investigation of two-dimensional unsteady stagnation-point flow and heat transfer impinging on an ...

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Investigation of Two-DimensionalUnsteady Stagnation-Point Flow andHeat Transfer Impinging on anAccelerated Flat Plate

Ali Shokrgozar AbbassiPh.D. Student

Asghar Baradaran Rahimi1

Professor

e-mail: [email protected]

Faculty of Engineering,

Ferdowsi University of Mashhad,

Mashhad 91775-1111, Iran

General formulation and solution of NavierStokes and energyequations are sought in the study of two-dimensional unsteadystagnation-point flow and heat transfer impinging on a flat platewhen the plate is moving with variable velocity and accelerationtoward main stream or away from it. As an application, amongothers, this accelerated plate can be assumed as a solidificationfront which is being formed with variable velocity. An externalfluid, along z-direction, with strain rate a impinges on this flatplate and produces an unsteady two-dimensional flow in whichthe plate moves along z-direction with variable velocity andacceleration in general. A reduction of NavierStokes and energyequations is obtained by use of appropriate similarity transforma-tions. Velocity and pressure profiles, boundary layer thickness,and surface stress-tensors along with temperature profiles arepresented for different examples of impinging fluid strain rate,selected values of plate velocity, and Prandtl number parameter.[DOI: 10.1115/1.4005742]

Keywords: stagnation-point flow and heat transfer, unsteady flow,

viscous fluid, accelerated plate, similarity solution

1 Introduction

There are many solutions of the NavierStockes and energyequations regarding the problem of stagnation-point flow and heattransfer in the vicinity of a flat plate or a cylinder. These studiesinclude: two-dimensional stagnation-point flow on a stationaryflat plate [1]; and axisymmetric stagnation flow on a circular cyl-inder [2]. Further studies are: heat transfer in an axisymmetricstagnation flow on a cylinder [3]; nonsimilar axisymmetric stagna-tion flow on a moving cylinder [4]; unsteady viscous flow in thevicinity of an axisymmetric stagnation-point on a cylinder [5];radial stagnation flow on a rotating cylinder with uniform transpi-ration [6]; axisymmetric stagnation flow toward a moving plate[7] in which the plate moves in its own plane. The more recent

studies in this area are: oscillating stagnation-point flow [8]; axi-symmetric stagnation-point flow and heat transfer of a viscousfluid on a moving cylinder with time-dependent axial velocity anduniform transpiration [9]; axisymmetric stagnation-point flow andheat transfer of a viscous fluid on a rotating cylinder with time-dependent angular velocity and uniform transpiration [10]; simi-larity solution of unaxisymmetric heat transfer in stagnation-pointflow on a cylinder with simultaneous axial and rotational move-

ments [11]; and recently, nonaxisymmetric three-dimensionalstagnation-point flow and heat transfer on a flat plate [12], three-dimensional stagnation flow and heat transfer on a flat plate withtranspiration [13], and exact solution of three-dimensionalunsteady stagnation flow on a heated plate [14]. Among all thestudies above only in Ref. [8], the flat plate oscillates verticallyand some particular solutions have been obtained by use of Four-iers expansions and in Ref. [14] which is an exact solution of aheated plate. Similarity solution of stagnation-point flow and heattransfer problem on a flat plate with arbitrary vertical movement

is nonexisting in the literature.In this study, the general two-dimensional unsteady viscousstagnation-point flow and heat transfer in the vicinity of a flat plateare investigated where this flat plate is moving toward or awayfrom the impinging flow with variable velocity and acceleration.The NavierStokes equations along with energy equation aresolved. One of the applications of this type of moving plate isencountered in solidification and melting problems. The externalfluid, along z-direction, with strain rate a impinges on a flat platewhile the plate is moving with variable velocity and accelerationalong z-direction. A similarity solution of the NavierStokes andenergy equations is derived in this problem. The obtained ordinarydifferential equations are solved by using finite-difference numeri-cal techniques. Velocity and pressure profiles, boundary layer thick-ness, and surface stress-tensors along with temperature profiles arepresented for different examples of impinging fluid strain rate,

selected values of plate velocity, and Prandtl number parameter.

