We consider an unsteady, two-dimensional, laminar, incompressible, Boussinesq, couple stress fluid over a semi- infinite vertical plate in a porous medium. Here, the X-axis is taken along the porous plate with direction opposite to the direction of gravity and the y-axis is taken exactly perpendicular to the porous plate. We assume the following assumptions in order to find the analytical solution for the above problem. x varies between 0 to +00; all physical quantities are independent of x except pressure; density is constant throughout the momentum equation except for body force; density pis a function of temperature only, hence P=Pa [1-B(T-T.)]; oscillatory suction velocity and the free stream velocity far away from the porous plate oscillates about a mean constant value in a direction parallel to the x-axis. du du 1 др +V at ду Po êx Fig.1: Physical model. Under these assumptions, the basic governing equations of motion are ди av +- = 0, dx ду Copyright to IJIRSET ar at y = aT +V=K Pr= == dt 20 at dy dy Voy V K * 8²T V du + Cay +U- u= d²u V dy² k do do a do +1== at dy 1 y2 The associated boundary conditions for the physical model are do 0²u y=0:u=0, T = T₁, dy dy² U U₁ IJIRSET International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization) Vol. 6, Special Issue 13, July 2017 -(1+Bee = . 150-A 1 CT 1-1. t' = yo: u=U (1), @→0. T→T. Making these equations dimensionless using the known physical quantities as vn v²k Vo @U.V. Gr=Bug (T-T.) Gr= + v²t V 00 10²0 by Pr oy u+ 2v² U . +E KC p do -u², == dy . 11 = E= 20 ду U² C, (T.-T.)' The fluctuating free-stream and suction velocities respectively defined as, U (1)=U₁ (1+ A&et) v(t) = -v₁ (1+ Bee) Bɛent) Vo do 8²u ду dy² yo: u(1+& Aet), @→0. + Bg(T-T₂). The +(U-u)+2- E-E₂0-B +: y = 0=1, www.ijirset.com 0→0. V Vo do -V₁ (1+Bɛ y Boy² dt -Vo(1+Bee)_10² The corresponding dimensionless boundary conditions takes the form, y = 0: u = 0, . ; k= and (7) Using Eq.(6) and (7), rewriting equations (2), (3) and (4) in the dimensionless form are, du до - ₁ (1+ Beem) ou du ¹²u 2 +Gr 0, ду dt oy² k ду D 0= (1) ISSN(Online): 2319-8753 ISSN (Print): 2347-6710 T-T T-T (2) 112 (3) (4) (5) (6) (8) (9) (10) (11)
We consider an unsteady, two-dimensional, laminar, incompressible, Boussinesq, couple stress fluid over a semi- infinite vertical plate in a porous medium. Here, the X-axis is taken along the porous plate with direction opposite to the direction of gravity and the y-axis is taken exactly perpendicular to the porous plate. We assume the following assumptions in order to find the analytical solution for the above problem. x varies between 0 to +00; all physical quantities are independent of x except pressure; density is constant throughout the momentum equation except for body force; density pis a function of temperature only, hence P=Pa [1-B(T-T.)]; oscillatory suction velocity and the free stream velocity far away from the porous plate oscillates about a mean constant value in a direction parallel to the x-axis. du du 1 др +V at ду Po êx Fig.1: Physical model. Under these assumptions, the basic governing equations of motion are ди av +- = 0, dx ду Copyright to IJIRSET ar at y = aT +V=K Pr= == dt 20 at dy dy Voy V K * 8²T V du + Cay +U- u= d²u V dy² k do do a do +1== at dy 1 y2 The associated boundary conditions for the physical model are do 0²u y=0:u=0, T = T₁, dy dy² U U₁ IJIRSET International Journal of Innovative Research in Science, Engineering and Technology (An ISO 3297: 2007 Certified Organization) Vol. 6, Special Issue 13, July 2017 -(1+Bee = . 150-A 1 CT 1-1. t' = yo: u=U (1), @→0. T→T. Making these equations dimensionless using the known physical quantities as vn v²k Vo @U.V. Gr=Bug (T-T.) Gr= + v²t V 00 10²0 by Pr oy u+ 2v² U . +E KC p do -u², == dy . 11 = E= 20 ду U² C, (T.-T.)' The fluctuating free-stream and suction velocities respectively defined as, U (1)=U₁ (1+ A&et) v(t) = -v₁ (1+ Bee) Bɛent) Vo do 8²u ду dy² yo: u(1+& Aet), @→0. + Bg(T-T₂). The +(U-u)+2- E-E₂0-B +: y = 0=1, www.ijirset.com 0→0. V Vo do -V₁ (1+Bɛ y Boy² dt -Vo(1+Bee)_10² The corresponding dimensionless boundary conditions takes the form, y = 0: u = 0, . ; k= and (7) Using Eq.(6) and (7), rewriting equations (2), (3) and (4) in the dimensionless form are, du до - ₁ (1+ Beem) ou du ¹²u 2 +Gr 0, ду dt oy² k ду D 0= (1) ISSN(Online): 2319-8753 ISSN (Print): 2347-6710 T-T T-T (2) 112 (3) (4) (5) (6) (8) (9) (10) (11)
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
