Con - ΠΑΝΕΠΙΣΤΗΜΙΟ...

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Transcript of Con - ΠΑΝΕΠΙΣΤΗΜΙΟ...

Page 1: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

Contents

�� CREEPING BIDIRECTIONAL FLOWS �

���� Plane Flow in Polar Coordinates �

���� Axisymmetric Flow in Cylindrical Coordinates ��

���� Axisymmetric Flow in Spherical Coordinates ��

���� Problems �

���� References �

i

Page 2: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

ii CONTENTS

Page 3: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

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� � �

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� 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� �����

Page 5: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

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� ���

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� 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�

Page 7: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

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� �� � � � �����

Page 8: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

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 �� � ������

Page 9: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

���� 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 � ��� � �

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� 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

Page 11: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

���� 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� �

Page 12: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

�� 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 � ���� �

Page 13: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

���� 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 � ���� � �

���

�� � � � ������

Page 14: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

�� 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 � ���� � � �� �� � � � ������

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���� 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

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�� 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� �

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���� 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

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� 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

� � � �

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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� ������

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�� 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� �

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���� 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

� � ��� �� � � �� � ��� �� � � � �����

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�� 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� � ����

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���� 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� �

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�� 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

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���� 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

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�� 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

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���� 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 � � � ������

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� 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 � �����

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���� 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�

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�� 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 � �������

Page 31: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

���� 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�����

Page 32: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

�� 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� �� ��� ������

Page 33: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

���� 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� �� ��� �����

Page 34: Con - ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥeuclid.mas.ucy.ac.cy/~georgios/courses/mas483/files/chap10.pdf · x and y The con tin uit ... sin r sin T able Stokes ow e quations in terms

�� Chapter ��� Creeping Bidirectional Flows

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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� ��� �

��