Flow of non–Newtonian fluids has attracted attention of many scientists and researchers because of their fundamental and practical importance in the industry as well as in the daily life. Shear stress of such fluids is non–linearly related with shear rate and it is very difficult to analyze their flow. Examples include food, rubber, gel, polymers, petrol, paper coating, plasma and grease etc. One such fluid is the power–law fluid (Ostwald–de Waele model) which has been used extensively in the industry especially as a lubricant.
Couple stress fluid is another important non–Newtonian fluid first examined by Stokes1 to describe the polar effects. The couple stress fluid can be described by a new type of tensor called couple stress tensor in addition to the Cauchy stress tensor. In such fluids, polar effects play a significant role which are present due to the couple stresses (moment per unit area) and body couples (moment per unit volume). Because of significant importance of couple stress fluids in the industrial and engineering applications, many researchers have analyzed these flows. Some applications are animal blood, liquid crystals, polymer thickened oil, fluid mechanics and polymeric suspensions. Devakar et al.,2 considered Stokes’ problems for the couple stress fluid. In another investigation, Devakar et al.,3 discussed properties of the couple stress fluid flowing between parallel plates. Heat transfer analysis for the flow of a couple stress fluids near a stagnation point has been carried out by Hayat et al.,4 Muthuraj et al.,5 studied viscous dissipation effects on MHD flow of a couple stress fluid in a vertical channel. Heat transfer analysis by Srinivasacharya et al.,6 has been carried out for couple stress flow due to expanding and contracting walls in a porous channel. Flow of couple stress fluid due to free convection through a porous channel was carried out by Hiremath and Patil7. Umavathi et al.,8 discussed heat transfer analysis for the channel flow of a couple stress fluid sandwiched between two viscous fluids. They showed that couple stress parameter is responsible for enhancing the fluid velocity.
A literature survey reveals that stagnation–point flow can be discussed in two ways either orthogonally or obliquely. Hiemenz9 provided an exact solution of stagnation–point flow for the first time. An oblique stagnation–point flow arises when a detached flow of fluid retouches the face of body. Non–orthogonal stagnation–point flow on a wall was examined by Stuart10 and Tamada11. Dorrepaal12 found an exact solution for the oblique stagnation–point flow of a viscous fluid. Effects of Weissenberg number on the flow and heat transfer due to stagnation–point was analyzed by Li et al.13 Labropulu et al.,14 discussed heat transfer analysis for the oblique flow impinging on a stretched sheet. Axisymmetric non–orthogonal stagnation–point flow over a circular cylinder has been considered by Weidman and Putkaradze15. Recently Ghaffari et al.,16–18 discussed different aspects for the flows towards oblique stagnation point.
Wang19 discussed the effects of slip parameter on the stagnation point flow of a viscous fluid. Devakar et al.,20 found an exact solution for a couple stress fluid by implementing slip condition at fluid–solid interface. Labropulu et al.,21 examined slip flow due to second grade fluid impinging orthogonally or obliquely on a surface. Blyth & Pozrikidis22 studied stagnation point flow by introducing slip condition at the interface of two viscous fluids. Axisymmetric stagnation–point flow near a lubricated stationary disc has been carried out by Santra et al.23 They used power–law fluid as a lubricant. Sajid et al.24 reconsidered the problem of Santra et al.,23 by applying generalized slip condition at fluid–lubricant interface introduced by Thompson & Troian.25 Recently Mahmood et el.,26 investigated oblique stagnation–point flow of a second–grade fluid over a plate lubricated by a power–law fluid. Some more recent investigations27–32 will also be fruitful for the readers.
Our aim in the present communication is to investigate the oblique flow of a couple stress fluids near a stagnation point over a lubricated plate. A power–law fluid has been utilized for the lubrication purpose. The flow problem consists of the set of coupled nonlinear ordinary differential equations along with nonlinear coupled boundary conditions. The Keller–box method33–36 has been implemented to solve the considered flow problem numerically. Influence of pertinent parameters on the flow characteristics is discussed through graphs and tables. The validity of present study has been checked by comparing results in the limiting case with that exist in the literature.
Consider the steady, two–dimensional, oblique flow of a couple stress fluids towards a stagnation point over a lubricated plate. A power–law fluid (Ostwald–de Waele model) is used as lubricant. The plate is fixed in xz–plane such that it is symmetric with respect to origin. The fluid impinges on the plate with an angle in the domain (Figure 1).
We assume that power–law lubricant spreads on the plate forming a thin coating with the flow rate given as
(1)
where
represents horizontal velocity component of the lubricant and denotes the variable thickness of the lubrication layer.
The flow problem is governed by the following equations37
(2)
(3)
(4)
where u and vrepresent, respectively horizontal and vertical velocity components of the couple stress fluid. Parameters ,
,
and respectively are density, pressure, kinematic viscosity and ratio of couple stress viscosity to the density.
