Unsteady free MHD convection flow past a vertical porous plate in slip-flow regime under fluctuating thermal and mass diffusion *

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Michael Hardy
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2 Unstead free MHD convection flow past a vertical porous plate in slip-flow regime under fluctuating thermal and mass diffusion * 4. Nomenclature A : Suction parameter C : Dimensionless species concentration C P : Specific heat at constant pressure C : Species concentration C : Concentration at the wall w C : Concentration in free stream D : Molecular diffusivit of the species g : Gravit Gc : Modified Grashof number Gr : Grashof number h : Rarefaction parameter k : Thermal conductivit K : Dimensionless permeabilit parameter K : Permeabilit parameter L : Constant M : Magnetic intensit Pr : Prandtl number q : Heat flux at the wall w Sc : Schmidt number t : Dimensionless time t : Time T : Temperature T : Temperature of wall w * Accepted for publication in Int. Jnl. of Advances in Theoretical and Applied Mathematics. 9

3 T : Temperature of fluid in free stream u : Dimensionless velocit component u : Velocit component V : Suction velocit V : Constant mean suction velocit Greek smbols α : Thermal diffusivit β : Coefficient of thermal expansion β : Coefficient of thermal expansion with concentration ε : Amplitude (<<) μ : Viscosit ν : Kinematic viscosit θ : Dimensionless temperature ρ : Densit σ : Stefan-Boltzmann constant τ : Dimensionless shearing stress τ : Shearing stress ω : Dimensionless frequenc ω : Frequenc 4. Introduction: The phenomenon of free convection arises in the fluid when temperature changes cause densit variation leading to buoanc forces acting on the fluid elements. This can be seen in our everda life in the atmospheric flow which is driven b temperature differences. Free convective flow past vertical plate has been studied extensivel and more intensivel b Ostrach [63] [64] [65]. The transient free convection from a vertical flat plate has been examined b Siegel [66] while the free convective heat transfer on a vertical semi-infinite plate has been investigated b Berezovsk et al. [67]. The laminar free convection from a vertical plate has been studied b Martnenko et al. [68]. 9

4 In all their papers the plate was assumed to be maintained at a constant temperature which is also the temperature of the surrounding stationar fluid. However in man applications that occur in industrial situations and at high temperatures quite often the plate temperature starts oscillating about a non-zero mean temperature. The free-convection flow is enhanced b superimposing oscillating temperature on the mean plate temperature. In situations demanding efficient transfer of heat and mass transfer the transient free convective flow occurs as such a flow acts as a cooling device. Further free convection is of interest in the earl stages of melting adjacent to a heated surface. There occurs several industrial and atmospheric applications where the transport processes occurring in nature due to temperature and chemical differences. Heat transfer in porous media is seen both in natural phenomena and in engineered processes. It is replete with the features that are influences of the thermal properties and volume fractions of the materials involved. These features are seen of course as responses to the causes that force the process into action. For instance man biological materials whose outermost skin is porous and pervious saturated or semi saturated with fluids give out and take in heat from their surroundings. The process of heat and mass transfer is encountered in aeronautics fluid fuel nuclear reactor chemical process industries and man engineering applications in which the fluid is the working medium. The applications are often found in situation such as fiber and granules insulation geothermal sstems in the heating and cooling chamber fossil fuel combustion energ processes and astrophsical flows. Further the magneto convection plas an important role in the control of mountain iron flow in the stead industrial liquid metal cooling in nuclear reactors and magnetic separation of molecular semi conducting materials. A classical example using the nuclear power stations is separation of Uranium U 35 from U 38 b gases diffusion. When mass transfer takes the place in a fluid rest the mass is transformed purel b molecular diffusion a result identified from concentration gradient. The wide range of its technological and industrial applications has stimulated considerable amount of interest in the stud of heat and mass transfer in convection flows. The natural convection flows adjacent to both vertical and horizontal surface which result from the combined buoanc effects of thermal and mass diffusion was first investigated b Gebhart and Pera [69] and Pera and Gebhart [7] while Soundalgekar [7] investigated the situation of unstead free convective flows 93

