# Some Exact  Solutions of Boundary  Layer  Flows along a Vertical Plate with  Buoyancy Forces Combined  with  Lorentz  Forces under Uniform Suction

Authors: Asterios Pantokratoras
Journal: Mathematical Problems in Engineering
 (2008)

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## Abstract

Exact analytical solutions of boundary layer flows along a vertical porous plate with uniform suction are derived and presented in this paper. The solutions concern the Blasius, Sakiadis, and Blasius-Sakiadis flows with buoyancy forces combined with either MHD Lorentz or EMHD Lorentz forces. In addition, some exact solutions are presented specifically for water in the temperature range of0C∘≤T≤8C∘, where water density is nearly parabolic. Except for their use as benchmarking means for testing the numerical solution of the Navier-Stokes equations, the presented exact solutions with
EMHD forces have use in flow separation control in aeronautics and hydronautics, whereas the MHD results have applications in process metallurgy and fusion technology. These analytical solutions are valid for flows with strong suction.


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## Body

## 1. Introduction

Free convection
along a vertical plate with mass transfer at the wall has been studied by many
authors in the past. Eichhorn [1] was the first to study the effect of
suction and injection on free convective flow. He considered power-law
variation of plate temperature and transpiration velocity under which
self-similar solutions of the governing equations are possible. For the case of
isothermal plate with uniform blowing or suction, similarity does not exist.
For the latter problem, Sparrow and Cess [2] provided approximate series
solutions valid near the plate leading edge for Pr=0.72. This problem was considered in more detail by Merkin [3]
who obtained asymptotic solutions, valid at large distances from the leading
edge. The next-order corrections to the boundary layer solution for this
problem, concerning gases, were obtained by Clarke [4] who did not invoke
the usual Boussinesq approximation. The solutions for strong suction and blowing
on general body shapes which admit a similarity solution have been given by
Merkin [5]. Parikh et al. [6] studied both numerically and experimentally
the problem of air (Pr=0.7) free
convection over an isothermal porous vertical plate with uniform transpiration
taking into account the air variable physical properties. Minkowycz and Sparrow
[7] using the local nonsimilarity method presented solutions for a wide
range of Pr numbers. Vedhanayagam et al. [8] presented a transformation of
the equations for general blowing and wall temperature variations, as well as results
for the isothermal plate with uniform blowing. A solution to the constant plate
temperature with uniform air blowing, based on the film model, has been derived
by Brouwers [9]. In a subsequent paper, Brouwers [10], using the film
model, derived a thermal correction factor and a novel friction correction
factor which were applied to free convection along a vertical porous plate. The
problem of blowing and suction on the free convection over a vertical plate
with a given wall heat flux has been considered by Chaudhary and Merkin [11]
who presented results valid for Pr=1.
Merkin [12] considered again the problem of free convection flow over a
vertical plate with prescribed temperature, and presented results for variable
transpiration velocities for Pr 1 and
7.In all
of the above works, the buoyant force is produced by the temperature difference
between the plate and the ambient fluid. Another kind of vertical force which
resembles the buoyant force is the electromagnetohydrodynamic (EMHD) Lorentz force which
acts parallel to the plate either assisting or opposing the flow.  The EMHD Lorentz force can be generated by a stripwise
arrangement of flush mounted electrodes and permanent magnets. Gailitis and Lielausis [13] were
probably the first to propose the use of the EMHD Lorentz force for flow
control over a plate.  The idea of using
the EMHD Lorentz force for flow control by Gailitis and Lielausis was later
abandoned and only recently attracted new attention [14–17].  In addition, in last years much investigation
on flow control using the EMHD Lorentz force was conducted at the Rossendorf
Institute and the Institute for Aerospace Engineering in Dresden, Germany
[18–22].Magnetohydrodynamics (MHD) is the field of
fluid mechanics that encompasses the phenomena arising when a magnetic field is
applied to an electrically conducting fluid. Air, water, and especially liquid
metals (lithium, mercury, and sodium) are electrically conducting fluids. The
effect of an applied magnetic field on heat transfer in external flows has been
investigated mainly for the cases of flat plate boundary layer and blunt body
stagnation point flows. The works published in these areas appeared in the late
1950’s and early 1960’s with application to space-vehicle surface heating upon
reentry. MHD is also applied to fusion technology [23].The problem of flow along a vertical,
stationary, isothermal plate of an electrically conducting fluid under a
horizontal magnetic field is a classical problem in magnetohydrodynamics and
has been treated for the first time by Sparrow and Cess [24]. Riley [25] and Kuiken [26] have
reexamined the problem in order to give exact solutions, but their attempts to
use the method of matched asymptotic expansions encountered difficulties. The
first complete exact results for this problem have been given by Wilks [27]
for Pr=0.72.In all of the above works, the vertical
plate and the ambient fluid were motionless; that is, the problem treated was
pure free convection. The problem of fluid flow along a stationary,
impermeable, horizontal plate situated in a fluid stream moving with constant
velocity is a classical problem of fluid mechanics that has been solved for the
first time in 1908 by Blasius [28]. In this
problem, the fluid motion is produced by the free stream. A similar problem
occurs when the plate moves with constant velocity in a calm fluid. This
problem has been treated for the first time by Sakiadis [29]. MHD flows along
a moving plate in viscoelastic and micropolar fluids are treated in [30–34]. The combination of a
moving vertical plate within a vertical free stream with the existence of a
vertical buoyancy force is called here Sakiadis-Blasius-buoyant flow.The purpose of the present paper is to
present some new exact analytical solutions for the Sakiadis-Blasius-buoyant
flow  over a vertical plate with uniform
suction with either EMHD Lorentz or
MHD Lorentz forces. Exact solutions of the Navier-Stokes equations are important for two
reasons [35]. Owing to their uniform validity, the basic phenomena
described by them can be more closely studied. In addition, the exact solutions
serve as standards for checking the accuracies of the many approximate methods,
whether they are numerical, asymptotic, or empirical. Current computer
technology makes the complete numerical integration of the Navier-Stoles
equations feasible. However, the accuracy of the results can only be ascertained
by comparison with an exact solution.


