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The effect of temperature-dependent viscosity on heat transfer over a continuous moving surface with variable internal heat generation
Applied Mathematics and Computation 153 (2004) 721–731 www.elsevier.com/locate/amc
The effect of temperature-dependent viscosity on heat transfer over a continuous moving surface with variable internal heat generation E.M.A. Elbashbeshy
a,*
, M.A.A. Bazid
b
a
b
Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt Department of Mathematics, Faculty of Education, El-Arish, Suez Canal University, Egypt
Abstract The effect of temperature-dependent viscosity on heat transfer over a continuous moving surface with variable internal heat generation is studied. The boundary layer equations are transformed to ordinary differential equations containing a Prandtl number, viscosity/temperature parameter and heat source (or sink) parameter. The velocity profiles, temperature profiles, the skin friction coefficient and the rate of heat transfer are computed and discussed in detail for various values of viscosity/temperature, heat source generation and Prandtl number. Ó 2003 Elsevier Inc. All rights reserved.
1. Introduction The continuous moving surface heat transfer problem has many practical applications in industrial manufacturing processes. Since the pioneering work of Sakiadis [1,2], various aspects of the problem have been investigated by many authors. Most studies have been concerned with constant surface velocity and temperature (see Tsou et al. [3]), but for
*
Corresponding author. E-mail address: [email protected] (E.M.A. Elbashbeshy).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00666-0
722
E.M.A. Elbashbeshy, M.A.A. Bazid / Appl. Math. Comput. 153 (2004) 721–731
many practical applications the surface undergoes stretching and cooling or heating that cause surface velocity and temperature variations. Crane [4], Vleggaar [5], and Gupta and Gupta [6] have analysed the stretching problem with constant surface temperature, while Soundalgekar and Ramana [7] investigated the constant surface velocity case with a power-law temperature variation. Grubka and Bobba [8] have analysed the stretching problem for a surface moving with a linear velocity and with variable surface temperature. Ali [9] has reported flow and heat characteristics on a stretched surface subject to power-law velocity and temperature distributions. The flow field of a stretching wall with a power-law velocity variation was discussed by Banks [10], Ali [11] and Elbashbeshy [12] extended Banks work for a porous stretched surface for different values of the injection parameter. Recently, Elbashbeshy and Bazid [13] have analysed the stretching problem which was discussed by Elbashbeshy [12] to include an uniform porous medium. More recently, Elbashbeshy and Bazid [14] have analysed the unsteady stretching problem. Also, Elbashbeshy and Bazid [15,16], have investigated the steady and unsteady stretching surface with internal heat generation. In all of the above mentioned studies the viscosity of the fluid was assumed to be constant. However it is known that this physical property may change significantly with temperature. To accurately predict the flow behaviour, it is necessary to take into account this variation of viscosity. Results [17,18] have shown that when this effect is included the flow characteristics may be substantially changed compared to the constant viscosity case. The present work is to study the influence of a variable viscosity on the heat transfer over a stretching surface with internal heat generation.