5 wherein the effects of viscous dissipation on the flow past an infinite vertical porous plate was highlighted. In the course of analsis it was assumed that the plate temperature oscillates in such a wa that its amplitude is small. Later Chen et al [7] studied the combined effect of buoanc forces from thermal and mass diffusion on forced convection. Subsequentl Ramanaiah and Malarvizhi [73] investigated the free convection on a horizontal plate in a saturated porous medium with prescribed heat transfer coefficient. Vighnesam and Soundalgekar [74] investigated the combined free and forced convection flow of water from a vertical plate with variable temperature. The transient free convection flow past an infinite vertical plate with periodic temperature variation was studied b Das et al [75]. In recent times Hossain et al. [76] studied the influence of fluctuating surface temperature and concentration on natural convection flow from a vertical flat plate. The present analsis discussed herein is based on the stud as referred and suggested b Soundalgekar and Wavre [77] [78]. In man practical applications the particle adjacent to a solid surface no longer takes the velocit of the surface. The particles at the surface have a finite tangential velocit; it "slips" along the surface. The flow regime is called the slip-flow regime and this effect cannot be neglected. Under the assumptions made b Sharma and Chaudhar [79] and Sharma and Sharma [8] have also discussed the free convection flow past a vertical plate in slip-flow regime. The had quoted several applications that occur in several engineering applications wherein heat and mass transfer occurs at high degree of temperature differences. Therefore in the fitness of the industrial and scientific applications and to be more realistic the effect of periodic heat and mass transfer on unstead free convection flow past a vertical flat porous plate under the influence of applied transverse magnetic field has been examined. The slip flow regime it is assumed that the suction velocit oscillates in time about a non-zero constant mean because in actual practice temperature species concentration and suction velocit ma not alwas be uniform. 4.3 Mathematical formulation: An unstead free convective flow of a viscous incompressible fluid past an infinite vertical porous flat plate in slip-flow regime is examined in this chapter. 94

6 Additionall a periodic temperature and concentration when variable suction velocit * * i t distribution V V ( ε Ae ) ω = + fluctuating with respect to time is considered. A rectamgular Cartesian co-ordinate sstem with wall ling verticall in x - plane is emploed. The x - axis is taken in verticall upward direction along the vertical porous plate and - axis is taken normal to the plate. Figure 4.: Schematic representation of the problem Since the plate is considered infinite in the x - direction hence all phsical quantities will be independent of x. Under these assumption the phsical variables are function of and t onl. Then neglecting viscous dissipation and assuming variation of densit in the bod force term (Boussinesq's approximation) the problem can be governed b the following set of equations: i t ( ) ( ) ( ) u ω u u σβ ν t ρ K * V + εae = gβ T T + gβ C C + ν u u iω t ( ε ) ρcp V Ae k t T * T T + = iω t ( ε ) t C * C C V + Ae = D (4.) (4.) (4.3) 95

7 The boundar conditions of the problem are: u = = + = + at = as * iω t iω t u L T Tw ε( Tw T ) e C Cw ε( Cw C ) e * u T T C C (4.4) We now introduce the following non-dimensional quantities into equations (4.) to (4.4 ) V tv u 4νω T T C C = t = u = = = C =. ω θ ν 4ν V V Tw T Cw C Gr = ( w ) gβν T T V 3 Gc = ( w ) o gβν C C V 3 μcp νρc Pr = = k k p ν Sc = D M σβ ν = ρv KV K = ν VL h = ν All phsical variables are defined in nomenclature where ( ) indicates the dimensional quantities. The subscript( ) denotes the free stream condition. Then equations (4.) to (4.3) reduce to the following non-dimensional form: 4 u iωt u u u ( + ε Ae ) = Grθ + Gc C + Mu t K iωt ( + ε Ae ) = 4 t Pr θ θ θ iωt ( + ε Ae ) = 4 t Sc C C C (4.5) (4.6) (4.7) The boundar conditions to the problem in the dimensionless form are: 96