## 2. The Mathematical Model

Consider
the flow along a vertical plate withu and v denoting, respectively, the
velocity components in the x and y directions, where x is the coordinate along the plate and y is the coordinate perpendicular to x (see Figure 1). For steady two-dimensional flow, the boundary
layer equations with constant fluid properties and linear relationship between
density and temperature are as follows:Figure 1
The flow configuration and coordinate system.continuity equation:(2.1)∂u∂x+∂v∂y=0,momentum equation:(2.2)u∂u∂x+v∂u∂y=ν∂2u∂y2+gβ(T−T∞),energy equation:(2.3)u∂T∂x+v∂T∂y=λ∂2T∂y2, where ν is the fluid
kinematic  viscosity, g is the gravitational acceleration, β is the volumetric expansion coefficient, T is the
fluid temperature, and λ is the fluid thermal diffusivity.When the suction velocity is very strong, (2.1)–(2.3) take the
following forms [36, page 297]:continuity equation:(2.4)∂v∂y=0,momentum equation:(2.5)v∂u∂y=ν∂2u∂y2+gβ(T−T∞),energy equation:(2.6)v∂T∂y=λ∂2T∂y2. It follows from (2.4) that v=vw=constant
and the momentum and energy equations take the following forms:momentum equation:(2.7)ν∂2u∂y2−vw∂u∂y=−gβ(T−T∞),energy equation:(2.8)λ∂2T∂y2−vw∂T∂y=0. For combined Sakiadis-Blasius flow, the
boundary conditions are as follows:(2.9)aty=0:u=uw,v=vw,T=Tw,asy→∞,u=u∞,T=T∞.  In the present work,  the suction velocity vw is always negative.The energy equation (2.8) is independent of the momentum equation, and its
solution [36, page 207] is (2.10)T−T∞Tw−T∞=exp(vwyλ).


