J. Appl. Environ. Biol. Sci., 8(1)217-224, 2018 | ISSN: 2090-4274 |
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1234,*
Asmat UllahYahya, DanialHabib,SajjadHussain,Shan E. Farooq
1Department of Mathematics National College of Business Administration & Economics Lahore, 2Department of Mathematics Virtual University Lahore, Pakistan,
3Department of Mathematics, Govt. Postgraduate College, Layyah, Pakistan, 4,*Department of Mathematics, University of the Punjab, Lahore, Pakistan.
Received: September 13, 2017 Accepted: December 1, 2017
ABSTRACT
This article addresses the time dependent flow of electrically conducting viscous fluid adjacent to a shrinking and stretching sheet. The flow is considered through porous medium and mixed convection heat transfer is added with thermal radiations. The flow is considered in the presence of applied magnetic field with slip boundary conditions. The mathematical formulation involves second order non-linear partial differential equations which are then correspondingly transformed to ordinary differential form for purpose of numerical solution. The results have been computed with computational technique NDsolve coded in Mathematica. Rigorous computations are meant to study
the influences of existing parameters namely the buoyancy parameter , the stretching/shrinking velocity
parameter, Prandtl number, R radiation parameter, velocity slip parameter, s unsteady parameter, M is
n
magnetic parameter, and f suction parameter. The momentum and thermal characteristics are mapped to represent
w the impacts of the above mentioned parameters on these physical quantities. KEYWORDS: Time Dependent Flow, Slip Flow, MHD Flow, Stretching/Shrinking Sheet, Thermal Radiation
INTRODUCTION
The study of time dependent flow of viscous incompressible fluid past vertical bodies has wide technological and engineering applications. Sukumar, et al. [1] an analysis is made to study the slip effects on MHD flow of Jeffrey fluid over an unsteady shrinking sheet with wall mss transfer. Samad and Rahman, [2] analyzed the effect of radiation on unsteady MHD free convection flow past a vertical porous plate which is immersed in a porous medium. Das et al. [3] analyzed an unsteady free convection flow past a vertical plate with heat and mass fluxes in the presence of thermal radiation by analytical method. Ali et al [4] considered unsteady, viscous, incompressible, electrically conducting blood flow and heat transfer through a parallel plate channel when the lower plate is stretching. Ahmad and Sajjad [5] investigated unsteady blood flow having micropolar fluid properties with heat source through parallel plates channel.
Thermal radiation is key to many fundamental phenomenon surrounding us, from solar radiation to fire incandescent lamp, cooling of towers, gas turbines and various propulsion devices for aircraft, energy utilization, temperature measurements, remote sensing for astronomy, space exploration, and play a major role in combustion and furnace design.
Khan, et al [6] analyzed Non-Newtonian MHD mixed convective power-law fluid flow over a vertical stretching sheet with thermal radiation, heat generation and chemical reaction effects. Reddy et al. [7] studied the thermal radiation and magnetic field effects on unsteady mixed convection flow and mass transfer over a porous stretching surface with heat generation. Seini and Makinde, [8] investigated the effects of heat radiation and first order homogeneous chemical reaction on hydromagnetic boundary layer flow of a viscous, steady, and incompressible fluid over an exponential stretching sheet.
Crane, [9] was the first who analyzed the steady two dimensional flow over a linearly stretching sheet and found the similarity solution in closed analytical form. Sharada and Shankar, [10] dealt mixed convection in MHD stagnation point flow due to stretching surface. Mahathaa et al. [11] suggested the stream of a viscous and incompressible nano fluid over a stretching sheet under the impact of transverse magnetic field. Ali et al. [12] considered magnato hydrodynamic flow of viscous fluid due to a sheet that stretches. Sajjad et al [13] investigated
*CorrespondingAuthor:Shan E Farooq, Department of Mathematics, University of the Punjab, Lahore, Pakistan. Email: shanefarooq@gmail.com
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MHD boundary layer flow and heat transfer for micropolar fluids over a shrinking sheet. Danial [14] analyzed magneto hydrodynamic flow of viscous fluids owing to moving boundary with thermal radiations, porosity of surfaces and medium. Thisworkbeenundertaken to examinetheunsteadyflow aspectof thepreviousstudybyZaimi and Ishsk [15]with extension for thermal radiation and applied magnetic field with an easy computational technique.
