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2016,28(5):873-885 DOI: 10.1016/S1001-6058(16)60688-1

Natural convective heat transfer and nanofluid flow in a cavity with top wavy wall and corner heater* Mikhail A. SHEREMET1,2, Ioan POP3, Hakan F. ÖZTOP4,5, Nidal ABU-HAMDEH5 1. Department of Theoretical Mechanics, Tomsk State University, Tomsk, Russia 2. Department of Nuclear and Thermal Power Plants, Tomsk Polytechnic University, Tomsk, Russia, E-mail: [email protected] 3. Department of Applied Mathematics, Babeş-Bolyai University, Cluj-Napoca, Romania 4. Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey 5. Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia (Received February 10, 2016, Revised May 5, 2016) Abstract: A numerical analysis of natural convection of nanofluid in a wavy-walled enclosure with an isothermal corner heater has been carried out. The cavity is heated from the left bottom corner and cooled from the top wavy wall while the rest walls are adiabatic. Mathematical model has been formulated using the single-phase nanofluid approach. Main efforts have been focused on the effects of the dimensionless time, Rayleigh number, undulation number, nanoparticle volume fraction and length of corner heaters on the fluid flow and heat transfer inside the cavity. Numerical results have been presented in the form of streamlines, isotherms, velocity and temperature profiles, local and average Nusselt numbers. It has been found that nanoparticle volume fraction essentially affects both fluid flow and heat transfer while undulation number changes significantly only the heat transfer rate. Key words: wavy cavity, natural convection, nanofluid, corner heater, numerical results

Introduction Natural convection heat transfer and fluid flow in a nanoparticle added fluid, namely nanofluid, filled cavities becoming very popular in recent years due to their heat transfer control potential in different energy systems. Most of these system may have curvilinear boundary conditions and the analysis is more difficult than the regular geometries (rectangle or square) due to grid inconsistency near the walls. Besides, these systems can be heated or cooled partially from different sides of their wall. Partial heater is mainly seen in electronical equipments of building heaters. These applications are reviewed by Öztop et al.[1] widely for pure fluid or nanofluid. Also, application of corner heater can be seen especially for some electronic equipments of building design. Ray and Chatterjee[2] made a work on mixed * Biography: Mikhail A. SHEREMET (1983-), Male, Ph. D., Professor

convection heat transfer in a lid-driven cavity with heat conducting circular solid object and corner heaters with Joule heating. They observed a major influence of the prevailing convection method and the applied magnetic field on the flow as well as the thermal field, while the effect of Joule heating was found to be of very small significance. Ahmed et al.[3] studied natural convection coupled with radiation in an inclined porous enclosure with corner heater. They observed that radiation parameter affects both heat transfer and fluid flow and heat transfer rate is an increasing function of radiation parameter and a decreasing function of Darcy number. In this context, Öztop et al.[4] made a computational work to obtain the laminar mixed convection and fluid flow in a lid-driven cavity under the magnetic field. In their work, the cavity was heated via corner heater. They used finite volume technique to solve the equations and found that temperature distribution inside the cavity mostly stems from the right side of corner due to impinging air and this distribution is affected by the changing of dimensions of heater lenght and magnetic force. Abu-Nada[5] studied the mixed convection heat transfer in a vertical