## 3. Results and Discussion

### 3.1. Sakiadis-Blasius Flow with Buoyancy
Forces and EMHD Lorentz Forces

It is assumed that, except for the buoyancy force, an EMHD Lorentz force acts
parallel to the plate. The EMHD Lorentz force can be generated by a stripwise
arrangement of flush mounted electrodes and permanent magnets as sketched in Figure2. For more information see Weier
[20]. Then, the momentum equation (2.7) takes the following
form: (3.1)ν∂2u∂y2−vw∂u∂y=−gβ(T−T∞)−πj0M08ρ∞exp(−πay), where j0 is the applied current
density, M0 is the
magnetization of the permanent magnets, α is the width of magnets and
electrodes, and ρ∞ is the ambient fluid density. It should be
noted here that the EMHD Lorentz force can either assist or oppose the flow and
is independent of the flow field. The last term in the momentum equation is the
EMHD Lorentz force which assists the flow and decreases exponentially with  y. Substituting the temperature difference from (2.10) into (3.1), we have (3.2)ν∂2u∂y2−vw∂u∂y=−gβ(Tw−T∞)exp(vwλy)−πj0M08ρ∞exp(−πay). Equation (3.2) is a linear
differential equation of second order and has the following exact analytical
solution for the Sakiadis-Blasius-buoyant-EMHD flow: (3.3)u−u∞uw−u∞=(1+G+Z)exp(vwνy)−Gexp(vwλy)−Zexp(−πay), where G is the buoyancy parameter: (3.4)G=λgβ(Tw−T∞)(uw−u∞)vw2(Pr−1) and Z is the Lorentz parameter: (3.5)Z=j0M0a28ρ∞(uw−u∞)(πν+avw). The Prandtl number is (3.6)Pr=νλ. Both parameters G and Z are dimensionless.
The absolute wall shear
stress is (3.7)τw=ρ∞(uw−u∞)vw−μgβ(Tw−T∞)vwPr+j0M0a8,
while
the dimensionless skin-friction coefficient is (3.8)cf=τwρ∞(uw−u∞)vw=1+G(1−Pr)+Z(πRe+1), where Re is the suction Reynolds number:  (3.9)Re=avwν.
It should be noted here that this is the first work on
fluid mechanics which uses both EMHD Lorentz forces and suction simultaneously,
and therefore the above Reynolds number, using as characteristic length the
distance between the magnets, appears for the first time in the literature.Figure 2
Arrangement of
electrodes and magnets for the creation of an EMHD Lorentz forceF in the flow
along a flat plate [20].When(3.10)Z=−1−G(1−Pr)(πRe+1)−1,
the wall shear stress is zero.  In Figure 3(a), two velocity profiles are shown for which the wall shear stress is zero (3.10)
is valid).  The transverse coordinate η is (3.11)η=−vwνy,
and the two curves meet the vertical axis orthogonally.Dimensionless
velocity distribution for different kinds of flow with buoyancy forces and
Lorentz forces. In cases (a) and (b), the wall shear stress is zero.
(a)(b)(c)(d)When the free-stream velocity is zero, we have the Sakiadis-buoyant-EMHD
flow:(3.12)uuw=(1+Gs+Zs)exp(vwνy)−Gsexp(vwλy)−Zsexp(−πay), where (3.13)Gs=λgβ(Tw−T∞)uwvw2(Pr−1),Zs=j0M0a28ρ∞uw(πν+avw). The  absolute wall shear stress  is (3.14)τw=ρ∞uwvw−μgβ(Tw−T∞)vwPr+j0M0a8,
while
the dimensionless skin-friction coefficient is (3.15)cf=τwρuwvw=1+Gs(1−Pr)+Zs(πRe+1).
When  (3.16)Zs=−1−Gs(1−Pr)(πRe+1)−1, the wall shear stress is  zero.When the plate is motionless, we have the Blasius-buoyant-EMHD
flow:(3.17)uu∞=1−(1−Gb−Zb)exp(vwνy)−Gbexp(vwλy)−Zbexp(−πay), where (3.18)Gb=λgβ(Tw−T∞)u∞vw2(Pr−1),Zb=j0M0a28ρ∞u∞(πν+avw).
The absolute wall shear stress is (3.19)τw=−ρ∞u∞vw−μgβ(Tw−T∞)vwPr+j0M0a8, while
the dimensionless skin-friction coefficient is (3.20)cf=τwρ∞u∞vw=−1+Gb(1−Pr)+Zb(πRe+1).
When  (3.21)Zb=1−Gb(1−Pr)(πRe+1)−1,
the wall shear stress is  zero.When the plate is motionless and the free-stream velocity is zero, we
have pure free convection and the analytical solution becomes(3.22)uvw=(G0+Z0)exp(vwνy)−G0exp(vwλy)−Z0exp(−πay), where (3.23)G0=λgβ(Tw−T∞)vw3(Pr−1),Z0=j0M0a28ρ∞vw(πν+avw).
In (3.22), the longitudinal velocity u is nondimensionalized with suction velocity
vw because in the above problem the suction velocity is the apparent
characteristic velocity of the flow.The absolute wall shear stress is(3.24)τw=−μgβ(Tw−T∞)vwPr+j0M0a8, while
the dimensionless skin-friction coefficient is (3.25)cf=τwρvw2=G0(1−Pr)+Z0(πRe+1).
When  (3.26)Z0=G0(Pr−1)(πRe+1)−1, the wall shear stress is zero.  In Figure 3(b), two velocity profiles are shown for which the wall shear stress is zero. The velocity profiles near the plate for cases (a) and (b) are shown in Figure 4.The velocity
profiles near the plate for cases (a) and (b) where it is clearly shown that the wall
shear stress is zero.
(a)(b)When there is no Lorentz force, we have convection due to buoyancy force
only and velocity is  given by(3.27)uvw=G0exp(vwνy)−G0exp(vwλy). The absolute wall shear stress is (3.28)τw=−μgβ(Tw−T∞)vwPr,
while
the dimensionless skin-friction coefficient is (3.29)cf=τwρvw2=G0(1−Pr).
When there is no buoyancy force, we have convection due to EMHD Lorentz
force only and velocity is given by (3.30)uvw=Z0exp(vwνy)−Z0exp(−πay). This is a new “strange” kind of
convection caused by the EMHD Lorentz force only. The physical meaning of the
above equation is as follows. Let us suppose that we have a porous plate where
fluid is removed with uniform suction. Although the plate is motionless and the
ambient fluid is at rest, a boundary layer forms along the plate due to the act
of the Lorentz force which is independent of the flow.The absolute value of wall shear
stress is(3.31)τw=j0M0a8,
while
the dimensionless skin-friction coefficient is (3.32)cf=τwρvw2=Z0(πRe+1). It is surprising that the absolute
wall shear stress is independent of viscosity and suction velocity.


