Applied Mathematical Sciences, Vol. 6,, no. 68, 47-65 Pressure Effects on Unsteady Free Convection and Heat Transfer Flow of an Incompressible Fluid Past a Semi-Infinite Inclined Plate with Impulsive and Uniformly Accelerated Motion Asma Begum Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-, Bangladesh Md. Abdul Maleque Department of Mathematics, Bangladesh University of Engineering and Technology, Dhaka-, Bangladesh M. Ferdows Department of Mathematics, University of Dhaka, Dhaka-, Bangladesh Masahiro Ota Department of Mechanical Engineering, Tokyo Metropolitan University - Minami-osawa, Hachioji, Tokyo 9-97, Japan ota-masahiro@c.metro-u.ac.jp Abstract A similar unsteady laminar boundary layer model is considered for free convection heat transfer flow of a Newtonian incompressible past an inclined plate with pressure effect. The similar boundary layer equations are solved numerically by applying Nachtsheim-Swigert shooting iteration technique along with Runge-Kutta sixth order integration method. Two cases of the motion of the flow have been considered such as the plate is impulsive and the other is the uniformly accelerated motion. The plate temperature is assumed to be a function
48 A. Begum, Md. Abdul Maleque, M. Ferdows, M. Ota of time. Solutions obtained in terms of dimensionless velocity temperature and pressure profiles as well as the local skin-friction and local Nusselt number for the values of governing parameters are presented for both cases. Keywords: Unsteady free convection, Inclined plate, Impulsive motion, Heat transfer Introduction Free convection flow adjacent to inclined surface bounded by an extensive body of fluid is of considerable importance in micrometeorological and industrial applications. Some of the earlier workers in this respect are Ostrach [], Stewartson [], Gill et al. []. Most of the existing analyses have used the similarity solutions for the vertical case with the buoyant force being the component of the body force along the plate. Many researchers such as Merkin [4,5] and Harvet and Blay [6] have investigated the problem of free convection over a vertical plate. Karkus [7] applied perturbation technique to study the natural convection flow adjacent to inclined isothermal and finite-length surfaces. Free convection boundary layer flow over a horizontal and slightly inclined surface has been studied by Pera and Gebhart [8]. Umermura and Law [9] developed a generalized formulation for the natural convection boundary layer flow over a flat plate of arbitrary inclination. Hossian et al. [] studied the free convection flow from an isothermal plate inclined at a small angle to the horizontal. Angel et al. [] presented numerical solution of free convection flow past an inclined surface. He studied the flow characteristics depended not only on the extent of inclination but also on the distance from the leading edge. The above works were on steady flows. The aim of the present works is to study the similarity solution on free convection boundary layer flow over a plate with different inclinations and pressure effects considering time dependent plate temperature. Governing equations of the flow Free convection boundary layer flow over a semi-infinite heated flat plate with π inclined arbitrary angle α ( α ) to the horizontal. For this purpose let us consider the unsteady motion of an electrically conducting viscous and incompressible fluid. The flow is assumed to be in the x -direction and y -axis is normal to it. At time t >, the plate temperature is instantly raised to T (> T ), where T be the temperature of the uniform flow and the plate starts with a velocity U (t) in its own plane. Consider u and ν be the velocity
Pressure effects on unsteady free convection 49 components in the rectangular co-ordinate system, U be the mean velocity of the plate in the x -direction and ρ be the fluid density and C p is the specific heat at constant pressure. The physical flow configuration is shown in the following Figure y T v u x g α T w Figure : Configuration of the problem The flow model considered is of unsteady free convection boundary layer flow over a flat plate with arbitrary inclination under the influence of applied field. We have considered a time dependent suction and plate temperature. It is assumed that the flow is onedimensional, unsteady state, laminar and the fluid is incompressible. The solutions of the governing equations have been done taking suitable similarity transformations. Two cases are considered ) Impulsively started plate moving in its own plane and ) Uniformly accelerated plate. The suction velocity is taken to be inversely proportional to the above length-scale. The continuity, momentum and energy equations for unsteady, viscous and incompressible flows are respectively given by.q = () q + ρ ( q. ) q = p + υ q + F t () T k + ( q. ) T = T t () ρc p Here q = q( u, v) is the velocity vector, F r is the body force per unit volume which defined as - ρ g, p is the pressure force, ρ is the density of the fluid, υ is the viscosity, β is the thermal expansion coefficients, g is the gravitational acceleration, T is the temperature inside the boundary layer,α is the inclination angle from horizontal direction, k is the thermal conductivity, c p is the specific