2 Problem Formulation

Flow is considered in Cartesian coordinates x; z with corre-sponding velocity components u; w, see Fig. 1. We consider thelaminar unsteady incompressible flow and heat transfer of a viscousfluid in the neighborhood of stagnation-point on a moving flat platelocated in the z 0 plane at t 0. An external fluid, along z-direc-tion, with strain rate a impinges on this accelerated flat plate alongz-direction and produces a two-dimensional flow on the plate. Obvi-ously, in the situation of moving plate, the boundary layer thicknessalong x-direction changes in contrary to the case of fix plate whenit is with constant thickness. As an application, this acceleratedplate can be assumed as a solidification front which is moving with

variable velocity along the z-axis in which the solid thickness isgrowing steadily in x direction, [15].

The unsteady NavierStokes and energy equations in Cartesiancoordinates governing the flow and heat transfer are given as

Fig. 1 Schematic problem graph

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the

JOURNAL OF HEAT TRANSFER. Manuscript received August 26, 2010; final manuscript

received November 17, 2011; published online April 26, 2012. Assoc. Editor:

Darrell W. Pepper.

Journal of Heat Transfer JUNE 2012, Vol. 134 / 064501-1CopyrightVC 2012 by ASME

wnloaded From: http://heattransfer.asmedigitalcollection.asme.org/ on 03/28/2013 Terms of Use: http://asme.org/terms


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@u

@x @w

@z 0 (1)

@u

@t u @u

@x w @u

@z 1

q

@P

@x @

2u

@x2 @

2u

@z2

!(2)

@w

@t u @w

@x w @w

@z 1

q

@P

@z @

2w

@x2 @

2w

@z2

!(3)

@T

@t u@T

@x w

@T

@z a

@2T

@x2 @2T

@z2 ! (4)

where P, q, , and a are the fluid pressure, density, kinematics vis-cosity, and thermal diffusivity.

3 Similarity Solution

3.1 Fluid Flow Solution. The classical potential flow solu-tion of the governing Eqs. (1)(3) is as follows:

U atx (5)W atf (6)

P1 P0 12

qV21u;w q 2

1

@V

@tdr

0 q at2

2x2 f2 q @

@tat

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 f2 q !dr (7)

where p0 is stagnation pressure, V is total velocity, r is the flowpassage, St is the amount of plate displacement in z-direction,and f = z St. By use of Eq. (7) on a streamline one obtainspotential pressure. The cross-section which is at infinity is insidethe potential region at which the variation of plate displacement isfelt by the external flow. Therefore, the flow strain rate changesaccording to the plate velocity and so it becomes a time-dependent quantity at. This is equivalent to the plate velocitybeing superimposed on the external flow, therefore f z St.

A reduction of the NavierStokes equations is sought by thefollowing coordinate separation in which the solution of the vis-

cous problem inside the boundary layer is obtained by composingthe inviscid and viscous parts of the velocity components as thefollowing:

u atxf0g (8)

w ffiffiffiffiffi

t

a0

ratfg (9)

g ffiffiffiffiffi

a0

t

rz St ; and f z St (10)

where terms involving fg in Eqs. (8) and (9) comprise the Carte-sian similarity form for unsteady stagnation-point flow and primedenotes differentiation with respect to g. Note, boundary layer isdefined here as the edge of the points where their velocity is 99%of their corresponding potential velocity. Transformations(8)(10) satisfy (1) automatically and their insertion into Eqs.(2)(3) yields an ordinary differential equation in terms of fgand an expression for the pressure

f000 f00 ~_St ~atf

~atf02 f0

~at@~at@s

1~atn

@~P

@n 0

(11)

in which

1

~atn@~P

@n ~at 1

~at@~at@s

2g~_S

n2 g2 ~_S

g

1 g

2

n2

(12)

@~at@s

~a0~_S

g0

~S

g0

~_S2

g20(13)

~P ~at f0 1 ~at~_S ffg d ~a2t f2 f2g d

2

Ad

g

fgdg ~a2t2

n2 g2

An2

g2

2 2~at~_S

2

1

g n2

g2

n n2 g2 dn ~at~_

S

n2

3g g !d

1

(14)

where

~Px; z; t Px; z; tqa0t

; ~St ffiffiffiffiffi

a0

t

rSt; ~at at

a0;