Related questions
Question
topic is fluid
use equation 1,2,3,4
and all given quantities if required
and obtain equation 8,9,10
![Effects of Oscillatory Flow of a Couple stress
Fluid over a Semi-Infinite Vertical Permeable
Plate in a Porous Medium with Viscous
Dissipation and Ohmic Effect
Suresh Babu R¹, Rushi Kumar B², Dinesh P.A¹
Department of Mathematics, Ramaiah Institute of Technology, Bengaluru, India¹
School of Advanced Sciences, Fluid Dynamics Division, VIT University, Vellore, India ²
ABSTRACT: This problem deals with unsteady, two-dimensional, Boussinesq, laminar, oscillatory flow of an
incompressible couple stress fluid over an infinite vertical permeable plate through the porous medium under the
influence of viscous dissipation and Ohmic effects is analytically. The highly non-linear coupled partial differential
equations are modelled for the boundary layer flow, heat and mass transfer using fully developed flow conditions and
are transferred into a set of linear PDE's using non-dimensional parameters. The analytical solutions of the resulting
boundary layer equations are assumed of oscillatory type and are solved using a two-term regular perturbation
technique. Numerical computation are performed to analyze the effect of various non-dimensional parameters of the
physical model like Grashof number Gr Prandtl number Pr, Viscous ratio, Permeability parameter k, Material
B
parameter and Frequency number on velocity, temperature and angular velocity profiles and are presented
graphically. The present results are compared with earlier works in the absence of some parameters of the model and
found to be in good arrangement.
KEYWORDS: Oscillatory flow, Couple stress fluid, Viscous Dissipation, Ohmic effect, Porous medium.
I. INTRODUCTION
In the last five decades, convective heat and mass transfer flow in a porous medium has been a subject of great
interest because of its numerous industrial, thermal and chemical engineering applications in different disciplines like
medicine, science and technology. For example, blood flow in a tube, packed-bed catalytic reactors, cooling of nuclear
reactors and underground energy transport, enhanced oil recovery system, thermal insulation, geothermal reservoirs,
drying of porous solids, tidal waves etc. It is very much interesting to study the effects of the free convection currents
on the oscillatory flow type of boundary layer flow and it is very important for technological point of view. Such a
study was first initiated by Lighthill by considering an incompressible, 2-dimensional viscous fluid flow by assuming
that the flow is a regular fluctuating superimposed flow in a boundary layer.
Soundalgekar [1] analysed a two-dimensional flow of an incompressible viscous fluid past an semi-infinite
vertical porous plate in a porous medium with constant suction. Rapits et al. [2] studied free convection currents of
mass transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant suction.
analytically. An extensive study of free convection effects on the oscillatory flow of a couple stress fluid through a
porous medium was discussed by Hiremath and Patil [3]. Raptis and Thakur [4] presented an analytical solutions for
couple stress fluid which is heated from below in porous medium. Magyari and Pop [5] also performed analytical
solutions for unsteady free convection flow over a vertical plate through a porous media.
Many transport processes occurring both in nature and in industries involves fluid flows with the combined heat
and mass transfer and it plays an essential role in transport phenomena. Such flows are driven by multiple buoyancy
effects which are arising from the density variations due to temperature and species concentration. Mixed convective
Copyright to UIRSET
www.ijirset.com
111
IJIRSET
ISSN(Online): 2319-8753
ISSN (Print): 2347-6710
International Journal of Innovative Research in Science,
Engineering and Technology
(An ISO 3297: 2007 Certified Organization)
Vol. 6, Special Issue 13, July 2017
flows in a porous medium have been examined during the last four decades due to its practical applications which can
be modelled or approximated of a boundary layer flow. Such a study was illustrated by Rudraiah et al.[6] on mixed.
convective oscillatory flow in a vertical porous stream and Ogulu [9] On the oscillating plate temperature flow of a
polar fluid past a vertical porous plate in the presence of couple stresses and radiation. Many authors like Youn Kim [7].