5 A. Begum, Md. Abdul Maleque, M. Ferdows, M. Ota heat at a constant pressure and is the vector differential operator which is defined by = Iˆ x + Iˆ y x y where Î x and Î y are the unit vectors along x and y axes respectively. The above mentioned equations (.)-(.) would serve as the governing equations of the problems under the Boussinesq s approximation v = (4) y u u u + v = υ + g β ( T T ) sinα t y y p = + g β ( T T )cosα ρ y T T T ρ c p ( + v ) = k (7) t y y Where the variables and related quantities are defined in the Nomenclature. The appropriate boundary conditions for the above problem are as follows: u = U (t), ν =ν (t), T = T (t) at y = (8a) u =, ν =, T = T, P = as y (8b) Two cases have been considered for the problem. They are: Case I: Impulsively started plate (ISP), i.e., when the plate is impulsively started and moves in its own plane. Case II: Uniformly accelerated plate (UAP), i.e., when the plate moves with a velocity taken to be a function of time. (5) (6). Similarity analysis Case I: Impulsive motion We introduce a similarity parameter σ as σ = σ (t) (9) where σ is the time dependent length scale. In terms of σ, a convenient solution of the equation (.4) is considered to be υ v = v () σ here the constant v represents a dimensionless normal velocity at the plate which is positive for suction and negative for blowing. Now we introduce the following dimensionless variables
Pressure effects on unsteady free convection 5 u = U ( t) = U T T θ ( ) = T T y, =, f ( ) = σ pσ, P( ) = U υρ u U () where U is the mean velocity, T is the mean temperature and p is the dimensionless pressure, all being constant. Using equations (9),() and () in equations (5)-(7), we obtain σ σ f = f + G r θ sinα () υ t / = P + G r θ cosα () σ σ θ v θ = θ υ t (4) where g β ( T T ) σ G r ( = ), is the Grashoff number, U υ Pr ( = υ ) k / ρc, is the Prandtl number p all are the dimensionless local parameters. The boundary conditions (8a) and (8b) then becomes f =, θ = at = (5) P r f =, θ = P = as So following the works of Sattar and Hossain, assuming that σ σ = c (a constant) (6) υ t σ σ The equations () and (4) are similar expect for the term, where time t υ t appears explicitly. Integrating (6) we obtain σ = cυt (7) where the constant of integration is determined through the condition that σ = when t =. It thus appears from (7) that by making a realistic choice of c to be equal to, then in (6) σ = υt which exactly corresponds to the usual scaling factor considered for various non steady boundary layer flow Schlichting []. Since σ is a scaling factor as well as a similarity parameter, one other value of c in (6) would not change the nature of the solution except that the scale would be different. Lastly, introducing (6) with c = in equations () and (4) respectively we have the following dimensionless ordinary differential equations:
5 A. Begum, Md. Abdul Maleque, M. Ferdows, M. Ota f = ζf G θ sinα (8) / r P = G r θ cosα (9) θ = ζp θ () r where v ζ = + Case II: Uniformly Accelerated Motion In this case U (t) is the free stream velocity and T (t) is the plate temperature are assumed to have the following forms : m+ U ( t) = U σ () * m T ( t) = T + ( T T ) σ * () σ where m is an integer and σ * = σ Now introducing m+ u = U ( t) f ( ) = U σ * f ( ) () T = T + ( T T ) σ m θ ( ) (4) * m+ ρu ( t) υ ρu σ * υ p = Pa ( ) = Pa ( ) σ σ (5) where P a is the dimensionless pressure for accelerated motion. Introducing the relations (9),() and (6)-() in equations (5),(6) and (7) and also introducing the following dimensionless parameters g β ( T T ) σ G r ( = ), is the Grashoff number, U υ υ and Pr ( = ), is the Prandtl number k / ρc p We obtain the following dimensionless differential equations : υ σ [( m + ) f f ] v f = f + Grθ sinα σ t (6) P a = G r θ cosα (7) υ σ [ mθ θ ] v θ = θ σ t P r (8) The boundary conditions (8a),(8b) now reduce to f =, θ = at = (9a) f =, θ =, P a = as (9b)
