1 Introduction

In recent years, fundamental problems of mixed convection heat transfer involving natural and forced convection in lid-driven cavities have received considerable attention in science and engineering. Practical applications of such cavity flows could be found in atmospheric flow^[1], lakes and reservoirs^[2], nuclear reactors^[3], chemical equipment^[4], lubrication grooves^[5], solar collectors^[6], solidification process^[7], and float grass production^[8].

Mixed convection in a lid-driven cavity is difficult to analyze since its governing equations are coupled partial differential equations. The flow governing dimensionless parameter is the Richardson number Ri=Gr/Re², which is defined as the ratio between the Grashof number Gr and the square of the Reynolds number Re, where they represent the relative strength of the natural and forced convection flow effects^[9]. The combined shear and buoyancy-driven convection is categorized into pure forced convection flow (Ri1), natural convection (Ri1), and mixed one for Ri=0.1~10, of which the former two regimes are the limiting cases. Therefore, it is of great significance to understand intensively the mechanism of mixed convection.

Various configurations of mixed convection cavity flows have been investigated. For example, the upper lid-driven case was investigated in Refs. [10]-[21]. Both the top and bottom lid-driven cases were discussed in Refs. [22]-[26]. The inclination case was studied in Refs. [27]-[31]. There were also many investigations on mixed convection flow with non-uniform heated boundaries that are commonly found in the industrial process. For example, Basak et al.^[32-34] carried out a series of studies on mixed convection in the lid-driven cavity with linear distribution of temperature at vertical sides or sinusoidal type^[35] at the bottom. More studies were carried out by Sivasankaran et al.^[36-37], Hsu and Wang^[38], Al-Amiri and Khanafer^[39], Abu-Nada and Chamkha^[40], and Nasrin and Parvin^[41].

In the above-mentioned studies, many different numerical techniques have been applied, such as the finite difference method^[9], the finite volume method (FVM)^{[33, 42]}, and the pseudo-spectral collocation method^[43], while no wavelet technique was used. It is known that the wavelet technique has great excellence for illustration of the local details of solutions, which has been successively applied in many different fields, including signal analysis^[44], fluid^[45], and solid^[46] mechanics. It is reasonable to suppose that this technique has better capability than other approaches to show the detailed information of vortexes and eddies. However, by now, the wavelet technique has rarely been used to solve nonlinear problems in fluid mechanics and heat transfer, owing to complex governing equations and boundary conditions. To overcome its major limitation, we introduce the homotopy idea to the Coiflets-type wavelet. This idea has been proved practicably. Several successful examples can be found in Refs. [47]-[49]. However, all of those examples were confined to handle nonlinear problems subject to homogeneous boundary conditions. None has been done on nonlinear problems with nonhomogeneous boundary conditions, which are obviously often encountered in the research of fluid mechanics and heat transfer.

In this paper, we shall extend the wavelet homotopy analysis method (wHAM) to the classical cavity flow and heat transfer problem. Similar studies have not been reported in the literature, as far as we know. The general case of the convective heat transfer in a cavity is considered. The cavity is placed on a slope with the upper and lower walls moving at constant speeds simultaneously. Both moving walls are subject to four different heating modes, including uniform, linear, exponential, and sinusoidal distributions. Comparisons with the previous results are given to testify the validity and correctness of our proposed technique. Detailed parametric investigations of nonuniform boundaries, amplitude ratio, phase deviation of temperature, and inclination of cavity are carried out for this generalized problem.

2 Mathematical description

A two-dimensional square cavity of length H is placed on a slope of inclined angle γ with its top and bottom walls moving at constant speeds U₀ and λU₀ in the same or opposite directions and heating non-uniformly. The sketch is shown in Fig. 1, where the origin of the Cartesian coordinate system is at the bottom left corner with the x- and y-axes along the length and height of the cavity, respectively. Non-uniform temperature distributions are imposed on both horizontal walls of the enclosure. λ=0 indicates that the bottom lid is stationary. λ=1 and λ=-1 mean that the horizontal walls move in the same or opposite directions, respectively. It is assumed that the viscous dissipation and heat radiation are neglected. It is also assumed that the fluid properties are constant and laminar so that the conservations of mass, momentum, and energy are satisfied. The density variation is modeled according to the Boussinesq approximation, in which the variations in fluid properties other than the density are ignored, and the density ρ only appears when it is multiplied by the gravitational acceleration linearized with the temperature,

(1)

where β is the thermal expansion coefficient, and ρ₀ and T₀ denote the reference state.

