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NUMERICAL STUDY OF FORCED CONVECTIVE HEAT GENERATION FLOW THROUGH A PERMEABLE WALLS WITH

SUCTION/INJECTION

1_{Ajala O. A. ,}2_{Obalalu A. M. ,} 3_{Abimbade S. F. and} 4_{Akinyemi O. T.}

1_{Department of Pure and Applied Mathematics, Ladoke Akintola University of}

Technology, Ogbomoso, Nigeria.

2,3,4_{Department of Statistics and Mathematical Sciences, Kwara State University,}

Malete, Kwara State, Nigeria.

1_{Corresponding Author E-mail: [email protected]}

Abstract

In this paper, forced convective heat generation in a steady flow of an incompressible viscous fluid through a channel permeable walls was studied. The nonlinear governing equations along with boundary are converted into or-dinary differential equations using appropriate similarity transformations. The nonlinear governing equations were solved numerically using weighted residual method and the residuals were minimized using the collocation method and computed using mathematica software for various values of physical parame-ters. The effect of various flow parameters on velocity and temperature profile are graphically discussed.

Keywords: Heat Generation, Forced Convective, Weighted Residual Method, Collocation Method.

1

a channel partially pulled with porous medium was studies by [4]. [5] investigated the radiation effect of heat generation and viscous dissipation on MHD free convec-tion flow along a stretching sheet. In [6-8], it was discovered that the layer value of buoyancy parameter can be used to control the temperature and concentration boundary layer and that suction stabilizes the boundary layer. Researchers have discussed heat and mass transfer fluid under various physical situations.The com-bined effect of free and forced convection on MHD flow in porous channel under the uniform magnetic field was investigated and the solutions were obtained for all heat absorption condition and restricted heat generation condition[9]. The study of hy-dromagnetic natural convective fluid flow between vertical parallel plates with time periodic boundary conditions using Adomian decomposition method was analyzed and the results of their computation shows that an increase in the magnetic field intensity has significantly influenced the fluid flow where the effect of heat genera-tion and sucgenera-tion are neglected [10]. Efficient energy utilizagenera-tion during the convecgenera-tion in fluid is one of the fundamental problem of engineering processes to improve the system, several researchers have theoretically studied heat generation in the flow systems under many physical situations, see [11-13]. However, no attempt has yet been presented for effect of variable heat generation on forced convective with surface boundary condition. Hence, in this study, the effect of heat generation on forced con-vective fluid flow through a channel with permeable walls was investigated and the model equations were obtained numerically using collocation method and the results were presented graphically to analyze the effect of various parameter on velocity and temperature profile.

2

V du

dy =−

1 ρ dP dx + 1 ρ d

dy µ(T) du dy ! − σB 2 0u ρ (1) V dT dy =α

d2_T

dy2 +

µ(T)

ρcp du dy !2 + σB 2 0u2

ρcp

+ Q0

ρcp

(T −Tf) (2)

The boundary conditions are

u(0) = 0, u(h) = 0,

−kdT

dy(0) =γ0(Tf −T(0)), −k dT

dy(0) =γ1(T(h)−T∞)

(3)

(x, y) is the axial and normal coordinates, u is the velocity of the fluid, p is the fluid

pressure, v is the uniform suction/injection velocity at the channel walls, γ0 is the heat transfer coefficient at the lower plate,γ1 is the heat transfer coefficient at the upper plate,

α is the thermal diffusivity, ρ is the fluid density,σ is the fluid electrical conductivity, k

is the thermal conductivity coefficient,cp is the specific heat at constant pressure, where

(G) is the pressure gradient parameter,Tf is the temperature of the hot fluid at the lower

permeable plate, T is the channel fluid temperature and T∞ is the ambient temperature

above the upper plate. The temperature dependent viscosityµ can be written as [14,17].

µ(T) =µ0`−m(T−T∞) (4)

where m is a viscosity variation parameter and µ0 is the fluid dynamic viscosity at the ambient temperature. Thus, the following non-dimensional quantities were introduced:

G= ∂p

∂x, µ= µ µ0

, α= k

ρcp

, µ(T −T∞) =, θ= (Tf−T∞)

Q= Q0 ρcP(

T −Tf), W =

u

v, X= x

h, P = ph µ0v

θ= T−T∞ Tf −T∞

, , P = P µ0v

h , η =

v

h, V = µ0v

ρ

(5)

Substituting equation (5) into (1)-(4) we obtain

d2w dη2 −

dθ dη

dw dη −`

θ_Redw

dη +Haw−G

= 0 (6)

d2_θ

dη2 −ReP r

dθ

dη +EcP r`

−θdw

dη 2

dθ

dη(0) =Bi0(θ(0)−1) and dθ

dη(0) =Bi0(1) (9)

Re= vh

v (Reynold−number) P r= v

α(P randtl−number)

