Stokes Flow – Invariant Representation of Solutions

Stokes Flow – Invariant Representation of SolutionsR. Shankar SubramanianThe term “invariant representation” implies that the results are written in aform that is independent of the coordinate system or, equivalently, the basisset used for expressing the components of vectors.In an early homework assignment, we learned about solutions of the Laplaceequation written in such invariant notation involving only the position vectorx iand its length r . These are called “vector harmonics” even though thesolutions are tensors, and are ordered such that at each level, the orderincreases by 1. There are two sets of such solutions. One is the set ofdecaying harmonics. The general decaying harmonic is given byn( )( n )φ 1 1( n 1)= − ∇∇∇⋅⋅⋅∇ ⎛ , n 0,1,2− + ⎜⎞⎟ =135 ⋅ ⋅ ⋅⋅⋅ 2 −1 ,...⎝r⎠The first few are written below.n times1r , x i3r , xxi jδijxxi jxk xiδ jk+ xjδki + xkδij− , −5 3 7 5r 3rr5rYou can see that these solutions approach 0 as r →∞. Growing harmonics,in contrast, grow with distance from the origin. These are obtained from the2n1decaying harmonics as r+ φ− ( n+1). The first few growing harmonics are givenbelow.2 2rr1, xi,xxi j− δij, xxi jxk − ( xiδ jk+ xjδki + xkδij), ⋅⋅⋅3 5Absolute and PseudotensorsWe’ll find these sets of harmonic functions very useful in constructingsolutions of Stokes problems. The important idea in each case will be thatwe’ll be looking for solutions of Laplace’s equation in one case for a scalarfield (pressure) and in the other for a vector field (homogeneous solution forthe velocity). The pressure and velocity fields are entities that are calledabsolute tensors. Absolute tensors are unaffected by whether the coordinatesystem (and therefore the basis set) is right or left-handed. On the other1

hand, we are familiar with vectors and tensors that change sign if the basisset is changed from one that is right-handed to one that is left-handed. Oneexample is the cross product of two absolute vectors. Another way to obtaina pseudovector is by taking the curl of an absolute vector field; an exampleis the vorticity vector field, defined as the curl of the velocity field.Performing a second such operation on a pseudovector such as taking a curlwill yield an absolute entity. Also, if the cross product of a pseudovector andan absolute vector is formed, the result with be an absolute vector. You willfind a more detailed discussion of the distinction between absolute andpseudotensors in the textbook by Leal (pages 178-179).Solution of the Stokes equationNow, let us begin with the Stokes equation for an incompressible flow.2µ ∇ u = ∇pAlso, an incompressible velocity field must satisfy ∇• u =0 . If we now takethe divergence of the Stokes equation, ∇• ( µ ∇ 2 u −∇ p)= 0 , which can be2rearranged to µ ∇ ( ∇• u)−∇•∇ p = 0. The first term is zero because the flow2is incompressible, leading to the result that ∇ p = 0. Therefore, the pressurefield in every Stokes problem must be a harmonic function, that is, afunction which satisfies the Laplace equation.We can construct a solution of the Stokes equation by adding ahomogeneous solution and a particular solution. The homogeneous solution( H )2u must also satisfy Laplace’s equation ∇(H )u =0. The particular solutionx( P)( P)ju , which satisfies the same equation as u, can be written as uj= p.2µLet us verify that this is indeed correct.( P) ∂ ( P) ∂ ⎛ xj⎞ 1 ⎛∂xj∂p ⎞ 1 ⎛ ∂p⎞∇ u = uj= ⎜ p⎟= ⎜ p+ xj⎟= ⎜δijp+xj⎟∂xi ∂xi ⎝2µ ⎠ 2µ ⎝ ∂xi ∂xi⎠ 2µ⎝ ∂x i ⎠Now, take the divergence.( ) 1 ⎡P ∂ ∂ ⎛ ∂p⎞ ⎤∇• ( ∇ u ) = ⎢ ( δijp)+ ⎜xj⎟ ⎥2µ⎣∂xi∂xi⎝ ∂x i ⎠ ⎦2

We’ll find that there are only a limited number of ways in which the vectors(or scalars) from the boundary conditions can be combined with the set ofdecaying or growing harmonics using operations such as the cross or dotproduct or simple scalar multiplication to generate scalars or vectors of order1. It is this fact that makes it relatively easy to construct the desiredsolutions in invariant form. We shall illustrate with three examples, flowcaused by a sphere rotating in place in a quiescent fluid, flow caused by asphere translating at a constant velocity through a quiescent fluid, and flowcaused by a point force in a quiescent fluid.4