Pressure effects on unsteady free convection 5 Now following the arguments in case I, equations () and () become respectively f + ζf 4( m + ) f + Grθ sinα = () P a = G r θ cosα () θ + P ζθ 4mP θ = () r r where v ζ = +. Skin- friction coefficient and Nusselt number : The dimensionless local wall shear stress and local surface heat flux for impulsively stared plate respectively are obtained as τ σ w = f () μu () and q w σ = θ () k( T T ) (4) Hence for impulsively started plate the dimensionless skinfriction coefficient and the Nusselt number are given by τ wσ τ i = = f () μ U (5) and qwσ N θ ui = = () k( T T ) (6) for accelerated plate they are obtained as τ wσ τ a = m+ μu σ = f () (7) * qwσ and N ua = (8) m+ k( T T ) σ * Thus the dimensionless values of the local skin-friction and the Nusselt number for impulsive as well as accelerated plate are obtained numerically. Results and discussion The system of nonlinear ordinary differential equations (8) () together with the boundary conditions (5) in the case I and ()-() together with the boundary condition (9) in the case II have been solved numerically by using sixth-order Runge-Kutta shooting method []. Various groups of the
54 A. Begum, Md. Abdul Maleque, M. Ferdows, M. Ota parameters α, Gr,Pr, ν, m were considered in different phases. In all the computations the step size Δ =.5 was selected that satisfied a convergence 6 criterion of in almost all of different phases mentioned above. However, different step sizes such as Δ =. to Δ =. were also tried and the obtained solutions have been found to be independent of the step sizes as observed in figure.. Gr =, Pr =.7, v =.5,α = o.8.6 Velocity Profiles -------- =. =.5 =..4. Temperature Profiles Fig. : Velocity & temperature profiles for different step sizes. The results for the two cases considered above are displayed graphically in Figures (.)-(.) respectively for dimensionless forms of velocity, temperature and pressure. Numerical computations have been carried out for the study of the effects of various parameters on the velocity, temperature and pressure distribution for both the case. For this purpose the effects of different parameters Gr, v, Pr, m and α on the fluid flow have been investigated. The value of Prandtl number Pr is taken equal to.7,. and 7. that corresponds physically.7 is suitable for air at c,. correspond to electrolyte solution as salt water and 7. corresponds to water. The value Grashoff number Gr is taken to be large., where larger values of Gr correspond to a cooling problem that is generally encountered in nuclear engineering in connection with the cooling of reactors. The positive or negatives values of Gr respectively represent cooling and heating of the plate. The
Pressure effects on unsteady free convection 55 suction velocity ν is taken to be equal to.5,. and. which are appropriate for the liquid metals. The values of α and m are chosen arbitrarily. Case I: The mentioned parameters the velocity, temperature and pressure distribution profiles are represented graphically in Figures (.)-(.6) for both the cooling and heating of impulsively started plate. From Figures (.) and (.) for the case when Gr > (in the presence of cooling of the plate by natural convection currents ) we observe that i) there is a rise in the velocity profiles due to an increase in α, ii) an increase in the suction parameter v causes a fall in the velocity fields. From Figures (.) for the case when Gr positive and negative (in the presence of heating and cooling of the plate by natural convection currents ) it is seen that there is fall in the velocity profiles due to an increase. Figures (.4) for the case when Gr > in the presence of cooling of the plate by natural convection currents. We also plotted temperature profiles in Figure (.5) in case of Gr > for a comparison in different Prandtl number Pr.We see that there is a decrease in temperature due to increases which is very large in case of water ( Pr =7.). Figure (.6 ) shows the pressure distribution profiles for two cases Gr < and Gr > for fixed values Pr, v and α. We observe that there is a rise in pressure due to increase for the casegr >, on the other hand, a reverse phenomenon occurs in case Gr <..4. Pr =.7, Gr =, ν ο =.5 f.8.6.4 α = Ο α = 4 O α = 6 O..5.5.5 Fig:.: Velocity profiles due to cooling of impulsively started plate for different values of α.
56 A. Begum, Md. Abdul Maleque, M. Ferdows, M. Ota. Gr =,Pr =.7, α = f.8.6.4 ν =.5 ν =. ν =...5.5.5 Fig:.: Velocity profiles due to cooling of impulsively started plate for different values ofν.. Pr =.7, ν ο =.5, α = o f.8.6.4 6 5 4 Gr = -5 Gr = - Gr = -5 4 Gr = 5 5 Gr = 6 Gr = 5. -. -.4.5.5.5 Fig:.: Velocity profiles due to cooling and heating of impulsively started plate for different values of Gr.
Pressure effects on unsteady free convection 57.4. Pr =.7, α = ο, ν ο =.5 f.8.6.4. Gr = 5 Gr = Gr = 5.5.5.5 Fig:.4: Velocity profiles due to cooling of impulsively started plate for different values of Gr..8 Gr =, α = ο, ν ο =.5.6 Pr =.7 Pr =. Pr = 7..4..5.5 Fig:.5: Temperature profiles for impulsively started plate for different values of Pr.