Ec= v

2

cp(

Tf −T∞)(Eckert−number) Q= Q0

ρcP(

T−Tf)(Heat−Generation)

Ha= σB

2 0H2

µu (Hartmann−number) =m(T−T∞)(V iscosity−parameter)

Bi0=

yoh

k (Biot−numberf orlower−plate) Bi1 = y1h

k (Biot−numberf orupper−plate)

3

Collocation Weighted Residual

The system of coupled non-linear ordinary differential equations (6) and (7) together with the boundary conditions (8) and (9) was solved numerically using Collocation Weighted Residual Method.

w=

10

X

K=0

akηk and θ=

10

X

K=0

bkηk (10)

Which can be interpreted as

w(η) =a0+a1η+η2a2+η3a3+a4η4+a5η5+a6η6+a7η7+a8η8+a9η9+a10η10 (11)

θ(η) =b0+b1η+η2b2+b3η3+b4η4+b5η5+b6η6+b7η7+b8η8+b9η9+b10η10 (12) Substituting equation (6) into (10), we obtain

w(η) = 2a2+ 6a3η+ 12a4η2+ 20a5η3+ 30a6η4+ 42a7η5+ 56a8η6+ 72a9η7+ 90a10η8−

` b0+b1η+η2b2+b3η3+b4η4+b5η5+b6η6+b7η7+b8η8+b9η9+b10η10

(a0+a1+a1η+ 2a2η+a2η2

+ 3a3η2+a3η3+ 4a4η3+a4η4+ 5a5η4+a5η5+ 6a6η5+a6η6+ 7a7η6+a7η7+ 8η7a8+η8a8+ 9η8+η9a9+ 10η9a10+η10a10)−(a1+ 2a2η+ 3a3η2+ 4a4η3+ 5a5η4+ 6a6η5+ 7a7η6+ 8a8η7+ 9a9η8+ 10a10η9)(b1+ 2b2η+ 3b3η2+ 4b4η3+ 5b5η4+ 6b6η5+ 7b7η6+ 8b8η7+ 9b9η8+ 10b10η9)

Substituting equation (7) into (11), we have

θ(η) =2b2+ 6ηb3+ 12η2b4+ 20η3b5+ 30η4b6+ 42η5b7+ 56η6b8+ 72η7b9+ 90η8b10

−

b1+ 2b2η+ 3b3η24b4η3+ 5b5η4+ 6b6η5+ 7b7η6+ 8b8η7+ 8b8η7+ 9b9η8+ 10b10η9

+

` b0+b1η+η2b2+b3η3+b4η4+b5η5+b6η6+b7η7+b8η8+b9η9+b10η10

a1+ 2a2η+ 3a3η2+ 4a4η3+ 5a5η4+ 6a6η5+ 7a7η6+ 8a8η7+ 9a9η8+ 10a10η9

2₊

a1η+η2a2+η3a3+a4η4+a5η5+a6η6+a7η7+a8η8+a9η9+a10η10

2

+

(b0+b1η+η2b2+b3η3+b4η4+b5η5+b6η6+b7η7+b8η8+b9η9+b10η10) (14) Using the boundary conditions,

w(0) = 0⇒a0 = 0 (15)

w(0) =a0+a1+a2+a3+a4+a5+a6+a7+a8+a9+a10= 0 (16)

θ(0) =−b0+b1+ 1 = 0 (17)

θ(1) =b0+ 2b1+ 3b2+ 4b3+ 5b4+ 6b5+ 7b6+ 8b7+ 9b8+ 10b9+ 11b10= 0 (18)

4

0.0 0.2 0.4 0.6 0.8 1.0

0.00 0.02 0.04 0.06 0.08 0.10 0.12

η

w

(

η

)

Ha=0.1 Ha=1

Ha=2 Ha=3

Fig. 1: velocity profile for different values ofHawhen

Re= 0.1,P r=0.7,Ec= 0.1,= 0.1,Q= 0.1,Bi0 =

0.1,Bi1= 0.1

0.0 0.2 0.4 0.6 0.8 1.0

0.1 0.2 0.3 0.4 0.5 0.6

η

θ

(

η

)

Ha=0.1 Ha=1

Ha=2 Ha=3

Fig. 2: Temperature profile for different values of

HawhenRe= 0.1,P r= 0.7,Ec= 0.1,= 0.1,Q=

0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 η W ( η )

Re=0.1 Re=1 Re=2 Re=3

Fig. 3: velocity profile for different values ofRewhen

Ha= 1,P r= 0.7,Ec= 0.1,= 0.1,Q= 0.1,Bi0 =

0.1,Bi0= 0.1

0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.6 0.7 0.8 0.9 η θ ( η )

Re=0 Re=1 Re=2 Re=3

Fig. 4: Temperature profile for different values ofRe

when Ha = 0.1,P r = 0.7,Ec = 0.1, = 0.1,Q =

0.1,Bi0= 0.1,Bi0= 0.1

0.0 0.2 0.4 0.6 0.8 1.0

0.00 0.05 0.10 0.15 η w ( η )