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lid-driven cavity by using dissipative particle dynamics (eDPD) and his results were compared with the finite volume solutions. He approved that eDPD simulations were benchmarked against finite volume solutions and it appropriately predicted the temperature and flow fields in mixed convection. In a series of recently published papers by Malvandi and Ganji[6], Malvandi[7], Malvandi et al.[8-10], several problems of nanofluid, such as, the effects of nanoparticle migration and asymmetric heating on magnetohydrodynamic forced convection of alumina/water nanofluid in microchannels, two-component heterogeneous mixed convection of alumina/water nanofluid in microchannels with heat source/sink, film boiling of magnetic nanofluids over a vertical plate in presence of a uniform variable-directional magnetic field, thermophoresis and Brownian motion effects on heat transfer enhancement at film boiling of nanofluids over a vertical cylinder, have been studied in detail. Most of the real engineering problems have curvilinear boundary. But it is difficult to solve these kinds of problems due to inconsistency between grid distribution and boundary. In this context, Öztop et al.[11] worked on natural convection in a cavity with internal heat generation. Varol and Öztop[12] made a comparative study on natural convection in inclined wavy and flat-plate solar collectors. Unsteady natural convection in an open wavy porous enclosure which is filled with a nanofluid problem was studied by Sheremet et al.[13]. They showed that the undulation parameter is highly effective on heat and fluid flow and the average Nusselt and Sherwood numbers decrease with an increase in the undulations number. Other related studies with undulated cavities can be found in refs. as Nikfar and Mahmoodi[14], Cho et al.[15], Adjlout et al.[16], and Das and Mahmud[17]. We mention to this end also, the recent book by Shenoy et al.[18] on convective flow and heat transfer from wavy surfaces: viscous fluids, porous media and nanofluids. The main aim of the present work is to make a computational analysis in a nanofluid filled enclosure with wavy-walled ceiling and corner heater on the left corner. Based on the above wide literature survey and authors’ knowledge the work is the first work on wavy walled cavity filled with nanofluid from a corner heater. 1. Basic equations Natural convection in a nanofluid based on water and solid nanoparticles located in a cavity with top wavy, right, bottom and top flat solid walls is analyzed. A domain of interest is shown in Fig.1, where x axis is the horizontal direction along the lower wall of the cavity and y axis is the vertical direction along the left wall of the cavity and D is the size of the square cavity. It is assumed that the left bottom corner is kept

at maximum temperature Th while the top wavy wall is kept at minimum temperature Tc . The height of the heater vertical wall is hy while the length of heater horizontal wall is hx . It is also assumed that the rest walls of the cavity are adiabatic. The walls of the cavity are assumed to be impermeable. It is considered that the top wavy wall is described by the following relation y1 = D[a + b cos(2 x / D)] .  = y1 = D[a + b cos(2  x / D)] is the distance between horizontal walls. Here a and b are constants that determine the shape and the wavy contraction ratio of the wavy wall (a + b = 1) . It should be noted that the nanofluid is Newtonian and incompressible. The base fluid and the nanoparticles are in thermal equilibrium and no relative motion occurs between them. The flow and heat transfer are considered to be two-dimensional and laminar, the radiation and viscous dissipation effects are negligible.

Fig.1 Physical model and coordinate system

Except for the density, the physical properties of the fluid are constant, while the Boussinesq approximation is valid. The governing equations for mass, momentum and thermal energy can be written as follows[18,19]  V = 0

 V

(1)



n f  + (V , )V  =  p + n f  2V   t   (  ) n f (T  Tc ) g   2T  2T   T  (  C p )n f  + (V , )T  = kn f  2 + 2  y   t   x

(2)

(3)

The utilized physical parameters of the nanofluid are presented in detail in Ref.[20]. Equations (1)-(3) can be written in Cartesian coordinates as[18,19]

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u  v + =0 x  y

(4)

 u u u  p +u +v + n f = x y  x  t

  2u  2u   2 + 2  (5) y   x

 v v v  p +u +v + n f = t x y    y  

 v  v   2 + 2 + y   x

n f 

n f 

2

2

(  )n f g (T  Tc )

(6)

T T T +u +v = nf t x y

  2T  2T   2 + 2 y   x

equations are given by

 = 0 ,  =1,  = 

 = 0,

 = 0,

Numerical solution is assumed to conduct using the following dimensionless variables x=

y x uD vD , y= ,  = 2 , u= , v= , D D D nf nf

=

T  Tc Th  Tc

(8)

with a dimensionless stream function ψ and vorticity



(9)

We can obtain the following equations

 2  2 + =  x 2 y 2

h   2 = 0 ,  =  2 on y = 0 , x  x  1 D y y (13e)

