role of movement of the walls with time-dependent velocity...

Scientia Iranica B (2021) 28(3), 1306{1317

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringhttp://scientiairanica.sharif.edu

Role of movement of walls with time-dependentvelocity on ow and mixed convection in verticalcylindrical annulus with suction/injection

A. Shakiba and A.B. Rahimi�

Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, P.O. Box 91775-1111, Iran.

Received 6 January 2020; received in revised form 6 June 2020; accepted 3 August 2020

KEYWORDSMixed convection;Vertical annulus;Time-dependentmoving walls;Transpiration;Exact solution.

Abstract. The unsteady mixed convection ow within a vertical concentric cylindricalannulus with time-dependent moving walls is investigated. Fluid is suctioned/injectedthrough the cylinder walls of the annulus. The e�ects of wall movement on ow and heattransfer characteristics inside the cylindrical annulus are sought. An exact solution ofNavier-Stokes and energy equations is obtained in this problem, for the �rst time. Here,the heat transfer occurs from the walls of the hot cylinder with constant temperature tothe moving uid with a cooling role. It is interesting to note that the results indicatethat the time-dependency of cylinder wall movements has no e�ect on the temperaturepro�le. The results also indicate that usage of an inverse of time for velocity movementcauses a signi�cant increase in velocity, stress tensor, and Nusselt number in the vicinity ofthe moving wall. Also, increasing the mixed convection and suction/injection parametersincreases the Nusselt number and decreases the stress tensor on the inner and outercylinders, respectively; no matter what velocity function is chosen for the moving wall.Therefore, using the velocity function of the wall movement and the change in other non-dimensional governing parameters, ow and heat transfer can be controlled.© 2021 Sharif University of Technology. All rights reserved.

1. Introduction

The study of a viscous uid ow in the space betweenvertical concentric cylinders whose cylindrical wallsmove in the axial direction with time-dependent veloc-ity is of great importance theoretically and practically,and it is the subject of extensive research in variousparts of engineering. Applications of an annulusinclude cylindrical tanks design, heat exchangers, newcooling systems in advanced nuclear reactors, thermalenergy storage cells, cooling of electronic devices, solarenergy collectors, drilling, the elimination of atheroscle-

*. Corresponding author.E-mail address: [email protected] (A.B. Rahimi)

doi: 10.24200/sci.2020.54784.3917

rosis, the fault detection of oil pipelines and so on.In laboratory work, Wu et al. [1] surveyed the mixedconvective ow of water inside a vertical annulus. Theresults revealed that the e�ect of buoyancy force isweaker than the other states when the constant uxboundary condition is considered on the inner wall andadiabatic on the outer wall. In an analytical work,Joshi [2] studied the free convective ow within an an-nulus channel. The results demonstrated that Nusseltnumber and ow rate are dependent on the annulus gapand the ratio of dimensionless temperature. Chen etal. [3] examined the ow and heat transfer in a verticalannulus numerically. The obtained results showed thatNusselt number depends on the radial ratio. Ianne�loet al. [4] studied the laminar ow and mixed convectiveheat transfer in an annulus section at the constanttemperature of the walls. The results of this research