2. Formulation of the problem Consider a steady, two-dimensional laminar flow on a continuous, stretching surface with uniform surface temperature Tx and velocity Ux moving axially through a stationary fluid. The x-axis runs along the continuous surface in the direction of the motion and y-axis is perpendicular to it. The fundamental equations of the laminar boundary layer are [15,18] ou ov þ ¼0 ox oy
ð1Þ
ou ou 1 o ou l u þv ¼ ox oy q1 oy oy
ð2Þ
u
oT oT k o2 T þv ¼ þ QðT T1 Þ ox oy q1 cp oy 2
ð3Þ
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723
with the associated boundary conditions y ¼ 0;
u ¼ Ux ;
y ! 1;
u ¼ 0;
v ¼ 0;
T ¼ Tx
T ¼ T1
ð4Þ
where u and v are the velocity components in the x and y directions, T is the temperature inside the boundary layer, q1 is the density away from the hot plate, l is the viscosity, k is the thermal conductivity, cp is the specific heat at constant pressure, T1 is the free stream temperature and Q is the volumetric rate of heat generation. For a viscous fluid, Ling and Dybbs [19] suggest a viscosity dependence on temperature T of the form l ¼ l1 =½1 þ cðT T1 Þ
ð5Þ
so that viscosity is an inverse linear function of temperature T . Eq. (5) can be written as 1=l ¼ aðT Tr Þ
ð6Þ
where a ¼ c=l1
and
Tr ¼ T1 1=c
ð7Þ
In the above relations (7), both a and Tr are constant and their values depend on the reference state and c a thermal property of the fluid. The equation of continuity is satisfied if we choose a stream function wðx; yÞ such that u¼
ow ; oy
v¼
ow ox
The mathematical analysis of the problem is simplified by introducing the following dimensionless coordinates: rffiffiffiffiffiffiffiffiffiffiffi Ux l g¼y ; m1 ¼ 1 2m1 x q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 wðx; yÞ ¼ 2m1 Ux x f ðgÞ ð8Þ T T1 Ux hðgÞ ¼ k ; Q¼ Tx T1 2x T Tr or hðgÞ ¼ þ hr Tx T1 where hr ¼
Tr T1 1 ¼ constant ¼ cðTx T1 Þ Tx T1
and its value is determined by the viscosity/temperature characteristics of the fluid and the operating temperature difference DT ¼ Tx T1 .
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Substituting (8) into Eqs. (2) and (3) we obtain f 000 þ
hr h 00 h0 f 00 ¼ 0 ff þ hr hr h
h00 þ Prðf h0 þ khÞ ¼ 0
ð9Þ ð10Þ
where Pr ¼ q1 m1 cp =k is the Prandtl number, the primes denote differentiation with respect to g and k is the heat source or sink parameters. The boundary conditions (4) now become f ¼ 0; f 0 ¼ 1; h ¼ 1 at g ¼ 0 f 0 ¼ 0; h ¼ 0 at g ! 1
ð11Þ
The shearing stress on the plate is defined by ou sx ¼ l oy y¼0 The skin friction coefficient is defined by 2sx q1 Ux2 pffiffiffi 2hr 00 1 f ð0; hr Þ Cf R2e ¼ hr 1
Cf ¼
The local Nusselt number for heat transfer in the present case is defined by x oT oy y¼0 Nu ¼ ðTx T1 Þ Nu pffiffiffiffiffi ¼ h0 ð0; hr Þ Re
3. Numerical method The coupled non-linear differential equations (9) and (10) are solved numerically by using the fourth-order Runge–Kutta integration scheme with Newton–Raphson shooting method on an IBM 586 personal computer. The solutions of the differential equations (9) and (10) subject to the boundary conditions (11) were obtained for different values of hr . For a given value of hr the values of f 00 ð0Þ and h0 ð0Þ were estimated and the differential equations (9) and (10) were integrated using Runge–Kutta method until the boundary conditions at infinity f 0 ðgÞ and hðgÞ decay exponentially to zero. If the boundary conditions at infinity are not satisfied then the numerical routine uses
E.M.A. Elbashbeshy, M.A.A. Bazid / Appl. Math. Comput. 153 (2004) 721–731
725
a half interval method to calculate corrections to the estimated values of f 00 ð0Þ and h0 ð0Þ. This process is repeated iteratively until exponentially decaying solutions in f 0 ðgÞ and hðgÞ were obtained. The numerical results were found to depend upon g1 and step size. A step size of Dg ¼ 0:1 gave sufficient accuracy for Pr ¼ 0:7 and 7 respectively. The value of g1 was chosen as large as possible between 3 and 12, depending upon hr , k and Pr, without causing numerical oscillations in the values of f 0 , f 00 and h. 4. Results and discussion In the case Tx T1 is positive, the hr is negative for liquid and positive for gases since c has the positive sign in each of these cases. Indeed the variable viscosity l can be rewritten in the form l1 l¼ 1 hh1 r Since h varies from zero at the edge of the boundary layer to one at the surface, the largest change in the fluid viscosity from its free-stream value l1 , occurs at the wall, where l1 l¼ 1 h1 r Then hr cannot take values between zero and one and that the constraints, hr > 1 for gases and hr < 0, for liquids. Figs. 1 and 2 show the variation of the dimensionless velocity f 0 ðgÞ and temperature hðgÞ for different values of heat source or sink heat k for hr ¼ 2
Fig. 1. Temperature distribution as a function of g for various values of k at hr ¼ 2:0, Pr ¼ 0:7.
