Fourth-Order Stationary Equations

9.4. Fourth-Order Stationary Equations

9.4.1. Biharmonic Equation ✛ ✛ ✜ =0

The biharmonic equation is encountered in plane problems of elasticity ( is the Airy stress function). It is also used to describe slow flows of viscous incompressible fluids ( is the stream function).

All solutions of the Laplace equation ✢ = 0 (see Sections 7.1 and 8.1) are also solutions of the biharmonic equation.

9.4.1-1. Two-dimensional equation. Particular solutions. In the rectangular Cartesian system of coordinates, the biharmonic operator has the form

1 ✣ . Particular solutions:

( ✄ , ✞ )=( ✤ cosh ✭ ✄ + ✥ sinh ✭ ✄ + ✦ ✄ cosh ✭ ✄ + ✧ ✄ sinh ✭ ✄ )( ✗ cos ✭ ✞ + ★ sin ✭ ✞ ), ( ✄ , ✞ )=( ✤ cos ✭ ✄ + ✥ sin ✭ ✄ + ✦ ✄ cos ✭ ✄ + ✧ ✄ sin ✭ ✄ )( ✗ cosh ✭ ✞ + ★ sinh ✭ ✞ ),

( ✄ , ✞ )= ✤ ✯ 2 ln ✯ + ✥ ✯ 2 + ✦ ln ✯ + ✧ , ✯ = ✰ ( ✄ − ✗ ) 2 +( ✞ − ★ ) 2 , ( ✄ , ✞ )=( ✤ ✄ + ✥ ✞ + ✦ )( ✧ cosh ✭ ✄ + ✱ sinh ✭ ✄ )( ✗ cos ✭ ✞ + ★ sin ✭ ✞ ), ( ✄ , ✞ )=( ✤ ✄ + ✥ ✞ + ✦ )( ✧ cosh ✭ ✞ + ✱ sinh ✭ ✞ )( ✗ cos ✭ ✄ + ★ sin ✭ ✄ ), ( ✄ , ✞

2 )=( 2 ✄ + ✞ )( ✧ cosh ✭ ✄ + ✱ sinh ✭ ✄ )( ✗ cos ✭ ✞ + ★ sin ✭ ✞ ),

( ✄ , ✞ )=( ✄ 2 + ✞ 2 )( ✧ cosh ✭ ✞ + ✱ sinh ✭ ✞ )( ✗ cos ✭ ✄ + ★ sin ✭ ✄ ),

where ✤ , ✥ , ✦ , ✧ , ✱ , ✗ , ★ , ✪ , ✬ , ✭ , and ✮ are arbitrary constants.

TABLE 30 Particular solutions of the biharmonic equation in some orthogonal curvilinear coordinate systems; ✤ , ✥ , ✦ , ✧ , ✗ , ★ , and ✒ are arbitrary constants

Transformation Particular solutions

− Polar coordinates ✳

(at ✒ = 0) Bipolar coordinates ✠ , ✡ :

cosh − cos ✡ ✴

= cosh ✠ − cos ✡

+ ✦ cosh( ✒ − 1) ✠ + ✧ sinh( ✒ − 1) ✠ ✵ ( ✗ cos ✒ ✡ + ★ sin ✒ ✡ ) Degenerate bipolar

cosh ✠ − cos ✡

cosh( ✒ ✶ )+ , ✥ sinh( ✒ coordinates ✶ )

+ ✵ ✷ 2 ✶ 2 + ✷ 2 + cosh( )+ sinh( )

cos( )+ sin( )

2 ✣ . Fundamental solution: ✸ ✸

2 ln ✯ ,

3 ✣ . Particular solutions of the biharmonic equation in some orthogonal curvilinear coordinate sys- tems are listed in ✹✻✺ Table 30 .

Reference : N. N. Lebedev, I. P. Skal’skaya, and Ya. S. Uflyand (1972).

9.4.1-2. Two-dimensional equation. Various representations of the general solution.

1 ✣ . Various representations of the general solution in terms of harmonic functions:

where ✶ 1 and ✶ 2 are arbitrary functions satisfying the Laplace equation ✢ ✶ ✼ =0( ✹✻✺ ✘ = 1, 2).

Reference : A. N. Tikhonov and A. A. Samarskii (1990).