~_St _St ffiffiffiffiffiffiffia0tp ; n ffiffiffiffiffi

a0

t

rx

(14a)

in which, dot denotes differentiation with respect to t, and quan-

tities ~Px; z; t, ~at, ~St, ~_St, n are nondimensional forms ofquantities P

x; z; t

, a

t

, S

t

, _S

t

, and x, respectively. Pressure

variation with respect to n is presented by Eq. (12) and strain vari-ation with respect to time which is the plate acceleration is pre-sented by Eq. (13). Relation (14) which represents pressure isobtained by integrating Eq. (3) in z-direction and by use of thepotential flow solution (5)(7) as boundary conditions. Here, Eq.(11) is in the most general form for any arbitrary flat plate verticalmovement and the boundary conditions for this equation are

g 0 : f 0; f0 0 (15)

g ! 1 : f0 1 (16)

Note that, when the plate velocity and acceleration tends to zero,the velocity profiles approaches to Hiemenzs Velocity profile,Ref. [1], and when the plate is stationary ( _S

0 and S

0), Eq.

(11) simplifies to the case of Hiemenz flow. In this case, the veloc-ity profile matches Hiemenz velocity profile exactly and this is theonly existing reference which can be used for validation.

3.2 Heat Transfer Solution. To transform the energy equa-tion into a nondimensional form for the case of defined wall tem-perature, we introduce

h Tg T1Tw T1 (17)

Making use of transformations (8)(10), this equation may bewritten as

h00

~_S

t

~a

tf Pr:h0 0 (18)

with the boundary conditions as

g 0 : h 1 (19)g ! 1 : h 0 (20)

where Pr =a, is Prandtl number and prime indicates differen-tiation with respect to g. Note that, for Pr 1 and steady-statecase the thickness of the fluid boundary layer and thermal bound-ary layer become the same but in unsteady-state case this conceptdoes not apply and Eq. (18) would not be obtained from substitu-tion ofh f0 in Eq. (11).

064501-2 / Vol. 134, JUNE 2012 Transactions of the ASME



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4 Shear-Stress

Shear-stress at the wall surface is calculated from

r l@u@zz0

(21)

where l is the fluid viscosity. Using the transformations (8)(10),the shear-stress at the flat plate surface becomes

~r rqat

nf00 (22)

This quantity is presented for different values of n in the Presenta-tion of Results section.

5 Presentation of Results

As example, the following three distinct velocity functions forconstant A are considered:

_St A expt; _St At2; _St Sint

The exponential velocity function above, for example, can beused to model the physical one or two-dimensional solidificationproblem. Equation (11) for a known plate velocity function is anordinary differential equation in which the variables n, _S

t

, and

St are known and can be solved numerically along with Eq. (18)by using a shooting method trial and error and based on theRunge-Kutta algorithm and the results are presented for selectedvalues of _St and Pr. This procedure is applied for maximumerror of less than 0.00001.

In Figs. 2 and 3, the boundary layer velocity profile has beenpresented at different values of n and for an exponential and poly-nomial plate velocity functions. As it can be seen, initially the ve-locity profile with a steep slope approaches the potential flowvelocity but as time increases, that is, when the plate velocity andacceleration approaches to zero, the velocity profile graduallytends toward the Hiemenz flow. Therefore, the effect of change ofplate velocity is felt more by the boundary layer velocity profilethan the effect of plate acceleration. In Fig. 4, one can see thethickness of the boundary layer velocity for sinusoidal plate veloc-

ity function. Figure 5 shows velocity in w-direction at selectedvalues ofn for different time values and exponential plate velocityfunction. Shear-stress is presented in Fig. 6 for different time andn values and for the sinusoidal plate velocity case. The shear-stress value on the plate surface becomes constant for large valuesof n at all times. It is interesting to note that the starting point ofshear-stress value on f-axis is in a fix range for all time values.Figures 7 and 8 show the nondimensional pressure along the

boundary layer thickness for selected values of n and differenttime values. It is interesting to note that this pressure change with