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Fig. 2. Velocity distribution as a function of g for various values of k at hr ¼ 2:0, Pr ¼ 0:7.
Fig. 3. Temperature distribution as a function of g for various values of k at hr ¼ 2:0, Pr ¼ 7:0.
and Pr ¼ 0:7 (air). It is found that, dimensionless velocity f 0 ðgÞ and the temperature hðgÞ increases with k. Figs. 3 and 4 show the variation of the dimensionless velocity f 0 ðgÞ and temperature hðgÞ for different values of k for hr ¼ 2 and Pr ¼ 7 (water). It is found that, the temperature increases, whereas the velocity decreases slightly as k increases. Figs. 5 and 6 show the variation of the dimensionless velocity f 0 ðgÞ and temperature hðgÞ for different values of hr for k ¼ 0:2 and Pr ¼ 7 (water). It is found that, dimensionless velocity f 0 ðgÞ increases slightly whereas the temperature hðgÞ decreases slightly as hr decreases.
E.M.A. Elbashbeshy, M.A.A. Bazid / Appl. Math. Comput. 153 (2004) 721–731
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Fig. 4. Velocity distribution as a function of g for various values of k at hr ¼ 2:0, Pr ¼ 7:0.
Fig. 5. Temperature distribution as a function of g for various values of hr at k ¼ 0:2, Pr ¼ 7:0.
Figs. 7 and 8 show the variation of the dimensionless velocity f 0 ðgÞ and temperature hðgÞ for different values of hr for k ¼ 0:2 and Pr ¼ 0:7 (air). It is found that, dimensionless velocity f 0 ðgÞ decreases whereas the temperature hðgÞ increases for air as hr increases. Figs. 9 and 10 show the variation of the dimensionless coefficient h0 ð0; hr Þ for different values viscosity/temperature parameter hr and heat source (or sink) parameter k. It is found that the temperature gradient h0 ð0; hr Þ at the stretched surface decreases with k and hr for Pr ¼ 0:7; 7. Figs. 11 and 12 show the variation of the dimensionless skin friction f 00 ð0Þ for different values k and hr at Pr ¼ 7. In these figures we clearly see that the
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Fig. 6. Velocity distribution as a function of g for various values of hr at k ¼ 0:2, Pr ¼ 7:0.
Fig. 7. Temperature distribution as a function of g for various values of hr at k ¼ 0:2, Pr ¼ 0:7.
skin friction increases with k and decreases with hr whereas at Pr ¼ 0:7, the dimensionless skin friction f 00 ð0Þ decreases with k and hr .
5. Conclusion In this study, the effect of temperature-dependent viscosity and the heat source or sink on the heat transfer over a continuous surface is analysed. Numerical results to the transformed boundary layer equations have been
E.M.A. Elbashbeshy, M.A.A. Bazid / Appl. Math. Comput. 153 (2004) 721–731
729
Fig. 8. Velocity distribution as a function of g for various values of khr at k ¼ 0:2, Pr ¼ 0:7.
Fig. 9. Variation of the rate of heat transfer coefficient as a function of k (Pr ¼ 7).
obtained by using the fourth-order Runge–Kutta integration scheme with Newton–Raphson shooting method. Of interest are the influences of the viscosity/temperature parameter and the source (or sink) parameter on the velocity, temperature, the heat transfer rate and the skin friction. The following results were obtained. 1. The velocity f 0 ðgÞ decreases slightly for water whereas increase for air with increases k. 2. The velocity f 0 ðgÞ decreases for both water and air with an increases of hr .
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Fig. 10. Variation of the rate of heat transfer coefficient as a function of k (Pr ¼ 0:7).
Fig. 11. Variation of the skin friction coefficient as a function of hr (Pr ¼ 7).