 = 0,  = 0 ,  = 

 2  2  on y = y1 , 0  x  1 y 2 x 2 (13f)

Here Pr =  f (  C p ) f /(  f k f ) is the Prandtl number, is the Rayleigh

number, and the functions H1 ( ) and H 2 ( ) are given by

H1 ( ) = (10)

  2  2        + = Pr H1 ( )  2 + 2  +  y x x y y   x

 x

(11)

      2  2  + = +  y x x y x 2 y 2

(12)

Ra Pr H 2 ( )

D (13a)

h  2 on y = 0 , 0  x  x 2 D y (13d)

Ra = g (  ) f (Th  Tc ) D 3 /( f  f )

  v y u= , = , v=  x y x y

hy

  2 = 0 ,  =  2 on x = 1 , 0  y  1 x x (13c)

 = 0 ,  =1,  = 

 = 0,

on x = 0 , 0  y 

hy   2 = 0 ,  =  2 on x = 0 ,  y 1 x x D (13b)

(7)

t n f

 2 x 2

Taking into account the considered dimensionless variables (8) the top wavy wall of the cavity is described by the following relation y1 = a + b cos(2  x) and  = y1 is the dimensionless distance between horizontal walls. The corresponding boundary conditions for these

1 +   p (1   )2.5 1   +  f 

 ( C p ) p ( C p ) f

  k p + 2 k f  2 (k f  k p )       k p + 2 k f +  (k f  k p )  (14) 2

  (  ) p    ( С p ) p  1   +  1   +  (  ) f   (  С p ) f   H 2 ( ) = 2   p   k p + 2k f  2 (k f  k p )  1 +     f   k p + 2k f +  (k f  k p )   (15) These functions include the Brinkman relation[21] for the effective dynamic viscosity and the Maxwell model[22] for the nanofluid thermal conductivity and depend on the nanoparticles concentration  and physical properties of the fluid and solid nanoparticles.

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Fig.2 Comparison of vorticity (on the left), streamlines (on the middle) and isotherms (on the right) at hx / D = hy / D = 0.5 ,

Ra = 105

2. Numerical methods and validation The cavity in the x and y plane, i.e., physical domain, is transformed into a rectangular geometry in the computational domain using an algebraic coordinate transformation by introducing new independent variables  and  . The physical geometry is mapped into a rectangle on the basis of the following transformation:

 =x,  =

y y = y1 a + b cos(2  x)

  2   2   2  2   2 +2 +  +    +  2 x    x   y    2 (17)

 = 0,  =1,  = 

 = 0,

hy  2 on  = 0 , 0    2 D  (20a)

hy   2 = 0 ,  =  2 on  = 0 ,  1 D   (20b)   2 = 0 ,  =  2 on  = 1 , 0    1   (20c) 2

    2  = 0 ,  = 1,  =   on  = 0 , 2  y  

2 2   2         2 +  + +     x   x   y    2

         +  + R a Pr H 2 ( )   2 x     x  

(19)

The corresponding boundary conditions of these equations are given by

 = 0,

  2         + = Pr H1 ( )  2 +  y   y     2

  2   2   2  2     2+ 2  + x   x   y   

(16)

Taking into account transformation (16) the governing Eqs.(10)-(12) will be rewritten in the following form:

 2  =  x 2 

        2   2  + = 2 +2 +  y   y    x 

0 

2

(18)

hx D

(20d)

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Fig.3 Comparison of vorticity (on the left), streamlines (on the middle) and isotherms (on the right) at hx / D = 0.25 , hy / D = 0.75 ,

Ra = 105 2

    2   = 0, = 0 ,  =  on  = 0 ,  2   y  

and the average Nusselt number Nu , which is given by

hx   1 D

D Nu = hx

(20e)

  2   2   2  = 0 ,  = 0 ,  =    +    2 on  x   y   

 =1, 0   1

(20f)

hx / D

 0

D Nu x d + hy

hy / D



Nu y d

(22)

0

The partial differential Eqs.(17)-(19) with corresponding boundary conditions (20) were solved using the finite difference method with the second order differencing schemes. Detailed description of the used numerical technique is presented by Shenoy et al.[18], and Sheremet et al.[20].