Bi0=0.1 Bi0=0.2 Bi0=0.5 Bi0=1

Fig. 5: velocity profile for different values ofBi0when P r= 0.7,Ec= 0.1,= 0.1,Q= 0.1,Ha= 0.1,Bi0=

0.1,Re= 1

0.0 0.2 0.4 0.6 0.8 1.0 0.55 0.60 0.65 0.70 0.75 0.80 η θ ( η )

Bi0=0.1 Bi0=0.2

Bi0=0.5 Bi0=1

Fig. 6: Temperature profile for different values ofBi0

when Ha = 0.1,P r = 0.7,Ec = 0.1, = 0.1,Q =

0.1,Bi0= 0.1,Re= 0.1

0.0 0.2 0.4 0.6 0.8 1.0

0.00 0.02 0.04 0.06 0.08 0.10 0.12 η w ( η )

Bi1=0.1 Bi1=0.2

Bi1=0.5 Bi1=1

Fig. 7: velocity profile for different values ofBi1when

Re= 1,P r= 0.7,Ec= 0.1,= 0.1,Q= 0.1,Bi0 =

0.1,Ha= 0.1

0.0 0.2 0.4 0.6 0.8 1.0

0.1 0.2 0.3 0.4 0.5 0.6 η θ ( η )

Bi1=0.1 Bi1=0.2

Bi1=0.5 Bi1=1

Fig. 8: Temperature profile for different values ofBi1

when Ha = 0.1,P r = 0.7,Ec = 0.1, = 0.1,Q =

0.0 0.2 0.4 0.6 0.8 1.0 0.00

0.05 0.10 0.15 0.20 0.25

η

W

(

η

)

Q=0 Q=0.25

Q=0.5 Q=1.0

Fig. 9: velocity profile for different values ofQ=

when Re= 1,P r = 0.7,Ec= 0.1, = 0.1, Bi1 =

0.1,Bi0= 0.1,Ha= 0.1

0.0 0.2 0.4 0.6 0.8 1.0 0.74

0.75 0.76 0.77 0.78 0.79 0.80 0.81

η

θ

(

η

)

Q=0 Q=.0.25 Q=0.5 Q=1.0

Fig. 10:Temperature profile for different values ofQ=

whenHa= 0.1,P r= 0.7,Ec= 0.1,= 0.1,Bi1 =

0.1,Bi0= 0.1,Ha= 0.1

0.0 0.2 0.4 0.6 0.8 1.0

-0.000010

-5.×10-6

0.000000

5.×10-6

0.000010 0.000015 0.000020

η

R

(

η

)

Fig 11: Minimized residual error (R(η))

Fig. (1) shows the effect of Hartmann number (Ha) on velocity profile and we observed

that decrease in Hartmann decreases the velocity profile and it serves as electromagnetic force to the viscous force. The presence of Lorentz force also act as resistance to flow. Fig. (2)shows the effect of magnetic (Ha) on the flow fluid, Increase in values of (Ha) decreases

suction Reynolds number (Re) increasing the fluid viscosity becomes lighter and viscous

heating increase due to increase convective heating at the lower plate increase leading to a rise in fluid in the temperature. Fig. (5) and (7) shows that the effect of (Bi0) rise in the fluid temperature is observed with increasing convective heating at the lower plate, (Bi1) in the fluid temperature decreases due to increase in convective heat loss at the upper plate.Figs. (6) and (8) graphically shows that the (Bi0) increases with increase in connectivity heating at lower plate and (Bi1) decreases with increase in convection cooling at the upper plate, this is expected since the fluid become lighter and flow faster with increasing temperature due to convective heating. Figs. (9) and (10) shows the heat generation due to the viscous heating increasing with the parameter values of heat generation and it is observed that fluid velocity and temperature profiles increase with increasing value of heat generation (Q), hence produces an increase in the heat transfer and flow faster.Fig. (11) shows the graph of the residual functions R(η) and it was

observed that the residuals are minimized in the domain (0to1)

5

This study investigated the effects of heat generation on variable viscosity channel flow with suction/injection together with convective heating/cooling at the walls have been investigated. The nonlinear model problem is tackled numerically using (WRM) collocation method. Based on the results presented above, the following conclusions are deduced.

The impact of different parameter such as effect of different parameter such as Hartman number (Ha), Biot number lower plate (Bi0), Biot number upper plate (Bi1), Reynold number (Re), Heat Generation (Q) are discussed graphically.

• An increase in (Q),(P r),(Bi0),(Ec),() and increases the velocity profiles, while an increase in (Ha) and (Bi1)decreases the velocity profile.

• An increase in (Q),(P r),(Bi0),(Re),(Ec),() and increases the temperature profiles, while an increase in (Ha) and (Bi1) decreases the temperature profile.

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