Fig. 12. Variation of the skin friction coefficient as a function of hr (Pr ¼ 0:7).
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3. The temperature hðgÞ increases for both water and air with an increases of k. 4. The temperature hðgÞ increases for both water and air with an increases of hr . 5. The skin friction f 00 ð0Þ increases with k and decreases with hr in the case Pr ¼ 7 whereas for air, the skin friction decreases with k and hr . 6. The temperature gradient at the surface h0 ð0; hr Þ decreases with k and hr for both water and air.
References [1] B.C. Sakiadis, Boundary layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow, AIChE J. 7 (1) (1961) 26–28. [2] B.C. Sakiadis, Boundary layer behavior on continuous solid surface: II. Boundary layer behavior on continuous flat surface, AIChE J. 7 (1) (1961) 221–225. [3] F.K. Tsou, E.M. Sparrow, R.J. Goldstein, Flow and heat transfer in the boundary layer on a continuous moving surface, Int. J. Heat Mass Transfer 10 (1967) 219–235. [4] L.J. Crane, Flow past a stretching plane, Z. Angew. Math. Phys. 21 (1970) 645–647. [5] J. Vleggaar, Laminar boundary layer behavior on continuous accelerating surfaces, Chem. Eng. Sci. 32 (1977) 1517–1525. [6] P.S. Gupta, A.S Gupta, Heat and mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Eng. 55 (6) (1977) 744–746. [7] V.M. Soundalgekar, T.V. Ramana, Heat transfer past a continuous moving plate with variable temperature, Warme-und Stoffubertragung 14 (1980) 91–93. [8] L.J. Grubka, K.M. Bobba, Heat transfer characteristics of a continuous stretching surface with variable temperature, J. Heat Transfer 107 (1985) 248–250. [9] M.E. Ali, Heat transfer characteristics of a continuous stretching surface, W€armme Stoff€ ubertragung 29 (1994) 227–234. [10] W.H.H. Banks, Similarity solutions of the Boundary layer equation for a stretching wall, J. Mech. Theor. Appl. 2 (1983) 375–392. [11] M.E. Ali, On thermal boundary layer on a power law stretched surface with suction or injection, Int. J. Heat Mass Flow 16 (1995) 280–290. [12] E.M.A. Elbashbeshy, Heat transfer over a stretching surface with variable heat flux, J. Phys. D: Appl. Phys. 31 (1998) 1951–1955. [13] E.M.A. Elbashbeshy, M.A.A. Bazid, Heat transfer over a continuously moving plate embedded in a non Darcian porous medium, Int. J. Heat Mass Transfer 43 (2000) 3087–3092. [14] E.M.A. Elbashbeshy, M.A.A. Bazid, Heat transfer over an unsteady stretching surface, Heat Mass Transfer, in press. [15] E.M.A. Elbashbeshy, M.A.A. Bazid, Heat transfer over an unsteady stretching surface, with internal heat generation, Appl. Math. Comput., in press. [16] E.M.A. Elbashbeshy, M.A.A. Bazid, Heat transfer over a steady stretching surface with internal heat generation, Can. J. Phys., in press. [17] E.M.A. Elbashbeshy, F.N. Ibrahim, Steady free convection flow with variable viscosity and thermal diffusivity along a vertical plate, J. Phys. D: Appl. Phys. 26 (1993) 2137–2143. [18] E.M.A. Elbashbeshy, M.A.A. Bazid, The effect of temperature-dependent viscosity on heat transfer over a continuous moving surface, J. Phys. D: Appl. Phys. 33 (2000) 2716–2721. [19] J.X. Ling, A. Dybbs, Forced convection over a flat plate submersed in a porous medium: variable viscosity case, Paper 87-WA/HT-23, ASME, New York, 1987.
