Con - ΠΑΝΕΠΙΣΤΗΜΙΟ...
Transcript of Con - ΠΑΝΕΠΙΣΤΗΜΙΟ...
Contents
�� CREEPING BIDIRECTIONAL FLOWS �
���� Plane Flow in Polar Coordinates �
���� Axisymmetric Flow in Cylindrical Coordinates ��
���� Axisymmetric Flow in Spherical Coordinates ��
���� Problems �
���� References �
i
ii CONTENTS
Chapter ��
CREEPING
BIDIRECTIONAL FLOWS
Consider the nondimensionalized� steady Navier�Stokes equation in the absence ofbody forces�
Re u � ru� � �rp � r�u � �����
where
Re � �V L
� �����
is the Reynolds number� When the motion of the �uid is �very slow�� the Reynoldsnumber is vanishingly small�
Re� � �
and the �ow is said to be creeping or Stokes �ow� In other words� creeping �owsare those dominated by viscous forces� the nonlinear inertia term� Re u � ru�� isnegligible compared to the linear viscous term� r�u� The Navier�Stokes equationmay then be approximated by the Stokes equation�
�rp � r�u � �� �����
This linear equation� together with the continuity equation�
r � u � � � �����
can be solved analytically for a broad range of problems� Flows at small� butnonzero� Reynolds numbers are amenable to regular perturbation analysis� TheStokes �ow solution can thus be viewed as the zeroth order approximation to thesolution� in terms of the Reynolds number�
Note that Eq� ����� also holds true for steady� unidirectional �ows which arenot necessarily creeping� in this case� the inertia term� u � ru� is identically zero� Asimilar observation applies to lubrication �ows� The inertia term is negligible when
� Re� � �
�
� Chapter ��� Creeping Bidirectional Flows
where � is the inclination of the channel and the Reynolds number is not necessarilyvanishingly small�
Reversibility is an important property of Stokes �ow� If u and p satisfy Eqs� �����and ������ then it is evident that the reversed solution� �u and �p� also satis�esthe same equations� The reversed �ow is obtained by using �reversed� boundaryconditions� e�g�� u��f r� instead of u�f r� along a boundary etc� Reversibility islost� once the nonlinear convective term� Re u � ru�� becomes nonzero�
Laminar� slow �ow approaching and de�ected by a submerged stationary sphereor cylinder� or� equivalently� �ow induced by a slowly traveling sphere or cylinderin a bath of still liquid� are representative examples of creeping �ow� These �owsare important in particle mechanics ��� and apply to air cleaning devices from parti�cles ���� to centrifugal or sedimentation separators� to �uidized�bed reactors� and tochemical and physical processes involving gas�bubbles or droplets ���� Slow �ows inconverging or diverging channels and conical pipes� are also examples of importantcreeping �ows ���� Finally� �ows in the vicinity of corners and other geometricalsingularities� are creeping� being retarded by the encounter with the solid bound�aries ����
This chapter is devoted to creeping� incompressible� bidirectional �ows� Anexcellent analytical tool for solving such �ows is the stream function� Consider� forexample� the creeping bidirectional �ow on the xy�plane� for which
ux � ux x� y� � uy � uy x� y� and uz � � �
For incompressible �ow� the continuity equation takes the following form�
�ux�x
��uy�y
� � � �����
the two non�zero components of the Navier�Stokes equation become
� �p
�x�
���ux�x�
���ux�y�
�� � � ����
and
� �p
�y�
���uy�x�
���uy�y�
�� � � �����
Hence� the �ow is governed by a system of three PDEs corresponding to the threeunknown �elds� p� ux and uy�
The continuity equation is automatically satis�ed by introducing Lagrange�s
stream function � x� y�� such that
ux � ����y
and uy ���
�x� �����
�
The pressure� p� can be eliminated by di�erentiating Eqs� ���� and ����� withrespect to y and x� respectively� and by subtracting one equation from the other�Substituting ux and uy � in terms of �� into the resulting equation leads to
���
�x�� �
���
�x��y��
���
�y�� � � ����
Recalling that the Laplace operator in Cartesian coordinates is de�ned by
r� � ��
�x��
��
�y�� ������
Eq� ���� can be written in the more concise form
r�� � r��r��
�� � � ������
The di�erential operator r�� de�ned by
r� � r��r�
�� ������
is called the biharmonic operator� Equation ������ is referred to as the biharmonicor Stokes equation�
The advantage of using the stream function is that� instead of a system of threePDEs for the three unknown �elds� ux� uy and p� we have to solve a single PDE forthe new dependent variable� �� The price we pay is that the highest derivatives ofthe governing equation are now fourth�order instead of second�order� Once � x� y�is calculated� the velocity components can be obtained by means of Eqs� ������ Thepressure� p� can then be calculated by integrating the momentum equations ����and ������
In the following three sections� we consider the use of the stream function forthree classes of creeping� incompressible� bidirectional �ows�
a� Plane �ows in polar coordinates�
b� Axisymmetric �ows in cylindrical coordinates� and
c� Axisymmetric �ows in spherical coordinates�
The various forms of the stream function and the resulting fourth�order PDEs aretabulated in Table ����� It should be noted that the use of the stream function isnot restricted to creeping �ows� The full forms of the momentum equation in termsof the stream function� for all the aforementioned classes of �ow� can be found inRef� ���
� Chapter ��� Creeping Bidirectional Flows
Plane �ow in Cartesian coordinates
Assumptions� ux � ux x� y�� uy � uy x� y�� uz � �
Stream function� ux � ����y
� uy ����x
Momentum equation� r�� � r��r��
�� �
r� � ��
�x�� ��
�y�� r� � ��
�x�� � ��
�x��y�� ��
�y�
Plane �ow in polar coordinates
Assumptions� ur � ur r� ��� u� � u� r� ��� uz � �
Stream function� ur � ��r����
� u� ����r
Momentum equation� r�� � r��r��
�� �
r� � ��
�r�� �r��r
� �r�
��
���
Axisymmetric �ow in cylindrical coordinates
Assumptions� uz � uz r� z�� ur � ur r� z�� u� � �
Stream function� uz � ��r���r
� ur ��r���z
Momentum equation� E�� � E��E��
�� �
E� � ��
�r�� �r��r
� ��
�z�
Axisymmetric �ow in spherical coordinates
Assumptions� ur � ur r� ��� u� � u� r� ��� u� � �
Stream function� ur � � �r� sin��
����
� u� ��
r sin����r
Momentum equation� E�� � E��E��
�� �
E� � ��
�r�� sin�
r����
��
sin����
�
Table ����� Stokes �ow equations in terms of the stream function�
Sec� ����� Plane Flow in Polar Coordinates �
���� Plane Flow in Polar Coordinates
In this section� we consider two�dimensional creeping incompressible �ows in polarcoordinates in which
ur � ur r� �� and u� � u� r� �� � ������
The continuity equation becomes
�
�r rur� �
�u���
� � � ������
and is automatically satis�ed by a Stokes stream function � r� ��� such that
ur � ��
r
��
��and u� �
��
�r� ������
Eliminating the pressure from the r� and ��components of the Navier�Stokes equa�tions� we obtain the biharmonic equation see Problem �����
r�� � r��r��
�� � � �����
Recall that� in polar coordinates� the Laplace operator is given by
r� � ��
�r��
�
r
�
�r�
�
r���
���� ������
As demonstrated by Lugt and Schwiderski ���� Eq� ����� admits separatedsolutions of the form
� r� �� � r��� f� �� � ������
where the exponent may be complex� For the Laplacian of �� we get
r�� � r���hf ��� �� � � ��� f� ��
i�
where the primes designate di�erentiation with respect to �� Another application ofthe Laplace operator yields
r�� � r���nf ����� �� �
h � ��� � � ���
if ��� �� � � ��� � ��� f� ��
o�
Due to Eq� ������
f ����� �� �h � ��� � � ���
if ��� �� � � ��� � ��� f� �� � � � �����
Chapter ��� Creeping Bidirectional Flows
This is a linear� homogeneous� fourth�order ordinary di�erential equation� the char�acteristic equation of which ish
m� � � ���i h
m� � � ���i� � �
Hence� the general solution for f� �� is
f� �� � A� cos �����B� sin ����� C� cos �����D� sin ���� � ������
where the constants A�� B�� C� and D� may be complex�Therefore� the general solution of Eq� ����� is
� r� �� � r��� �A� cos � ��� �B� sin � ��� � C� cos � ��� �D� sin � ���� � ������
The two velocity components are now easily obtained�
ur r� �� � �r� ��A� � �� sin � ��� � B� � �� cos � ���
� C� � �� sin � ��� � D� � �� cos � ���� � ������
u� r� �� � ���r� �A� cos � ��� � B� sin � ���
� C� cos � ��� � D� sin � ���� � ������
The pressure p is calculated by integrating the r� and ��momentum equations seeProblem ������
p � ��� r��� �C� sin � ��� � D� cos � ���� � ������
In general� there are in�nitely many admissible values of which depend on thegeometry and the boundary conditions� Since the problem is linear� these solutionsmay be superimposed� The particular solutions to Eq� ����� for ���� � and �are considered in Problem ����� A particular solution independent of � is given inProblem �����
Example ������� Flow near a cornerConsider �ow of a viscous liquid between two rigid boundaries �xed at an angle �� Fig� ������ Since the velocity on the two walls is zero� inertia terms are negligiblenear the neighborhood of the corner� Therefore� the �ow can be assumed to belocally creeping� The solution to this �ow problem was determined by Dean andMontagnon ���� The stream function is expanded in a series of the form
� r� �� ��Xk��
��k ��Xk��
a�k r�k�� f�k �� � ������
���� Plane Flow in Polar Coordinates
where the polar coordinates r� �� are centered at the vertex� The exponents k aresuitably ordered so that
� Re �� Re �� � � � �The �rst of the inequalities ensures that the velocity vanishes at the corner� Thesecond one ensures that the �rst term in the summation will dominate� unless a�����
r
����
���
����
�������� ��
���� ����
��
r����
Figure ����� Creeping �ow near a corner�
As pointed out by Dean and Montagnon ���� a disturbance far from the cornercan generate either an antisymmetric or a symmetric �ow pattern near the corner�and the corresponding stream function is an even or odd function of �� respectively�Taking advantage of the linearity of the Stokes equation� we study the two types of�ow separately�
Antisymmetric �ow near a corner
For this type of �ow� f� �� is even B��D���� and
f� �� � A� cos � ��� � C� cos � ��� � �����
The boundary conditions ur�u��� on ���� demand that
f� �� � f �� �� � � on � � �� �which gives the following two equations�
A� cos � ��� � C� cos � ��� � �
A� � �� sin � ��� � C� � �� sin � ��� � �
� Chapter ��� Creeping Bidirectional Flows
For a nontrivial solution for A� and C� to exist� the determinant of the coe�cientmatrix must be zero������ cos � ��� cos � ���
� �� sin � ��� � �� sin � ���
����� � � �
With a little manipulation� we get the following eigenvalue equation
sin �� � � sin �� � ������
Figure ����� Sketch of Mo�att�s vortices in a sharp corner�
With the obvious exception of the trivial solution ��� the eigenvalues arenecessarily complex� when �� is less than approximately ����o� This implies theexistence of an in�nite sequence of eddies near the corner� These were predictedanalytically by Mo�att ��� A sketch of Mo�att�s vortices is shown in Fig� ����� Thestrength of these vortices decays exponentially as the corner is approached� Theratio of the distance of the centers of successive vortices is given by
riri��
� e��q� �
where q� is the imaginary part of the leading eigenvalue� ��p��iq�� Table ����provides the real and imaginary parts of � for various values of ���
For values of �� greater than ����o� all the solutions of Eq� ������ are real� Asshown in Table ����� the value of the leading exponent � decreases with the angle��� When ������o� ��� which corresponds to simple shear �ow� For values of ��greater than ���o� � is less than unity� From Eqs� ������ and ������� we deducethat� in such a case� the velocity derivatives and� consequently� the pressure and
���� Plane Flow in Polar Coordinates �
�� p� q� in o�
���� ����� �������� ����� �������� ���� �����
����� ����� �������� ����� ����������� ���������� ����������� ����������� ���������� ����������� ����������� ���������� ������
Table ����� Real and imaginary parts of the leading exponent� ��p��iq�� forantisymmetric �ow near a corner�
the stress components go to in�nity at the corner� This is an example of a stress
singularity that is caused by the nonsmoothness of the boundary� The strongestsingularity appears at �����o� In this case� ����� which corresponds to �owaround a semi�in�nite �at plate�
Symmetric �ow near a corner
For this type of �ow� f� �� is odd A��C���� and
f� �� � B� sin � ��� � D� sin � ��� � ������
In this case� the eigenvalue equation is
sin �� � sin �� � �����
The real and imaginary parts of the leading exponent� ��p��iq�� are tabulatedin Table �� for various values of the angle ��� The trivial solutions ��� and �are not taken into account�� For �� greater than approximately ����o� Eq� �����admits only real solutions� For ������o� ��� which corresponds to orthogonalstagnation�point �ow� �
�� Chapter ��� Creeping Bidirectional Flows
�� p� q� in o�
���� ������� ������� ������ ��������� ������ ��������� ����� ����������� ���� ��������� ������ ����������� ����������� ����������� ���������� ���������� ���������� ��������� ������
Table ����� Real and imaginary parts of the leading exponent� ��p��iq�� forsymmetric �ow near a corner�
Example ������� Intersection of a wall and a free surfaceDue to symmetry� the preceding analysis of creeping �ow near a corner also holdsfor �ow near the intersection of a rigid boundary and a free surface positioned at��� Fig� ������ Along the free surface�
u� � � and �r� � �
�r�
�r
u�r
�
�
r
�ur��
�� � �
consequently�
u� ��ur��
� � �� � ����
���� � �
Using physical arguments� Michael ���� showed that the angle of separation� ��cannot take arbitrary values� He showed that� when the surface tension is zero� �mus be equal to �� Therefore� we focus on the case of �ow near a wall and a freesurface meeting at an angle �� as in Fig� �����
Even set of solutions
Even solutions�
�� � r��� �A� cos � ��� � C� cos � ���� �
���� Plane Flow in Polar Coordinates ��
r
����
�������
����
��
���� ������
��
r����
Free surface
Figure ����� Creeping �ow near the intersection of a wall and a free surface�
r
�
���� ���� �� ���� ���
�����
r����
Wall Free surface
Figure ����� Creeping �ow near a wall and a free surface meeting at an angle ��
satisfy automatically the conditions ur�� at ��� and u��� at ��� see Fig� �����for the de�nition of ��� The condition u��� at ��� demands that
A� cos � ��� � C� cos � ��� � � at � � � �
which yields C���A�� Finally� the condition �r��� at ��� leads to the followingequation
� ��� cos � ��� � � ��� cos � ��� � � �
From standard trigonometric identities� we get
� � � �� sin� sin � � � cos� cos� � � ��
cos� � � �� ��
���
�� � � �
Therefore� even solutions are given by
�� � a� r��� �cos � ��� � cos � ���� � �
�
���
�� � � � ������
�� Chapter ��� Creeping Bidirectional Flows
where a��A���C�� The corresponding velocity components and the pressure are�
ur � �a� r� �� � �� sin � ��� � � �� sin � ���� � ������
u� � a� � �� r� �cos � ��� � cos � ���� � ������
p � �a�� r��� sin � ��� � ������
Note that the leading term of the pressure is characterized by an inverse�square�rootsingularity�
p � �a��� ��prsin
�
��
This is also true for the velocity derivatives and the stress components� This isan example of a stress singularity caused by the sudden change of the boundarycondition along a smooth boundary�
Odd set of solutions
Odd solutions�
�� � r��� �B� sin � ��� � D� sin � ���� �
satisfy automatically the conditions u��� at ��� and �r��� at ���� The conditionur�� at ��� requires that
� ��B� � � ��D� � � �
The remaining condition u��� at ��� leads to
� �� sin � ��� � � �� sin � ��� � � ��
� cos� sin� � � sin � sin � � � ��
sin � � � �� � �� �� � � �Note that the trivial solution �� has been omitted� Therefore� odd solutions areof the form
�� � a� r��� � � �� sin � ��� � � �� sin � ���� � � �� �� � � � ������
���� Plane Flow in Polar Coordinates ��
where � � ��a�� � ��B��� � ��D�� The corresponding solutions for ur� u�and p are�
ur � �a� � � �� r� �cos � ��� � cos � ���� � ������
u� � a� � �� r� � � �� sin � ��� � � �� sin � ���� � �����
p � ��a�� � �� r��� cos � ��� � ������
�
Wallux�uy��
Free surface�yx�uy��
Symmetry plane
Fullydeveloped�ow
Uniform�ow
x
ySingular point
DIE EXTRUDATE
Figure ����� The plane stick�slip problem��
It should be noted that the solutions discussed in the previous two exampleshold only locally� The constants a� and b� are determined from the boundaryconditions of the global problem� Consider� for example� the so�called plane stick�
slip problem� illustrated in Fig� ����� This problem owes its name to the fact thatthe boundary conditions change suddenly at the exit of the die� from no�slip to slip�The stick�slip problem is the special case of the extrudate�swell problem in the limitof in�nite surface tension which causes the free surface to be �at� The singularsolution obtained in Example ������ holds in the neighborhood of the exit of thedie� The leading term of the local solution is
���� � a��� r���
cos
��
�� cos
�
�
�
where the polar coordinates r� �� are centered at the exit of the die� The planestick�slip problem was solved analytically by Richardson ����� It turns out thata����
p� ���������
As already mentioned� the velocity derivatives and the stresses corresponding tothe leading term of the local solution are characterized by an inverse�square�root
�� Chapter ��� Creeping Bidirectional Flows
singularity� This has a negative e�ect on the performance of standard numericalmethods used to model the stick�slip or the extrudate�swell� �ow� The rate ofconvergence with mesh re�nement and the accuracy are� in general� poor in theneighborhood of stress singularities� The strength of the singularity may be allevi�ated by using slip along the wall which leads to nonsingular �nite stresses ��������Alternatively� special numerical techniques� such as singular �nite elements ��������or special mesh re�nement methods ��� must be employed� in order to get accurateresults in the neighborhood of the singularity�
���� Axisymmetric Flow in CylindricalCoordinates
Consider a creeping axisymmetric incompressible �ow in cylindrical coordinates suchthat
uz � uz r� z� � ur � ur r� z� and u� � � � ������
It is easily shown that the stream function � r� z�� de�ned by
uz � ��
r
��
�rand ur �
�
r
��
�z� �����
satis�es the continuity equation identically� Substituting uz and ur into the z� andr�components of the Navier�Stokes equation� and eliminating the pressure lead tothe following equation Problem ������
E�� � E��E��
�� � � ������
where the di�erential operator E� is de�ned by
E� � ��
�r�� �
r
�
�r�
��
�z�� ������
Separating the axial from the radial dependence and stipulating a power�law func�tional dependence on r� we seek a solution to Eq� ������ of the form
� � r� f z� � ������
Applying the operator E� to the above solution yields
E�� � � �� r��� f z� � r� f �� z� �
���� Axisymmetric Flow in Cylindrical Coordinates ��
where the primes denote di�erentiation with respect to z� Applying the operatorE� once again� we get
E�� � � ��� � �� r��� f z� � � � �� r��� f �� z� � r� f ���� z� �
Due to the Stokes �ow Eq� ������� the only admissible values of are � and �� Forboth values� we get the simple fourth�order ODE
f ���� z� � � �
the general solution of which is
f z� � c� � c�z � c�z� � c�z
� � ������
The value �� corresponds to the solution ��f z� which is independent of r�Let us� however� focus on the more interesting case of ��� in which we have
� r� z� � r��c� � c�z � c�z
� � c�z��� ������
The values of the constants c�� c�� c� and c� are determined from the boundaryconditions� For the velocity components� we get�
uz r� z� � ��
r
��
�r� �� f z� � ��
�c� � c�z � c�z
� � c�z�� ������
ur r� z� ��
r
��
�z� r f � z� � r
�c� � �c�z � �c�z
��
�����
It can be shown that the z� and r�components of the Navier�Stokes become
��p�z
� ���uz�z�
� � and � �p
�r� �
��ur�z�
� �
or�p
�z� ��� f �� z� and
�p
�r� � r f ��� z� �
respectively� Integration of the above two equations yields
p r� z� � ��� c���z� � r�
�� �� c� z � c � ������
where c is a constant�
Example ������� Axisymmetric squeezing �owSqueezing �ows are induced by externally applied normal stresses or vertical ve�locities by means of a mobile boundary� The induced normal velocity propagates
� Chapter ��� Creeping Bidirectional Flows
within the liquid due to incompressibility� and changes direction� due to obstaclesto normal penetration� The most characteristic example is Stefan�s squeezing �ow