2 ✣ . Complex form of representation of the general solution:

( ✄ , ✞ ) = Re

where ( ) and ✿ ✾ ✽ ( ✽ ) are arbitrary analytic functions of the complex variable ✽ = ✄ + ❀❁✞ ; ✽ = ✄ − ❀❁✞ ,

= −1. The symbol Re[ ✤ ] stands for the real part of the complex quantity ✤ .

Reference : A. V. Bitsadze and D. F. Kalinichenko (1985).

9.4.1-3. Two-dimensional boundary value problems for the upper half-plane.

1 ✣ . Domain: − ❂ < ✄ < ❂ ,0≤ ✞ < ❂ . The desired function and its derivative along the normal are prescribed at the boundary:

= ✾ ( ✄ ) at ✞ = 0. Solution:

Reference : G. E. Shilov (1965).

2 ✣ . Domain: − ❂ < ✄ < ❂ ,0≤ ✞ < ❂ . The derivatives of the desired function are prescribed at the boundary:

= ✿ ( ✄ ) at ✞ = 0. Solution:

where ✦ is an arbitrary constant.

Example. Let us consider the problem of a slow (Stokes) inflow of a viscous fluid into the half-plane through a slit of width 2 ❉ with a constant velocity ❊ that makes an angle ❋ with the normal to the boundary (the angle is reckoned from the normal counterclockwise).

and ❍ ◆ are the fluid velocity components), the problem is reduced to the special case of the previous problem with

With the stream function ● introduced by the relations ❍❏■

0 for | P |> ❉ . Dean’s solution:

0 for | P |> ❉ ,

( P , ❙ )= ( P − ❉ ) cos ❋ + ❙ sin ❋ ❱ arctan ❲

( P + ❉ ) cos ❋

+ sin

arctan P

Reference : I. Sneddon (1951).

9.4.1-4. Two-dimensional boundary value problem for a circle. Domain: 0 ≤ ✯ ≤ ✗ ,0≤ ✲ ≤2 ❩ . Boundary conditions in the polar coordinate system:

= ✿ ( ✲ ) at ✯ = ❭ . Solution:

Reference : A. N. Tikhonov and A. A. Samarskii (1990).

9.4.1-5. Three-dimensional equation. In the rectangular Cartesian coordinate system, the three-dimensional biharmonic operator is ex-

pressed as

. Particular solutions in the Cartesian coordinate system:

( ✈ ❜ , ❝ , ❧ )= ♥♦❡ ❜ sin( ♣ ❜ )+ ❣ sin( ♣ ❜ )+ ❤ ❜ cos( ♣ ❜ )+ cos( ♣ ❜ ) q sin( r ❝ ) exp s✉t ❧ ❥ ♣ 2 + r 2

( ❜ , ❝ , ❧ )= ♥ ❡ ❜ sin( ♣ ❜ )+ ❣ sin( ♣ ❜ )+ ❤ ❜ cos( ♣ ❜ )+ cos( ♣ ❜ ) q cos( r ❝

) exp + ,

st ❧ ❥ ♣

( ❜ , ❝ , ❧ )= ♥ ❡ ❜ sin( ♣ ❜ )+ ❣ sin( ♣ ❜ )+ ❤ ❜ cos( ♣ ❜ )+ cos( ♣ ❜ ) q sinh( r ❝

) exp − ,

st ❧ ❥ ♣

( ❜ , ❝ , ❧ )= ♥ ❡ ❜ sin( ♣ ❜ )+ ❣ sin( ♣ ❜ )+ ❤ ❜ cos( ♣ ❜ )+ cos( ♣ ❜ ) q cosh( r ❝ ✐ ) exp s t ❧ ❥ ♣ 2 − r 2 ✈ ,

( ❜ , ❝ , ❧ )= ♥ ❡ ❜ sinh( ♣ ❜ )+ ❣ sinh( ♣ ❜ )+ ❤ ❜ cosh( ♣ ❜ )+ ✐ cosh( ♣ ❜ ) q sinh( r ❝ ) sin s ❧ ❥ ♣ 2 + r 2 ✈ ,

( ❜ , ❝ , ❧ )= ♥ ❡ ❜ sinh( ♣ ❜ )+ ❣ sinh( ♣ ❜ )+ ❤ ❜ cosh( ♣ ❜ )+ cosh( ♣ ❜ ) q cosh( r ❝ ) cos s ❧ ❥ ♣ 2 ✐ + r 2 ✈ , where ❡ , ❣ , ❤ , ✐ , ♣ , and r are arbitrary constants.