It should be noted here that  2  by sin(2  x) = , x [a + b cos(2  x)]2  1  2 = , =0, y a + b cos(2  x) y 2  2 4 2 2 by[a cos(2  x) + b + bsin 2 (2  x)] = x 2 [a + b cos(2  x)]3 The physical quantities of interest are the local Nusselt numbers Nu x and Nu y , which are defined as

Nu x = 

kn f      kn f       , Nu y =    k f  y     =0 k f    =0 (21)

Fig.4 Variation of the average Nusselt number of the heat source surface versus the dimensionless time and the mesh parameters

The performance of corner heater part of the model was tested against the results of Varol et al.[23] for steady-state natural convection in a differentially

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Fig.5 Streamlines  and isotherms  for Ra = 105 ,  = 2.0 ,  = 0.03 , hx / D = hy / D = 0.5

heated square cavity filled with the regular fluid for Pr = 0.7 . Figures 2 and 3 show a good agreement between the obtained fields of vorticity, stream function and temperature for Ra = 105 and different values of the heat source sizes and the numerical data of Varol et al.[23]. For the purpose of obtaining grid independent solution, a grid sensitivity analysis was performed. The grid independent solution was performed by preparing the solution for unsteady free convection in a square wavy cavity filled with a Cu-water nanofluid at

Ra = 105 , Pr = 6.28 ,  = 0.02 ,  = 2.0 , a = 0.9 , hx / D = 0.5 , hy / D = 0.5 . Three cases of the uniform grid are tested: a grid of 100100 points, a grid of 150150 points, and a much finer grid of 300 300 points. Figure 4 shows an effect of the mesh parameters on the average Nusselt number of the heat source surface. On the basis of the conducted verifications the uniform grid of 150150 points has been selected for the following analysis.

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3. Results and discussion Numerical analysis has been conducted for the following values of the governing parameters: Rayleigh number ( Ra = 104 -105 ) , undulation number ( = 1.0 - 4.0) , solid volume fraction parameter of nanoparticles ( = 0 - 0.05) , shape parameter of the wavy wall (a = 0.9) , dimensionless time ( = 0 - 0.27) , di-

mensionless heater length in x direction (hx / D = 0.25 - 0.75) and dimensionless heater length in y direction (hy / D = 0.25 - 0.75) . Particular efforts have been focused on the effects of the dimensionless time, Rayleigh number, undulation number, nanoparticle volume fraction and the heater length in x and y directions on the fluid flow and heat transfer inside the cavity. Streamlines, isotherms, velocity and temperature profiles as well as local and average Nusselt numbers for different values of key parameters mentioned above are illustrated in Figs.5-16. Figure 5 illustrates contours of the stream function and temperature for Ra = 105 ,  = 2.0 ,  = 0.03, hx / D = hy / D = 0.5 and different values of the dimensionless time. At initial time ( = 0.001) one can find isotherms of low temperature close to the top wavy wall and isotherms of high temperature close to the left bottom corner where the isothermal heat source is located. The dominant heat transfer mechanism is heat conduction. As a result of such temperature distributions several convective cells appear inside the cavity. The main vortex forms close to the corner heat source due to temperature differences between the heater and initial temperature of the domain of interest and the secondary vortices form inside the wavy troughs due to the differences between the wavy wall temperature and also initial temperature of the cavity. Growth of the dimensionless time leads to both combination of the major vortex with secondary ones and an intensification of convective flow inside the main circulation. At the same time at  = 0.013 (Fig.5(d)) one can find an interaction of the high temperature wave with low one in the left top part of the cavity, namely over the vertical side of the heat source. Such interaction illustrates intensification of convective heat transfer with a formation of a thermal plume of high temperature in this zone and low temperature thermal plume close to the right vertical wall (see Fig.5(e)). Also it should be noted that fluid flow rate is a nonlinear function of the dimensionless time, namely one can find an intensification of convective flow inside the major circulation up to  = 0.021 and an attenuation of fluid flow in the case of   0.021 (see Figs.5(f), 5(g)). It is worth noting an evolution of orientation and location of the main convective cell core. At initial time the main convective core is elongated