The effect of temperature-dependent viscosity on heat transfer over a continuous moving surface with variable internal heat generation E.M.A. Elbashbeshy
a,*
, M.A.A. Bazid
b
a
b
Department of Mathematics, Faculty of Science, Ain Shams University, Abbassia, Cairo, Egypt Department of Mathematics, Faculty of Education, El-Arish, Suez Canal University, Egypt
Abstract The effect of temperature-dependent viscosity on heat transfer over a continuous moving surface with variable internal heat generation is studied. The boundary layer equations are transformed to ordinary differential equations containing a Prandtl number, viscosity/temperature parameter and heat source (or sink) parameter. The velocity profiles, temperature profiles, the skin friction coefficient and the rate of heat transfer are computed and discussed in detail for various values of viscosity/temperature, heat source generation and Prandtl number. Ó 2003 Elsevier Inc. All rights reserved.
1. Introduction The continuous moving surface heat transfer problem has many practical applications in industrial manufacturing processes. Since the pioneering work of Sakiadis [1,2], various aspects of the problem have been investigated by many authors. Most studies have been concerned with constant surface velocity and temperature (see Tsou et al. [3]), but for
*
Corresponding author. E-mail address: [email protected] (E.M.A. Elbashbeshy).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00666-0
722
E.M.A. Elbashbeshy, M.A.A. Bazid / Appl. Math. Comput. 153 (2004) 721–731
many practical applications the surface undergoes stretching and cooling or heating that cause surface velocity and temperature variations. Crane [4], Vleggaar [5], and Gupta and Gupta [6] have analysed the stretching problem with constant surface temperature, while Soundalgekar and Ramana [7] investigated the constant surface velocity case with a power-law temperature variation. Grubka and Bobba [8] have analysed the stretching problem for a surface moving with a linear velocity and with variable surface temperature. Ali [9] has reported flow and heat characteristics on a stretched surface subject to power-law velocity and temperature distributions. The flow field of a stretching wall with a power-law velocity variation was discussed by Banks [10], Ali [11] and Elbashbeshy [12] extended Banks work for a porous stretched surface for different values of the injection parameter. Recently, Elbashbeshy and Bazid [13] have analysed the stretching problem which was discussed by Elbashbeshy [12] to include an uniform porous medium. More recently, Elbashbeshy and Bazid [14] have analysed the unsteady stretching problem. Also, Elbashbeshy and Bazid [15,16], have investigated the steady and unsteady stretching surface with internal heat generation. In all of the above mentioned studies the viscosity of the fluid was assumed to be constant. However it is known that this physical property may change significantly with temperature. To accurately predict the flow behaviour, it is necessary to take into account this variation of viscosity. Results [17,18] have shown that when this effect is included the flow characteristics may be substantially changed compared to the constant viscosity case. The present work is to study the influence of a variable viscosity on the heat transfer over a stretching surface with internal heat generation.
2. Formulation of the problem Consider a steady, two-dimensional laminar flow on a continuous, stretching surface with uniform surface temperature Tx and velocity Ux moving axially through a stationary fluid. The x-axis runs along the continuous surface in the direction of the motion and y-axis is perpendicular to it. The fundamental equations of the laminar boundary layer are [15,18] ou ov þ ¼0 ox oy
ð1Þ
ou ou 1 o ou l u þv ¼ ox oy q1 oy oy
ð2Þ
u
oT oT k o2 T þv ¼ þ QðT T1 Þ ox oy q1 cp oy 2
ð3Þ
E.M.A. Elbashbeshy, M.A.A. Bazid / Appl. Math. Comput. 153 (2004) 721–731
723
with the associated boundary conditions y ¼ 0;
u ¼ Ux ;
y ! 1;
u ¼ 0;
v ¼ 0;
T ¼ Tx
T ¼ T1
ð4Þ
where u and v are the velocity components in the x and y directions, T is the temperature inside the boundary layer, q1 is the density away from the hot plate, l is the viscosity, k is the thermal conductivity, cp is the specific heat at constant pressure, T1 is the free stream temperature and Q is the volumetric rate of heat generation. For a viscous fluid, Ling and Dybbs [19] suggest a viscosity dependence on temperature T of the form l ¼ l1 =½1 þ cðT T1 Þ
ð5Þ
so that viscosity is an inverse linear function of temperature T . Eq. (5) can be written as 1=l ¼ aðT Tr Þ
ð6Þ
where a ¼ c=l1
and
Tr ¼ T1 1=c
ð7Þ
In the above relations (7), both a and Tr are constant and their values depend on the reference state and c a thermal property of the fluid. The equation of continuity is satisfied if we choose a stream function wðx; yÞ such that u¼
ow ; oy
v¼
ow ox
The mathematical analysis of the problem is simplified by introducing the following dimensionless coordinates: rffiffiffiffiffiffiffiffiffiffiffi Ux l g¼y ; m1 ¼ 1 2m1 x q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 wðx; yÞ ¼ 2m1 Ux x f ðgÞ ð8Þ T T1 Ux hðgÞ ¼ k ; Q¼ Tx T1 2x T Tr or hðgÞ ¼ þ hr Tx T1 where hr ¼
Tr T1 1 ¼ constant ¼ cðTx T1 Þ Tx T1
and its value is determined by the viscosity/temperature characteristics of the fluid and the operating temperature difference DT ¼ Tx T1 .
