It is shown that for several classes of generalized analytic functions arising in linearized equations of hydrodynamics and magnetohydrodynamics, the Cauchy integral formulae follow from the one for generalized holomorphic vectors in a uniform fashion. If hydrodynamic fields (velocity, pressure and vorticity) admit representations in terms of corresponding generalized analytic functions, those representations and the Cauchy integral formulae form two essential parts of the generalized analytic function approach, which readily yields either closed-form solutions or boundary integral equations. This approach is demonstrated for problems of axisymmetric and asymmetric Stokes flows, two-phase axisymmetric Stokes flows, two-dimensional and axisymmetric Oseen flows.
(a) Vector fields in linear hydrodynamics
A vector field Ω=Ω(x) and a scalar field Ψ=Ψ(x) related by 1.1where is a constant real-valued vector and x is the position vector (multiplier 2 is introduced for convenience), arise in linearized equations of hydrodynamics and magnetohydrodynamics (MHD) (Happel & Brenner 1983; Zabarankin & Krokhmal 2007; Zabarankin 2010, 2011a). For , (1.1) is known as the Moisil–Theodorescu system (Moisil & Theodorescu 1931; Bitsadze 1969), whereas for Ψ=0 and , (1.1) simplifies to the classical potential flow equations (Mises 1944; Bitsadze 1969).
Example 1.1 (Ideal fluid)
The velocity field u of an ideal fluid is irrotational and incompressible (solenoidal), i.e. 1.2which corresponds to (1.1) with Ω=u, Ψ≡0 and .
Example 1.2 (Stokes flows)
Under the assumption of negligible inertial and thermal effects, the time-independent velocity field u of a viscous incompressible fluid is governed by the Stokes equations 1.3where p is the pressure in the fluid, μ is shear viscosity and Δu=∇(div u)−curl(curl u) (Happel & Brenner 1983). The Stokes equations (1.3) imply that the vorticity ω=curl u and pressure p are related by 1.4which corresponds to (1.1) with Ψ=p, Ω=μω and .
Example 1.3 (Oseen flows)
Suppose a solid body translates with constant velocity v in a quiescent viscous incompressible fluid. If the Reynolds number is sufficiently small, the time-independent velocity field u with partially accounted inertial effects can be described by the Oseen equations 1.5where p is the pressure, and μ and ρ are fluid shear viscosity and density, respectively (Happel & Brenner 1983). Let v⋅ω=0 with ω=curl u. Then the Oseen equations (1.5) can be recast in two equivalent forms: 1.6and 1.7which are both particular cases of (1.1): Ψ=p, Ω=μω+ρ[v×u] and in (1.6), and Ψ=p−ρ(v⋅u), Ω=μω and in (1.7).
Let (x,y,z) be a Cartesian coordinate system with basis (i,j,k), and let (r,φ,z) be a cylindrical coordinate system with basis (er,eφ,k). The both coordinate systems have the same z-axis and are related in the ordinary way.
Example 1.4 (Magnetohydrodynamics)
Let a non-magnetic solid body of revolution translate at constant velocity in an electrically conducting viscous incompressible fluid under the presence of an initially constant and uniform magnetic field. It is assumed that body’s axis of revolution, body’s velocity and the direction of the undisturbed magnetic field are all parallel to the z-axis. The fluid velocity u, fluid pressure p and magnetic field disturbances h+ and h− inside and outside the body, respectively, can be described by linearized dimensionless equations of MHD, provided that u and h− are small: 1.8where R is the Reynolds number, M is the Hartmann number and Rm is the magnetic Reynolds number. In this case, the electric field is zero everywhere, and u=ur(r,z)er+uz(r,z)k, ω=curl u=ω(r,z)eφ, p=p(r,z) and . Let . It is shown in Zabarankin (2011a) that (1.8) can be recast in the form of (1.1) with and , j=1,2.