A. Shakiba and A.B. Rahimi/Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 1306{1317 1307

indicated the dependency of the friction coe�cient andNusselt number on the wall on the mixed convectiveparameter. Malvandi et al. [5] concluded that thebuoyancy force has a negative e�ect on the e�ciencyof the system by examining the mixed convective ow of nano uid in the vertical annulus and in theconstant heat ux boundary conditions on the walls.Avc� and Ayd�n [6] analytically investigated the mixedconvective ow within an annulus in a fully developedstate. They revealed that the enhancement of mixedconvective parameters increases heat transfer. Jha etal. [7] investigated the behavior of mixed convective ow in a micro-annulus for a viscous incompressible uid. They came to the conclusion that as thesuction and uid injection into the walls increased, thetemperature and velocity of the uid heightened. Jhaet al. [8] also studied the e�ects of the suction/injectionof the uid in a vertical micro-porous annulus. Theyconcluded that when the uid is injected, the frictionis reduced at the inner cylinder surface. It should benoted that these results are reversed for the outer cylin-der. In an experimental-numerical work, Husain andSiddiqui [9] examined transient natural convection owwithin an annulus. The results indicated that a dropin the transient period appeared with the enhancementof heat input. Moreover, they showed that annulusheight causes the gradual augmentation of the transientperiod. In another study, Husain et al. [10] concludedthat the Nusselt number increases and decreases withthe enhancement of radius ratio and aspect ratio,respectively, by investigation of geometrical parametersa�ecting the natural convection ow in a verticalannulus. Some other relative studies have also beenconducted in the �eld of exploring ow and heattransfer characteristics in annulus geometry [11{13].

Fluid ow and heat transfer within concentriccylinders with movable walls have always fascinatedresearchers owing to their usage in manufacturing andthe cooling processes of thermal equipment. More-over, forecasting the distribution of transient statetemperature and heat transfer rate from the initialstate to a steady state is of great importance insome engineering issues. It needs to be stated thatthe application of such cases are as follows: Extru-sion processes in productive industries, polymer pipecoating, simulation of a passing train through a longtunnel, drilling operations, cooling processes such ashot rolling, nuclear reactors and nuclear fuel channels[14{16]. Shigechi and Lee [17] investigated the uid ow and heat transfer of an annulus with a movablewall. They discovered that the friction decreased byincreasing the velocity of the movable wall and theNusselt number changes are strongly dependent on thethermal boundary conditions. In an analytical work,Chamkha [18] surveyed the ow and heat transfer ofthe uid in a vertical moving plate in an unsteady

state and with the existence of a magnetic �eld. Theresults represented that Nusselt number and the skinfriction coe�cient declined by augmenting the thermalabsorption coe�cient. In a numerical work, Abediniand Rahimi [19] scrutinized the mixed convective heattransfer in an annulus consisting of two horizontal androtating concentric cylinders. They investigated thee�ect of di�erent rotation functions as time-dependentboundary conditions on inner and outer cylinders.Saleh and Rahimi [20] inspected the unsteady owand heat transfer around the stagnation point ofan in�nite and moving cylinder with time-dependentaxial velocity and uniform normal transpiration as anexact solution. In a numerical work, Jha et al. [21]examined the natural convection ow in a transientstate in a vertical porous annulus. Temperatureand velocity are independent of Prandtl number in asteady state when there is no suction/injection, but aredependent on Prandtl number if the suction/injectionexists. Jha et al. [22] dealt with the fully developedsteady/unsteady ow and the natural convection owin a vertical annulus. They found that with theenhancement of dimensionless time, the dimensionlesstemperature and velocity increase until they reach asteady state. Through acquiring the results of anexact solution, Shakiba and Rahimi [23] studied the uid ow and heat transfer features within a verticalcylindrical annulus by considering the e�ects of radialmagnetic �eld and suction/injection. Busedra andTavoularis [24] investigated natural convection owby considering a vertical annular channel. Theirnumerical solution results indicated that the heattransfer rate for the concentric annular channel washigher than the eccentric annular channel under thesame thermal conditions. Moreover, the mass owrate in eccentric channel mode is higher than in theconcentric mode. Hekmat and Ziarati [25] studied thee�ect of volume fraction changes and magnetic �eldgradients on Ferro uid mixed convection ow within avertical annulus. The numerical results revealed thatwhen the magnetic �eld gradient is negative, the skinfriction coe�cient increases and decreases respectivelyon the inner and outer walls by enhancing the volumefraction and magnetic �eld gradient. Jha et al. [26]investigated the natural convection ow in a verticalannulus whose walls are subject to periodic heat ux,which is a function of time. The analytical resultsshowed that by augmenting the Strouhal number andPrandtl number, the heat transfer rate increased. Jhaet al. [27] surveyed the natural convection ow within amicro-channel. Through obtaining analytical solutionresults, they concluded that the increase of Hall cur-rent heightens the uid velocity in both primary andsecondary directions of ow. Abbas et al. [28] studiedthe natural convection ow of Ferro uid in a verticalmicro cylindrical tube. The e�ects of a radial magnetic