Chamkha [8], Sahin Ahmed [10], Umamaheswari [11] and Raju et al. [12] have done their extensive research work by
considering an additional MHD effect past a vertical plate in a porous medium. Comprehensive literature surveys,
references regarding the subject of porous medium can be hand in recent books by Nield and Bejan [13].
The main objective of this work is, therefore, to study effects of oscillatory flow of a couple stress fluid over a
semi-infinite vertical permeable plate in a porous medium along the viscous dissipation and Ohmic effects analytically
with oscillatory suction at the vertical porous plate and oscillatory free stream velocity away from the plate in a porous.
medium. To achieve the objective of the present work, the plan of this work is as follows. We consider in the next
section a vertical porous plate embedded in a fluid saturated porous medium with oscillatory suction velocity at the
vertical plate. In this section we also give the conservation equations for momentum and energy with suitable boundary
and inertia effect by considering Darcy-Lapwood-Brinkman equation. The relevant physical parameters are also
discussed in this section. These basic equations involve variable coefficients which are solved analytically using regular
perturbation technique. These solutions are numerically computed and the results are discussed in the last section.
II. MATHEMATICAL FORMULATION
We consider an unsteady, two-dimensional, laminar, incompressible, Boussinesq, couple stress fluid over a semi-
infinite vertical plate in a porous medium. Here, the X-axis is taken along the porous plate with direction opposite to
the direction of gravity and the v-axis is taken exactly perpendicular to the porous plate. We assume the following](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1981090d-ce0a-4c1b-8fc8-54ed7e4c9c9f%2F7f88c5cb-de4a-4bcb-bc68-967daf5a189e%2Fpx9ys9l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Effects of Oscillatory Flow of a Couple stress
Fluid over a Semi-Infinite Vertical Permeable
Plate in a Porous Medium with Viscous
Dissipation and Ohmic Effect
Suresh Babu R¹, Rushi Kumar B², Dinesh P.A¹
Department of Mathematics, Ramaiah Institute of Technology, Bengaluru, India¹
School of Advanced Sciences, Fluid Dynamics Division, VIT University, Vellore, India ²
ABSTRACT: This problem deals with unsteady, two-dimensional, Boussinesq, laminar, oscillatory flow of an
incompressible couple stress fluid over an infinite vertical permeable plate through the porous medium under the
influence of viscous dissipation and Ohmic effects is analytically. The highly non-linear coupled partial differential
equations are modelled for the boundary layer flow, heat and mass transfer using fully developed flow conditions and
are transferred into a set of linear PDE's using non-dimensional parameters. The analytical solutions of the resulting
boundary layer equations are assumed of oscillatory type and are solved using a two-term regular perturbation
technique. Numerical computation are performed to analyze the effect of various non-dimensional parameters of the
physical model like Grashof number Gr Prandtl number Pr, Viscous ratio, Permeability parameter k, Material
B
parameter and Frequency number on velocity, temperature and angular velocity profiles and are presented
graphically. The present results are compared with earlier works in the absence of some parameters of the model and
found to be in good arrangement.
KEYWORDS: Oscillatory flow, Couple stress fluid, Viscous Dissipation, Ohmic effect, Porous medium.
I. INTRODUCTION
In the last five decades, convective heat and mass transfer flow in a porous medium has been a subject of great
interest because of its numerous industrial, thermal and chemical engineering applications in different disciplines like
medicine, science and technology. For example, blood flow in a tube, packed-bed catalytic reactors, cooling of nuclear
reactors and underground energy transport, enhanced oil recovery system, thermal insulation, geothermal reservoirs,
drying of porous solids, tidal waves etc. It is very much interesting to study the effects of the free convection currents
on the oscillatory flow type of boundary layer flow and it is very important for technological point of view. Such a
study was first initiated by Lighthill by considering an incompressible, 2-dimensional viscous fluid flow by assuming
that the flow is a regular fluctuating superimposed flow in a boundary layer.
Soundalgekar [1] analysed a two-dimensional flow of an incompressible viscous fluid past an semi-infinite
vertical porous plate in a porous medium with constant suction. Rapits et al. [2] studied free convection currents of
mass transfer flow through a porous medium bounded by an infinite vertical limiting surface with constant suction.
analytically. An extensive study of free convection effects on the oscillatory flow of a couple stress fluid through a
porous medium was discussed by Hiremath and Patil [3]. Raptis and Thakur [4] presented an analytical solutions for
couple stress fluid which is heated from below in porous medium. Magyari and Pop [5] also performed analytical
solutions for unsteady free convection flow over a vertical plate through a porous media.