Fig.6 Velocity profiles versus the dimensionless time for Ra =

105 ,  = 2.0 ,  = 0.03 , hx / D = hy / D = 0.5

Fig.7 Temperature profiles at middle cross-sections versus the dimensionless time for Ra = 105 ,  = 2.0 ,  = 0.03 ,

hx / D = hy / D = 0.5

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tion of quasi circular core at  = 0.013 . Further increase in time leads to a clockwise rotation of this core and at the moment of horizontal orientation ( = 0.021) we have an elongation of the major convective core along the horizontal axis and central location of this core. Also it is interesting to analyze an evolution of the thermal plumes at the moment of their interaction ( = 0.016) . An augmentation of dimensionless time leads to a deformation of low temperature thermal plume and a displacement of this plume close to the right wall with a reduction of a thickness of this structure. In the case of steady-state mode ( = 0.270) it is possible to note that the considered length of the heat source does not allow to heat the cavity because the heat sink rate is greater than the heat supply intensity taking into account the length of the wavy wall and a position of the isotherm  = 0.5 inside the cavity. Velocity and temperature profiles for different values of the dimensionless time are presented in Figs.6 and 7. As it has been mentioned above fluid flow arises at initial time close to the heat source taking into account values of the velocity components. An intensification of fluid flow up to  = 0.021 occurs with time and after that one can find attenuation of the circulation. Also it should be noted that close to the middle wavy crest we have a weak recirculation that can be confirmed by a presence of small negative horizontal velocity. Distribution of the vertical velocity presented in Fig.6(b) illustrates more intensive motion close to the heat source with non-linear effect of the dimensionless time on fluid flow rate. Temperature variations with time in Fig.7 reflect heating of the zone near the heat source and central part where the thermal plume of high temperature is located and cooling of the rest zones. Moreover in the case of steady-state mode one can find an interaction of the thermal plumes with high and low temperatures.

Fig.8 Streamlines  and isotherms  for  = 2.0 ,  = 0.03 ,

 = 0.27 , hx / D = hy / D = 0.5

along the vertical axis. A displacement of this core along the line connecting the left bottom corner and the center of the middle wavy trough occurs with time. Also one can find widening of the core and a forma-

Fig.9 Variation of the average Nusselt number versus the dimensionless time and Rayleigh number for  = 2.0 ,  = 0.03 , hx / D = hy / D = 0.5

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Fig.10 Streamlines  and isotherms  for Ra = 105 ,  = 0.03 ,  = 0.27 , hx / D = 0.5 , hy / D = 0.5

It is worth noting that obtained time-dependent data physically illustrate an evolution of temperature field and fluid flow patterns and the main reason for such behavior is the presence of temperature difference between the heat source and sink as well as the wavy shape of the top wall. Figures 5(h) and 8 show streamlines and isotherms at  = 2.0 ,  = 0.03 ,  = 0.27 , hx / D = hy / D = 0.5 and for different values of the Rayleigh number. Regardless of the Rayleigh number value main convective cell with one core is formed inside the cavity with some secondary vortices in wavy troughs. A rise in Ra leads to an intensification of the convective flow

(

Ra =104 max

= 5.32  

Ra =5104 max

= 13.03  

Ra =105 max

= 17.36)

and an elongation of the convective cell core along the horizontal axis. The latter illustrates a reduction of the velocity boundary layers thickness formed close to the vertical walls. Taking into account the temperature distributions both a formation of more essential thermal plumes and a decrease in the thermal boundary layers thickness close to the heat source surface occur with time. Figure 9 demonstrates the dependences of the average Nusselt number at the heat source surface on the Rayleigh number and dimensionless time. First of all it is necessary to note that the average Nusselt number essentially increases with Ra due to the ine-