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Substituting (8) into Eqs. (2) and (3) we obtain f 000 þ
hr h 00 h0 f 00 ¼ 0 ff þ hr hr h
h00 þ Prðf h0 þ khÞ ¼ 0
ð9Þ ð10Þ
where Pr ¼ q1 m1 cp =k is the Prandtl number, the primes denote differentiation with respect to g and k is the heat source or sink parameters. The boundary conditions (4) now become f ¼ 0; f 0 ¼ 1; h ¼ 1 at g ¼ 0 f 0 ¼ 0; h ¼ 0 at g ! 1
ð11Þ
The shearing stress on the plate is defined by ou sx ¼ l oy y¼0 The skin friction coefficient is defined by 2sx q1 Ux2 pffiffiffi 2hr 00 1 f ð0; hr Þ Cf R2e ¼ hr 1
Cf ¼
The local Nusselt number for heat transfer in the present case is defined by x oT oy y¼0 Nu ¼ ðTx T1 Þ Nu pffiffiffiffiffi ¼ h0 ð0; hr Þ Re
3. Numerical method The coupled non-linear differential equations (9) and (10) are solved numerically by using the fourth-order Runge–Kutta integration scheme with Newton–Raphson shooting method on an IBM 586 personal computer. The solutions of the differential equations (9) and (10) subject to the boundary conditions (11) were obtained for different values of hr . For a given value of hr the values of f 00 ð0Þ and h0 ð0Þ were estimated and the differential equations (9) and (10) were integrated using Runge–Kutta method until the boundary conditions at infinity f 0 ðgÞ and hðgÞ decay exponentially to zero. If the boundary conditions at infinity are not satisfied then the numerical routine uses
E.M.A. Elbashbeshy, M.A.A. Bazid / Appl. Math. Comput. 153 (2004) 721–731
725
a half interval method to calculate corrections to the estimated values of f 00 ð0Þ and h0 ð0Þ. This process is repeated iteratively until exponentially decaying solutions in f 0 ðgÞ and hðgÞ were obtained. The numerical results were found to depend upon g1 and step size. A step size of Dg ¼ 0:1 gave sufficient accuracy for Pr ¼ 0:7 and 7 respectively. The value of g1 was chosen as large as possible between 3 and 12, depending upon hr , k and Pr, without causing numerical oscillations in the values of f 0 , f 00 and h. 4. Results and discussion In the case Tx T1 is positive, the hr is negative for liquid and positive for gases since c has the positive sign in each of these cases. Indeed the variable viscosity l can be rewritten in the form l1 l¼ 1 hh1 r Since h varies from zero at the edge of the boundary layer to one at the surface, the largest change in the fluid viscosity from its free-stream value l1 , occurs at the wall, where l1 l¼ 1 h1 r Then hr cannot take values between zero and one and that the constraints, hr > 1 for gases and hr < 0, for liquids. Figs. 1 and 2 show the variation of the dimensionless velocity f 0 ðgÞ and temperature hðgÞ for different values of heat source or sink heat k for hr ¼ 2
Fig. 1. Temperature distribution as a function of g for various values of k at hr ¼ 2:0, Pr ¼ 0:7.