Many transport processes occurring both in nature and in industries involves fluid flows with the combined heat
and mass transfer and it plays an essential role in transport phenomena. Such flows are driven by multiple buoyancy
effects which are arising from the density variations due to temperature and species concentration. Mixed convective
Copyright to UIRSET
www.ijirset.com
111
IJIRSET
ISSN(Online): 2319-8753
ISSN (Print): 2347-6710
International Journal of Innovative Research in Science,
Engineering and Technology
(An ISO 3297: 2007 Certified Organization)
Vol. 6, Special Issue 13, July 2017
flows in a porous medium have been examined during the last four decades due to its practical applications which can
be modelled or approximated of a boundary layer flow. Such a study was illustrated by Rudraiah et al.[6] on mixed.
convective oscillatory flow in a vertical porous stream and Ogulu [9] On the oscillating plate temperature flow of a
polar fluid past a vertical porous plate in the presence of couple stresses and radiation. Many authors like Youn Kim [7].
Chamkha [8], Sahin Ahmed [10], Umamaheswari [11] and Raju et al. [12] have done their extensive research work by
considering an additional MHD effect past a vertical plate in a porous medium. Comprehensive literature surveys,
references regarding the subject of porous medium can be hand in recent books by Nield and Bejan [13].
The main objective of this work is, therefore, to study effects of oscillatory flow of a couple stress fluid over a
semi-infinite vertical permeable plate in a porous medium along the viscous dissipation and Ohmic effects analytically
with oscillatory suction at the vertical porous plate and oscillatory free stream velocity away from the plate in a porous.
medium. To achieve the objective of the present work, the plan of this work is as follows. We consider in the next
section a vertical porous plate embedded in a fluid saturated porous medium with oscillatory suction velocity at the
vertical plate. In this section we also give the conservation equations for momentum and energy with suitable boundary
and inertia effect by considering Darcy-Lapwood-Brinkman equation. The relevant physical parameters are also
discussed in this section. These basic equations involve variable coefficients which are solved analytically using regular
perturbation technique. These solutions are numerically computed and the results are discussed in the last section.
II. MATHEMATICAL FORMULATION
We consider an unsteady, two-dimensional, laminar, incompressible, Boussinesq, couple stress fluid over a semi-
infinite vertical plate in a porous medium. Here, the X-axis is taken along the porous plate with direction opposite to
the direction of gravity and the v-axis is taken exactly perpendicular to the porous plate. We assume the following
![We consider an unsteady, two-dimensional, laminar, incompressible, Boussinesq, couple stress fluid over a semi-
infinite vertical plate in a porous medium. Here, the X-axis is taken along the porous plate with direction opposite to
the direction of gravity and the y-axis is taken exactly perpendicular to the porous plate. We assume the following
assumptions in order to find the analytical solution for the above problem. x varies between to + all
physical quantities are independent of x except pressure; density is constant throughout the momentum equation
except for body force; density pis a function of temperature only, hence P=Pa [1-B(T-T₂)].. oscillatory
suction velocity and the free stream velocity far away from the porous plate oscillates about a mean constant value in
a direction parallel to the x-axis.
T
--50-A²)
|₁
Fig.1: Physical model.
Under these assumptions, the basic governing equations of motion are
du ду
+
= 0,
dx ду
(1)
du du
1 op
d²u
do
+V =
ct dy
+ Bg(T-T).
Po ox
-u+2v
dy² k
ду
Copyright to UJIRSET
112
IJIRSET
ISSN(Online): 2319-8753
ISSN (Print): 2347-6710
International Journal of Innovative Research in Science,
Engineering and Technology
(An ISO 3297: 2007 Certified Organization)
Vol. 6, Special Issue 13, July 2017
ar OT 8²T
U du
U
+v- =K
+
+
Ət
dy dy² Cay
kC
P
(3)
do
do a d¹o
+V
=
ôt ду
I by²
(4)
The associated boundary conditions for the physical model are
do
y=0u = 0,
==
T = T₁,
dy
dy²
yoo: u=U (1),
@→0, T→T.
(5)
Making these equations dimensionless using the known physical quantities as
u
v²t
v²k
vn
W
T-T
V
y² = 10)
u U =
t' =
n' =
k=
.
y =
2
V
V
Vo
Vo
T-T
Bug (T-T)
Pr=0,
Uo
Gr=
@U.V.