1308 A. Shakiba and A.B. Rahimi/Scientia Iranica, Transactions B: Mechanical Engineering 28 (2021) 1306{1317

�eld, as well as the e�ects of suction/injection, werealso considered in this study. The results of theanalytical solution showed that uid velocity decreasedwith enhancement of the injection parameter, and uidvelocity increased with augmentation of the suctionparameter. Furthermore, the opposite of these resultsis con�rmed for temperature distribution.

Although much research has been conducted inthis area, the mixed convective ow of the uid in anannulus, whose walls move at a time-dependent veloc-ity, has not been studied so far. Accordingly, the uid ow passing through an annulus and its heat transferin a transient state is scrutinized in this article by se-lecting time-dependent functions for the cylinder wall.

In this study, the e�ect of cylinder walls that movein an axial direction with constant velocity, on variousparameters, was investigated. The results revealed thatthe increase of the mixed convective parameter and thesuction/injection parameter without the presence of amagnetic �eld led to the growth of shear stress andheat transfer rate on the inner cylinder wall. Moreover,the movement of each of the inner and outer cylindersgives rise to the highest and lowest heat transfer rate,respectively, on the inner cylinder wall.

2. Problem statement

In the present study, the behavior of the ow and heattransfer of the uid within concentric cylinders is exam-ined when the walls of the cylinders move in the axialdirection with time-dependent velocity. According toFigure 1, a cylindrical coordinate system (r; �; z) isused, where the geometry of the annulus consists of twovertical and concentric cylinders with a circular sectionwhere the z axis is along with the annulus axis and ther axis is perpendicular to it. It should be stated thatthe values of radius and temperature are considered aand T1, respectively, for the inner cylinder, and thesevalues are assumed b and T0, respectively, for the outercylinder. In addition, the temperature of the innercylinder is higher than the outer cylinder. The walls ofinner and outer cylinders are considered perforated and uid is suctioned/injected from/into them. To solvethis problem, the equations of continuity, momentumand energy need to be solved, along with the boundaryconditions in a transient state. By doing so andchanging the parameters governing the problem, the uid ow can be controlled and the heat transfer ofthe system can be increased.

As a matter of note, the relevant results will bepresented in this article through obtaining the dimen-sionless equations related to the ow and heat transferand solving them. The present study is a fundamentalstep for better understanding of the e�ects of wallmotion, along with the suction/injection of the uid,on improving the e�ciency of heat exchangers. If this

Figure 1. Problem geometry.

method leads to the dramatic increase of heat transferin heat exchangers, its industrial use will satisfy a basicneed in the industry, which is an increase in the heattransfer of heat exchangers while their volume reduces.

3. Governing equations and boundaryconditions

The uid ow is taken into account as laminar, tran-sient, fully developed, Newtonian and incompressible.By exerting a no slip condition to the walls and employ-ing a Boussinesq approximation, the conservation equa-tions, which include the equations of continuity, mo-mentum and energy, are written according to Eqs. (1){(3). The viscous dissipation and axial conduction of the uid and wall are foregone in writing these equations.With regard to annulus length, which is in�nite, all thephysical parameters of the problem are a function of r,except the pressure, which is a function of z.

Continuity equation:

1r@@r

(rVr) = 0: (1)

Momentum equation:

@Vz@t

+ Vr@Vz@r

= �1�@p@z

+�r@@r

�r@Vz@r

�+g� (T � T0) : (2)

Energy equation:


@T@t

+ Vr@T@r

=�Tr

@@r

�r@T@r

�: (3)

In Eqs. (1) to (3), �T , g and � are thermal di�usivity,gravity and the thermal expansion coe�cient of the uid, respectively.