### 3.2. Sakiadis-Blasius Flow with Buoyancy
Forces and EMHD Lorentz Forces in Water Near 4°C

The momentum equation with EMHD Lorentz
force and the buoyancy force expressed in density is(3.33)ν∂2u∂y2−vw∂u∂y=−gρ∞−ρρ∞−πj0M08ρ∞exp(−πay). The buoyancy term can be calculated from the following
equation [37]: (3.34)ρ∞−ρρ∞=γ(T−T∞)2, where γ=0.8×10−5(C°)−2.  Equation (3.34) is valid for the temperature
range of 0°C≤T≤8°C,
and T∞=4°C
where the variation of water density is nearly parabolic. Therefore, (3.33)
becomes (3.35)ν∂2u∂y2−vw∂u∂y=−gγ(T−T∞)2−πj0M08ρexp(−πay). Substituting the temperature difference
from (2.10) into (3.35), we have (3.36)ν∂2u∂y2−vw∂u∂y=−gγ(Tw−T∞)2exp(2vwλy)−πj0M08ρexp(−πay). The
exact solution of this equation is (3.37)u−u∞uw−u∞=(1+W+Z)exp(vwνy)−Wexp(2vwλy)−Zexp(−πay), where (3.38)W=λgγ(Tw−T∞)22(uw−u∞)vw2(2Pr−1).
The absolute wall shear stress is (3.39)τw=ρ∞(uw−u∞)vw−μgγ(Tw−T∞)22vwPr+j0M0a8,
while
the dimensionless skin-friction coefficient is (3.40)cf=τwρ∞(uw−u∞)vw=1+W(1−2Pr)+Z(πRe+1).
The wall shear stress becomes zero when (3.41)Z=−1+W(2Pr−1)(πRe+1)−1. When the free-stream velocity is zero, we have the Sakiadis-buoyant-EMHD
flow: (3.42)uuw=(1+Ws+Zs)exp(vwνy)−Wsexp(2vwλy)−Zsexp(−πay), where (3.43)Ws=λgγ(Tw−T∞)22uwvw2(2Pr−1).
The absolute wall shear stress is (3.44)τw=ρ∞uwvw−μgγ(Tw−T∞)22vwPr+j0M0a8, while
the dimensionless skin-friction coefficient is (3.45)cf=τwρ∞uwvw=1+Ws(1−2Pr)+Zs(πRe+1).
The wall shear stress becomes zero when (3.46)Zs=−1+Ws(2Pr−1)(πRe+1)−1. When the plate is motionless, we
have the Blasius-buoyant-EMHD flow: (3.47)uu∞=1−(1−Wb−Zb)exp(vwνy)−Wbexp(2vwλy)−Zbexp(−πay), where (3.48)Wb=λgγ(Tw−T∞)22u∞vw2(2Pr−1). The absolute wall shear stress is (3.49)τw=−ρ∞u∞vw−μgγ(Tw−T∞)22vwPr+j0M0a8,
while
the dimensionless skin-friction coefficient is (3.50)cf=τwρu∞vw=−1+Wb(1−2Pr)+Zb(πRe+1). The wall shear stress becomes zero when (3.51)Zb=1+Wb(2Pr−1)(πRe+1)−1. When the plate is motionless and the
free-stream velocity is zero, we have pure free convection and the analytical
solution becomes (3.52)uvw=(W0+Z0)exp(vwνy)−W0exp(2vwλy)−Z0exp(−πay), where (3.53)W0=λgγ(Tw−T∞)22vw3(2Pr−1).
The absolute wall shear stress is (3.54)τw=−μgγ(Tw−T∞)22vwPr+j0M0a8, while
the dimensionless skin-friction coefficient is (3.55)cf=τwρvw2=W0(1−2Pr)+Z0(πRe+1).
The wall shear stress becomes zero when (3.56)Z0=W0(2Pr−1)(πRe+1)−1. When there is no EMHD Lorentz force,
we have convection due to buoyancy force only and velocity is given by (3.57)uvw=W0exp(vwνy)−W0exp(2vwλy). The absolute
wall shear stress is (3.58)τw=−μgγ(Tw−T∞)22vwPr, while
the dimensionless skin-friction coefficient is (3.59)cf=τwρvw2=W0(1−2Pr).