Since the flow transients are always smooth, the steady mixed convection flow in a square cavity makes sense compared with the unsteady one which plays an important role in the configurations^[10]. With those assumptions, the governing equations are written as follows.

The continuity equation is

(2)

The momentum equations in the X- and Y-directions

(3)

(4)

The thermal energy transport equation without internal heat production is

(5)

The appropriate boundary conditions are

(6)

Here, U and V are the velocity components in the X- and Y-directions, respectively, P is the pressure, T is the temperature, T_t(X) and T_b(X) are the temperature distributions on the top and bottom walls, respectively, g is the acceleration of gravity, and k and c_p denote the thermal conductivity and the specific heat, respectively.

Substituting the following vorticity-stream function formulation

(7)

and the following dimensionless variables

(8)

into the above governing equations, the dimensionless governing equations are obtained,

(9)

(10)

where the Reynolds number Re, the Grashof number Gr, and the Prandtl number Pr are defined by

(11)

in which α, ν, and ΔT denote the thermal diffusivity, the dynamic viscosity, and the reference temperature difference, respectively.

Finally, we eliminate the vortices ω to obtain the final coupled dimensionless differential equations,

(12)

(13)

subject to the boundary conditions

(14)

where Ri=Gr/Re² is the Richardson number, ▽² is the Laplace operator, ▽⁴ is the Biharmonic operator, elaborated in the Cartesian coordination system by

and the nonlinear operator for arbitrary functions F and G is

It is known that the no-slip boundary conditions (14) have been widely used to describe one side or two-side lid-driven cavities. However, there are non-physical singularities at four corners, which have to be considered in the computation.

The local Nusselt number is a key factor to measure the heat transfer rate, which can be defined as

(15)

where Nu_b and Nu_t are the Nusselt numbers on the bottom and top walls, respectively.

The average Nusselt numbers along the horizontal surfaces demonstrating the level of heat transfer rate are defined by

(16)

3 Wavelet-homotopy technique

In the generalized orthogonal Coiflet system, when an arbitrary function f(x) is approximated by the Coiflets, the approaching precision only depends on its resolution level and wavelet properties irrelevant to f(x). Hence, the generalized orthogonal Coiflet basis can be employed as the expression functions for approximating solutions of nonlinear problems to be solved. In doing so, the governing equations (12) and (13) are first transformed into a group of linear ones by the homotopy analysis method (HAM) technique. The highly accurate generalized orthogonal Coiflet series solutions are given by the wavelet Galerkin method.

3.1 Linearization

The linearization of the nonlinear systems (12) and (13) is fulfilled by means of the HAM via constructing the zero deformation equations,

(17a)

(17b)

where p∈[0, 1] is an embedding parameter, and are the linear operators, c₁ and c₂ are the convergence-control parameters, Φ(x, y; p) and Θ(x, y; p) are the mappings of ψ(x, y) and θ(x, y), and [Φ, Θ] and [Φ, Θ] are the nonlinear operators defined by

(18a)

(18b)

For p=0 and p=1, we have

(19)

It should be pointed that as p varies from 0 to 1, the initial guesses ψ₀ and θ₀ are transformed to the desired solutions ψ and θ. Therefore, the Taylor expansions for Ψ and Θ with respect to p are illustrated as

(20)

(21)

where

(22)

The auxiliary linear operators, the initial guesses, and the convergence control parameters are appropriately selected so that the Taylor series converge at p=1. The homotopy series solutions are determined as

(23)

(24)

Differentiating Eqs. (17a) and (17b) M times with respect to p, then dividing them by M!, and finally setting p=0, we obtain the Mth-order deformation equations,

(25)

(26)

where

with

(27)

3.2 Generalized orthogonal Coiflets selection

Following Liu et al.^[50] and Zhou and Wang^[51], an arbitrary function is approximated by Coiflets,

(28)

where N is the vanishing moment, M₁ is the first-order vanishing moment, and ϕ is the wavelet basis. Note that the coefficients of Coiflets series are the approximate values of the function middle points. In order to include nonhomogeneous boundary information into the Coiflets, inspired by Liu et al.^[50] and Zhou and Wang^[51], we construct the boundary Coiflets by adding the spline functions to make up for the missing derivative information on boundaries, which can be written as

(29)

where T_{0, k}(x) and T_{1, k}(x) are the modification functions

(30)

in which p_{i, k}⁰ and p_{i, k}¹ are coefficients, determined by the following coefficient matrices P₀^c and P₁^c:

On substitution of Eq. (29) into Eq. (28), we finally obtain

(31)

where

Taking Eq. (31) into account, the targeted approximation of f(x) is divided into two parts. One is the generalized Coiflets with the polynomial interpolation modifications, written by

(32)

The other is the boundary wavelets expressed by

(33)

where the superscripts 0 and 1 correspond to the left and right boundaries, respectively, and the subscript a relates to orders.