2 ❞ . Particular solutions in the cylindrical coordinate system s ❢ = ❥ ❜ 2 + ❝ 2 ✈ :

( ❢ , ✇ , ❧ )= ① ② ( r ❢ )( ❡ ❢ cos ✇ + ❣ ❢ sin ✇ + ❤ )( ❭ 1 cos ③ ✇ + ❦ 1 sin ③ ✇ )( ❭ 2 cosh r ❧ + ❦ 2 sinh r ❧ ),

( ❢ , ✇ , ❧ )= ④ ② ( r ❢ )( ❡ ❢ cos ✇ + ❣ ❢ sin ✇ + ❤ )( ❭ 1 cos ③ ✇ + ❦ 1 sin ③ ✇ )( ❭ 2 cosh r ❧ + ❦ 2 sinh r ❧ ),

( ❢ , ✇ , ❧ )= ⑤⑥② ( r ❢ )( ❡ ❢ cos ✇ + ❣ ❢ sin ✇ + ❤ )( ❭ 1 cos ③ ✇ + ❦ 1 sin ③ ✇ )( ❭ 2 cos r ❧ + ❦ 2 sin r ❧ ),

( ❢ , ✇ , ❧ )= ⑦ ② ( r ❢ )( ❡ ❢ cos ✇ + ❣ ❢ sin ✇ + ❤ )( ❭ 1 cos ③ ✇ + ❦ 1 sin ③ ✇ )( ❭ 2 cos r ❧ + ❦ 2 sin r ❧ ),

( ❢ , ✇ , ❧ )= ① ② ( r ❢ )( ❡ cos ③ ✇ + ❣ sin ③ ✇ )( ❭ 1 cosh r ❧ + ❦ 1 sinh r ❧ + ❭ 2 ❧ cosh r ❧ + ❦ 2 ❧ sinh r ❧ ),

( ❢ , ✇ , ❧ )= ④ ② ( r ❢ )( ❡ cos ③ ✇ + ❣ sin ③ ✇ )( ❭ 1 cosh r ❧ + ❦ 1 sinh r ❧ + ❭ 2 ❧ cosh r ❧ + ❦ 2 ❧ sinh r ❧ ),

( ❢ , ✇ , ❧ )= ⑤⑥② ( r ❢ )( ❡ cos ③ ✇ + ❣ sin ③ ✇ )( ❭ 1 cos r ❧ + ❦ 1 sin r ❧ + ❭ 2 ❧ cos r ❧ + ❦ 2 ❧ sin r ❧ ),

( ❢ , ✇ , ❧ )= ⑦ ② ( r ❢ )( ❡ cos ③ ✇ + ❣ sin ③ ✇ )( ❭ 1 cos r ❧ + ❦ 1 sin r ❧ + ❭ 2 ❧ cos r ❧ + ❦ 2 ❧ sin r ❧ ), where ③ = 0, 1, 2, ⑧▼⑧▼⑧ ; ❡ , ❣ , ❤ , ❭ 1 , ❭ 2 , ❦ 1 , ❦ 2 , and r are arbitrary constants; the ① ② ( ⑨ ) and ④ ② ( ⑨ ) are the Bessel functions; and the ⑤ ② ( ⑨ ) and ⑦ ② ( ⑨ ) are the modified Bessel functions.