(b) Generalized analytic functions
A generalized analytic (pseudoanalytic) function with the real and imaginary parts u=u(ξ,η) and v=v(ξ,η), respectively, and with is defined by the Bers–Vekua system (Bers 1953; Vekua 1962): 1.9where a=a(ξ,η), b=b(ξ,η), c=c(ξ,η) and d=d(ξ,η) are known real-valued functions. For a≡b≡c≡d≡0, (1.9) simplifies to the Cauchy–Riemann system.
The relationship (1.1) leads to several classes of generalized analytic functions. It can be readily seen that with the substitution (1.1) takes the form 1.10so that , where ΔΥ=div(∇Υ). The system (1.10) defines a generalized holomorphic vector (Υ,Λ) introduced by Obolashvili (1975) (see also Liede (1990)) and is a particular case of the quaternionic equation (Kravchenko & Shapiro 1996; Kravchenko 2003) related to time-harmonic Maxwell’s equations. Next two corollaries show that under certain assumptions on the symmetry of Λ and Υ, the system (1.10) defines h-analytic and H-analytic functions.
Corollary 1.5 (h-analytic functions)
If where λ is a real-valued constant, and 1.11then (1.10) reduces to the system 1.12which defines an h-analytic function . In this case, both u and v satisfy the modified Helmholtz equation and .
Corollary 1.6 (H-analytic functions)
Let and 1.13where λ is a real-valued constant and n is a nonnegative integer. In this case, the vectorial relationship (1.10) simplifies to two equations 1.14which define an nth-order H-analytic function and imply that
Obviously, (1.14) is a particular case of (1.9). Also, if u and v satisfy (1.14), then is a so-called p-analytic function (with p=e−2λzr2n+1) introduced by Polozhii (1973) for arbitrary real-valued characteristic p=p(r,z) as yet another generalization of ordinary analytic functions.
The system (1.14) has several important cases:
For n=0 and λ=0, (1.14) defines a zero-order r-analytic function and arises in axially symmetric problems of Stokes flows (Zabarankin & Krokhmal 2007; Zabarankin 2008a) and isotropic elastic medium (Polozhii 1973; Alexandrov & Soloviev 1978).
(c) Generalized Cauchy integral formula and its application
The theories of p-analytic functions and generalized analytic functions defined by (1.9) furnish general forms for the Cauchy integral formula, which often need to be specialized and refined for particular classes of generalized analytic functions (Chemeris 1995; Kravchenko 2008; Zabarankin 2008a). For example, the theory of p-analytic functions facilitated obtaining the Cauchy integral formula for zero-order H-analytic functions (Zabarankin 2010). However, it does not readily yield an explicit-form Cauchy kernel for nth-order r-analytic functions, which was derived via an integral representation involving ordinary analytic functions similar to the representation for zero-order r-analytic functions (Alexandrov & Soloviev 1978; Zabarankin 2008a). Section 2 shows that the Cauchy integral formulae for h-analytic functions and H-analytic functions with particular cases of n=0 and λ=0 follow in a uniform fashion from the Cauchy integral formula for a generalized holomorphic vector defined by (1.10).
In applications, the generalized Cauchy integral formula is indispensable only if involved physical fields (e.g. velocity, pressure, vorticity, etc.) admit representations in terms of corresponding generalized analytic functions. In some cases as for an ideal fluid, the velocity u is already a generalized analytic function, whereas, for example, for a two-dimensional Stokes flow, the fields u, p and ω are represented by Kolosov’s formulae with two ordinary analytic functions involving their derivatives. Analogues of Kolosov’s formulae with generalized analytic functions are available for three-dimensional Stokes flows (Zabarankin 2008a,b), axisymmetric Oseen flows (Zabarankin 2010) and axisymmetric linearized MHD (Zabarankin 2011a). However, in contrast to Kolosov’s formulae, they contain no derivatives of the involved generalized analytic functions, which considerably simplifies obtaining closed-form solutions and deriving boundary-integral equations based on the generalized Cauchy integral formula (Zabarankin 2008a,b, 2010, 2011a). Thus, the Cauchy integral formula and the analogues of Kolosov’s formulae for u, p, and ω form two essential parts of the generalized analytic function approach to linear hydrodynamics. Section 3 demonstrates this approach in problems of three-dimensional Stokes flows, two-phase axisymmetric Stokes flows, two-dimensional Oseen flows and axisymmetric Oseen flows.
