have established an analytic solution based on the OHAM to venture further into the regime of nonlinear fluids.

2. Analytical approach

We start with the basic idea of OHAM, the governing equations to solved are classified into two velocity and temperature that is

(1)

where are linear operators, are non-linear operators, are known function and is a boundary operator.

Construct an optimal homotopy which satisfies the following equations:

(2)

(3)

where and is an embedding parameter, is a nonzero auxiliary function for and , and are unknown functions

Clearly, when and it holds that

(4)

Choose the auxiliary function in the form

(5)

where are constants to be determined.

Construct a Taylor's series solution of equations (2) and (3) in the form

(6)

substituting equation (6) into equations (2) and (3) and boundary and equating the coefficients of like powers of to obtain

(7)

and the th order equation is defined by

(8)

where is the coefficient of obtained by expanding in series with respect to the embedding parameter

(9)

The solution of equation (1) can be approximately obtained in the form

(10)

(11)

Substitute equations (10) and (11) into equation (1) it results the following expression for residual:

(12)

(13)

If , then (10) and (11) will be the exact solution. Generally such a case will not happen for nonlinear problems, but we can minimize the functional by the method of lease squares

(14)

where and belong to the domain of the problem. Finally, the unknown constants can be optimally identified from the conditions

(15)

With these known values of , the approximate solution of equation (10) and (11) are well determined.

Statement of the problem

In the present investigation, we consider convective heat transfer and flow in a rectangular enclosure driven by a horizontal wall, the horizontal plate divides the rectangular enclosure into equal halves while being cooled from one horizontal and vertical wall, with the other vertical wall thermally isolated as described by Waheed [1] Figure 1 show the physical configuration.

Figure 1. Physical configuration

(16)

(17)

(18)

(19)

where is the temperature of ambient medium

In the above equations we will describe the velocity components and as the derivatives of the stream function

(20)

the vorticity equation

(21)

Introduced the following non-dimensional variables

(22)

where is the stream function, is the vorticity and is the weight of the enclosure.

The results of the analysis (after dropping the -notation) are the stream function equation

(23)

and the vorticity transport equation

(24)

and also, the energy equation

(25)

where is the Reynolds number, is the Richardson number and is the Prandtl number.

The boundary conditions and initial condition for two enclosure and are as follows:

(26)

Special cases for the boundary condition:Case 1: Upper wall of enclosure

(27)

Case 2: Lower wall of enclosure

(28)

where is the fluid density, is the pressure, is the specific heat capacity at constant pressure, is the volumetric coefficient of thermal expansion, is the fluid viscosity, is the thermal conductivity, is the time, is the temperature and , are the fluid velocity components, in the and directions, respectively.

The aim of the present investigation is to discuss the analytical approach for the steady two-dimensional incompressible and laminar in rectangular domain.Applications

Choosing the linear and nonlinear operator defined by

(29)

(30)

(31)

(32)

and the stream function

(33)

The correspond boundary conditions are

(34)

The zeroth order deformation:

(35)

(36)

(37)

(38)

whose solutions are

(39)

(40)

where

Using equation (39), the stream function (33) becomes

(41)

subject to the the boundary conditions

(42a)

(42b)

where

Our first approach to find the stream function is to split problem (41) in two problems that are easier to solve:

(43)

where is a particular solution of the stream function and is the solution of the corresponding homogeneous stream function. That is,

(44)

non-homogeneous equation (44) corresponding to the homogeneous boundary condition in (42a)

(45)

homogeneous equation (45) corresponding to the nonhomogeneous boundary condition in (42b)

Using the method of eigenfunction expansion we expand in a series of eigenfunctions

(46)

By differentiating (46) twice with respect to then replacing in (44) we obtain

(47)

Again, differentiating term-by-term of (46) twice with respect to we obtain:

(48)

so that equation (47) becomes

(49)

where

Multiplying equation (49) by and integrating with respect to from to and interchanging the order of summation and integration, we get

(50)

Using the orthogonality property, this equation reduce to

(51)

where

(52)

The series (46) satisfy the boundary condition (42a) if

(53)

Equation (51) subject to the condition (53) is a second order linear ODE which can be solve by using the method of variation of parameters.

The complementary function for the homogeneous function is . Taking and as function of let

(54)

(55)

we choose and such that

(56)

Therefore,

(57)

Substituting equations (54) and (57) into equation (51), we obtain

(58)

Solving equations (56) and (58) for and we obtain

(59)

(60)

Integrating, we get

(61)

(62)

Thus

(63)

where is a dummy variable.

Hence, the solution of the nonhomogeneous equation describe by equations (41) and (42a), using the superposition principle is

(64)

By using the superposition principle, the general solution of equation (45) is found to be

(65)

Using the nonhomogeneous boundary condition in (42b)

(66)

It follows from (66)

(67)

Equation (67) reduces to

(68)

Hence, the series solution of the stream function (41) is

(69)

After evaluating the integral, we have

(70)

In view of equations (20) and (21), the velocity components are the derivatives of the stream function (70)

(71)

(72)

The first order deformation:

(73)

(74)

(75)

(76)

where and are defined by equations (39), (40), (71), and (72) , respectively.

We assume the solutions of equations (73) and (75) of the form

(77)

(78)

where and are the particular solutions while and are the solutions of the corresponding homogeneous equations. That is

(79)

(80)

using the method of eigenfunction and variation of parameters the solution of and satisfying the homogeneous boundary condition (74) and (76) respectively, are express in the following form the following form

(81)

(82)

where

(83)

(84)

Now, we shall find and from equation (80) and (82) satisfying

(85)

(86)

where and are the only non-zero boundary conditions.

The above boundary conditions are obtained by using equations (77) and (78) and the given boundary condition (74) and (76).

By using the superposition principle, the general solution of equations (80) and (82) are found

(87)

(88)

where and are constants to determined.

Now, applying the nonhomogeneous boundary conditions (85) and (86) into equations (87) and (88), respectively, we get, after renaming the constants, the equations

(89)

Also, applying homogeneous boundary conditions (85) and (86), equations (87) and (88), respectively can be written as

(90)

which gives

(91)

Also, from equations (87) and (88), we have

(92)

therefore,

(93)

Hence, the series solutions of equations is

(94)

(95)

3. Conclusions Remarks

In this paper, an optimal homotopy asymptotic method is applied for solving rectangular enclosures driven by a continuously moving horizontal plate. This procedure is explicit, efficient and has a distinct advantage over usual approximation methods in that the approximate solution obtained here is valid not only for weakly nonlinear equations, but also for strongly nonlinear ones. This approach seems to be useful and can be used to obtain other analytical solutions for other couple flow problems.