3 ❞ . Particular solutions in the spherical coordinate system s ❢ = ❥ ❜ 2 + ❝ 2 + ❧ 2 ✈ :

(cos ⑩ )( ❭ cos ❸ ✇ + ❦ sin ❸ ✇ ), where ③ = 0, 1, 2, ⑧▼⑧▼⑧ ; ❸ = 0, 1, 2, ⑧▼⑧▼⑧ , ③ ; ❡ , ❣ , ❤ , ✐ , ❭ , and ❦ are arbitrary constants; the

( ⑨ ) are the Legendre polynomials; and the ❶ ② ❷ ( ⑨ ) are the associated Legendre functions defined by

. Fundamental solution:

5 ❞ . Representations of solutions to the biharmonic equation in terms of harmonic functions:

( ❜ , ❝ , ❧ )=( ❜ 2 + ❝ 2 + ❧ 2 ) ❼ 1 ( ❜ , ❝ , ❧ )+ ❼ 2 ( ❜ , ❝ , ❧ ), where ❼ 1 and ❼ 2 are arbitrary functions satisfying the three-dimensional Laplace equation ❽ 3 ❼ ❾ =0

( ➀✻➁ ❿ = 1, 2). The coefficient ❜ of ❼ 1 in the first formula can be replaced by ❝ or ❧ .

Reference : A. V. Bitsadze and D. F. Kalinichenko (1985).

9.4.1-6. ③ -dimensional equation.

1 ❞ . Particular solutions:

where the ➃ , , , , , ➌ , ➑ , ➒ , ➆ ❾ , ➓ ❾ , and ➐ ❾ are arbitrary constants.

2 ❞ . Fundamental solution:

For ➙ ➠✻➡ = 2, see Paragraph 9.4.1-1, Item 2 ❞ .

Reference : G. E. Shilov (1965).

3 ❞ . Various representations of solutions to the biharmonic equation in terms of harmonic functions:

(x) = ❜ ➢➥➤ 1 (x) + ➤ 2 (x),

2 (x) = |x| 2

1 ➤ 2 (x) + 2 (x), | x| =

where ➤ 1 and ➤ 2 are arbitrary functions satisfying the ➙ -dimensional Laplace equation ➩ ➜ ➤ ➫ =0 ( ➭ = 1, 2).

Reference : A. V. Bitsadze and D. F. Kalinichenko (1985).

9.4.2. Equations of the Form ➯ ➯ ➲ = ➳ ( ➵ , ➸ )

Nonhomogeneous biharmonic equation. It is encountered in plane problems of elasticity and hydrodynamics.

9.4.2-1. Domain: − ➺ < ➻ < ➺ ,− ➺ < ➼ < ➺ . Solution: ➽

Reference ➪ : A. V. Bitsadze and D. F. Kalinichenko (1985).

9.4.2-2. Domain: − ➺ < ➻ < ➺ ,0≤ ➼ < ➺ . Boundary value problem. The upper half-plane is considered. The derivatives are prescribed at the boundary: ➽ ➽

= ❐ ( ➻ ) at ➼ = 0. Solution: ➽

is an arbitrary constant,

Reference : I. Sneddon (1951).

9.4.2-3. Domain: 0 ≤ ➻ ≤ Ò 1 ,0≤ ➼ ≤ Ò 2 . The sides of the plate are hinged.

A rectangle is considered. Boundary conditions are prescribed: ➽ ➽ ➽ ➽

=0 at ➼ = Ò 2 . Solution: ➽

) sin( Ú Ø 2 Ü ) sin( Ù

9.4.3. Equations of the Form á á â – ã â = ä ( å , æ )

9.4.3-1. Homogeneous equation ( ➽ ç ≡ 0). This equation describes the shapes of two-dimensional free transverse vibrations of a thin elastic

plate; the function defines the deflection (transverse displacement) of the plate’s midplane points relative to the original plane position and è

1 ê 4 = 2 é is the frequency parameter. Here, ë ë = ë is the biharmonic operator and ë is the Laplace operator defined as

2 + 2 in the Cartesian coordinate system, =

+ 2 í 2 in the polar coordinate system.

. Particular solutions ( ➽ Û ò Û 1 , ò 2 , ò 3 , ò 4 Û , ó 1 , and ó 2 are arbitrary constants):

, Ü )= ô✻ò sin( è 1 ( Û 1 )+ ó 1 cos( è 1 Û ) õ ô✻ò 2 sin( è 2 Ü )+ ó 2 cos( è Ü õ ,

)= )+ ó 1 cos( è 1 ) õ ô✻ò 2 sinh( è 2 Ü )+ ó 2 cosh( è 2 Ü ) õ ,

1 sinh( è 1 Û )+ ó 1 cosh( è 1 Û ) õ ô✻ò 2 sin( è 2 Ü )+ ó 2 cos( è 2 Ü ) õ ,

1 sinh( 1 )+ 1 cosh( 1 )

2 sinh( 2 )+ 2 cosh( 2 ) , =( 1 + 2 ) ,

where the Û

( ) and ø ( ) are the Bessel functions of the first and second kind, the ù ( ) and ú ( ) are the modified Bessel functions of the first and second kind, ➉ = û

2 + Ü 2 , and = 0, 1, 2, ü▼ü▼ü

. General solution:

where ý 1 and ý 2 are arbitrary functions satisfying the Helmholtz equations

For solutions to these equations, see Section 7.3.