3.3 Coiflets selection and approaching process

Taking the inhomogeneous boundary conditions (14) into consideration, we need to construct the appropriate Coiflets to approximate ψ and θ. The procedure is listed below.

Firstly, we determine the quantities of needful types of Coiflets by defining the following boundary matrices:

(34)

(35)

Thus, the boundary Coiflets related to the different orders of derivative are subject to the nonhomogeneous Neumann boundaries,

(36)

As a result, the dimensionless stream function and the temperature for Mth-order solutions are expressed by Coiflets as

(37)

(38)

where the boundary parts B_ψ(x, y) and B_θ(x, y) correspond to the Dirichlet and Neumann types,

(39)

According to the general Gaussian integration method^[51], the linear and nonlinear operators and acting on the approximating functions are converted to

(40)

Therefore, based on the above criterion, the fields of vorticity and velocity in the x- and y-directions are reconstituted by

(41)

(42)

(43)

where the boundary parts are

Finally, we substitute Eq. (38) into Eqs. (15) and (16) to define Coiflets constitution of local and average Nusselt numbers,

(44)

(45)

where

Finally, linear operators are selected as and . We construct the iterating equations by substituting Eqs. (37) and (38) into the high order HAM deformation equations (25) and (26), with the consideration of criterion Eq. (40). Detailed precesses are illustrated in {Appendix B} along with some definitions in Ref. [49].

In order to accelerate the solution convergence, the iteration HAM (iHAM) technique^[52] is introduced. In doing so, we continuously replace the initial guesses by Mth-order approximations to adjust the trajectory of solutions from the initial guesses to the exact ones. The iteration formula is

(46)

To evaluate the validity of our solutions, an error function E_Res is defined as an indicator for checking convergence of Mth-order solutions. It works well for nonlinear problems without analytical solutions and is written by

(47)

If analytical solutions exist, we define the error distribution function as

(48)

where F(x, y) denotes analytical solutions, and F_e(x, y) denotes computational results. It can help us know exactly the error distribution at a certain point. The absolute error between the analytical solutions and our approximations is calculated by using the following error function:

(49)

where F=ψ, θ are the Coiflets approximations, and F_e=ψ_e, θ_e are the analytical ones.

4 Validation

At the beginning of our analysis, we validate the efficiency of our proposed technique by three steps. First, we compare our Coiflets solutions with the exact ones that exist in very particular cases. Then, a comparison of our solutions for the pure cavity flow with the previous published results is made for check of the accuracy. Finally, a further comparison of our solutions with other numerical ones available in the literature for general cases of mixed convection in the cavity (Gr=10³~10⁶, Re=1~10³) is given for testification of the generalization of our proposed technique.

We first check the convergence of our solutions by comparing them with the exact ones. Note that, for Re=100, Gr=100, and λ=0, append the following additional items P and Q defined in Appendix A on the right-hand side of Eqs. (12) and (13).

As shown in Fig. 2, the absolute error E_SQ between our solutions and the exact ones reduces very quickly as the iteration increases. We also notice that the convergence rate can be improved as the resolution level j enlarges. Further to check the accuracy of our solutions, we list the relative and absolute errors of ψ(x, y) and θ(x, y) in Table 1. It is found that all errors reduce rapidly as the resolution level j increases from 3 to 5. It is also noticed from the table that the error for j=6 is not obviously superior to that for j=5, but the computational time is much longer. In order to balance precision and efficiency, the resolution level j=5 is selected in the following computations so that the Coiflets series solutions contain 31×31 and 31×33 items for the stream and temperature approximations, respectively.