Fig.11 Variation of the average Nusselt number versus the dimensionless time and undulation number for Ra = 105 ,  = 0.03 , hx / D = hy / D = 0.5

rtia effect. An evolution of the analyzed process characterizes an approach of the steady-state heat transfer mode. Therefore the time moment  = 0.27 reflects the steady-state regime. Moreover one can find that an approach of the steady-state regime is rapid for high values of the Rayleigh number. Also an increase in Ra leads to a reduction of the duration of the heat conduction zone where Nu decreases with time and approaches the first extremum. The second extremum that is formed for Ra  5  104 illustrates an evolution of the convective heat transfer mode and after that

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oscillating level characterizes an interaction between thermal plumes of high and low temperatures. Such description reflects a physical evolution of the considered process due to the buoyancy force effect.

Fig.13 Profiles of the local Nusselt number versus versus the nanoparticle volume fraction for Ra = 105 ,  = 2.0 , hx / D = 0.5 , hy / D = 0.5

Fig.14 Variation of the average Nusselt number versus the dimensionless time and nanoparticle volume fraction for Ra = 105 ,  = 2.0 , hx / D = 0.5 , hy / D = 0.5

Figure 10 presents profiles of stream function and temperature for Ra = 105 ,  = 0.03 ,  = 0.27 , hx / D = hy / D = 0.5 and different values of the undulation

Fig.12 Streamlines  and isotherms  for Ra = 105 ,  = 2.0,  = 0.27 , hx / D = hy / D = 0.5

number. An increase in the undulation number leads to modification of the convective core, namely, for high values of  major convective core displaces to the central part of the cavity with weak counter-clockwise rotation of this core. At the same time the fluid flow rate changes insignificantly. Also it should be

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Fig.15 Streamlines  and isotherms  for Ra = 105 ,  = 2.0 ,  = 0.27 ,  = 0.03

noted that the effect of the waviness of the top wall is reflected only in several streamlines located near the walls. Such hydrodynamic features are typical also for isotherms. It is worth noting that an increasing of the undulation number leads to more intensive cooling of the domain of interest taking into account the positions of isotherms  = 0.4 and  = 0.5 . Such effect can be explained by an essential growth of the cooling surface length with undulation number. Figure 11 illustrates an effect of  and  on the average Nusselt number for Ra = 105 ,  = 0.03 , hx / D = hy / D = 0.5 . A rise in the undulation number leads to an enhancement of the heat transfer rate due to cooling surface elongation. It is interesting to note that modification of the form of the top wall reflects in a variation of the heat transfer rate from the isothermal heat source located in the bottom part of the cavity. At the same time due to more essential cooling of the wavy troughs with the undulation number a dimensionless heat flux along this wall decreases and results in a reduction of the average Nusselt number at wavy heat sink with  . An effect of the nanoparticle volume fraction is presented in Fig.12 for Ra = 105 ,  = 2.0 ,  = 0.27 , hx / D = hy / D = 0.5 . An increment of the solid nanoparticle concentration leads to an attenuation of the convective flow taking into account values of  max : (

 =0.0 max

= 18.59  

 =0.03 max

= 17.36  

 =0.05 max

= 16.57) .

The latter can be explained by a growth of the flow resistance due to a presence of solid particles inside