### 3.3. Magnetohydrodynamic Free Convection with
Uniform Suction

The momentum equation for this
problem is [24] (3.60)ν∂2u∂y2−vw∂u∂y−σB2ρ∞u=−gβ(Tw−T∞)exp(vwλy), where
σ is the fluid electrical
conductivity and B is the strength of magnetic field which is applied
transversely to the flow (see Figure 1). The MHD Lorentz force σB2u/ρ∞ always opposes the flow.The analytical solution of the Sakiadis-buoyant-MHD
flow for a moving plate has been produced by Vajravelu [38]. Here we will
give the analytical solution for a motionless plate and motionless ambient
fluid (pure free convection). The exact solution is (3.61)uvw=Gm(exp(Avwνy)−exp(vwλy)), Where A=0.5(1+(1+4M)1/2), M is the Hartmann
number: (3.62)M=σB2νρ∞vw2, and Gm is the
buoyancy parameter: (3.63)Gm=νgβ(Tw−T∞)vw3(Pr2−Pr−M). The
absolute wall shear stress is (3.64)τw=μgβ(Tw−T∞)(A−Pr)vw(Pr2−Pr−M), while
the dimensionless skin-friction coefficient is (3.65)cf=τwρ∞vw2=Gm(A−Pr).


### 3.4. Magnetohydrodynamic Sakiadis Flow in Water Near 4°C

The momentum equation with magnetic
field and buoyancy force expressed in density is(3.66)ν∂2u∂y2−vw∂u∂y−σB2ρ∞u=−gρ∞−ρρ∞. Taking into account (3.34), we have (3.67)ν∂2u∂y2−vw∂u∂y−σB2ρ∞u=−gγ(T−T∞)2. Substituting the temperature difference
from (2.10) into (3.67), we have (3.68)ν∂2u∂y2−vw∂u∂y−σB2ρ∞u=−gγ(Tw−T∞)2exp(2vwλy). Equation (3.68) has the following
analytical solution for Sakiadis-buoyant-MHD flow: (3.69)uuw=(1+Gsw)exp(Avwνy)−Gswexp(2vwλy), where (3.70)Gsw=νgγ(Tw−T∞)2uwvw2(4Pr2−2Pr−M).
The
absolute wall shear stress is (3.71)τw=ρ∞uwvwA+μgγ(Tw−T∞)2(A−2Pr)vw(4Pr2−2Pr−M), while
the dimensionless skin-friction coefficient is (3.72)cf=τwρ∞uwvw=A+Gsw(A−2Pr).
When the plate is motionless, we
have pure free convection and velocity is given by (3.73)uvw=G0wexp(Avwνy)−G0wexp(2vwλy), where (3.74)G0w=νgγ(Tw−T∞)2vw3(4Pr2−2Pr−M).
The
absolute wall shear stress is (3.75)τw=μgγ(Tw−T∞)2(A−2Pr)vw(4Pr2−2Pr−M), while
the dimensionless skin-friction coefficient is (3.76)cf=τwρ∞vw2=Gsw(A−2Pr).