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Fig. 2. Velocity distribution as a function of g for various values of k at hr ¼ 2:0, Pr ¼ 0:7.
Fig. 3. Temperature distribution as a function of g for various values of k at hr ¼ 2:0, Pr ¼ 7:0.
and Pr ¼ 0:7 (air). It is found that, dimensionless velocity f 0 ðgÞ and the temperature hðgÞ increases with k. Figs. 3 and 4 show the variation of the dimensionless velocity f 0 ðgÞ and temperature hðgÞ for different values of k for hr ¼ 2 and Pr ¼ 7 (water). It is found that, the temperature increases, whereas the velocity decreases slightly as k increases. Figs. 5 and 6 show the variation of the dimensionless velocity f 0 ðgÞ and temperature hðgÞ for different values of hr for k ¼ 0:2 and Pr ¼ 7 (water). It is found that, dimensionless velocity f 0 ðgÞ increases slightly whereas the temperature hðgÞ decreases slightly as hr decreases.
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Fig. 4. Velocity distribution as a function of g for various values of k at hr ¼ 2:0, Pr ¼ 7:0.
Fig. 5. Temperature distribution as a function of g for various values of hr at k ¼ 0:2, Pr ¼ 7:0.
Figs. 7 and 8 show the variation of the dimensionless velocity f 0 ðgÞ and temperature hðgÞ for different values of hr for k ¼ 0:2 and Pr ¼ 0:7 (air). It is found that, dimensionless velocity f 0 ðgÞ decreases whereas the temperature hðgÞ increases for air as hr increases. Figs. 9 and 10 show the variation of the dimensionless coefficient h0 ð0; hr Þ for different values viscosity/temperature parameter hr and heat source (or sink) parameter k. It is found that the temperature gradient h0 ð0; hr Þ at the stretched surface decreases with k and hr for Pr ¼ 0:7; 7. Figs. 11 and 12 show the variation of the dimensionless skin friction f 00 ð0Þ for different values k and hr at Pr ¼ 7. In these figures we clearly see that the
728
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Fig. 6. Velocity distribution as a function of g for various values of hr at k ¼ 0:2, Pr ¼ 7:0.
Fig. 7. Temperature distribution as a function of g for various values of hr at k ¼ 0:2, Pr ¼ 0:7.
skin friction increases with k and decreases with hr whereas at Pr ¼ 0:7, the dimensionless skin friction f 00 ð0Þ decreases with k and hr .
5. Conclusion In this study, the effect of temperature-dependent viscosity and the heat source or sink on the heat transfer over a continuous surface is analysed. Numerical results to the transformed boundary layer equations have been
E.M.A. Elbashbeshy, M.A.A. Bazid / Appl. Math. Comput. 153 (2004) 721–731
729
Fig. 8. Velocity distribution as a function of g for various values of khr at k ¼ 0:2, Pr ¼ 0:7.
Fig. 9. Variation of the rate of heat transfer coefficient as a function of k (Pr ¼ 7).
obtained by using the fourth-order Runge–Kutta integration scheme with Newton–Raphson shooting method. Of interest are the influences of the viscosity/temperature parameter and the source (or sink) parameter on the velocity, temperature, the heat transfer rate and the skin friction. The following results were obtained. 1. The velocity f 0 ðgÞ decreases slightly for water whereas increase for air with increases k. 2. The velocity f 0 ðgÞ decreases for both water and air with an increases of hr .
730
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Fig. 10. Variation of the rate of heat transfer coefficient as a function of k (Pr ¼ 0:7).
Fig. 11. Variation of the skin friction coefficient as a function of hr (Pr ¼ 7).
Fig. 12. Variation of the skin friction coefficient as a function of hr (Pr ¼ 0:7).
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731
3. The temperature hðgÞ increases for both water and air with an increases of k. 4. The temperature hðgÞ increases for both water and air with an increases of hr . 5. The skin friction f 00 ð0Þ increases with k and decreases with hr in the case Pr ¼ 7 whereas for air, the skin friction decreases with k and hr . 6. The temperature gradient at the surface h0 ð0; hr Þ decreases with k and hr for both water and air.
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