Boundary conditions are also de�ned as in Eq. (4).With regard to the boundary conditions, the cylinderwalls were initially stationary and the temperature ofall parts was constant and equal to T0. Then, the wallof the inner cylinder is set at the temperature of T1 atthe moment of (t > 0), and it moves in the directionof the cylinder axis with the velocity of F (t) which istime dependent.

t = 0 T (r; 0) = T0Vz = (a; 0) = 0Vz(b; 0) = 0

t > 0 T (a; t) = T1T (b; t) = T0

Vz(a; t) = F (t)Vz(b; t) = 0 (4)

The velocity component in the radial direction can beobtained in the form of Vr = �aV0

r through integrationof Eq. (1) and using the boundary condition of Vr =�V0 at r = a. In the following, the velocity componentin the axial direction, Vz, will be displayed as u.

The dimensionless parameters related to thepresent problem are de�ned as in the following:

�=t�a2 ; R =

ra; � =

ba; � =

T � T0

T1 � T0;

U =uu0; S =

V0a�; Gr =

g��TD3h

�2 ;

Pr =��T

; Re =u0Dh

�; � = Gr=Re: (5)

In which u0, Dh, �, �T , and �T are the reference ve-locity, hydraulic diameter, kinematic viscosity, thermaldi�usivity and temperature di�erence, respectively,and they are de�ned according to:

u0=�D2

h�

dpdz; Dh = 2 (b� a) ; � =

��;

�T=k�Cp

; �T = T1 � T0: (6)

Regarding Figure 1 and Eq. (5), S is de�ned asthe suction/injection parameter and S < 0 representsa situation in which the suction takes place in thedirection of the annulus to the walls. S > 0 indicatesthe opposite of the above situation.

The dimensionless format of conservationequations, including momentum and energy, alongwith the boundary conditions are expressed in termsof Eqs. (7){(9).

The boundary conditions are:

� = 0 �(R; 0) = 0 u(1; 0) = 0U(�; 0) = 0

� > 0 �(1; �) = 1�(�; �) = 0

U(1; �) = F (�)U(�; �) = 0 (7)

The momentum equation becomes:

@U (R; �)@�

� SR

@@R

U (R; �)� 1R

@@R

�R@@R

U (R; �)�

� 14(�� 1)2 [1 + �� (R; �)] = 0: (8)

The energy equation is:

@� (R; �)@�

� SR

ddR

� (R; �)

=1

Pr1R

ddR

�RddR

� (R; �)�: (9)

Dimensional shear stress and Nusselt number onthe walls of inner and outer cylinders are de�nedaccording to the following relations:

�j@wall =@U@R

��@wall

: (10)

NujR=1;� = �2 (�� 1)@�@R

��R=1;�

� (1; �)� �b ;

NujR=�;� = 2 (�� 1)@�@R

��R=�;�

� (�; �)� �b : (11)

In Eq. (11), �b is the dimensionless bulk temperatureof the uid and is de�ned according to the following:

�b =Tb � T0

T1 � T0=

�R1RU (R;�) � (R;�) dR

�R1RU (R;�) dR

: (12)

In the following, the solution results are presented.