Following Tooke & Blythe38 the free stream velocity components can be written as
(5)
where a and b are constants. Furthermore
is the parameter that supervises the pressure gradient along x–axis which generates the shear flow incident to the orthogonal stagnation–point and the parameter
represents the boundary layer displacement produced on the lubricated surface. It is worth to mention that the flow field (5) displays the combined effects of both the horizontal shear flow and the orthogonal stagnation–point flow.
Eliminating the pressure between Eqs. (2) and (3) one obtains
(6)
The expression for the skin friction or wall shear stress is given as
(7)
where
and are viscosity and couple stress viscosity respectively. The usual no–slip boundary condition at the solid–lubricant interface implies
(8)
As the power–law coating is very slim, therefore
(9)
We assume that velocity and shear stress of both the fluids are continuous at the interface
Thus continuity of shear stress implies
(10)
In which represents the viscosity of the lubricant. Letting the viscosity of lubricant can be written as
(11)
in which
is dynamic coefficient of viscosity and
is the consistency index. Fluid behaves as viscous, shear thinning and shear thickening, respectively for
and
.
We further assume that
(12)
It is worth to point out that
is interfacial velocity component of both fluids. The thickness
of the power–law lubricant is given by
(13)
The continuity of horizontal velocity components of both the fluids gives
(14)
Substituting Equations (11)–(14) in Equation (10) we get
(15)
Similarly implementing the continuity of interfacial velocity components of bulk fluid and lubricant along y–axis we get
(16)
Employing Equation (9) we get
(17)
Following Santra et al.,23 the boundary conditions (15) and (17) can be imposed at the fluid–solid interface. Boundary conditions at free stream have been mentioned in equation (5).
Introducing
(18)
The governing Equations (6), (8), (15), and (17) reduce to
(19)
(20)
(21)
(22)
(23)
Where
is called the couple stress parameter and
denotes the free stream shear. The parameter
in Equations (21) and (22) is given as
(24)
Integrating Equations (19) and (20) and using free stream conditions, we get
(25)
(26)
Where
is a free parameter and
. In order to eliminate
from Equation (26)we let
to obtain
(27)
The boundary conditions in new variables become
(28)
Equation (24) suggests that to obtain similar solution, one should have
. The parameter
given in Equation (24) measures slip produced on the surface and can be written as
(29)
As clear from Equation (29),
is a representation of ratio of viscous length scale
to the lubrication length scales
. For a highly viscous bulk fluid (i.e. when
is large) and a very thin lubricant (i.e. when
is small), the parameter
is increased. As the parameter
approaches to infinity, the traditional no–slip conditions
, andcan be recovered from Equations (21) and (28). On the other hand when the bulk fluid is less viscous and
attains a massive value,
and consequently the full slip boundary conditions , and are achieved.Therefore
interprets the inverse measure of slip called slip parameter.
Employing (18), the dimensionless wall shear stress is given by
(30)
To find the stagnation–point
on the surface, we set
Therefore
(31)
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Equations (21), (25), (27) and (28) are solved using Keller–box method33–36 which is based on an implicit finite difference approach. This numerical scheme is very effective to solve non–linear and coupled boundary value problems directly without converting them into initial value problems. As a first step, a system of first order ordinary differential equations is obtained in the following way:
(32)
Therefore, Equations (25) and (27) imply
(33)
The transformed boundary conditions n=0.5 for imply
(34)
(35)
The obtained first–order system is approximated with central–difference for derivatives and averages for the dependent variables. The reduced algebraic system is given by
, ,, , (36)
, , (37)
(38)
(39)
where
etc. Equations (38) and (39) are nonlinear algebraic equations and therefore, have to be linearized before the factorization scheme can be used. We write the Newton iterates in the following way:
For the
iterates:
(40)
for all dependent variables. By substituting these expressions in Equations (36)–(39) and dropping the quadratic and higher–order terms in
, a linear tridiagonal system of equations will be obtained as follows:
(41)
(42)
(43)
(44)
(45)
(46)
subject to boundary conditions
(47)
(48)
where
etc. The resulting linearized system of algebraic equations is solved by the block–elimination method. In matrix–vector form, the above system can be written as
(49)
in which
(50)
where the elements inA are of matrices and that of and are respectively of order
. Now, we let
(51)
where L is a lower and U is an upper triangular matrix.
Equation (51) can be substituted into Equation (49) to get
(52)
Defining
(53)
Equation (52) becomes
(54)
where the elements of are
column matrices. The elements of
can be solved from Equation (54). Once the elements of
are found, Equation (53) then gives the solution. When the elements of
are found, Equation (49) can be used to find the next iteration.
None.