9.4.3-2. Domain: 0 ≤ ≤ ÿ 1 ,0≤ Ü ≤ ÿ 2 . Boundary value problem.

A rectangle is considered. Boundary conditions are prescribed: ➽

9.4.3-3. Domain: 0 ≤ ➉ ≤ ✒ ,0≤ ö ≤2 ß . Eigenvalue problem with ✓ ≡ 0. The unknown and its normal derivative are zero on the boundary of a circular domain:

=0 at ➉ = ➉ ✒ .

Eigenvalues: ✡

= 0, 1, 2, ü▼ü▼ü , ✑ = 1, 2, 3, ü▼ü▼ü , where the ☞ are positive roots of the transcendental equation ✡ ✡ ✡ ✡ ✕

Numerical values of some roots:

Reference : V. V. Bolotin (1978).

9.4.3-4. Domain: ( ☎ ✧ ★ ) 2 +( ✆ ✧ ✩ ) 2 ≤ 1. Eigenvalue problem with ✓ ≡ 0. The unknown and its normal derivative are zero on the boundary of an elliptic domain:

2 =0 2 on ( ☎ ✧ ★ ) +( ✆ ✧ ✩ ) =1

Eigenvalues and eigenfunctions (approximate formulas): ✫

(c)

11 5 1 2 ✔ (c)

11 ) cos ✣ ,

(s)

11 1 5 2 ✔ (s)

)= 1 ( ✕ 11 ) 1 ( ✕ 11 )− 1 ( ✕ 11 ) 1 ( ✕ 11 ) sin ✣ , )= 1 ( ✕ 11 ) 1 ( ✕ 11 )− 1 ( ✕ 11 ) 1 ( ✕ 11 ) sin ✣ ,

The above formulas were obtained with the aid of generalized (nonorthogonal) polar coordinates

, ✣ defined by

(0 ≤ ≤ 1, 0 ≤ ✣ ≤2 ß ) and the variational method.

The maximum error in the eigenvalue ✎ 1 for ✲ = ✰ 1−( ✩✳✧ ★ ) 2 ≤ 0,86 is less than 1%. The errors in ✎ (c) 11 and ✎ (s) 11 for ✲ ≤ 0,6 do not exceed 2%. In the limit case ✲ = 0 that corresponds to a circular domain, the formulas provide exact results.

Reference : L. D. Akulenko, S. V. Nesterov, and A. L. Popov (2001).

9.4.4. Equations of the Form ✴ 4 + ✴ æ 4 = ✷ ( ✶ , æ )

9.4.4-1. Homogeneous equation ( ✓ ≡ 0).

1 ✸ . Particular solutions:

( 1 , )= sin( )+ cos( )+ sinh( )+ cosh( ) exp

2 sin 2 ✎ ✿ ❀ ,

( ❁ , ✿ )= ✭✮✹ sin( ✎ ❁ )+ ✺ cos( ✎ ❁ )+ ✻ sinh( ✎ ❁

)+ 1 cosh( ) exp

2 cos 2 ✎ ✿ ❀ ,

( ❁ , ✿ )= ✭✮✹ sin( ✎ ❁

1 )+ 1 ✺ cos( ✎ ❁ )+ ✻ sinh( ✎ ❁ )+ ✼ cosh( ✎ ❁ ) ✯ exp ✽ − ✾

2 ✎ ✿ ❀ sin ✽ ✾ 2 ✎ ✿ ❀ ,

2 ✎ ✿ ❀ cos ✽ ✾ 1 2 ✎ ✿ ❀ , where ✹ , ✺ , ✻ , ✼ , and ✎ are arbitrary constants.