respect to n is the cause of change of all the quantities like bound-ary layer thickness, shear-stress, velocity profile, and temperatureboundary layer thickness. In Fig. 9, the thermal boundary layerprofile for selected values of Prandtl numbers and different valuesof time are presented for the case of exponential plate velocityfunction. It can be seen that for large time values, the slope of thethermal boundary layer profile decreases and this shows that theslope of the profile increases sharply with increase of Prandtlnumber and plate velocity and acceleration toward the impingingflow and sharp increase in heat transfer rate is obvious for the caseof the earlier time values and for Pr 20. In Fig. 10, the thermalboundary layer profile is shown for different values of n and forexponential plate velocity function. As it is expected, the profilechanges are very slight with respect to n and that is because thefunction fg is the cause of these changes and in higher plate ve-locity or acceleration and also for higher Prandtl numbers thisfunction diminishes rapidly. Figure 11 depicts the variations ofthermal boundary layer thickness with respect to Prandtl numberfor different values of time and for the case of exponential func-tion of plate velocity. As it is seen from this figure, this thickness

Fig. 2 Velocity in x-direction at selected values of n for differ-ent time values and exponential plate velocity function

Fig. 3 Velocity in x-direction at selected values of n for differ-ent time values and polynomial plate velocity function

Fig. 4 Boundary layer thickness at different time values and si-nusoidal plate velocity function

Journal of Heat Transfer JUNE 2012, Vol. 134 / 064501-3



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Fig. 5 Velocity in w-direction at selected values of n for differ-ent time values and exponential plate velocity function

Fig. 6 Shear-stress at different time values and all values of nfor sinusoidal plate velocity function

Fig. 7 Pressure variation in boundary layer at certain time anddifferent values of n for exponential plate velocity function

Fig. 8 Pressure variation in boundary layer at n 0:1 and dif-ferent values of time for exponential plate velocity function

Fig. 9 Thermal boundary layer profile for Pr 0:1 and differenttime values for exponential plate velocity function

Fig. 10 Thermal boundary layer profile for Pr 1:0 and differenttime values at n 0:1; 2:2 for exponential plate velocity function

064501-4 / Vol. 134, JUNE 2012 Transactions of the ASME



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decreases with the increase of Pr number and increase of plate ve-locity and acceleration. Figure 12 presents the same trend ofchanges of thermal boundary layer profile for polynomial and si-nusoidal plate velocity plate.

6 Conclusions

General formulation and similarity solution of the Navier

Stokes and energy equations have been derived in the study oftwo-dimensional unsteady stagnation-point flow and heat transferimpinging on a flat plate where this plate is moving with arbitrary

velocity and acceleration functions of time toward main stream oraway from it. The results of the stagnation-point flow and heattransfer of the case of stationary plate, Hiemenz flow, is reachedby simplifications of this formulation. Presented examples of platevelocity functions are exponential, polynomial and sinusoidal inthis paper. All kinds of applications of plate movement is encoun-tered in industry where the stagnation-point flow and heat transferis involved but our main reason is use of exponential plate veloc-ity function to formulate solidification and melting in these kindsof studies. Velocity and pressure profiles, boundary layer thick-

ness, and surface stress-tensors along with temperature profileshave been presented for different examples of impinging fluidstrain rate, selected values of plate velocity, and Prandtl numberparameter. The effect of variations of plate velocity is felt moreby the boundary layer velocity profile than the effect of plateacceleration. The minimum boundary layer thickness happens atthe maximum value of plate velocity and acceleration effect playsa secondary role. When the ratio of nondimensional velocity tothe nondimensional acceleration is a linear function, then theboundary layer thickness and shear-stress approach to a constantvalue as the distance from z-axis is increased and this constantvalue depends on the plate velocity function but when this ratio isa nonlinear function, then the boundary layer thickness and shear-stress would not be a constant.

References[1] Hiemenz, K., 1911, Die grenzchicht an einem in den gleichformingen Flussig-

keitsstrom eingetauchten geraden KreisZylinder, Dinglers Polytech. J., 326,pp. 321410.

[2] Wang, C. Y., 1974, Axisymmetric Stagnation Flow on a Cylinder, Q. Appl.