4. Validation

To prove the accuracy of the results, the velocity pro�leobtained from the present study and the analyticalresults of Jha and Aina [7] are plotted in Figure 2(a)when the Knudsen number is equal to zero (Kn =0). In this comparison, the inner and outer cylindersare stationary and the suction takes place from theirwalls. Furthermore, the mixed convective parameterand Prandtl number are considered 50 and 7, respec-tively. According to Figure 2(b), by considering the


Figure 2. Validation: (a) The comparison of velocity pro�le in the annulus with Jha and Aina's [7] work in � = 50, andS = �0:5, (b) the Nusselt number on inner cylinder of annulus compared with Ref. [29], and (c) Comparison of results ofaverage Nusselt number in terms of Rayleigh number with Ref. [30].

suction/injection, the Nusselt number values achievedon the inner cylinder wall in di�erent radius ratioshave been compared with the results of Jha et al. [29].As can be seen, the results in Figure 2(a) and (b)overlap one another completely. In Figure 2(c), acomparison has been carried out between the resultof the present study and the experimental results ofWrobel et al. [30]. In this �gure, the average Nusseltnumber is plotted in terms of Rayleigh number in avertical annulus for conditions where the horizontalwalls are considered insulated and the vertical wallshave a constant temperature. As observed, there is arelatively close correspondence between both results.

5. Results and discussions

Generally, �nding the exact solution for Navier-Stokesequations has many mathematical complexities. How-ever, Eqs. (8) and (9) render exact solutions for thisproblem. It is worth mentioning that the exact solutionresults of the steady state case of the present problem,when the cylinder walls move in axial directions at

constant velocity, are presented in references [23,31].As mentioned, this study aims at investigating asituation in which the cylinder wall moves with time-dependent velocity in a transient state. There aretwo categories of answers for Eqs. (8) and (9). IfS = � 1

Pr , the energy equation turns into the classicalheat equation with known results for �(R; �) which canbe replaced into the momentum equation to obtain thevelocity distribution. The second situation is whenS 6= � 1

Pr where numerical methods are used to attainthe desired results for �(R; �) and which are then usedin momentum equations.

5.1. Temperature variation analysisThe diagrams and contours of dimensionless tempera-ture variations in the distance between the inner andouter cylinders for two states of S = �2 at di�erenttimes are illustrated in Figure 3. The inner cylinder hasa higher temperature than the uid and, therefore, heattransfer occurs due to the temperature di�erence. Asobserved, when the uid is suctioned from the annulusaxis to the outer cylinder (S = �2), the uid tempera-


Figure 3. Diagrams and contours of dimensionless temperature in the distance between two cylinders at di�erent times inS = �2; �= 2; and �= 5: (a) Pr = 2:00 and (b) Pr = 7:00.

ture �eld penetrates into the outer wall direction, andthe uid temperature increases in the distance betweenthe two cylinders as time increases. Furthermore, whenthe uid is injected into the cylinder axis direction(S = 2), the temperature increases near the innercylinder as time enhances. It should be noted that theobtained results indicate the ine�ectiveness of cylinderwall motion on temperature pro�le. Prandtl numberis a dimensionless number that is obtained from theratio of momentum di�usivity to thermal di�usivity. InFigure 3(a) and (b), the diagrams and contours of thedimensionless temperature are compared for the twoPrandtl numbers of 2 and 7. As can be observed, thethickness of the thermal boundary layer decreases andthe slope of the dimensionless temperature diagramincreases by increasing the Prandtl number.

5.2. Fluid ow analysisVariations of dimensionless velocity pro�le and contourin the distance between the inner and outer cylin-ders at di�erent times and for two di�erent rates ofsuction/injection S = �2 are displayed in Figure 4.The movement of the inner cylinder wall is time-

dependent in the axial direction and corresponds toan exponential motion of F (�) = e�� . As can beseen, the movement of the inner cylinder wall a�ectsthe uid velocity pro�le. With regard to the fact thatthe selected velocity pro�le is of a slowing down type,the velocity heightens initially as time increases andthen it reduces. When the uid is suctioned from thedirection of the annulus axis toward its external part(S = �2), the uid is drawn into the outer cylinder walland the velocity pro�le takes a parabolic form. Thisprocess causes the velocity to increase in the middlearea between the two cylindrical pipes. Moreover, asthe uid is injected toward the annulus axis, the volumeof uid increases near the inner cylinder wall.