8 u iωt iωt u = h θ = + εe C = + εe at = u θ C as (4.8) 4.4 Method of Solution: Assuming the small amplitude oscillations ( ε << ) we can represent the velocit u temperatureθ and concentration C near the plate as follows: ( ) ( ) ε ( ) u t u u e ω i t = + (4.9) ( ) ( ) ( ) θ t θ εθ e ω i t = + (4.) ( ) ( ) ε ( ) C t C C e ω i t = + (4.) Substituting (4.9) to (4.) in (4.5) to (4.7) equating the coefficients of harmonic and non harmonic terms neglecting the coefficients ofε we get: u + u M + u = Grθ GcC K (4.) iω u + u M + + u = Grθ GcC Au K 4 (4.3) θ + Prθ = (4.4) iω Prθ θ + Prθ = A Prθ (4.5) 4 C + = (4.6) ScC iω ScC C + ScC = AScC (4.7) 4 The corresponding boundar conditions reduce to: 97

9 u u u = h u = h θ = θ = C = C = at = (4.8) u = u = θ = θ = C = C = as where primes denote differentiation with respect to. Solving the equations (4.) to (4.7) under the boundar conditions (4.8) we get: θ Pr ( ) e = (4.9) C ( ) = e Sc (4.) ( ) Pr u = me + me + me (4.) θ m Sc 3 4 m 6 Pr ( ) m e m e = (4.) 9 ( ) C me m e m Sc = 8 + (4.3) m 6 9 ( ) m Pr u me me me me me me = (4.4) m m Sc where M M K = + m = m ( + h ) + m ( + hsc) Pr hm m + + 4M = 3 m = Pr Gr Pr M m Gc = Sc Sc M 4 m 5 4iA Pr = ω m 6 Pr+ Pr + iω Pr = m7 = 4iAPr ω m 8 4iASc = ω m Sc + Sc + iωsc 9 = m = 4iASc ω m = m ( + hm ) + m ( + hm ) m ( + hm ) m ( + hpr) m ( + hsc) hm 98

10 m + + 4M + iω = m mgr = m m M + iω ( /4) m m mgc = m m M iω mm A 5 ( + /4) m m ( M + iω /4) ma 3 Pr mgr 7 = Pr Pr / 4 m = 7 ( M + iω ) Sc Sc ( M + iω ) m = masc mgc 4 /4 The important characteristics of the problem are the skin-friction and heat transfer at the plate. Skin-friction: The dimensionless shearing stress on the surface of a bod due to a fluid motion is known as skin-friction and is defined b the Newton's law of viscosit. τ u = μ (4.5) Substituting equations (4.) and (4.4) into (4.9) we can calculate the shearing stress component in dimensionless form as τ u τ = = ρv = ( Pr ) = mm m Pr msc+ εe m m + mm + mm mm m m Sc iωt (4.6) 99

11 4.5 Results and conclusions: 4.5. The effect of Grashof number on velocit profiles is illustrated in figure 4.. It is noticed that as Grashof number increases the velocit of the fluid medium increases. Further it is noticed in the boundar laer region the velocit increases and thereafter it decreases. Also far awa from the plate not much of significant effect of Grashof number is noticed..5 VELOCITY.5 Gr =. Gr =. Gr = 3. Gr = 5. Gc =. Pr =.7 Sc =.6 M =. K =.6 A =. h =. ω =. ε =. t = Figure 4.: Effect of Grashof number on velocit profiles

12 4.5. The variation of velocit with respect to modified Grashof number is noticed in figure 4.3. It is observed that as modified Grashof number increases in general the velocit increases. As was seen in earlier case the rise in velocit is noticed in the boundar laer region and thereafter it decreases. VELOCITY Gc =. Gc =. Gc = 3. Gc = 5. Gr =. Pr =.7 Sc =.6 M =. K =.6 A =. h =. ω =. ε =. t = Figure 4.3: Effect of modified Grashof number on velocit profiles