## 3.1. Sakiadis-Blasius Flow with Buoyancy
Forces and EMHD Lorentz Forces

It is assumed that, except for the buoyancy force, an EMHD Lorentz force acts
parallel to the plate. The EMHD Lorentz force can be generated by a stripwise
arrangement of flush mounted electrodes and permanent magnets as sketched in Figure2. For more information see Weier
[20]. Then, the momentum equation (2.7) takes the following
form: (3.1)ν∂2u∂y2−vw∂u∂y=−gβ(T−T∞)−πj0M08ρ∞exp(−πay), where j0 is the applied current
density, M0 is the
magnetization of the permanent magnets, α is the width of magnets and
electrodes, and ρ∞ is the ambient fluid density. It should be
noted here that the EMHD Lorentz force can either assist or oppose the flow and
is independent of the flow field. The last term in the momentum equation is the
EMHD Lorentz force which assists the flow and decreases exponentially with  y. Substituting the temperature difference from (2.10) into (3.1), we have (3.2)ν∂2u∂y2−vw∂u∂y=−gβ(Tw−T∞)exp(vwλy)−πj0M08ρ∞exp(−πay). Equation (3.2) is a linear
differential equation of second order and has the following exact analytical
solution for the Sakiadis-Blasius-buoyant-EMHD flow: (3.3)u−u∞uw−u∞=(1+G+Z)exp(vwνy)−Gexp(vwλy)−Zexp(−πay), where G is the buoyancy parameter: (3.4)G=λgβ(Tw−T∞)(uw−u∞)vw2(Pr−1) and Z is the Lorentz parameter: (3.5)Z=j0M0a28ρ∞(uw−u∞)(πν+avw). The Prandtl number is (3.6)Pr=νλ. Both parameters G and Z are dimensionless.
The absolute wall shear
stress is (3.7)τw=ρ∞(uw−u∞)vw−μgβ(Tw−T∞)vwPr+j0M0a8,
while
the dimensionless skin-friction coefficient is (3.8)cf=τwρ∞(uw−u∞)vw=1+G(1−Pr)+Z(πRe+1), where Re is the suction Reynolds number:  (3.9)Re=avwν.
It should be noted here that this is the first work on
fluid mechanics which uses both EMHD Lorentz forces and suction simultaneously,
and therefore the above Reynolds number, using as characteristic length the
distance between the magnets, appears for the first time in the literature.Figure 2
Arrangement of
electrodes and magnets for the creation of an EMHD Lorentz forceF in the flow
along a flat plate [20].When(3.10)Z=−1−G(1−Pr)(πRe+1)−1,
the wall shear stress is zero.  In Figure 3(a), two velocity profiles are shown for which the wall shear stress is zero (3.10)
is valid).  The transverse coordinate η is (3.11)η=−vwνy,
and the two curves meet the vertical axis orthogonally.Dimensionless
velocity distribution for different kinds of flow with buoyancy forces and
Lorentz forces. In cases (a) and (b), the wall shear stress is zero.
(a)(b)(c)(d)When the free-stream velocity is zero, we have the Sakiadis-buoyant-EMHD
flow:(3.12)uuw=(1+Gs+Zs)exp(vwνy)−Gsexp(vwλy)−Zsexp(−πay), where (3.13)Gs=λgβ(Tw−T∞)uwvw2(Pr−1),Zs=j0M0a28ρ∞uw(πν+avw). The  absolute wall shear stress  is (3.14)τw=ρ∞uwvw−μgβ(Tw−T∞)vwPr+j0M0a8,
while
the dimensionless skin-friction coefficient is (3.15)cf=τwρuwvw=1+Gs(1−Pr)+Zs(πRe+1).
When  (3.16)Zs=−1−Gs(1−Pr)(πRe+1)−1, the wall shear stress is  zero.When the plate is motionless, we have the Blasius-buoyant-EMHD
flow:(3.17)uu∞=1−(1−Gb−Zb)exp(vwνy)−Gbexp(vwλy)−Zbexp(−πay), where (3.18)Gb=λgβ(Tw−T∞)u∞vw2(Pr−1),Zb=j0M0a28ρ∞u∞(πν+avw).
The absolute wall shear stress is (3.19)τw=−ρ∞u∞vw−μgβ(Tw−T∞)vwPr+j0M0a8, while
the dimensionless skin-friction coefficient is (3.20)cf=τwρ∞u∞vw=−1+Gb(1−Pr)+Zb(πRe+1).
When  (3.21)Zb=1−Gb(1−Pr)(πRe+1)−1,
the wall shear stress is  zero.When the plate is motionless and the free-stream velocity is zero, we
have pure free convection and the analytical solution becomes(3.22)uvw=(G0+Z0)exp(vwνy)−G0exp(vwλy)−Z0exp(−πay), where (3.23)G0=λgβ(Tw−T∞)vw3(Pr−1),Z0=j0M0a28ρ∞vw(πν+avw).
In (3.22), the longitudinal velocity u is nondimensionalized with suction velocity
vw because in the above problem the suction velocity is the apparent
characteristic velocity of the flow.The absolute wall shear stress is(3.24)τw=−μgβ(Tw−T∞)vwPr+j0M0a8, while
the dimensionless skin-friction coefficient is (3.25)cf=τwρvw2=G0(1−Pr)+Z0(πRe+1).
When  (3.26)Z0=G0(Pr−1)(πRe+1)−1, the wall shear stress is zero.  In Figure 3(b), two velocity profiles are shown for which the wall shear stress is zero. The velocity profiles near the plate for cases (a) and (b) are shown in Figure 4.The velocity
profiles near the plate for cases (a) and (b) where it is clearly shown that the wall
shear stress is zero.
(a)(b)When there is no Lorentz force, we have convection due to buoyancy force
only and velocity is  given by(3.27)uvw=G0exp(vwνy)−G0exp(vwλy). The absolute wall shear stress is (3.28)τw=−μgβ(Tw−T∞)vwPr,
while
the dimensionless skin-friction coefficient is (3.29)cf=τwρvw2=G0(1−Pr).
When there is no buoyancy force, we have convection due to EMHD Lorentz
force only and velocity is given by (3.30)uvw=Z0exp(vwνy)−Z0exp(−πay). This is a new “strange” kind of
convection caused by the EMHD Lorentz force only. The physical meaning of the
above equation is as follows. Let us suppose that we have a porous plate where
fluid is removed with uniform suction. Although the plate is motionless and the
ambient fluid is at rest, a boundary layer forms along the plate due to the act
of the Lorentz force which is independent of the flow.The absolute value of wall shear
stress is(3.31)τw=j0M0a8,
while
the dimensionless skin-friction coefficient is (3.32)cf=τwρvw2=Z0(πRe+1). It is surprising that the absolute
wall shear stress is independent of viscosity and suction velocity.