����� illustrated in Fig� ���� The vertically moving fronts meet the resistance ofthe inner liquid layers and are de�ected radially� For small values of the velocityV of the two plates� the gap �H changes slowly with time and can be assumed tobe constant� that is the �ow can be assumed to be quasi�steady� If in addition� the�uid is highly viscous� then the creeping �ow approximation is a valid assumption�
r
z
V
V
H
R
Figure ���� Squeezing �ow�
Introducing the cylindrical coordinates shown in Fig� ��� and employing thestream function de�ned in Eq� ������ we can make use of the previous analysis�The stream function is thus given by Eq� �������
� r� z� � r� f z� � r��c� � c�z � c�z
� � c�z���
The four constants are determined from the boundary conditions� At z��� symmetryrequires that uz��ur �z��� and therefore
f z� � f �� z� � � at z � � �
consequently c��c���� At z�H � uz�V and ur�� which gives
c�H � c�H� �
V
�and c� � �c�H
� � � �
Sec� ����� Axisymmetric Flow in Spherical Coordinates �
Solving for c� and c�� we get�
c� ��V
�Hand c� � � V
�H��
Therefore�
f z� �V
�
��
z
H
�z
H
� �
The stream function and the two velocity components are given by
� �V
�r�
��
z
H
�z
H
� � ������
uz � �V�
��
z
H
�z
H
� � �����
ur � � �V
�Hr
���
z
H
� � ������
Finally� from Eq� ������� the pressure distribution is
p r� z� ���V
�H�
��z� � r�
�� c �
Assuming that p�p� at r�R and z��� we �nd that
p r� z� ���V
�H�
��z� � R� � r�
�� p� � ������
�
���� Axisymmetric Flow in SphericalCoordinates
In this section� we consider the case of axisymmetric �ow in spherical coordinates�such that
ur � ur r� �� � u� � u� r� �� and u� � � � ������
The Stokes stream function� de�ned by
ur � � �
r� sin�
��
��and u� �
�
r sin�
��
�r� ������
�� Chapter ��� Creeping Bidirectional Flows
satis�es the continuity equation identically� Substituting Eqs� ������ into the r�and ��momentum equations and eliminating the pressure� we obtain Problem �����
E�� � E��E��
�� � � ������
where the di�erential operator E� is de�ned by
E� � ��
�r��
sin�
r��
��
�
sin�
�
��
� ������
Example ������� Creeping �ow past a �xed sphereAs already mentioned� �ows around submerged bodies are of great importance in aplethora of applications� The most important example of axisymmetric �ow is thevery slow �ow past a �xed sphere� illustrated in Fig� ����� A practically unboundedviscous incompressible �uid approaches� with uniform speed U � a sphere of radius R�The sphere is held stationary by some applied external force� Clearly� the resulting�ow is axisymmetric with u����
r
�
R
x
y
U
Figure ���� Creeping �ow past a sphere�
The boundary conditions for this �ow are as follows� a� On r�R� ur�u���� In terms of the stream function de�ned in Eq� ������� weget
��
���
��
�r� � on r � R � �����
b� As r��u � U i � U cos� er � sin� e��
which givesur � U cos� and u� � �U sin� as r� �
���� Axisymmetric Flow in Spherical Coordinates ��
In terms of �� we get
��
��� �Ur� sin� cos� and
��
�r� �Ur sin�� as r � �
Integrating the above two equations� we get
� � �U�r� sin�� as r � � ������
The above condition suggests seeking a separated solution to Eq� ������ of theform
� r� �� � Uf r� sin�� � ������
In terms of f r�� the boundary conditions ����� and ������ become�
f r� � f � r� � � at r � R �����
and
f r� � ��
�r� as r � � �����
Applying the operator E� to the separated solution ������ yields
E�f r� sin�� � sin��
�d�
dr�� �
r�
�f r� �
and thus
E�f r� sin�� � sin��
�d�
dr�� �
r�
��
f r� �
From Eq� ������� we get
�d�
dr�� �
r�
��
f r� � � � �����
This equation is homogeneous in r and is known to have solutions of the formf r��r�� Substituting into Eq� ������ we get
� � ��� �� � � �� � ��� �� r��� � � �
The admissible values of are� therefore� the roots of the equation
� � ��� �� � � �� � ��� �� � � � �����
�� Chapter ��� Creeping Bidirectional Flows
i�e�� ���� �� � and �� The general solution for f is then
f r� �A
r� Br � Cr� � Dr� � �����
For the boundary condition ����� to be satis�ed� we must have
C � ��
�and D � � �
From the boundary conditions ������ we then get
AR � BR � �
�R�
� AR� � B � R
������� �� A � ��
�and B �
�
�R �
Therefore�
f r� � �UR�
�
��
r
R
�� �
r
R
�
R
r
�����
and
� r� �� � �UR�
�
��
r
R
�� �
r
R
�
R
r
sin�� � �����
������
������
���
Figure ����� Calculated streamlines of creeping �ow past a sphere�
The two velocity components become�
ur �U
�
�� � �
R
r
�
R
r
� cos� � ����
���� Axisymmetric Flow in Spherical Coordinates ��
u� � �U�
�� � �
R
r
�R
r
� sin� � �����
Note that reversing the direction of the �ow leads to a change of the sign of ueverywhere� In Fig� ����� we show streamlines as predicted by Eq� ������ Theseare symmetric fore and aft of the sphere�
A quantity of major interest is the drag force� FD� on the sphere� For its cal�culation� we need to know the pressure and the stress components� To obtain thepressure� we �rst substitute ur and u� into the r� and ��components of the Navier�Stokes equation which yields
�p
�r� �� RU
cos�
r�and
�p
���
�
�� RU
sin�
r��
We then integrate the above equations getting
p r� �� � p� � �
�� RU