In Figure 5, the e�ect of the changes of dimension-less mixed convective parameters on the dimensionlessvelocity pro�le of the uid and its contour between twoinner and outer cylinders is depicted at di�erent times.The outer cylinder wall is stationary and the innercylinder moves in the axial direction exponentially andslows down according to the time-dependent functionof F (�) = e�� . As in Eq. (5), the mixed convectiveparameter is a dimensionless number and is de�ned


Figure 4. Diagram and contour of dimensionless velocity in the distance between the two cylinders at di�erent times inS = �2; Pr = 7:0, �= 2; �= 5; and F (�) = e�� .

Figure 5. Diagram and contour of dimensionless velocity in the distance between two pipes at di�erent times � = 5; 500;Pr = 7:0; � = 2; S = 2; and F (�) = e�� .

as the ratio of Grashof number to Reynolds number.Since the inner cylinder wall has a higher temperature,the natural convection term becomes stronger for largermixed convective parameters and, thus, causes the uidvelocity to increase near the hot wall (of the inner cylin-der). Furthermore, the velocity accelerates initiallynear the inner cylinder wall as time increases and thevelocity function reduces again because of being sloweddown. A key point to note in Figure 5 is that when �equals 500, the in uence of the uid velocity �eld onthe range between two cylinders enhances owing to theincrease of buoyancy force, and it creates a parabolicpro�le between the two cylinders.

As mentioned earlier, various functions with dif-ferent application modes, such as slowing down expo-nent (F (�) = e�� ), slowing down fraction (F (�) =1=�) and periodic trigonometry (F (�) = sin �), areemployed to study the e�ect of wall motion on the ow and heat transfer of the uid in the space betweentwo cylinders. In Figure 6, the changes of uid owpattern caused by the movement of the inner cylinder

wall in the axial direction are presented by employingeach of these functions. With regard to Figure 6(a)and (b), the uid velocity heightens on the wall of theinner cylinder as time increases and then it declinesand reaches the damping state. As can be observed,the amount of velocity augmentation is abrupt andconsiderable in the state of (F (�) = 1=�).

Moreover, according to Figure 6(c), the uidvelocity near the inner cylinder wall is periodicallyincreased and decreased by using the trigonometricfunction. In Figure 6(d), a comparison is made betweenthe three employed functions simultaneously. Clearly,by comparing velocity pro�les at the time of � = 1,functions F2, F3 and F1 apply the maximum velocityto the uid adjacent to the wall, respectively. As aconsequence, more velocity changes are applied to the uid adjacent to the inner cylinder wall by employingthe F2 function. In addition, as the dimensionless timeincreases to � = 3, the wall velocity decreases andthe uid velocity also slows down. It is evident thatthe functions F1, F2 and F3 have the highest velocity


Figure 6. Diagram of dimensionless velocity in the distance between two cylinders for the case that the inner cylinderwall moves with time-dependent velocity: (a) F1 = e�� ; (b) F2 = 1

� , (c) F3 = Sin(�), and (d) Comparison in Pr = 7:0;�= 2; � = 5, S = 2.

reduction, respectively, at the dimensionless time of� = 3. Accordingly, function F1 reaches the steadystate sooner than the other functions.

5.3. Heat transfer analysisThe Nusselt number is a dimensionless parameterused to measure the amount of heat transfer at theboundaries of a system. According to Eq. (11), theNusselt number is de�ned as the ratio of convectionheat transfer rate to the conduction heat transfer rate.In Figure 7, the Nusselt number is plotted on the wallsof inner and outer cylinders at di�erent times when theinner cylinder wall moves with various time-dependentfunctions. According to Figure 7, the highest Nusseltnumbers on the inner and outer walls are related to thefunctions of F (�) = 1=� and F (�) = e�� , respectively.Besides, the rate of Nusselt number change on bothwalls is incremental with increasing time, so that theenhancement of Nusselt number on the inner cylinderwall is noticeable during early times.