## 3.2. Sakiadis-Blasius Flow with Buoyancy
Forces and EMHD Lorentz Forces in Water Near 4°C

The momentum equation with EMHD Lorentz
force and the buoyancy force expressed in density is(3.33)ν∂2u∂y2−vw∂u∂y=−gρ∞−ρρ∞−πj0M08ρ∞exp(−πay). The buoyancy term can be calculated from the following
equation [37]: (3.34)ρ∞−ρρ∞=γ(T−T∞)2, where γ=0.8×10−5(C°)−2.  Equation (3.34) is valid for the temperature
range of 0°C≤T≤8°C,
and T∞=4°C
where the variation of water density is nearly parabolic. Therefore, (3.33)
becomes (3.35)ν∂2u∂y2−vw∂u∂y=−gγ(T−T∞)2−πj0M08ρexp(−πay). Substituting the temperature difference
from (2.10) into (3.35), we have (3.36)ν∂2u∂y2−vw∂u∂y=−gγ(Tw−T∞)2exp(2vwλy)−πj0M08ρexp(−πay). The
exact solution of this equation is (3.37)u−u∞uw−u∞=(1+W+Z)exp(vwνy)−Wexp(2vwλy)−Zexp(−πay), where (3.38)W=λgγ(Tw−T∞)22(uw−u∞)vw2(2Pr−1).
The absolute wall shear stress is (3.39)τw=ρ∞(uw−u∞)vw−μgγ(Tw−T∞)22vwPr+j0M0a8,
while
the dimensionless skin-friction coefficient is (3.40)cf=τwρ∞(uw−u∞)vw=1+W(1−2Pr)+Z(πRe+1).
The wall shear stress becomes zero when (3.41)Z=−1+W(2Pr−1)(πRe+1)−1. When the free-stream velocity is zero, we have the Sakiadis-buoyant-EMHD
flow: (3.42)uuw=(1+Ws+Zs)exp(vwνy)−Wsexp(2vwλy)−Zsexp(−πay), where (3.43)Ws=λgγ(Tw−T∞)22uwvw2(2Pr−1).
The absolute wall shear stress is (3.44)τw=ρ∞uwvw−μgγ(Tw−T∞)22vwPr+j0M0a8, while
the dimensionless skin-friction coefficient is (3.45)cf=τwρ∞uwvw=1+Ws(1−2Pr)+Zs(πRe+1).
The wall shear stress becomes zero when (3.46)Zs=−1+Ws(2Pr−1)(πRe+1)−1. When the plate is motionless, we
have the Blasius-buoyant-EMHD flow: (3.47)uu∞=1−(1−Wb−Zb)exp(vwνy)−Wbexp(2vwλy)−Zbexp(−πay), where (3.48)Wb=λgγ(Tw−T∞)22u∞vw2(2Pr−1). The absolute wall shear stress is (3.49)τw=−ρ∞u∞vw−μgγ(Tw−T∞)22vwPr+j0M0a8,
while
the dimensionless skin-friction coefficient is (3.50)cf=τwρu∞vw=−1+Wb(1−2Pr)+Zb(πRe+1). The wall shear stress becomes zero when (3.51)Zb=1+Wb(2Pr−1)(πRe+1)−1. When the plate is motionless and the
free-stream velocity is zero, we have pure free convection and the analytical
solution becomes (3.52)uvw=(W0+Z0)exp(vwνy)−W0exp(2vwλy)−Z0exp(−πay), where (3.53)W0=λgγ(Tw−T∞)22vw3(2Pr−1).
The absolute wall shear stress is (3.54)τw=−μgγ(Tw−T∞)22vwPr+j0M0a8, while
the dimensionless skin-friction coefficient is (3.55)cf=τwρvw2=W0(1−2Pr)+Z0(πRe+1).
The wall shear stress becomes zero when (3.56)Z0=W0(2Pr−1)(πRe+1)−1. When there is no EMHD Lorentz force,
we have convection due to buoyancy force only and velocity is given by (3.57)uvw=W0exp(vwνy)−W0exp(2vwλy). The absolute
wall shear stress is (3.58)τw=−μgγ(Tw−T∞)22vwPr, while
the dimensionless skin-friction coefficient is (3.59)cf=τwρvw2=W0(1−2Pr).