Further to assess the accuracy of our solutions, we compare them with the previously published results for the pure lid driven cavity flow^[53-56]. It is known from Eq. (12) that for the case of Gr=0, the problem is reduced to the classical cavity flow. Many numerical computations are available for various configurations of cavity flow with upper lid driven. We compare our results with theirs, as illustrated hereinafter. The comparisons of our results for the values of the minimum stream function ψ, the minimum and maximum velocity components of the flow field at midsections x, y=0.5, and their locations with those given by Rubin et al.^[57], Schreiber and Keller^[58], Vanka^[59], Hou et al.^[60], Goyon^[61], Barragy and Carey^[62], Zhang^[63], Gupta and Kalita^[64], Erturk et al.^[65], and Marchi et al.^[56] are shown in Tables 2 and 3. Excellent agreement is found. Furthermore, by comparing our results with those given by Ghia et al.^[53], Botella and Peyret^[54], Bruneau and Saad^[55], and Marchi et al.^[56] for distributions of velocity at sections x, y=1/2, it is found that our results agree well with theirs, showing good precision of our proposed technique, as illustrated in Fig. 3.

The average Nusselt numbers, which are physically important quantities to measure the ratio of the conductive thermal resistance to the convective thermal resistance of the fluid, are also calculated via the reconstitution algorithm denoted in Eq. (45) when Gr=100 and Re=1~1 000, as shown in Table 4. It is clearly shown from the table that our solutions match all of those previous results given by Iwatsu et al.^[9], Khanafer and Chamkha^[11], Waheed^[67], Tiwari and Das^[16], Ismael et al.^[24], Sheremet and Pop^[25], Ahmed^[68], Sharif^[27], Sivasankaran et al.^[30], and Abu-Nada and Chamkha^[40]. Besides, the local Nusselt number Nu_b and the velocity at midsections x, y=0.5 by our approach are also in excellent agreement with those by the FVM using SIMPLEC algorithms^[69], as shown in Fig. 4.

In the following, we make comparisons for Gr≠0. In this more general case, following the previous studies^[9], it is assumed that the top wall uniformly moves and heats, while the bottom wall keeps fixed, and both the left and right walls are adiabatic and stationary. Note that in this configuration, the jump discontinuity of the velocity and temperature at the corner points exists, which corresponds to computational singularities contributing to the boundaries of the Dirichlet and Neumann types for θ and ψ. Different from the previous numerical results with special treatment by assuming the average temperature of two walls at the corner to keep the adjacent grid-nodes at respective wall temperatures^[66], or obtaining optimal grid size invariant to grid by the intersection of the two differentially heated boundaries^[33], we do not need any particular treatment for boundary conditions. The only thing we need to do is to adjust the vanishing moment and resolution level of wavelet. The highly accurate solutions can be readily obtained using our proposed technique. Further comparisons of our results with the previously published ones by Iwatsu et al.^[9], Waheed^[67], Khanafer et al.^[13], and Abdelkhalek^[14] for the minimum and maximum velocity components U_x and V_y at x=0.5 and y=0.5 in the cases of Gr=100 and Re=100, 400 are presented in Tables 5 and 6. An excellent agreement is found for various resolution level j from 3 to 6.

5 Effects of alternative temperature distributions

In the above analysis, our proposed approach has been verified via a series of comparisons. Here, we apply it to several different cases overlooked in the previous studies.

The mixed convection cavity flow considered here is subject to the stationary and thermally isolated boundary conditions on the left and right walls, and the movable and variously thermal boundary conditions on the top and bottom walls. This is to say, the temperature distribution on the bottom boundary is settled as T_b=T₀+A₁, while on the top boundary, it is given by various forms of temperature distribution including constant, linear, exponential, and sinusoidal cases,

(50)

(51)

(52)

(53)

where T₀ is a reference temperature, A₁ and A₂ are constants, and ξ is the phase deviation.

By introducing an amplitude ratio =A₂/A₁∈[0, 1] and setting the reference temperature difference ΔT=A₁, we can obtain that the dimensionless temperature distribution on the bottom is θ_b=1, and the dimensionless temperature distribution on the top is

(ⅰ) uniform type: θ_t=,

(ⅱ) linear type: θ_t=1+(-1)x,

(ⅲ) exponential type: θ_t=(e-e^x+(e^x-1))/(e-1),

(ⅳ) sinusoidal type: θ_t=sin(2πx+ξ).