In Figure 8, the e�ect of mixed convection param-eter changes on Nusselt number is presented at di�erent

Figure 7. Diagram of Nusselt number on the walls ofinner and outer cylinders in terms of dimensionless time inthree di�erent states of F1 = e�� ; F2 = 1

� ; andF3 = Sin(�) as a time dependent function of inner cylinderwall motion in Pr = 7:0; �= 2; � = 5, and S = 2.


Figure 8. Diagram of Nusselt number in terms of mixed convection parameter: (a) On the wall of inner cylinder (R = 1)and (b) on the wall of inner cylinder (R = � = 2) in three di�erent states of F1 = e�� ; F2 = 1

� ; and F3 = Sin(�) as atime-dependent function of the inner cylinder wall motion in Pr = 7:0 and S = 2.

times for three di�erent states of wall motion with timedependent velocity. As can be seen, the buoyancy forceof the uid increases near the hot wall (of the innercylinder) with the augmentation of � and, consequently,the Nusselt number increases on the wall of the innercylinder and decreases on the wall of the outer cylinder.By comparing the di�erent states of the inner cylinderwall motion, it is found that, for example, at the timeinterval of (� = 2:00), the highest Nusselt numberson the inner and outer cylinder walls are related to thefunctions of F (�) = sin � and F (�) = e�� , respectively.

5.4. Surface tension analysisThe variations of dimensionless shear stress on thewalls of inner and outer cylinders in terms of time fordi�erent states of wall motion with various functionsare shown in Figure 9. It is apparent that thehighest shear stress on the inner cylinder wall is relatedto the function of F (�) = 1=� , whose amount isnoticeable. According to Figure 6(b), since shearstress corresponds to the rate of velocity variations,the sharp increase of shear stress for this function canbe attributed to the growth of the velocity gradientnear the inner cylinder wall. Moreover, the processof shear stress changes, in terms of time, is di�erent,depending on the type of function used. For instance,when the trigonometric function of F (�) = sin � isutilized, its shear stress diagram is also in periodicformat. According to the diagram, it is clear thatwhere the velocity gradient has negative values, theshear stress also shows negative values.

The e�ects of S parameter changes on shearstress on the walls of the inner and outer cylindersfor di�erent functions applied to the inner cylinderwall are illustrated in Figure 10. As can be seen,regardless of the type of function used, the shear stressdecreases and increases, respectively, on the walls ofthe inner and outer cylinders by enhancing S fromnegative to positive values. For example, by injecting

Figure 9. Diagram of dimensionless shear stress in termsof time on the walls of inner and outer cylinders in threedi�erent states of F1 = e�� ; F2 = 1

� ; and F3 = Sin(�) as atime dependent function of inner cylinder wall motion inPr = 7:0; �= 2; � = 5, and S = 2.

the uid toward the annulus axis, the velocity gradientdiminishes near the inner cylinder wall and causes theshear stress to reduce on the inner cylinder wall. Theopposite of this process occurs for the shear stress onthe outer cylinder. As already mentioned, when thefunction of F (�) = 1=� is utilized, the amount of shearstress is greater on the wall of the inner cylinder thanthe other situations, due to the drastic changes of thevelocity gradient.

6. Conclusions

In this study, the role of movement of walls with time-dependent velocity on ow and mixed convection in avertical cylindrical annulus with suction/injection has


Figure 10. Diagram of dimensionless shear stress basedon suction/injection parameter on the walls of inner andouter cylinders in three di�erent states of F1 = e�� ;F2 = 1

� ; and F3 = Sin(�) as a time dependent function ofinner cylinder wall motion in � = 1:0;Pr = 7:0; �= 2; and�= 5.