2. Generalized Cauchy integral formula
This section shows that for several classes of generalized analytic functions, the Cauchy integral formulae follow from the one for generalized holomorphic vectors defined by (1.10). The next theorem is the vector-form restatement of the matrix-form Cauchy integral formula for generalized holomorphic vectors given by (8) and (16) in Obolashvili (1975).
Theorem 2.1 (Cauchy integral formula for generalized holomorphic vectors)
Let D be a bounded open region in with a piecewise smooth boundary ∂D, and let Υ and Λ satisfy (1.10) in D, be continuously differentiable in D and continuous in D∪∂D. Then Υ and Λ can be represented in D via their boundary values by 2.1and 2.2where dS(y) is the surface area element, is the outward normal of ∂D at point y, ∇y is the gradient with respect to y, and is the fundamental solution of the modified Helmholtz equation2 with Km(⋅) being the mth-order modified Bessel function of the second kind.3 If then in the two-dimensional case, .
Let D− be the complement of D∪∂D in (or in ) and let (Υ,Λ) be a generalized holomorphic vector continuously differentiable in D− and continuous in D−∪∂D. Then the Cauchy integral formula (2.1) and (2.2) takes the form: , x∈D− and , x∈D−, provided that Υ and Λ vanish at infinity (Liede 1990).
Let f and g be scalar and vector functions, respectively, that are continuous on ∂D. Then and determine a generalized holomorphic vector for x∉∂D. If f and g are Hölder continuous4 on ∂D and x0∈∂D, then the limits of and as x=x+ and x=x− approach x0 from inside and outside D, respectively, are determined by and , which are analogues of the Sokhotski–Plemelj formulae (Obolashvili 1975, eqn 19).
Corollary 2.4 (Ideal fluid)
The equations (1.2) governing the velocity field u of an ideal fluid correspond to (1.10) with Υ=0, Λ=u, and so that for (1.2) in the three-dimensional case, (2.2) simplifies to the formula (Mises 1944; Bitsadze 1969; Morgunov 1974) 2.3whereas in the two-dimensional case, the components ux and −uy form an ordinary analytic function and the linear combination (2.2)(2.2)⋅j with reduces to the ordinary Cauchy integral formula.
Corollary 2.5 (Stokes flow)
The equations (1.4) that relate the pressure p and vorticity ω in a Stokes (creeping) flow correspond to (1.10) with p=Υ, Λ= μ ω and . Thus, for (1.4), the Cauchy integral formula (2.1)–(2.2) yields and where is the triple product of vectors and . These formulae are a vector-form restatement of the matrix-form Cauchy integral formula for the generalized analytic functions defined by the Moisil–Theodorescu system (Bitsadze 1969).
The Cauchy integral formula for h-analytic functions defined by (1.12) is stated in theorem 18 in Duffin (1971) for λ≥0. The next corollary shows that it follows from (2.1) and (2.2) and holds for positive and negative λ.
Corollary 2.6 (Cauchy integral formula for h-analytic functions)
Let D be a bounded open region in the xy-plane with a piecewise smooth positively oriented boundary ℓ, and let G be an h-analytic function in D and Hölder continuous on ℓ, then 2.4where denotes the Cauchy operator for h-analytic functions, and σ=|λ| |τ−ζ|. For (2.4) reduces to the Cauchy integral formula for ordinary analytic functions.