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Cylinder, Int. J. Eng. Sci., 16, pp. 392400.[5] Gorla, R. S. R., 1979, Unsteady Viscous Flow in the Vicinity of an Axisym-

metric Stagnation-Point on a Cylinder,, Int. Sci. Technol., 17, pp. 8793.[6] Cunning, G. M., Davis, A. M. J., and Weidman, P. D., 1998, Radial Stagnation

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[7] Wang, C. Y., 1973, Axisymmetric Stagnation Flow Towards a Moving Plate,

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ceeding of Royal Society of London, Ser. A, 384, pp. 175190.[9] Saleh, R., and Rahimi, A. B., 2004, Axisymmetric Stagnation-Point Flow and

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[10] Rahimi, A. B., and Saleh, R., 2007, Axisymmetric Stagnation-Point Flow andHeat Transfer of a Viscous Fluid on a Rotating Cylinder With Time-Dependent

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Heat Transfer in Stagnation-Point Flow on a Cylinder with Simultaneous Axial

and Rotational Movements, Trans. ASME J. Heat Transfer, 130, p. 054502.[12] Shokrgozar Abbasi, A., and Rahimi, A. B., 2009, Non-Axisymmetric Three-

Dimensional Stagnation-Point Flow and Heat Transfer on a Flat Plate, Trans.

ASME J. Fluids Eng., 131(7), p. 074501.[13] Shokrgozar Abbasi, A., and Rahimi, A. B., 2009, Three-Dimensional

Stagnation-Point Flow and Heat Transfer on a Flat Plate With Transpiration, J.

Thermophys. Heat Transfer, 23(3), pp. 513521.[14] Shokrgozar Abbasi, A., and Rahimi, A. B., 2011, Exact Solution of Three-

Dimensional Unsteady Stagnation Flow on a Heated Plate, J. Thermophys.Heat Transfer, 25(1), pp. 5558.[15] Bian, X., and Rangel, R. H., 1996, The Viscous Stagnation-Flow Solidification

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Fig. 11 Thermal boundary layer thickness for different valuesof time and for all values of Prandtl number and exponentialplate velocity function

Fig. 12 Thermal boundary layer profile for Pr 1:0 and differ-ent time values for polynomial plate velocity function

Journal of Heat Transfer JUNE 2012, Vol. 134 / 064501-5
http://dx.doi.org/10.1007/BF00385923http://dx.doi.org/10.1016/0020-7225(78)90082-4http://dx.doi.org/10.1016/0020-7225(78)90082-4http://dx.doi.org/10.1023/A:1004243728777http://dx.doi.org/10.1023/A:1004243728777http://dx.doi.org/10.1002/aic.v19:5http://dx.doi.org/10.1098/rspa.1982.0153http://dx.doi.org/10.1098/rspa.1982.0153http://dx.doi.org/10.1115/1.1845556http://dx.doi.org/10.1115/1.2375132http://dx.doi.org/10.1115/1.2885173http://dx.doi.org/10.1115/1.3153366http://dx.doi.org/10.1115/1.3153366http://dx.doi.org/10.2514/1.41529http://dx.doi.org/10.2514/1.41529http://dx.doi.org/10.2514/1.48702http://dx.doi.org/10.2514/1.48702http://dx.doi.org/10.1016/0017-9310(96)00042-7http://dx.doi.org/10.1016/0017-9310(96)00042-7http://dx.doi.org/10.1016/0017-9310(96)00042-7http://dx.doi.org/10.2514/1.48702http://dx.doi.org/10.2514/1.48702http://dx.doi.org/10.2514/1.41529http://dx.doi.org/10.2514/1.41529http://dx.doi.org/10.1115/1.3153366http://dx.doi.org/10.1115/1.3153366http://dx.doi.org/10.1115/1.2885173http://dx.doi.org/10.1115/1.2375132http://dx.doi.org/10.1115/1.1845556http://dx.doi.org/10.1098/rspa.1982.0153http://dx.doi.org/10.1098/rspa.1982.0153http://dx.doi.org/10.1002/aic.v19:5http://dx.doi.org/10.1023/A:1004243728777http://dx.doi.org/10.1016/0020-7225(78)90082-4http://dx.doi.org/10.1007/BF00385923

investigation of two-dimensional unsteady stagnation-point flow and heat transfer impinging on an ...

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