cos�
r�� �����
where p� is the uniform pressure at in�nity� Therefore� on the sphere�
p R� �� � p� � �
�
�U
Rcos� � ����
The rr� and r��components of the total stress tensor on the surface of the sphereare�
Trr � �p � �rr � �p � ���ur�r
� �p� ��
�
�U
Rcos� � � � �p� �
�
�
�U
Rcos�
��rr�� due to Eq� ������ and
Tr� � �r� � �
�r�
�r
u�r
�
�
r
�ur��
�� ��
�
�U
Rsin� �
The force per unit area exerted on the sphere is given by
f � er �T � Trr er � Tr� e� � Trr cos� i� sin� j� � Tr� � sin� i� cos� j� ��
f � Trr cos� � Tr� sin�� i � Trr sin� � Tr� cos�� j �
Due to symmetry� the net force� f � on the sphere is in the direction i of the uniform�ow� i�e��
f � i � f � Trr cos� � Tr� sin� �
�� Chapter ��� Creeping Bidirectional Flows
Substitution of Trr and Tr� leads to
f � �p� cos� ��
�
�U
R� ������
To obtain the drag force� we integrate f over the sphere surface�
FD �
Z��
�
Z �
�
f R� sin� d�d� � ��R�
Z �
�
�p� cos� �
�
�
�U
R
sin� d� ��
FD � � �RU � ������
Note that the term �p� cos� does not contribute to the drag force� due to symmetry�Equation ������ is the famous Stokes law for creeping �ow past a sphere �����
Equation ������ can also be cast in the general form
FD � �RS � U i� � ������
where RS denotes the shape tensor� In the case of an isotropic sphere� the shapetensor is obviously given by
RS � �R I � ������
Shape tensors for several bodies are given in Ref� ����The drag coecient� CD� is generally obtained by dividing the drag force by
� ��U� and by the area of the body projected on a plane normal to the direction ofthe uniform velocity �eld� Therefore� in the present case�
CD � FD���U
� �R��� ������
which takes the form
CD ���
Re� ������
where the Reynolds number� Re� is de�ned by
Re � ��UR
�� �����
It should be noted that the creeping �ow assumption�
ju � ruj �����r�u
��� �is not valid far from the sphere� where the velocity gradients are vanishing and�consequently� inertia forces become comparable to viscous forces ���� The failure
���� Axisymmetric Flow in Spherical Coordinates ��
of Stokes �ow is more striking in the case of two�dimensional �ow past a circularcylinder� In this �ow problem� the assumption of a separated solution of the form� r� ���Uf r� sin� leads to Problem �����
� r� �� � U
A
r� Br � Cr� � Dr ln r
sin� �
The trouble with the above solution is that there is no choice of the arbitraryconstants with which all the boundary conditions are satis�ed� Historically� thisfailure is known as the Stokes paradox�
To overcome the failure of Stokes �ow far from the sphere� Oseen ���� used thesubstitution
u � U i � u� � ������
with which the Navier�Stokes equation becomes
U i � ru� � u� � ru� � ��
�rp � �r�u� � ������
The nonlinear inertia term� u� � ru�� is vanishingly small and can be neglected�Therefore�
U i � ru� � ��
�rp � �r�u� � �����
Equation ����� is known as Oseen�s equation� and its solution is called Oseen�s
approximation� Lamb ���� obtained an approximate solution to Eq� ����� for thesphere problem which yields
FD � � �RU
�� �
�
�Re� O
�Re�
��� ������
Proudman and Pearson ���� solved the full Navier�Stokes equation at smallReynolds number using a singular perturbation method and obtained the follow�ing expression for the drag force�
FD � � �RU
�� �
�
�Re�
��Re� lnRe�O
�Re�
��� ������
�
Example ������� Creeping �ow around a translating sphereConsider creeping �ow around a sphere translating steadily with velocity U i throughan incompressible� Newtonian liquid which is otherwise undisturbed� Setting the
�� Chapter ��� Creeping Bidirectional Flows
origin of the spherical coordinate system at the instantaneous position of the centerof the sphere� we obtain the velocity of the liquid by adding
�U i � �U cos� er � U sin� e�
to the velocity vector found in the previous example� We thus get
ur �U
�
���
R
r
�
R
r
� cos� � ������
and
u� �U
�
��
R
r
�
R
r
� sin� � ������
The corresponding stream function is given by
� r� �� �UR�
�
��
r
R
�R
r
�sin�� � ������
���
���
���
���
���
Figure ����� Calculated streamlines of creeping �ow around a translating sphere�
In Fig� ���� we show streamlines predicted by Eq� ������� Note that the distur�bance due to the motion of the sphere propagates to a considerable distance fromthe sphere� �
Example ������� Creeping �ow around bubbles and dropletsThe analysis for �ow around a gas bubble of zero viscosity is the same as that for �ow
���� Axisymmetric Flow in Spherical Coordinates ��
past a solid sphere studied in Example ������� except that the boundary conditionsat the liquid�gas interface become�
ur � � at r � R no penetration in gas volume� �
�r� � � at r � R no traction on the free surface� �
The second condition implies that u� is� in general� nonzero on the interface� Thedrag force turns out to be Problem ����
FD � �� �RU � ������
Hence� the corresponding shape tensor is RS���RI�In the case of creeping �ow of a Newtonian liquid of viscosity �o past a spherical
droplet of another Newtonian liquid of viscosity �i� the boundary conditions on theinterface are�
uo � ui at r � R continuity of velocity� �
�or� � � ir� at r � R continuity of shear stress� �
The drag force is given by Problem ������
FD � �� �o�o �
���i
�o � �iRU � �����
Equation ����� contains the case of creeping �ow past a solid sphere� in the limit�i �� and the case of creeping �ow past a gas bubble� in the limit �i � ��
The preceding analyses apply to bubbles and droplets of small size� so thatsurface tension forces are su�ciently strong to suppress the deforming e�ect ofviscous forces� and to keep the bubbles or droplets approximately spherical ���� �
Example ������� Terminal velocityConsider a solid spherical particle of radius R and density �p falling under gravityin a bath of a Newtonian �uid of density � and viscosity �� The sphere attainsa constant velocity Ut� called the terminal velocity� once the gravitational force iscounterbalanced by the hydrodynamic forces exerted on the sphere� i�e�� the buoy�ancy and drag forces�
�
�� R��pg � �
�� R��g � � �RUt � � � ������
� Chapter ��� Creeping Bidirectional Flows
Solving for Ut yields
Ut ��R� �p � �� g
�� ������
Note that when the particle is less dense than the �uid� �p � � �� the terminalvelocity is negative which obviously means that the particle would be rising in thesurrounding �uid�
From Eq� ������� we deduce that Stokes law holds when
Re � ��UR
��
�R� �p � ��� g
��� � �
i�e� when
R��
��
� �p � ��� g
���� �����
�
���� Problems
����� Show that the stream function� �� de�ned in Eq� ������ satis�es the bihar�monic equation� Eq� ������ in polar coordinates� for creeping plane incompressible�ow�
����� By integrating the r� and ��momentum equations� show that the generalform of the pressure p� in creeping plane incompressible �ow in polar coordinates�is given by Eq� �������
����� Show that� in the particular cases ���� � and �� the function f� �� inEq� ������ degenerates to the following forms�
f�� �� � A cos�� � B sin�� � C� � D �����
f� �� � A cos� � B sin � � C� cos� � D� sin� �����
f� �� � A cos�� � B sin�� � C� � D �����
����� Show that Eq� ����� has the following particular solution which is indepen�dent of ��
� r� � Ar� ln r � B ln r � C r� � D � �����
Show that� in this case� the pressure p is given by
p � �A � � � c � �����
���� Problems �
where c is the integration constant�
���� Consider the creeping �ow of a Newtonian liquid in a corner formed by twoplates� one of which is sliding on the other with constant speed U � as shown inFig� ������ The angle � between the two plates is constant�
U
�
Figure ������ Creeping �ow near a corner with one sliding plate�
a� Introducing polar coordinates centered at the corner� write down the governingequation and the boundary conditions in terms of the stream function � r� ��� b� Show that the particular solution
� r� �� � Urf� �� � Ur A cos� � B sin� � C� cos� � D� sin�� �����
found in Problem ������ satis�es all the boundary conditions� c� Show that the stream function is given by
� r� �� �Ur
�� � sin���� � � �� sin� � � sin� sin � � ��� � ����
d� Calculate the velocity components� e� Determine the shear stress at ��� and show that it exhibits a � r singularitywhich suggests that an in�nite force is required in order to maintain the motion ofthe sliding plate� What is the origin of this nonphysical result�
���� Show that the stream function � de�ned in Eq� ����� satis�es Eq� ������for creeping� axisymmetric� incompressible �ow in cylindrical coordinates�
����� Show that the stream function � de�ned in Eq� ������ satis�es Eq� ������for creeping� axisymmetric� incompressible �ow in spherical coordinates�
�� Chapter ��� Creeping Bidirectional Flows
����� Consider the creeping �ow of a Newtonian liquid past a �xed circular cylinderof radius R assuming that� far from the cylinder� the �ow is uniform with speed U � a� Introducing polar coordinates centered at the axis of symmetry� write down thegoverning equation and the boundary conditions for this �ow in terms of the streamfunction � r� ��� b� In view of the boundary condition at r �� assume a solution of the form
� r� �� � Uf r� sin� � �����
and show that this leads to
� r� �� � U
A
r� Br � Cr� � Dr ln r
sin� � �����
c� Show that there is no choice of the constants A� B� C and D for which allthe boundary conditions are satis�ed Stokes paradox�� Why does the Stokes �owassumption fail� Explain how a well�posed problem can be obtained�
��� � Consider the creeping �ow of an incompressible Newtonian liquid approach�ing� with uniform speed U � a �xed spherical bubble of radius R� a� Introducing spherical coordinates with the origin at the center of the bubble�write down the governing equation and the boundary conditions for this �ow interms of the stream function � r� ��� b� Show that the stream function is given by
� � �UR�
�
�r
R
�� r
R
sin�� � r R � ����
c� Calculate the two nonzero velocity components and the pressure� d� Show that the drag force exerted on the bubble is given by Eq� �������
������ Consider the creeping �ow of an incompressible Newtonian liquid of viscosity�o approaching� with uniform speed U � a �xed spherical droplet of viscosity �i andradius R� a� Introducing spherical coordinates with the origin at the center of the bubble�write down the governing equation and the boundary conditions for this �ow interms of the stream function � r� ��� b� Show that the stream function is given by