Highly accurate results obtained by our proposed technique are presented in terms of the streamline and isotherm patterns, and the velocity and temperature distributions, as well as the local Nusselt number. The parameters used here are the amplitude ratio =0.5, the inclined angle γ=π/6, and the phase difference ξ=π/4. Since the Richardson number provides a measure of the importance of buoyancy driven natural convection relative to lid-driven forced one, Re=100, Ri=1, and Pr=0.71 are selected which are commonly used in the previous studies.

Taking Figs. 5(a) and 5(b) into consideration, the velocity profiles at x, y=0.5 are nearly the same for the uniform, linear, and exponential cases, while this profile is totally different for the sinusoidal one. The flow fluctuation in the top and bottom half parts of the cavity is contrary. The local Nusselt number is an important physical quantity to evaluate the active conduction and transfer rate of losing energy. It is seen from Fig. 5(c) that on the top wall, the local Nusselt number decays gradually along the x-coordinate in the uniformly heated case. For the linearly and exponentially cases, the trend of the local Nusselt number is quite similar. It first increases as x enlarges, and after reaching a peak value, it decreases as x evolves. In the left part of the top wall, the flow gains heat from the wall (Nu < 0), and the local Nusselt number reduces to zero nearly at x=0.25 and x=0.31. Then, the conduction direction reverses that the flow releases heat (Nu>0) and the local Nusselt number reaches a maximum value around x=0.8. For the sinusoidally heated case, the local Nusselt number fluctuates as x evolves. In the ranges of x=0~0.2 and x=0.75~1, the local Nusselt number reduces to two minimum values respectively, while it reaches the maximum value at x ≈ 0.5. As seen in Fig. 5(d), the distributions of local Nusselt numbers along x on the bottom wall for the uniformly, linearly, and exponentially heated cases are quite similar. The uniform one is slightly higher than the others. The sinusoidally heated case exhibits different trend, which performs a much higher rate of absorbing heat from the wall except a small area near the left edge.

To make contrast with the average Nusselt numbers, for the uniform, linear, exponential, and sinusoidal cases, their results by Eq. (45) are, respectively, 0.946, 0.683, 0.610, and 1.591 6 for the top wall, and -0.918, -0.747, -0.677, and -1.611 for the bottom wall. The actual heat absorption rate of fluid Nu_b+Nu_t is 0.031 5, -0.064, -0.067, and -0.02 for those cases. From the point of view of energy conservation, when Nu_b+Nu_t is negative, most of the energy generated from the bottom wall is transferred to the top wall, while only a small amount is absorbed by fluid flow consumed in dissipation of viscosity along with input energy by lid-driven wall. However, when Nu_b+Nu_t is positive, it indicates that the fluid flow absorbs the kinetic energy transferred from the top moving wall. In summary, the sinusoidally heated top lid performs the highest heat transfer rate no matter on the bottom or top boundary. That is superior to others in the same amplitude ratio, which changes the flow and temperature fields, showing superiority in heat transfer as compared with the other heating modes.

6 Effects of parameters

It is known from Eqs. (12) and (13) that there are several important non-dimensionless parameters that can affect flow and heat transfer in the cavity. Therefore, parametric studies need to be considered to cover the scale of these important parameters.

In this section, the non-dimensional temperature distribution for the sinusoidal case is given as θ_b=sin(2πx) and θ_t=sin(2πx+ξ), which are simplified from

(54)

Most of our results are captured at Pr=0.71 by the control variable method for different values of the amplitude ratio , the phase deviation ξ, and the inclined angle γ. The first-order iHAM technique is applied in all calculations with the convergence criterion that the relative error E_Res < 10^-8. In Subsections 6.1 and 6.2, we figure out the effects of the amplitude ratio and the phase deviation respectively on the top wall compared with the bottom one. In Subsection 6.3, various inclined angles are investigated.

6.1 Part Ⅰ: effect of amplitude ratio

In the following analysis, we simulate the uniformly heated case for =0, 0.25, 0.5, 0.75, 1, which will be used as a reference for comparison of the influence of the amplitude ratio with the sinusoidally heated case. Assume that both the top and bottom walls are heated sinusoidally for the non-uniformly heated case.

Table 7 lists the minimum ψ, the minimum velocities at x=0.5, and the minimum and maximum velocities at y=0.5, together with the average Nusselt numbers. It is noticed that for the uniformly heated case, as enlarges, ψ_min and V_min first decrease and then increase, while the variation of V_max is opposite. U_min and Nu_b increase gradually, while Nu_t decreases instantly. However, for the sinusoidally heated case, V_max and Nu_t increase, but the others decrease as evolves. Besides, we notice that the heat transfer rate on the top wall (Nu_t) is larger than that on the bottom wall (Nu_b) for the sinusoidal type.