been considered. An exact solution of the Navier-Stokes and energy equations has been obtained. It hasbeen shown that the energy equation turns to a heatequation in the case of a transpiration function equalto the inverse of the Prandtl number. Otherwise, thebehavior of the ow and heat transfer of the uid hasbeen investigated by considering selected types of time-dependent functions for the movement of the cylinderwalls, as slowing down exponent F (�) = e�� , slowingdown fraction F (�) = 1=� and periodic trigonometryF (�) = sin � . The walls of inner and outer cylinders areperforated and the e�ects of suction/injection on thesewalls have also been studied. The obtained results haverevealed that:

� The movement of cylinder walls has no e�ect on thedimensionless temperature pro�le;

� The comparison of three time-dependent motionfunctions indicated that the uid reaches the steadystate earlier than the other states when the functionof F (�) = e�� is used in the inner cylinder wall.Furthermore, the maximum Nusselt number is ob-tained on the outer wall through using this function;

� The comparison of time-dependent functions shedlight on the fact that the function of F (�) = 1=�causes greater velocity changes than the other func-tions. This function also brings about a signi�cantincrease of the uid velocity near the inner cylinderwall. Also, the maximum Nusselt number and shearstress are created on the inner cylinder wall by usingthis function;

� The rate of Nusselt number changes is incrementalon the wall of both cylinders;

� With the enhancement of �, Nusselt number in-creases and decreases, respectively, on the inner andouter walls;

� Regardless of the type of the function used, theshear stress decreases and increases, respectively,on the inner and outer cylinder walls by enhancingthe suction/injection rate from negative to positivevalues.

Nomenclature

a Radius of the inner cylinder (m)b Radius of the outer cylinder (m)Cp Speci�c heat (J/kgK)Dh Hydraulic diameter, Dh = 2(b� a)Gr Grashof numberg Gravitational acceleration (m/s2)k Thermal conductivity (W/mK)Nu Nusselt numberp Pressure (Pa)Pr Prandtl numberr Axis in the cylindrical coordinatesR Dimensionless radiosRe Reynolds numberS Suction/injection parametert Time (S)T Temperature of uid (K)T0 Wall temperature of outer cylinder (K)T1 Wall temperature of inner cylinder (K)�T Temperature di�erence, �T = T1 � T0

u Velocity in z direction (m/s)u0 Reference constant velocity (m2/s)U Dimensionless velocityz Axis in the cylindrical coordinates

Greek letters

�T Thermal di�usivity (m2/s), �T =k=�Cp

� Thermal expansion coe�cient (1/K)� Mixed convection parameter� Dimensionless temperature of uid� Ratio of the radius between two

cylinders� Dynamic viscosity (kg/(m.s))� Density (kg/m3)� Kinematic viscosity (m2/s), � = �=�


� Dimensionless time� Dimensionless shear stress

Subscripts

l Value on inner wallb Bulk temperature� Value on outer wall

References

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Biographies

Ali Shakiba was born in Isfahan, Iran, in 1990. Hereceived his BS degree in Mechanical Engineering fromthe Islamic Azad University, South Tehran Branch,Iran in 2012, and his MS degree in Mechanical Engi-neering from Mazandaran Institute of Technology, Iranin 2014. He is currently a PhD student of MechanicalEngineering in the Faculty of Mechanical Engineeringat Ferdowsi University of Mashhad, Iran. His researchinterests include heat transfer, uid mechanics, math-ematics, nano uid, MHD, FHD, CFD simulation andrenewable energy.

Asghar Baradaran Rahimi was born in Mashhad,Iran, in 1951. He received his BS degree in MechanicalEngineering from Tehran Polytechnic, in 1974, anda PhD degree in Mechanical Engineering from theUniversity of Akron, Ohio, USA, in 1986. He has beenProfessor in the Department of Mechanical Engineeringat Ferdowsi University of Mashhad since 2001. His re-search and teaching interests include heat transfer and uid dynamics, gas dynamics, continuum mechanics,applied mathematics and singular perturbation.

role of movement of the walls with time-dependent velocity...

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