Detail. Let Λ and Υ be represented by (1.11) with , where λ is a real-valued constant, and let x=(x,y) and y=(x1,y1) be vectors in the xy-plane. In this case, the formula (2.2), projected onto k, and the formula (2.1) simplify to 2.5and 2.6respectively, where Φ(y−x)=K0(|λ| |τ−ζ|)/(2π) and ds(y) is the curve length element. Let . Then with and the expression (2.5)(2.6) reduces to 2.7
Finally, the limits and imply that (2.4) reduces to the Cauchy integral formula for ordinary analytic functions as .
Theorem 2.7 (Cauchy integral formula for H-analytic functions)
Let be a bounded open region in the rz-half plane (in the cylindrical coordinates (r,φ,z), r≥0) with a piecewise smooth positively oriented boundary ℓ, which is either closed or an open curve with the endpoints lying on the z-axis (if contains a segment of the z-axis). Also, let G be an nth-order H-analytic function in and Hölder continuous on ℓ. The Cauchy integral formula for G is given by 2.9where is the generalized Cauchy operator, and Ω+(ζ,τ) and Ω−(ζ,τ) are determined by 2.10with 2.11
Let D be the axially symmetric region obtained by revolving around the z-axis in the cylindrical coordinate system (r,φ,z). For nth-order H-analytic functions, Λ and Υ are represented by (1.13) with , where λ is a real-valued constant. Since u(r,z) and v(r,z) in (1.13) do not depend on the angular coordinate φ, the formulae (2.1) and (2.2) can be considered in the rz-half plane corresponding to φ=0. In this case, x=(r er+z eφ)|φ=0=ri+zk, and the coordinate φ will be used to describe the vector y as , so that 2.12Then in the identified rz-half plane, (2.1) and (2.2) projected onto j take the form 2.13and 2.14respectively, where Φ≡Φ(y−x)=Φ(r,z,r1,z1,φ), dS(y)=r1 ds dφ and ds=ds(r1,z1) is the curve length element.
Let , so that and . In this case, where y−x is given by (2.12). Then, with the relationship , which holds for the outward normal and positively oriented ℓ, and with the identity the combination (2.13)(2.14) simplifies to 2.15where
Changing the variable φ=π−2t and using ϱ(ζ,τ,t) in (2.11), we have 2.16and 2.17where the middle integrals in (2.16) and (2.17) are integrated by parts with the relationships respectively. Now, (2.15) with (2.16) and (2.17) yields (2.9)–(2.11). □
The generalized Cauchy integral formula (2.9)–(2.11) extends the one for zero-order H-analytic functions derived in Zabarankin (2010, theorem 1) via the theory of p-analytic functions (Polozhii 1973), whereas the following result was obtained in Zabarankin (2008a) based on an integral representation of nth-order r-analytic functions through ordinary analytic functions.
Corollary 2.8 (Cauchy integral formula for nth-order r-analytic functions)
For λ=0, the kernels (2.10) become real-valued functions determined by 2.18where Γ(⋅) is the gamma function, is the hypergeometric function, and .
Detail. With , the formula (2.18) follows from the relationships 2.19and 2.20see the appendix A, and the fact that .
Corollary 2.9 (Cauchy integral formula for zero-order r-analytic functions)
The case of λ=0 and n=0 in (1.14) corresponds to zero-order r-analytic functions, for which (2.18) simplifies to 2.21where and are complete elliptic integrals of the first and second kinds, respectively. The function has a logarithmic singularity as : .
Corollary 2.10 (Cauchy integral formula for first-order r-analytic analytic functions)
The case of λ=0 and n=1 in (1.14) corresponds to first-order r-analytic functions, for which (2.18) simplifies to 2.22where .
3. Application to linear hydrodynamics
This section demonstrates the approach of generalized analytic functions to problems of linear hydrodynamics. Advantages of this approach are in the convenience of representations of hydrodynamic fields (velocity, vorticity and pressure) and key characteristics (drag, torque and lift) in terms of generalized analytic functions and in the simplicity of obtaining closed-form solutions and boundary-integral equations via the generalized Cauchy integral formulae; compare to the methods, e.g. in Pozrikidis (1992) and Bardzokas et al. (2007).