( ❁ , ✿ )= ✭✮✹ sin( ✎ ❁ )+ ✺ cos( ✎ ❁ )+ ✻ sinh( ✎ ❁ )+ ✼ cosh( ✎ ❁ ) ✯ exp ✽

2 ✸ . General solution:

( ❁ , ✿ ) = Re ✭✦❂ ( ❃ 1 )+ ❄ ( ❃ 2 ) ✯ .

Here, ❂ ( ❃ 1 ) and ❄ ( ❃ 2 ) are arbitrary analytic functions of the complex variables ❃ 1 ❁

2 (1 + ❅ ) ✿ . The symbol Re[ ✹ ] stands for the real part of the complex quantity ✹ ✤✦✥ .

and 1 ❃ 2 = ❁ + ✾

Reference : A. V. Bitsadze end D. F. Kalinichenko (1985).

3 ✸ . Domain: − ❆ < ❁ < ❆ ,0≤ ✿ < ❆ . Boundary value problem. The upper half-space is considered. Boundary conditions are prescribed:

Reference : G. E. Shilov (1965).

9.4.4-2. Nonhomogeneous equation. Boundary value problems in a rectangle. We consider problems in a rectangular domain 0 ≤ ❁ ≤ ❘ 1 ,0≤ ❍ ≤ ❘ 2 with different homogeneous

boundary conditions. The solution can be expressed in terms of the Green’s function as

Below are the Green’s functions for two types of boundary conditions.

1 ✸ . The function and its first derivatives are prescribed at the sides of the rectangle:

= 2 . Green’s function:

( ❩ )= sinh( 1 ) − sin( 1 ) cosh( ❁ ) − cos( ❁ ) ❝

− ❵ cosh( ❪

1 ) − cos( ❪

1 ) ❝ ❵ sinh( ❭ ❁ ) − sin( ❭ ❁ ) ❝ ,

( ❍ )= ❵ sinh( ❬ ❘ ❪ ❭ 2 ) − sin( ❬ ❘ ❭ 2 ❪ ) ❝ ❵ cosh( ❬ ❍ ❪ ❭ ) − cos( ❬ ❭ ❍ ❪ ) ❝

− ❵ cosh( ❬ ❘ 2 ) − cos( ❬ ❘ 2 ) ❝ ❵ sinh( ❬ ❍ ) − sin( ❬ ❍ ) ❝ , where the

and ❬ are positive roots of the transcendental equations ❪ ❪

cosh( ❘ 1 ) cos( ❘ 1 ) = 1, cosh( ❘ 2 ) cos( ❘ 2 ) = 1.

2 ✸ . The function and its second derivatives are prescribed at the sides of the rectangle: ❭ ❭

=0 at ❁ = 0,

=0 at ❁ = ❘ 1 ,

= 2 . Green’s function:

4 4 sin( ❁ ) sin( ❍ ) sin( ❋ ) sin( ❱ ),

9.4.5. Equations of the Form ❤

9.4.5-1. Particular solutions of the homogeneous equation ( ≡ 0):

, ❍ )= ❵✮♥ sin( ♦ ❁ )+ ♣ cos( ♦ ❁ )+ q sinh( ♦ ❁ )+ r cosh( ♦ ❁ ) ❝ exp( s ❍ ) sin( s ❍ ),

( ❁ , ❍ )= ❵✮♥ sin( ♦ ❁ )+ ♣ cos( ♦ ❁ )+ q sinh( ♦ ❁ )+ r cosh( ♦ ❁ ) ❝ exp( s ❍ ) cos( s ❍ ),

( ❁ , ❍ )= ❵✮♥ sin( ♦ ❁ )+ ♣ cos( ♦ ❁ )+ q sinh( ♦ ❁ )+ r cosh( ♦ ❁ ) ❝ exp(− s ❍ ) sin( s ❍ ),

( ❁ , ❍ )= ❵✮♥ sin( ♦ ❁ )+ ♣ cos( ♦ ❁ )+ q sinh( ♦ ❁ )+ r cosh( ♦ ❁ ) ❝ exp(− s ❍ ) cos( s ❍ ),

where s

2 + ✉ ) 1 ✈ 4 ; ♥ , ♣ , q , r , and ♦ are arbitrary constants.