MathematicalAnalysis
The time dependent radiative flow problem is formulated mathematically with assumptions as follows: The fluid is incompressible and viscous. The flow is unsteady, two dimensional Cartesian coordinates are being used. The fluid flows towards a stagnation point. The flow is due to stretching or shrinking vertical sheet that is placed in plan y = 0. The flow is confined to y > 0. The fluid velocity is v = v(u, v)as function of t, x, y coordinates. The fluid temperature is T, The equations of flow and heat transfer are given as
u v
0 (1)
x y
22
u v U u g (T T ) (2)
y
2 *32
T T T T 16 T T
u v (3)
t x y y23 Cp y2
The boundary conditions are
u
u cx L( ), v v0, T Tw at y 0 y (4)
u U () , T T , as y
where v0 is suction/injection velocity, where is the buoyancy parameter, ɛ is the stretching parameter, is the Prandtl number and is the velocity slip parameter, The fluid temperature is T, The surface temperature (x) = + , where is free stream temperature and b is positive constant, e electrical conductivity, for
kinematics viscosity, c for specific heat capacity,µ dynamic viscosity coefficient, * is the Stephan-Boltz-man
p
constant, *is Rosseland constant α =k/ρcp is thermal diffusivity, is the magnetic permeability, g is the
0
acceleration of gravity and is the thermal expansion coefficient.
We use similarity function to convert the above mathematical model into ordinary differential form as below:
a
(x, yt ,)
xf ( )
(1 t)
T T
()
TwT
u ; v
yx a
u xf
1 t
J. Appl. Environ. Biol. Sci., 8(1)217-224, 2018
1
v ( a )2 f
1 t The continuity equation (1) is satisfied here identically. By inserting the above relations in to equations (2) and (3), we get
1 (5)
2 4
3
The boundary condition (4) become f (0) fw , f (0) f (0), (0) 1
f () 1, 0 (7)
() as 0 Where
g(1t)2 c(1 t)
(T T ) is the buoyancy parameter , is the stretching/shrinking parameter, Pr
2 w
aax
12
4T 3(1 t) a
= is the Prandtl number, R is a radiation parameter, L is the velocity slip
12
c abx 1 2 (1 t)
∝ n p
2H2
e 00 1 t 12
parameter, s unsteady parameter, M (1t)is magnetic parameter, and f V ()
a a wa
suction parameter.
RESULTSANDDISCUSSION
Mathematical formulation for the unsteady MHD flow of viscous fluid due to stretching / shrinking surface with radiative heat transfer and slip boundary resulted in the form of a set of non linear ordinary differential equation namely Eq.(5) to Eq.(7). This difficult system of equations has been firstly reduced to set of first order differential equations which is coded for ND solve command of mathematical version 11.1 Rigorous effort has been carried out for sufficient ranges of the pertinent parameters involved in the resulting model equations in order to
have a look in to the physical nature of the problem. Results for horizontal velocity f ( )and temperature function
( )have been plotted and presented for some representative values of the influential parameters.
Table 1 indicates that magnitude of f (0) and θˊ(0) reduces with increase in M when λ=1 but f (0)
increases in magnitude and θˊ(0) decreases when = -1. Table 2 depicts that f (0) and θˊ(0) decreases in
magnitude with increase in δ when λ=1 or λ= -1. Table 3 shows that magnitude of f (0) decreases but θˊ(0) increases in magnitude with increase in Pr when λ=1 or λ = -1. The increase in unsteadiness parameters caused increase in magnitude of f (0) and θˊ(0) increase when λ=1 and decrease in f (0) but increase in θˊ(0) for λ= -1
as presented in table 4. The increase in value of parameter Rn shows increase in magnitude of f (0) but decrease in magnitude of θˊ(0)when λ=1 or λ= -1 as shown in table 5. The increase in the values of parameter ε (ɛ >0), decrease the magnitude of f (0) but increase in(0) for λ=1 or λ= -1 as depicted in table 6.
theparameter ɛ(ɛ>0 /ɛ<0)on f ( )is indicated in the Fig.2. The magnitude of
Fig.1 shows that the curve of f ( )rises up with increase in the value of slip parameter δ. The effect of f ( )increases with increase in the magnitude ɛ. Fig.3 demonstrates the pattern of curve for different values of λ (λ < 0). It is seen that
f ()buoyancy opposing phenomena (λ < 0) causes reduction in value of f ( ). The increase in magnetic field strength
causes significant decrease in fluid flow velocity as shown in Fig.4. The increase
f ( )in the values of
unsteadiness parameter s causes increase in flow speed f ( )as increase in Fig.5. Similarly Fig.6 shows that the flow speed increase in magnitude as the increase in the value of suction parameter fw As usual the increase in values of Prandtle number Pr, as demonstrated in Fig.7. Fig.8 has been presented to show the influence of λ on θ(ƞ). The curve of θ(ƞ) rise with increase in λ (λ < 0). Similarly the radiation parameter Rn and ɛ, velocity parameter negative
shows rise in temperature ( )as illustrated respectively in Fig.9 and Fig.10. The increase in suction velocity at
wall causes decreases in θ(ƞ)as demonstrated in Fig.11, for increase in values of fw. The Fig.12 indicated the influences of unsteady parameters ‘s’ on θ(ƞ). It is seen that θ(ƞ) decreases with increasing in ‘s’.