For clarity, , and will denote the Cauchy operator in (2.9) for zero-order r-analytic, first-order r-analytic and zero-order H-analytic functions, respectively, whereas is the Cauchy operator in (2.4) for h-analytic functions.
(a) Ideal fluid
In the two-dimensional case, equations (1.2) for the velocity field u=ux(x,y)i+uy(x,y)j of an ideal fluid reduce to the Cauchy–Riemann system for ux and −uy, so that is an ordinary analytic function of a complex variable . If a solid infinitely long airfoil is aligned with the z-axis and is immersed into a uniform flow , then and on the boundary ℓ of the airfoil cross section in the xy-plane, where is the outward normal for ℓ. In this case, is the airfoil lifting force (Blasius–Chaplygin formula), where ρ is the fluid density. It simplifies to the Kutta–Joukowski formula .
In the three-dimensional case, u satisfying (1.2) is itself a generalized analytic function for which the Cauchy integral formula is given by (2.3); see Mises (1944) and Morgunov (1974) for application of (2.3) to the ideal fluid. In particular, in an axisymmetric flow with the z-axis of revolution, uz and ur are independent of the angular coordinate φ and form a zero-order r-analytic function with Im[G∂ζ/∂n]=0 on any fixed solid boundary, where .
(b) Stokes flows
Under the zero Reynolds number assumption, the time-independent velocity field u and pressure p of a viscous incompressible fluid are governed by the Stokes equations (1.3). The relationship (1.4) plays a pivotal role in constructing solution forms for (1.3) in terms of generalized analytic functions; see example 1.2.
(i) Two-dimensional case
In the two-dimensional case, u and p depend on the Cartesian coordinates x and y only, i.e. u=ux(x,y)i+uy(x,y)j, ω=ω(x,y)k and p=p(x,y), and consequently, the system (1.4) implies that is an ordinary analytic function. In this case, 3.1where is a complex variable, and g1(ζ) and g2(ζ) are analytic functions; see (4.1) and (4.2) in Richardson (1995). In fact, the representation (3.1) is Kolosov’s formulae (Kolosov 1909) for an incompressible elastic medium (with Poisson’s ratio 1/2). It is used to reduce two-dimensional Stokes flow problems to boundary-integral equations via the Cauchy integral formula (Muskhelishvili 1977, 1992). If a solid infinitely long cylinder of arbitrary cross section is aligned with the z-axis, then is the total flow reaction force per unit length of cylinder’s span, where ℓ is the cross section’s boundary in the xy-plane. However, the problem of unbounded two-dimensional Stokes flow with solid obstacles has no solution bounded at infinity (Happel & Brenner 1983). This well-known paradox is resolved by the Oseen approximation of the Navier–Stokes equations.
(ii) Three-dimensional axially symmetric flows
If a flow is axisymmetric with the z-axis of revolution, then in the cylindrical coordinates (r,φ,z), u and p are independent of the angular coordinate φ: u=ur(r,z) er+uz(r,z)k, uφ≡0, p=p(r,z) and ω=ω(r,z) eφ (ω=eφ⋅curl u). In this case, a complex variable is introduced by , and proposition 7 in Zabarankin (2008a) states that 3.2where G1 and G2 are zero-order r-analytic functions. The representation (3.2) is a three-dimensional analogue of (3.1), but in contrast to the latter, it involves no derivatives of G1 and G2.