9.4.5-2. Domain: 0 ≤ ❁ ≤ ❘ 1 ,0≤ ❍ ≤ ❘ 2 . Boundary value problems.

1 ✇ . We consider problems in a rectangular domain with different homogeneous boundary conditions. The solution can be expressed in terms of the Green’s function as

Below are the Green’s functions for two types of boundary conditions.

2 ✇ . The function and its first derivatives are prescribed at the sides of the rectangle:

=0 at ❍ = ❘ 2 . Green’s function:

)= ❩ sinh( 1 ) − sin( ❘ 1 ) ❝ ❵

2 ) − sin( ❬ ❘ ❭ 2 ) ❝ ❵ cosh( ❬ ❭ ❍ ❪ ) − cos( ❬ ❪ ❭ ❍ ) ❝

− ❵ cosh( ❬ ❘ 2 ) − cos( ❬ ❘ 2 ) ❝ ❵ sinh( ❬ ❍ ) − sin( ❬ ❍ ) ❝ , where the

and ❬ are positive roots of the transcendental equations ❪ ❪

cosh( ❘ 1 ) cos( ❘ 1 ) = 1, cosh( ❘ 2 ) cos( ❘ 2 ) = 1.

3 ✇ . The function and its second derivatives are prescribed at the sides of the rectangle: ❭ ❭

= ❘ 2 . Green’s function:

) sin( ) sin( ),

9.4.6. Stokes Equation (Axisymmetric Flows of Viscous Fluids)

9.4.6-1. Stokes equation for the stream function in the spherical coordinate system. The Stokes equation for the stream function in the axisymmetric case is written as

sin ③ ❞

2 ② ▼ ❞ ③ sin ③ ❞ ③ ◆ . It governs slow axisymmetric flows of viscous incompressible fluids, with ❙ being the stream

function, ② and ③ the spherical coordinates. The components of the fluid velocity are related to the ❙ ❙

stream function by ❞ =

and ④ ⑥

sin

sin ③ ❞ ②

General solution ( ♥ , ♣ , q , r

, and ❩ ❩ r ⑦

are arbitrary constants): ❩ ❩ ❩

( 3− , ③ )= ❉ ♥ ② + ♣ ② + q ② + r ② ❀ ⑨ (cos ③ )

where the ❩

( ❶ ) and ⑩ ( ❶ ) are the Gegenbauer functions of the first and second kind, respectively. These are linearly related to the Legendre functions ❩ ❩ ❷ ( ❶ ) and ❩ ❩ ❩ ❸ ( ❶ ❩ ) by

The Gegenbauer functions of the first kind are represented in the form of a finite power series as ❩

. In particular,

8 ❶ (1 − ❶ 2 )(7 ❶ 2 − 3). The Gegenbauer functions of the second kind are defined as ❩ ❩

where the functions ❻

( ❩ ❶ ) are expressed in terms of the Gegenbauer functions of the first kind as

the series start with ⑨ 0 or ⑨

1 ❢ , depending on whether is odd or even. In particular,

4 ( )= 24 (15 ❶ − 13), ❻ 5 ( ❶ )= 120 (105 ❶ − 115 ❶ + 16). For ❢ ≥ 2, the Gegenbauer functions of the second kind assume infinite values at the points

= ❏ 1, which correspond to = 0 and = . Therefore, if physically there are no singularities in the problem, then the quantities in (1) labeled with a tilde must be set equal to zero. In the

overwhelming majority of problems on the flow about particles, drops, or bubbles, the stream function in the spherical coordinates is given by formula (1) with ❩ ❩ ❩

1 ❢ = 0 = 1 = 0 = 1 = 0 = 1 = 0 = 0; = = = =0 for = 2, 3, ❹✳❹✳❹

Example 1. In the problem on the translational Stokes flow about a solid spherical particle, the following boundary conditions are imposed on the stream function ❾ :

, where ➀ is the radius of the particle and ➅ is the unperturbed fluid velocity at infinity.

Stokes solution:

Here, only the terms for ➊ = 2 in the first sum of (1) remain.

Example 2. In the problem on the axisymmetric straining Stokes flow about a solid spherical particle, the following boundary conditions are imposed on the stream function ❾ :

, where ➀ is the radius of the particle and ➋ is the shear coefficient.