## 3.3. Magnetohydrodynamic Free Convection with
Uniform Suction

The momentum equation for this
problem is [24] (3.60)ν∂2u∂y2−vw∂u∂y−σB2ρ∞u=−gβ(Tw−T∞)exp(vwλy), where
σ is the fluid electrical
conductivity and B is the strength of magnetic field which is applied
transversely to the flow (see Figure 1). The MHD Lorentz force σB2u/ρ∞ always opposes the flow.The analytical solution of the Sakiadis-buoyant-MHD
flow for a moving plate has been produced by Vajravelu [38]. Here we will
give the analytical solution for a motionless plate and motionless ambient
fluid (pure free convection). The exact solution is (3.61)uvw=Gm(exp(Avwνy)−exp(vwλy)), Where A=0.5(1+(1+4M)1/2), M is the Hartmann
number: (3.62)M=σB2νρ∞vw2, and Gm is the
buoyancy parameter: (3.63)Gm=νgβ(Tw−T∞)vw3(Pr2−Pr−M). The
absolute wall shear stress is (3.64)τw=μgβ(Tw−T∞)(A−Pr)vw(Pr2−Pr−M), while
the dimensionless skin-friction coefficient is (3.65)cf=τwρ∞vw2=Gm(A−Pr).


## 3.4. Magnetohydrodynamic Sakiadis Flow in Water Near 4°C

The momentum equation with magnetic
field and buoyancy force expressed in density is(3.66)ν∂2u∂y2−vw∂u∂y−σB2ρ∞u=−gρ∞−ρρ∞. Taking into account (3.34), we have (3.67)ν∂2u∂y2−vw∂u∂y−σB2ρ∞u=−gγ(T−T∞)2. Substituting the temperature difference
from (2.10) into (3.67), we have (3.68)ν∂2u∂y2−vw∂u∂y−σB2ρ∞u=−gγ(Tw−T∞)2exp(2vwλy). Equation (3.68) has the following
analytical solution for Sakiadis-buoyant-MHD flow: (3.69)uuw=(1+Gsw)exp(Avwνy)−Gswexp(2vwλy), where (3.70)Gsw=νgγ(Tw−T∞)2uwvw2(4Pr2−2Pr−M).
The
absolute wall shear stress is (3.71)τw=ρ∞uwvwA+μgγ(Tw−T∞)2(A−2Pr)vw(4Pr2−2Pr−M), while
the dimensionless skin-friction coefficient is (3.72)cf=τwρ∞uwvw=A+Gsw(A−2Pr).
When the plate is motionless, we
have pure free convection and velocity is given by (3.73)uvw=G0wexp(Avwνy)−G0wexp(2vwλy), where (3.74)G0w=νgγ(Tw−T∞)2vw3(4Pr2−2Pr−M).
The
absolute wall shear stress is (3.75)τw=μgγ(Tw−T∞)2(A−2Pr)vw(4Pr2−2Pr−M), while
the dimensionless skin-friction coefficient is (3.76)cf=τwρ∞vw2=Gsw(A−2Pr).


## 4. Conclusions

In this paper, the
boundary layer buoyant flow along a vertical porous plate with uniform suction
has been treated. Exact analytical solutions have been found for Blasius,
Sakiadis, and combined Blasius-Sakiadis flows with MHD and EMHD Lorentz forces.
A series of new dimensionless parameters (G, Z, W, Re) are introduced for the
presentation of the results in elegant form. The author believes that the results
of the present work will enrich the list with the existing exact solutions of
the Navier-Stokes equations and may help the investigation of flow separation
control in aeronautics and hydronautics as well as the application of
magnetohydrodynamics in industry.



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*Source: 149272-2008-10-21.xml*