13 4.5.3 The contribution of Schmidt number on the velocit profiles is observed in figure 4.4. For a constant Schmidt number the velocit increases initiall in the boundar laer region and thereafter it decreases. In general it can be seen that increase in the Schmidt number causes the velocit of the fluid medium to decrease..4 VELOCITY..8.6 Sc =.4 Sc =.6 Sc =.94 Sc =.5 Gr =. Pr =.7 Gc =. M =. K =.6 A =. h =. ω =. ε =. t = Figure 4.4: Effect of Schmidt number on velocit profiles

14 4.5.4 The influence of magnetic field on the velocit field is illustrated graphicall in figure 4.5. It is observed that as the magnetic intensit is increasing the fluid velocit is found to be decreasing. Therefore it can be concluded that the magnetic field suppresses the motion of the fluid to a great extent. VELOCITY M =. M =.4 M =.9 M =.6 Gr =. Pr =.7 Gc =. Sc =.6 K =.6 A =. h =. ω =. ε =. t = Figure 4.5: Effect of magnetic field on velocit 3

15 4.5.5 The contribution of the porosit of the fluid bed over the velocit is observed in figure 4.6. It is observed that as the pore size of the fluid bed increases the fluid velocit increases. Such a phenomena is due to the percolation of fluid through such pores resulting in the increase of fluid velocit..6 VELOCITY K =. K =.3 K =.6 K =. Gr =. Pr =.7 Gc =. Sc =.6 M =. A =. h =. ω =. ε =. t = Figure 4.6: Effect of permeabilit parameter on velocit 4

16 4.5.6 Variation in the velocit of the fluid medium with respect to suction parameter is illustrated in figure 4.7. It is noticed that as the suction parameter is increased the velocit of the fluid medium is found to be decreasing. Further as we move far awa from the plate the contribution of such suction parameter is not that significant as expected. VELOCITY A =. A =. A = 3. A = 4. Gr =. Pr =.7 Gc =. Sc =.6 M =. K =.6 h =. ω =. ε =. t = Figure 4.7: Effect of suction parameter on velocit 5

17 4.5.7 The influence of rarefaction parameter on the velocit field is illustrated in figure 4.8. It is seen that as the parameter is increased the velocit of the fluid medium is found to be increasing. An interesting observation is that on the boundar the dispersion is found to be more distinguishable while we move far awa from the plate the effect seems to be consolidated and not much of variation is noticed.. h =. h =. h =.4 h =. VELOCITY Gr =. Pr =.7 Gc =. Sc =.6 M =. K =.6 A =. ω =. ε =. t = Figure 4.8: Effect of rarefaction parameter on velocit 6

18 4.5.8 The influence of Prandtl number on the temperature field is illustrated in figure 4.9. It is observed that the prandtl number has significant contribution over the temperature field. From the illustrations it is seen that as the Prandtl number increases the temperature decreases. Further not much of significant contribution b the Prandtl number is noticed as we move far awa from the plate. TEMPERATURE Pr =.4 Pr =.5 Pr =.7 Pr =. Gr =. Gc =. Sc =.6 M =. K =.6 A =. h =. ω =. ε =. t = Figure 4.9: Effect of Prandtl number on temperature profiles 7

19 4.5.9 The influence of suction parameter on the temperature field is illustrated in figure 4.. It is seen that as the suction parameter increases the temperature field decreases. Further far awa from the plate it is noticed that the parameter has no effect at all.. TEMPERATURE A =. A =. A = 3.5 A = 5. Gr =. Gc =. Pr =.7 Sc =.6 M =. K =.6 h =. ω =. ε =. t = Figure 4.: Effect of suction parameter on temperature 8

20 4.5. The influence of Schmidt number on concentration is illustrated in figure 4.. It is noticed that as Schmidt number increases the fluid concentration decreases. Such a decrease is found to be in the exponential order. It is observed that as Schmidt number increases the concentration field profiles are found to be more parabolic in nature. CONCENTRATION Sc =.4 Sc =.6 Sc =.94 Sc =.5 Gr =. Gc =. Pr =.7 A =. M =. K =.6 h =. ω =. ε =. t = Figure 4.: Effect of Schmidt number on concentration 9