Suppose a solid axisymmetric finite body with the z-axis of revolution translates in the quiescent fluid at constant velocity vzk. Then u=vzk on body’s surface S (no-slip boundary condition), and u and p vanish at infinity. Let open regions and be the interior and exterior of the body’s cross section in the rz-half plane (r≥0) with common positively oriented boundary ℓ (cross section of S). Then (3.2) implies that on ℓ and that G1 and G2 vanish at infinity.5 If G1 on ℓ is known, then on ℓ, and with the Cauchy integral formula (2.9) for zero-order r-analytic functions (see corollary 2.9), (3.2) yields representations for u, p and ω in : where the operator is determined by 3.3
Note that has no Cauchy-type singularity on ℓ, and thus, is continuous as ζ approaches ℓ from within . This fact and the boundary condition on ℓ imply that G1 on ℓ satisfies the boundary-integral equation 3.4
The operator (3.3) has only a logarithmic-type singularity, and (3.4) is solved as follows. Let ℓ be parametrized by ζ=ζ(t), t∈[t1,t2], then G1 on ℓ is approximated by a finite functional series , t∈[t1,t2], with basis functions , , where none of ϕ1k(t) is constant, and coefficients ak, bk are found by minimizing the total quadratic error on [t1,t2]:
The representation (3.2) and the boundary condition on ℓ yield , ζ∈ℓ, and proposition 11 in Zabarankin (2008a) shows that the drag exerted on the body is given by Note that Fz is unaffected if G1 is added a real constant; so Fz holds for any solution of (3.4). For example, if in the rz-half plane, a sphere of radius a is parametrized by , t∈[−1,1], then , t∈[−1,1] and Fz=−6πμ vza, which is the well-known Stokes formula for the sphere drag. Zabarankin (2008a) showed that for prolate and oblate spheroids, biconvex lens and torus of circular cross section, solutions of the boundary-integral equation (3.4) coincide with corresponding analytical solutions in Happel & Brenner (1983), Zabarankin & Krokhmal (2007) and Zabarankin &Ulitko (2006). Also, Zabarankin (2008a) solved (3.4) for solid bi-spheroids (two separate spheroids of equal size) and torus of elliptical cross section, whereas Zabarankin & Molyboha (2010) used (3.4) to find minimum-drag shapes for solid bodies of revolution subject to constraints on body’s volume and body’s shape.
(iii) Two-phase three-dimensional axially symmetric flow
Suppose an initially spherical liquid drop with radius a is placed into an extensional flow at the origin (r=z=0), where ς is the share rate, and suppose D+ and D− are the regions occupied by the deformed drop and the ambient fluid, respectively, with common boundary S. Let u+ and u− be the actual velocity in the drop and the velocity disturbance of the extensional flow, respectively, and let p± be the pressure in D±. It is assumed that the drop and the ambient fluid are incompressible and viscous with the same viscosity μ and that for fixed S, u± and p± satisfy the Stokes equations (1.3) in D±. For convenience, let the linear dimensions, u±, and p± be rescaled by a, a ς and μ ς, respectively, and let Ca=μa ς/γ be the capillary number, where γ is the interfacial tension.
This problem is axisymmetric with the z-axis of revolution, and the boundary conditions on S are given by 3.5where is the outward normal for S and ω±=eφ⋅curl u± (Zabarankin & Nir 2011, proposition 3.1). Also, u− and p− vanish at infinity.
At the steady state, on S (kinematic condition), and in this case, u±, ω± and p± are time-independent and have a representation similar to that of (3.2): 3.6and 3.7where and are zero-order r-analytic functions in D± with and with G−1 and G−2 vanishing at infinity, and c is a real constant.
Let open regions be the interior of cross sections of D± in the rz-half plane, and let ℓ be common positively oriented smooth boundary of . With the Cauchy integral formula for zero-order r-analytic functions (corollary 2.9), the representations (3.6) and (3.7) and the boundary conditions (3.5) yield a closed-form solution for u±, p± and ω± (Zabarankin & Nir 2011, theorem 3.3): 3.8where , f−=0 and the operator is defined by (3.3). Zabarankin & Nir (2011) used (3.8) to find the drop’s steady shape from the kinematic condition.