Here, only the terms for ➊ = 3 in the first sum of (1) remain. Example 3. Solving the problem of the translational Stokes flow about a spherical drop (or bubble) is reduced to solving

the Stokes equation outside and inside the drop. The boundary condition at infinity is specified in example 1. Conjugate boundary conditions are set at the drop surface; these conditions can be found in the references cited below and are not written out here.

Hadamard–Rybczynski solution:

where ➀ is the radius of the drop, ➅ the unperturbed fluid velocity at infinity, ➌ the ratio of the dynamic viscosities of the fluids inside and outside the drop (the value ➌ = 0 corresponds to a gas bubble and ➌ = ➇

to a solid particle). Example 4. Solving the problem of the axisymmetric straining Stokes flow about a spherical drop (or bubble) is reduced

to solving the Stokes equation outside and inside the drop. The boundary condition at infinity is specified in example 2. Conjugate boundary conditions are set at the drop surface; these conditions can be found in the references cited below and are not written out here.

Taylor solution:

where ➀ is the drop radius, ➋ the shear coefficient, ➌ the ratio of the dynamic viscosities of the fluids inside and outside the

➍✦➎ drop (the value = 0 corresponds to a gas bubble and

to a solid particle).

References : G. I. Taylor (1932), V. G. Levich (1962), J. Happel and H. Brenner (1965), A. D. Polyanin, A. M. Kutepov, A. V. Vyazmin, and D. A. Kazenin (2001).

9.4.6-2. Stokes equation in the bipolar coordinate system. When studying axisymmetric problems of a flow about two spherical particles (drops, bubbles),

one uses the bipolar coordinates , ➐ ; these are related to the cylindrical coordinates ➑ = ② cos ③ , = ② sin ③ by

cosh ➐ − cos ➏

2 The general solution of the equation ❙ ( ) = 0 in the bipolar coordinate system has the form

(cosh ➐ − cos ➏ ↔ ) 3 ✈ 2 ➔ →

2 ) ➐ ➠ + ♣ ↔ sinh ❵ ( ➟ − 2 ) ➐ ➠ + q ↔

cosh ↔ ❵ ( ↔ ➟ − ↔ 2 ) ➐ ➠ ↔ + ♣ ➢ ↔ sinh ↔ ❵ ( ➟ − 2 ↔ ) ➐ ➠ + q ➢ cosh ❵ ( ➟ + 2 ) ➐ ➠ + ➡ ➢ sinh ↔ ❵ ( ➟ + 2 ) ➐ ➠ , ↔

where the ➞ , ♣ , q , ➡ , ➞ ➢ , ♣ ➢ , q ➢ , and ➡ ➢ are arbitrary constants and the ( ➤ ) and ( ➤ ) are the Gegenbauer functions.

Reference : J. Happel and H. Brenner (1965).

9.4.6-3. Stokes equation in the oblate spheroidal coordinate system. When studying axisymmetric problems of flows about spheroidal particles, one uses the oblate ➒

spheroidal coordinates ➏ , ➐ ; these are related to the cylindrical coordinates ➑ = ➥ cos ➦ , = ➥ sin ➦ by

= ➒ ➧ cosh ➏ sin ➐ , = ➧ sinh ➏ cos ➐ .

The solution of the equation ➨ 2 ( ➨ 2 ➩ ) = 0 that describes the flow of a fluid about a prolate spheroid in the direction parallel to the spheroid axis is expressed as

[ ➳ ➵ ( ➳ + 1)] − [( ➳ − 1) ➵ ( ➳ + 1)] arccot ➳

cosh sin

= sinh , ➳ 0 = sinh 0 .

2 [ 0 ( 0 + 1)] − [( ➳ 2 0 − 1) ➵ ( ➳ 2 0 + 1)] arccot ➳ 0 ➸

Here, ➩ is the stream function, is the fluid velocity at infinity, ➧ and ➳

0 are the constants related ➽✦➾ to the spheroid semiaxes

and ➻ ( ➺ > ➻ ) by ➧ = ➼ ➺ 2 − ➻ 2 and ➳ 0 = ➻✳➵ ➧ .

Reference : J. Happel and H. Brenner (1965).