�o � �UR�
�
��
r
R
�� ��o � ��i
�o � �i
r
R�
��i�o � �i
R
r
sin�� � r R � �������
���� References ��
outside the droplet� and by
�i �UR�
�
�o�o � �i
�r
R
��r
R
� sin�� � r � R � �������
inside the droplet� c� Calculate the two nonzero velocity components and the pressure� d� Show that the drag force exerted on the droplet is given by Eq� ������
������ Calculate the terminal velocity of a� a spherical bubble rising under gravity in a pool of a Newtonian liquid� and b� a spherical droplet of density �i and viscosity �i falling under gravity in a New�tonian liquid of density �o and viscosity �o�
������ Consider the creeping �ow of an incompressible Newtonian liquid of viscosity� and density � approaching� with uniform speed U � a �xed solid sphere of radiusR and introduce spherical coordinates with the origin at the center of the sphere�Assume that slip occurs on the sphere surface according to
�r� � � u� at r � R � �������
where � is a slip parameter� a� Show that the drag force exerted on the sphere by the liquid is given by
FD � � �RU�� � �R
�� � �R� �������
and identify limiting cases of the above result� b� Calculate the terminal velocity of a solid sphere in a pool of Newtonian liquidwhen slip occurs on the sphere surface according to Eq� ��������
���� References
�� S� Kim and S�J� Karrila� Microhydrodynamics Principles and Selected Applica�
tions� Butterworth�Heinemann� Boston� ���
�� K� Wark and C�F� Warner� Air Pollution Its Origin and Control� Harper �Row� New York� ����
�� J�H� Seinfeld� Atmospheric Chemistry and Physics of Air Pollution� Wiley �Sons� New York� ����
�� S� Middleman� Fundamentals of Polymer Processing� McGraw�Hill� New York�����
�� Chapter ��� Creeping Bidirectional Flows
�� G�K� Batchelor� An Introduction to Fluid Dynamics� Cambridge UniversityPress� Cambridge� ���
� R�B� Bird� R�C� Armstrong and O� Hassager� Dynamics of Polymeric Liquids
Volume �� Fluid Mechanics� Wiley � Sons� New York� ����
�� H�J� Lugt and E�W� Schwiderski� �Flows around dihedral angles� I� Eigenmotionanalysis�� Proc� Roy� Soc� London A���� ����� ����
�� W�R� Dean and P�E� Montagnon� �On the steady motion of viscous liquid in acorner�� Proc� Camb� Phil� Soc� ��� ����� ����
� H�K� Mo�att� �Viscous and resistive eddies near a sharp corner�� J� Fluid Mech�
��� ���� ����
��� D�H� Michael� �The separation of a viscous liquid at a straight edge�� Mathe�
matica �� ����� �����
��� S� Richardson� �A �stick�slip� problem related to the motion of a free jet at lowReynolds numbers�� Proc� Camb� Phil� Soc� � ������ �����
��� W�J� Silliman and L�E� Scriven� �Separating �ow near a static contact line�Slip at the wall and shape of a free surface�� J� Comp� Phys� ��� ������� �����
��� T�R� Salamon� D�E� Bornside� R�C� Armstrong and R�A� Brown� �Local simi�larity solutions in the presence of a slip boundary condition�� Physics of Fluids�� ��������� ����
��� G�C� Georgiou� L�G� Olson� W�W� Schultz and S� Sagan� �A singular �niteelement for Stokes �ow� the stick�slip problem�� Int� J� Numer� Methods
Fluids �� �������� ����
��� G�C� Georgiou and A� Boudouvis� �Converged solutions of the Newtonianextrudate swell problem�� Int� J� Numer� Methods Fluids ��� ������ ���
�� T�R� Salamon� D�E� Bornside� R�C� Armstrong and R�A� Brown� �The roleof surface tension in the dominant balance in the die well singularity�� Phys�
Fluids � ���� ����
��� M�J� Stefan� �Versuch uber die Scheinbare Adh asion�� Akad� Wissensch�
Wien� Math��Natur� �� ��� ������
���� References ��
��� G�G� Stokes� �On the e�ect of the internal friction of �uids on the motion ofpendulums�� Trans� Camb� Phil� Soc� �� � ������
�� H� Brenner and R�G� Cox� �The resistance to a particle of arbitrary shapein translational motion at small Reynolds numbers�� J� Fluid Mech� �� �� ����
��� C�W� Oseen� � Uber die Stokessche formel and uber die verwandte aufgabe inder hydrodynamik�� Arkiv Mat� Astron� Fysik � � �����
��� H� Lamb� �On the uniform motion of a sphere through a viscous �uid�� Phil�
Mag� ��� ��� �����
��� I� Proudman and J�R�A� Pearson� �Expansions at small Reynolds number forthe �ow past a sphere and a circular cylinder�� J� Fluid Mech� �� ��� �����
�� Chapter ��� Creeping Bidirectional Flows
Index
axisymmetric �ow� �� ��� ��
bidirectional �ows� �biharmonic equation� �biharmonic operator� �buoyancy� ��
creeping �ow� �
drag coe�cient� ��drag force� ��� ��� ��� �
extrudate�swell problem� ��
Lagrange�s stream function� �lubrication� �
Oseen�s equation� ��Oseen�s Oseen�s approximation� ��
plane �ow� �
reversibility of Stokes �ow� �Reynolds number� �
shape tensor� ��slip� �squeezing �ow� �Stefan�s squeezing �ow� �stick�slip problem� ��Stokes equation� �� �
Stokes �ow� �� �� � ��� ��� ��reversibility of� �
Stokes law� ��Stokes paradox� ��� ��Stokes stream function� �� ��stream function� �� �� �� ��� ��stress singularity� � ��� ��
terminal velocity� ��� �
��