21 4.5. The influence of suction parameter on the concentraion of the fluid medium is shown in figure 4.. It is observed that as the suction parameter increases the concentration of the fluid medium decreases. Further as we move far awa from the plate it is observed that the effect of such suction parameter to be diminishing rapidl. CONCENTRATION A =. A =. A = 3.5 A = 5. Gr =. Gc =. Pr =.7 Sc =.6 M =. K =.6 h =. ω =. ε =. t = Figure 4.: Effect of suction parameter on concentration

22 4.5. The combined influence of magnetic intensit and Grashof number over the skin-friction is noticed in figure 4.3. It is noticed that for a fixed magnetic intensit as Grashof number increases the skin-friction is found to be increasing. Further when Grashof number is varied it is seen that the skinfriction decreases as the magnetic intensit increases. SKINFRICTION.5.5 Gr =. Gr =. Gr = 3. Gr = 5. Gc =. Pr =.7 Sc =.6 K =.6 A =. h =. ω =. ε =. t = Magnetic Intensit(M) Figure 4.3: Effect of Grashof number on skin-friction

23 4.5.3 The consolidated effected of Schmidt number with respect to the magnetic intensit on skin-friction is observed in figure 4.4. It is noticed that in general as Schmidt number increases the skin-friction decreases. Further as the magnetic influence is increased over the sstem the skin-friction decreases. Also it is noticed that the influence of Schmidt number at higher values of the magnetic intensit reduces drasticall..6 SKINFRICTION.4..8 Sc =. Sc =.5 Sc =. Sc = 3. Gr =. Pr =.7 Gc =. K =.6 A =. h =. ω =. ε =. t = Magnetic Intensit(M) Figure 4.4: Effect of Schmidt number on skin-friction

24 4.5.4 The combined influence of Schmidt number with respect to the rarefaction parameter is illustrated in figure 4.5. It is observed that as Schmidt number increases the skin-friction reduces with respect to the rarefaction parameter. Also it is noticed that both the parameters do not exhibit an significant influence on skin-friction at higher values. 4 SKINFRICTION Sc =. Sc =.5 Sc =. Sc = 3. Gr =. Pr =.7 Gc =. K =.6 A =. M =. ω =. ε =. t = Rarefaction Parameter (h) Figure 4.5: Effect of Schmidt number on skin-friction 3

25 4.5.5 The influence of Grashof number on skin-friction is noticed in figure 4.6. It is observed that as Grashof number increases the skin-friction is found to be increasing with respect to the rarefaction parameter. SKINFRICTION Gr =. Gr =. Gr = 3. Gr = 5. Sc =.6 Pr =.7 Gc =. K =.6 A =. M =. ω =. ε =. t = Rarefaction Parameter (h) Figure 4.6: Effect of Grashof number on skin-friction 4

26 4.5.6 The combined influence of Grashof number and suction parameter on skinfriction is shown in figure 4.7. In general it is observed that for a constant suction parameter as Grashof number increases the skin-friction is found to be increasing. Interestingl the relation is found to be nearl linear. 7 SKINFRICTION Gr =. Gr =. Gr = 3. Gr = 5. Sc =.6 Pr =.7 Gc =. K =.6 h =. M =. ω =. ε =. t = Suction Parameter(A) Figure 4.7: Effect of Grashof number on skin-friction 5

27 4.5.7 The influences of suction parameter and Schmidt number on skin-friction are noticed in figure 4.8. It is seen that as Schmidt number increases for a constant suction parameter the skin-friction decreases. Further the relation is found to inverse and is linear..4. Sc =.3 Sc =.5 Sc =. Sc = 3. Gr =. Pr =.7 Gc =. K =.6 h =. M =. ω =. ε =. t =. SKINFRICTION Suction Parameter(A) Figure 4.8: Effect of Schmidt number on skin-friction **** 6

Transient free convective MHD flow through porous medium in slip flow regime

IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 11, Issue 5 Ver. 5 (Sep. - Oct. 2015), PP 52-58 www.iosrjournals.org Transient free convective MHD flow through porous