Table 1: Theresult of f (0) and θˊ(0)for different valuesof M with fixed valuesof ε =1, Pr =0.72.
1 | 1 | 1 | 1 | |||
0.1 | 0.3066 | -0.3023 | -1.07603 | -1.010 | ||
0.2 | 0.2513 | -0.3771 | -1.06871 | -1.002 | ||
0.3 | 0.1975 | -0.4288 | -1.06154 | -0.994 | ||
0.4 | 0.1450 | -0.4790 | -1.0545 | -0.987 |
Table2: The result of f (0) and fordifferent values of with fixed valuesof =1, =0.72.
1 | 1 | 1 | 1 | |||
0.1 | 0.3066 | -0.3237 | -1.0160 | -1.01007 | ||
0.2 | 0.2678 | -0.2863 | -1.0816 | -1.0034 | ||
0.3 | 0.2375 | -0.2569 | -1.0859 | -0.9979 | ||
0.4 | 0.2133 | -0.2330 | -1.0893 | -0.9935 |
Table 3: Theresult of f (0) and for different values ofS with fixed valuesof =1, Pr=0.72.
1 | 1 | 1 | 1 | |||
0.1 | 0.3066 | -0.3237 | -1.0760 | -1.0100 | ||
0.2 | 0.3602 | -0.2649 | -1.0988 | -1.0352 | ||
0.3 | 0.4152 | -0.2046 | -1.1215 | -1.0602 | ||
0.4 | 0.4716 | -0.1428 | -1.1442 | -1.0850 |
Table4: The result of and for different valuesof Rn with fixed valuesof =1, =0.72.
Rn | ||||||
---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | |||
0.1 | 0.3066 | -0.3237 | -1.0760 | -1.010 | ||
0.2 | 0.3129 | -0.3304 | -1.0258 | -0.962 | ||
0.3 | 0.3183 | -0.3360 | -0.9838 | -0.923 | ||
0.4 | 0.3229 | -0.3409 | -0.9480 | -0.890 |
J. Appl. Environ. Biol. Sci., 8(1)217-224, 2018
Table5: The result of f (0) and fordifferent values ofPr with fixed valuesof =1
Pr | |||||||
---|---|---|---|---|---|---|---|
1 | 1 | 1 | 1 | ||||
1 | 0.28647 | -0.3020 | -1.2472 | -1.1739 | |||
3 | 0.21049 | -0.2187 | -2.1177 | -2.0352 | |||
5 | 0.17695 | -0.1824 | -2.7224 | -2.6410 | |||
7 | 0.15663 | -0.1607 | -3.2138 | -3.1332 |
Table 6: Theresult of and for different values of with fixed valuesof =1, =0.72.
1 | 1 | 1 | 1 | |||
0.1 | 1.4390 | 0.6421 | -0.8661 | -0.7653 | ||
0.3 | 1.2195 | 0.4705 | -0.9172 | -0.8270 | ||
0.5 | 0.9803 | 0.2716 | -0.9651 | -0.8830 | ||
0.7 | 0.7231 | 0.0489 | -1.0112 | -0.9367 |
Fig. 1: Graph of f undertheeffect of δ. Fig.2: Graph of f undertheeffect ofɛ. Fig.3: Graph of f undertheeffect of λ. Fig. 6: Graph of f underthe effect of M.
Fig.4: Graph of f undertheeffect of s. Fig.7: Graph ofθ(ƞ) undertheeffect ofPr.
Fig. 5: Graph of f under the effect of Fw. Fig. 8: Graph ofθ(ƞ) undertheeffect ofλ.
J. Appl. Environ. Biol. Sci., 8(1)217-224, 2018
This article examined unsteady stagnation point flow of viscous fluid under the effects of magnetic field, radiative heat source and boundary slip. Moreover, the flow is continuous due to vertical sheet with moving boundary that is stretching or shrinking. Some of the important results are as follows:
The curve of f ( )rises up with increase in the value of slip parameter δ.
The magnitude of
f ( )increases with increase in the magnitude of ɛ, stretching parameter (ɛ > 0 / ɛ < 0). The buoyancy opposing phenomena (λ < 0) causes reduction in f / (ƞ). The increase in magnetic field strength causes significant decrease in fluid flow velocity .
f () The increase in the values of unsteadiness parameter s causes increase in flow speed
f ( ). The flow speed increases in magnitude as the increase in the value of suction parameter fw. The increase in Prandtl number Pr, causes reduction in temperature
() The curve of
( )rise with increase in λ (λ < 0).
The radiation parameter Rn and the parameter ɛ, (ɛ < 0) shows rise in temperature
() The increase in suction velocity at wall causes decreases in
() The temperature function
( )decreases with increase in unsteady parameter s.
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