From the energy point of view, it is pointed out that the energy absorbed by flow from boundaries is separated into two parts. One is from the boundary moving lid by the shear force (E_tk), and the other is by heat conduction from the bottom wall (E_br). The energy is transferred into two directions. One part is to add the internal energy of the top wall (E_tr), while the other is absorbed by the flow for the kinetic energy (E_fk) and the internal energy (E_fr) with the conservation relation E_tk+E_br=E_tr+E_kf+E_fr. When Nu_t+Nu_b < 0, it indicates E_br>E_tr contributing to E_tk < E_fk+E_fr, showing that the energy absorbed from the bottom wall is transformed into the energy of flow. If Nu_t+Nu_b>0, the situation is quite inverse that the part of the kinetic and internal energy of flow is used to be transferred into the top wall. By increasing , Nu_t+Nu_b decreases, showing that the heat conduction is dominant, and the heat energy absorbed by fluid flow from boundary also enhances gradually at this time.

Figures 6(a) and 6(b) illustrate the variations of the local Nusselt number with x on the top and bottom walls, respectively. It is found from Fig. 6 that is only associated with the top wall and nearly has no effect on the heat transfer on the bottom wall. Moreover, the local Nusselt number on the top wall fluctuates along with x, where the direction of heat transfer at both ends is inwards (Nu < 0) contrary to that is outwards (Nu>0) in the centre. In summary, for the sinusoidal case, the heat transfer rate increases with evolving, while the reverse point Nu=0 is fixed irrelevant to as >0. When =0, the top wall is heated uniformly resulting in totally different distribution of energy transfer rate, as shown in Fig. 6(a). In this situation, not only the location of the reverse point Nu=0 shifts right, but also the variational amplitude anomalously increases, which is equaling to the scale in contrast with =0.5.

6.2 Part Ⅱ: effect of phase deviation

The amplitude ratio and the inclined angle are selected as =1/2 and γ=π/6, respectively. Figure 7 illustrates the effect of the phase deviation on velocities at midsection x=0.5 and y=0.5, respectively. It is seen from the figure that the change tendency along the section is almost not affected by ξ. However, the minimum velocity at x=0.5 and the maximum and minimum velocities at y=0.5 first increase and then decrease, reaching a maximum value around ξ=3π/4. In conclusion, the variation in the phase deviation affects the maximum velocity value rather than its distribution to a large extent.

Figures 8(a) and 8(b) demonstrate the effect of the phase deviation on the local Nusselt numbers along the top and bottom walls with the inclined angle π/6. It is seen that the local Nusselt number along the top wall is significantly influenced with the phase deviation evolving. Its profile along the top wall is approximately sinusoidal distribution and moves towards the left wall with ξ increasing, indicating that the area of heat transfer moves inwards and outwards periodically, alternately changing with each other. The maximum and minimum transfer rates are slightly affected on the bottom wall so that its variation trend is not perceived.

Figure 9 illustrates the effect of the phase deviation on the average Nusselt number for different inclined angles. As shown in Figs. 9(a) and 9(b), the fluid flow near the bottom wall acquires heat energy from the wall (Nu_b < 0) and then dissipates the energy to the top wall (Nu_b>0) for arbitrary phases ξ∈[0, 2π]. As the phase deviation varies from 0 to 2π, which is detailed in Table 8, both the absolute values of Nu_b and Nu_t perform sinusoidal periodic variation, which first reduce to the minimum nearly at 7π/8. Thereafter, they simultaneously and gradually increase. It indicates that heat transfer on bottom and top walls could be suppressed to the greatest extent, while the phase deviation is selected properly. Meanwhile, increasing γ is conducive to weaken the energy transfer rate on both the bottom and top boundaries, because the efficient buoyancy effect by gcosγ gradually decreases. As shown in Fig. 9(c), the heat absorb rate of fluid Nu_b+Nu_t < 0 is affected by the phase deviation and the inclined angle. When ξ evolves from 0 to 2π, its variation trend is similar to those of Nu_b and Nu_t. However, it is irrelevant to alter the endothermic rate of fluid flow from the boundaries by increasing the inclined angles.