E=
2
©* =
Uovo
C, (T.-T.)
D
(6)
The fluctuating free-stream and suction velocities respectively defined as,
U(t)=U₁ (1+ A&ent) v(t)=-v₁ (1+ Beeint)
(7)
and
Using Eq.(6) and (7),
du du du λ
rewriting equations (2), (3) and (4) in the dimensionless form are,
du
- V₁ (1+ Bε e¹¹)
Bɛ
=
до
+ +- (U−u)+2
dt oy² k
ду
+Gr 0,
Ot
ду
(8)
2
де
80
-V
Vo(1+ BE 08
18²0
+E
by Proy2 dv
=
+
at
do
do
1d²w
-Vo(1+Beeint)
=
di
by Boy²
The corresponding dimensionless boundary conditions takes the form,
do
y=0:
u = 0,
==
0 = 1,
ду
Ôy ²
yo: u(1+& Ae), @→ 0,
0→0.
III. METHOD OF SOLUTION
+U
CT
↑
T-T.
2
Vo
Okty
E-E₂0-B²)
ý
www.ijirset.com
.
0=
(2)
(9)
(10)
(11)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1981090d-ce0a-4c1b-8fc8-54ed7e4c9c9f%2F7f88c5cb-de4a-4bcb-bc68-967daf5a189e%2F9ffb3t5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:We consider an unsteady, two-dimensional, laminar, incompressible, Boussinesq, couple stress fluid over a semi-
infinite vertical plate in a porous medium. Here, the X-axis is taken along the porous plate with direction opposite to
the direction of gravity and the y-axis is taken exactly perpendicular to the porous plate. We assume the following
assumptions in order to find the analytical solution for the above problem. x varies between to + all
physical quantities are independent of x except pressure; density is constant throughout the momentum equation
except for body force; density pis a function of temperature only, hence P=Pa [1-B(T-T₂)].. oscillatory
suction velocity and the free stream velocity far away from the porous plate oscillates about a mean constant value in
a direction parallel to the x-axis.
T
--50-A²)
|₁
Fig.1: Physical model.
Under these assumptions, the basic governing equations of motion are
du ду
+
= 0,
dx ду
(1)
du du
1 op
d²u
do
+V =
ct dy
+ Bg(T-T).
Po ox
-u+2v
dy² k
ду
Copyright to UJIRSET
112
IJIRSET
ISSN(Online): 2319-8753
ISSN (Print): 2347-6710
International Journal of Innovative Research in Science,
Engineering and Technology
(An ISO 3297: 2007 Certified Organization)
Vol. 6, Special Issue 13, July 2017
ar OT 8²T
U du
U
+v- =K
+
+
Ət
dy dy² Cay
kC
P
(3)
do
do a d¹o
+V
=
ôt ду
I by²
(4)
The associated boundary conditions for the physical model are
do
y=0u = 0,
==
T = T₁,
dy
dy²
yoo: u=U (1),
@→0, T→T.
(5)
Making these equations dimensionless using the known physical quantities as
u
v²t
v²k
vn
W
T-T
V
y² = 10)
u U =
t' =
n' =
k=
.
y =
2
V
V
Vo
Vo
T-T
Bug (T-T)
Pr=0,
Uo
Gr=
@U.V.
E=
2
©* =
Uovo
C, (T.-T.)
D
(6)
The fluctuating free-stream and suction velocities respectively defined as,
U(t)=U₁ (1+ A&ent) v(t)=-v₁ (1+ Beeint)
(7)
and
Using Eq.(6) and (7),
du du du λ
rewriting equations (2), (3) and (4) in the dimensionless form are,
du
- V₁ (1+ Bε e¹¹)
Bɛ
=
до
+ +- (U−u)+2
dt oy² k
ду
+Gr 0,
Ot
ду
(8)
2
де
80
-V
Vo(1+ BE 08
18²0
+E
by Proy2 dv
=
+
at
do
do
1d²w
-Vo(1+Beeint)
=
di
by Boy²
The corresponding dimensionless boundary conditions takes the form,
do
y=0:
u = 0,
==
0 = 1,
ду
Ôy ²
yo: u(1+& Ae), @→ 0,
0→0.
III. METHOD OF SOLUTION
+U
CT
↑
T-T.
2
Vo
Okty
E-E₂0-B²)
ý
www.ijirset.com
.
0=
(2)
(9)
(10)
(11)
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