B.Sri Padmavati·T.Amaranath

A note on general solution of Stokes equations

B.Sri Padmavati1·T.Amaranath1

©The Chinese Society of Theoretical and Applied Mechanics;Institute of Mechanics,Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016

Li et al.(2015)claim that it is sufficient to use two harmonic functions to express the general solution of Stokes equations.In this paper,we demonstrate that this is not true in a general case and that we in fact need three scalar harmonic functions to represent the general solution of Stokes equations(Venkatalaxmi et al.,2004).

Stokes equations·General solution·Harmonic functions

DOI 10.1007/s10409-016-0601-3

1 Introduction

In Ref.[1],it was shown that any divergence-free vector field q can be expressed in terms of two scalar functions as

It was further shown in Ref.[2]that if A and B are biharmonic and harmonic functions,respectively,then q is a general solution of Stokes equations

which are the equations of motion governing the flow of a viscous,incompressible fluid,where q is the velocity,p isthe pressure,andµis the coefficient of dynamic viscosity of the fluid.Using a result due to Almansi[3]that every biharmonic can be expressed in terms of two harmonic functions,the general solution of Stokes equations can therefore be represented in terms of three harmonic functions.

2 Plane boundaries

In Ref.[2],it was shown that when the problem involves a plane boundary,the solution of Stokes equations can be expressed in terms of two scalar functions A(x,y,z)and B(x,y,z)as

where A is a biharmonic function and B is a harmonic function.

It was further shown that A can be expressed effectively in terms of only two harmonic functions A1and A2as A= A1+z A2,so that the general solution can be expressed in terms of three harmonic functions.

However,in a recent paper[4],the authors claim that it is enough to use one biharmonic function G to write

if q represents the velocity in a Stokes flow.Since G involves two harmonic functions using Almansi’s result[3],this claim implies that we can express the velocity in any Stokes flow in terms of only two harmonic functions.

Even though it may be adequate in some cases,such as the one considered by the authors in Ref.[4],it is not true in general as was claimed by the authors in Ref.[4].Weclaim that we need one more scalar function B that is also a harmonic function to represent the velocity of a Stokes flow in the general case.We demonstrate this with the following example of a Stokes flow given by a rotlet[5].

3 Example

Consider a rotlet of strength F located at(0,0,c)whose axis is along the positive z-direction,i.e.,perpendicular to the plane boundary z=0.If the corresponding velocity vector is q=u i+v j+w k,then the velocity components in this case are given by

If we can express this velocity vector only in terms of one biharmonic function G(i.e.,in terms of two harmonic functions),then

By equating the corresponding velocity components,this implies that

But on differentiating Eq.(10)with respect to y and Eq.(11) with respect to x,we get

respectively,which are inconsistent.Hence,there is no G such that q=CurlCurl(k G)for the q given in Eq.(9).

However,if we substitute

into Eq.(4),we obtain the required velocity components.

4 Conclusion

In conclusion,it is not true that all possible solutions of Stokes equations can be represented in terms of only two scalar (harmonic)functions.In fact,a minimum number of three independent harmonic functions is required to represent any solution of Stokes equations.

References

1.Padmavathi,B.S.,Amaranath,T.:A note on the decomposition of solenoidal fields.Appl.Math.Lett.15,803–805(2002)

2.Venkatalaxmi,A.,Padmavathi,B.S.,Amaranath,T.:Complete general solution of Stokes equations for plane boundaries.Mech.Res. Comm.31,465–475(2004)

3.Almansi,E.:Sull’integrazione dell’equazione differenziale∇2nF=0.Ann.di Mat.Ser.3,1(1899).(in Italian)

4.Li,X.-Y.,Ren,S.-C.,He,Q.-C.:A general solution for Stokes flow and its application to the problem of a rigid plate translating in a fluid.Acta Mech.Sin.31,32–44(2015)

5.Batchelor,G.K.:The stress system in a suspension of force-free particles.J.Fluid Mech.41,545(1970)

✉ T.Amaranath amaranath.t@gmail.com B.Sri Padmavati bs.padmavathi@gmail.com

1School of Mathematics&Statistics,University of Hyderabad, Hyderabad 500046,India

29 May 2015/Accepted:1 July 2016/Published online:12 October 2016