6.3 Part Ⅲ: effect of inclined angle

Here, we consider an enclosure with top and bottom walls moving simultaneously with the reverse direction λ=-1 with different inclination angles. The phase deviation and the amplitude ratio are selected as ξ=π/ 4 and =1/2. The overall average Nusselt number on the top and bottom boundaries, which is used to measure the overall hear transfer rate, is plotted in Fig. 10. Here, it is considered as a function of the cavity inclination angle from 0~π/2 with Ri=0.1, 1, 10, respectively. For the forced convection regime (Ri=0.1), the absolute value of the average Nusselt number decreases slowly with the inclined angle. For the mixed convection (Ri=1) and the natural convection (Ri=10), Nu_b and Nu_t decrease dramatically as the inclined angle enlarges. This is because the buoyancy effect gradually prevails in the heat transfer process. The sum Nu _b+Nu_t is negative, indicating that the absolute value of average Nusselt on the top surface is smaller than that on the bottom one, which is different from the one that is identical with the uniform temperature distribution by Sharif^[27]. The energy absorption rate of fluid from walls Nu_b+Nu_t is irreverent to the inclined angles when Ri≤1, but it slightly decreases when the natural convection regime (Ri=10) dominates in the heat transfer mode caused by the increase in γ.

7 Conclusions

In the present work, the wavelet-homotopy technique is developed successfully to give Coiflets wavelet solutions for the classic cavity flow and heat transfer problem with nonhomogeneous boundary conditions. It is the first time for application of the proposed technique to the problems of fluid mechanics and heat transfer. Comprehensive validations are elaborated with other numerical methods. Excellent agreement is found. Parametric physical analysis including the amplitude ratio, the inclinations, and different phase deviations is given. In addition, the following physical findings are observed.

(ⅰ) In comparison with uniformly, linearly, and exponentially heated boundaries, sinusoidally heated horizontal walls perform the best properties of heat transfer rate under the same condition of amplitude ratio, which greatly alters the flow and temperature fields.

(ⅱ) As the amplitude ratio increases from 0 to 1, the heat transfer rate of top wall increases, while on the bottom, it keeps the same and the reverse point Nu=0 is also fixed.

(ⅲ) Adding inclination efficiently reduces the buoyancy effect so as to weaken heat transfer rate on both walls, but it is irrelevant to alter the rate of gaining energy by the fluid from the boundaries.

(ⅳ) Periodic variation of amplitude ratio resulting from different phase deviations contributes to approximate periodic change of heat transfer properties, while an extreme phase nearly 7π/8 exists where heat transfer is suppressed to the greatest extent.

Acknowledgements

Thanks to the anonymous reviewer for their constructive comments and suggestions. The authors also wish to express their thanks to the very competent reviewers for the valuable comments and suggestions.

Appendix A

(A1)

with the exact solutions

(A2)

Appendix B

(A3)

(A4)

Eventually, by multiplying h_{j, n'}(x)h_{j, m'}(y) and φ_{j, n}(x)φ_{j, m'}(y) on both sides of Eqs. (A3) and (A4) and then integrating on the domain [0, 1] by applying the wavelet Galerkin method, some definitions were illustrated in Ref. [49]. We obtain the following coupled iterating algebra equations:

(A5)

(A6)

where the iterating tensors are

the straight vectors are

and the suffixes are defined as

Then, we calculate the tensors constituted by the connection coefficients, which are split into three categories for computation. By approaching the Coiflets h_{j, k} with each other, the formula is given by

(A7)

By the standard original Coiflets φ_{j, k} with each other, the formula is given by

(A8)

By the approaching Coiflets with the standard original Coiflets, they are written as

(A9)

(A10)

By the approaching Coiflets with the boundary Coiflets , it is defined by

(A11)

The straight vectors of the nonlinear parts and are approximated by product of calculated vectors of ψ and θ, which are expressed by

(A12)

It is worth mentioning that the effects of the nonhomogeneous boundaries involve two aspects. One is that it affects the wavelet approaching precision for ψ₀ and θ₀ via Eqs. (37) and (38). The other is that it adds the integration correction items in Eqs. (A5) and (A6),

(A13)

Then, we take the sum according to Eqs. (23) and (24),

(A14)

where the approximation of the nonhomogeneous boundary part is

(A15)

It is pointed out that the approximate tensors of dimension are (2^j+1)×(2^j+1),