the cavity. Also one can find more intensive cooling of the cavity with high concentration of nanoparticles owing to an increase in the nanofluid thermal conductivity and more developed cooling surface. Figures 13 and 14 illustrate an effect of the nanoparticles concentration on the local Nusselt number along horizontal and vertical parts of the heat source and average Nusselt number. Taking into account profiles for Nu x it is possible to conclude that this local parameter increases with the horizontal axis. Moreover an essential augmentation appears at the end of the horizontal heat source that can be explained by an essential interaction between high temperature from the heat source and low temperature from the thermal plume developed from the cold top wavy wall. At the same time the local Nusselt number along the vertical heat source is non-linear due to a behavior of the thermal plume. One can find here a growth of Nu y up to y = 0.3 due to a reduction of the thermal boundary layer thickness. Further decrease in the local Nusselt number can be explained by an expansion of the thermal plume and as a result the temperature gradient diminishes. An increment in the local Nusselt number with the nanoparticle volume fraction occurs regardless of the position of the heat source. The abovementioned variations of Nu x and Nu y with  reflect a rising of the average Nusselt number with the nanoparticles concentration (see Fig.14). It should be noted that variation of the nanoparticle volume fraction from 0.0 to 0.05 illustrates the heat transfer enhancement up to 7.6%.

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Figure 15 demonstrates an effect of the heat source length on the streamlines and isotherms for Ra = 105 ,  = 2.0 ,  = 0.27 ,  = 0.03 . An intensification of convective flow inside the cavity (  15.51  

hx / D =0.5 max

= 17.36  

hx / D =0.75 max

hx / D =0.25 max

= 18.94)

=

and

more essential heating of the enclosure occur with a growth of the heat source length along the horizontal axis at hy / D = 0.5 . Moreover it is possible to find a diminution of the thermal boundary layer thickness with the length of the heat source along the horizontal axis. Such changes lead to a decrease in the average Nusselt number (see Fig.16(a)). At the same time an increment of the heat source length along the vertical axis at hx / D = 0.5 leads to insignificant changes in the fluid flow rate inside the cavity (  17.24  

hy / D =0.5 max

= 17.36  

hy / D =0.75 max

hy / D =0.25 max

= 17.26)

=

and

also more essential heating of the enclosure. It should be noted that in the case of hx / D = 0.5 , hy / D = 0.5 heating of the cavity is more essential in comparison with the case of hx / D = 0.75 , hy / D = 0.75 .

Fig.16 Variation of the average Nusselt number versus the dimensionless time and dimensionless heater length in x direction for Ra = 105 ,  = 2.0 ,  = 0.03

Such behavior can be explained by the presence of more essential thermal obstacle from the long bo-

ttom heat source (hx / D = 0.75) to the penetration of low temperature wave from the top cold wall, because the low temperature wave moves along the right vertical wall. Therefore an elongation of the heat source can essentially prevent the cooling of the domain of interest. An increase in hy / D leads to non-linear changes of the average Nusselt number (see Fig.16(b)). 4. Conclusions The unsteady problem of natural convection of a Cu/water nanofluid within a wavy-walled enclosure having the isothermal heat source located along the left bottom corner was formulated in dimensionless stream function, vorticity and temperature using the singlephase nanofluid model and solved numerically on the basis of finite difference method of a second-order accuracy. Streamlines, isotherms, profiles of velocity and temperature as well as local and average Nusselt number distributions in a wide range of governing parameters have been obtained. The main findings can be listed as: (1) An increase in the Rayleigh number leads to an intensification of convective flow and heat transfer, and an elongation of the convective cell core along the horizontal axis with a decrease in the velocity boundary layers thickness formed close to the vertical walls. (2) A growth of the undulation number illustrates a modification of the convective core, namely, for high values of  major convective core displaces to the central part of the cavity with weak counter-clockwise rotation of this core. At the same time the fluid flow rate changes insignificantly while the heat transfer enhances. (3) An attenuation of the convective flow and an intensification of the heat transfer occur with the nanoparticle volume fraction. (4) A rise in the heat source length along the horizontal axis leads to an augmentation of the convective flow inside the cavity and more essential heating of the enclosure. Moreover it is possible to find a decrease in the thermal boundary layer thickness with the length of the heat source along the horizontal axis. Such changes lead to a diminution of the average Nusselt number. Acknowledgements This work of Sheremet M. A. was conducted as a government task of the Ministry of Education and Science of the Russian Federation (Grant No. 13.1919.2014/K. The authors also wish to express their thank to the very competent Reviewers for the valuable comments and suggestions.

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