58 A. Begum, Md. Abdul Maleque, M. Ferdows, M. Ota p 8 6 4 Pr =.7, ν ο =.5, α = o Gr = -5 Gr = - Gr = -5 4 Gr = 5 5 Gr = 6 Gr = 5 - -4-6 4 5 6-8.5.5 Fig:.6: Pressure distribution profiles for impulsively started plate for different values of Gr. Case II: Figures (.7-.) show the variation of the velocity profiles due to accelerated motion of the plate when it is cooled and heated by natural convection currents respectively. Figure (.) shows the temperature profiles. We observe from the figures that the effects of various parameters on velocity and temperature are similar to those of the impulsively started plate. Thus the discussion of the results in this case is not produced for brevity.
Pressure effects on unsteady free convection 59.8 Pr =.7, Gr =, ν ο =.5, m =. f.6.4 α = o α = 4 o α = 6 o..5.5 Fig:.7: Velocity profiles due to cooling of uniformly accelerated plate for different values of α..8 Pr =.7, m =., ν ο =.5, α = ο f.6.4. Gr = -5 Gr = - Gr = -5 -..5.5 Fig:.8: Velocity profiles due to heating of uniformly accelerated plate for different values of Gr.
6 A. Begum, Md. Abdul Maleque, M. Ferdows, M. Ota.8 Gr =, Pr =.7, α = ο, ν ο =.5 f.6.4 m =. m =. m =...5.5 Fig:.9: Velocity profiles due to cooling of uniformly accelerated plate for different values of m..8 Gr = -5, pr =.7, m =., α = o.6 curves ν ο =.5 f.4. ν ο =. ν ο =. -..5.5 Fig:.: Velocity profiles due to heating of uniformly accelerated plate for different values of v.
Pressure effects on unsteady free convection 6.8 Gr =, m =., α = ο, ν ο =.5 θ.6.4 Pr =.7 Pr =. Pr = 7...5.5 Fig:.: Temperature profiles for uniformly accelerated plate for different values of Pr. 6 4 Pr =.7, m =., ν ο =.5, α = o Gr = -5 Gr = - Gr = -5 4 GR = 5 5 Gr = 6 Gr = 5 p - 5 4-4 6-6.5.5 Fig:.: Pressure distribution profiles for uniformly accelerated plate for different values of α.
6 A. Begum, Md. Abdul Maleque, M. Ferdows, M. Ota Finally in Tables & numerical values of the skin friction and Nusselt / / Number respectively proportional to f () and θ () are given for impulsive as well as uniformly accelerated motion of the plate. In Table, it appears that the skin friction coefficients increase with the increase of α and Gr but decreases with the increase of ν and Pr. On the other hand, Nusselt number decreases with the increase of ν and Pr. The Table Indicates that the skin friction coefficients increase with the increase of α,gr, m but decreases with the increase of ν and Pr. The coefficient of Nusselt number increases with the increase of m, ν and Pr. We see from both the table that the wall shear stress has a larger effect in case of impulsively started plate as compared to the uniformly accelerated plate. Table Numerical values of skin friction coefficient, τ i and nusselt number N ui for impulsively started plate Gr Pr α v - 5-5 5.7 7..7.7.7.7.7 4 6.5.5.5.5..5.5 τ i.658 -.87.96 -.565 -.869 -.5.8 N ui.5 5.55.8..4594.99.5
Pressure effects on unsteady free convection 6 Table Numerical values of skin friction coefficient, τ a and nusselt number uniformly accelerated plate N ua for Gr Pr m α v - 7..7.7.7.7.7...... 4.5.5.5.5..5 τ a -.855 -.76 -.7969 -.976 -.5674-4.988 N ua 8..89.997.95.9.96 Conclusion Unsteady free convection boundary layer flow over a heated plate with different inclinations has been studied. The present work is Time dependent free convection analysis over an inclined heated plate. The mentioned parameters the velocity profiles are represented graphically for both the cooling and heating of impulsively started plate. When Gr>,there is a rise in the velocity profiles due to an increase in α and an increase in the suction parameter ν causes a fall in the velocity fields. When Gr positive and negative the velocity profiles due to an increase.on the other hand the variation of the velocity profiles due to accelerated motion of the plate when it is cooled and heated by natural convection currents respectively. The temperature profiles in case of Gr> for a comparison in different Prandtl number Pr. There is a decrease in temperature due to increase which is large in case of water (Pr = 7.). The pressure distribution profiles due to increase for the case Gr>, on the other hand, a reverse phenomenon occurs in case Gr>. The skin friction coefficients increase with the increase of α and Gr but decreases with the increase of ν and Pr and Nusselt number decreases with the increase of ν and Pr.
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