(iv) Three-dimensional asymmetric flows
Suppose a solid axisymmetric finite body with the z-axis of revolution either translates in the quiescent fluid along the x-axis (x-translation) or rotates around the y-axis (y-rotation). Proposition 2 in Zabarankin (2008b) shows that in this case, u and p are represented by 3.9where is a complex variable, G1=G1(ζ) and G2=G2(ζ) are zero-order r-analytic functions, G3=G3(ζ) is a first-order r-analytic function, and all three G1, G2 and G3 vanish at infinity. On the body’s surface S, the boundary conditions are given by u=vxi for the x-translation and by u=[ϖy j×(x i+zk)] for the y-rotation, where vx and ϖy are constants.
Let and be open regions corresponding to the interior and exterior of the body’s cross section in the rz-half plane, and let ℓ be common positively oriented boundary of (cross section of body’s surface S). Then and on ℓ, where and f2=vx for the x-translation and and f2=ϖyz for the y-rotation. If G1, G2 and G3 on ℓ are determined, then representing ur, uφ and uz in with (3.9) and with the Cauchy operators and is straightforward (see corollaries 2.9 and 2.10).
Suppose that G1 and Re G3 on ℓ are known, then and on ℓ, where f3=vx for the x-translation and for the y-rotation. Theorem 3 in Zabarankin (2008b) proves that G1 and Re G3 on ℓ are found from a system of two boundary-integral equations: 3.10where f4=−vx for the x-translation and for the y-rotation (2Im G3=−rIm G1, ζ∈ℓ, is retained for brevity of notation). Observe that , ζ∈ℓ, is the Sokhotski–Plemelj formula for G3 on ℓ, which is equivalent to a single real-valued equation in the sense that if Re G3 on ℓ is known then Im G3 on ℓ is found from this formula, and vice versa. Thus, the system (3.10) is viewed as three real-valued equations with three real-valued unknowns Re G1, Im G1 and Re G3. Zabarankin (2008b) solved (3.10) with the quadratic error minimization technique for the x-translation of bi-spheroids and for the y-rotation of torus of elliptical cross section and showed that the solutions for bi-spheres and torus of circular cross section coincide with corresponding analytical solutions in Goren & O’Neill (1980), Wakiya (1967) and Zabarankin (2007).
Propositions 7 and 8 in Zabarankin (2008b) state that the drag exerted on the body in the x-translation and the torque in the y-rotation are determined by which compared with (48a) and (54a) in Zabarankin (2008b), respectively, have additional multiplier 2 because multiplier 1/2 was omitted from the right-hand side in (3.9). Observe that Fx and Fz in the axisymmetric translation have similar expressions.
(c) Oseen flows
Suppose a solid body translates in a viscous incompressible fluid with constant velocity. Under the low Reynolds number assumption, the time-independent velocity field and pressure are governed by the Oseen equations (1.5). Example 1.3 implies that u and p can be represented by h-analytic functions in the two-dimensional case and by H-analytic functions in the three-dimensional case.
(i) Two-dimensional case
If the body is a solid infinitely long cylinder of arbitrary cross section, which is aligned with the z-axis and translates in the quiescent fluid at constant velocity v=−vxi, then u=ux(x,y)i+uy(x,y)j, ω=ω(x,y)k and p=p(x,y), and u and p vanish at infinity in the xy-plane. In this case, (1.6) and (1.7) determine an ordinary analytic function g and h-analytic function h, respectively, with a complex variable , and thus, u, p and ω have a representation 3.11where λ=ρvx/(2μ)≠0, and g and h vanish as .
The functions g and h are uniquely determined. Indeed, let pairs g1, h1 and g2, h2 be two different solutions, then g=g2−g1 and h=h2−h1 solve the exterior homogeneous two-dimensional Oseen flow problem, which with u and p vanishing as has only zero solution (Finn 1965). Thus, (3.11) with p≡0 and ω≡0 yields Im g≡0 and Re h≡0, and then (3.11) and u≡0 imply Re g≡0 and Im h≡0.
Let open regions D+ and D− be cross sections of the cylindrical body and fluid in the xy-plane, respectively, with common boundary ℓ (positively oriented Jordan curve). The boundary condition u=−vxi on ℓ and the representation (3.11) imply on ℓ. Suppose the value of g on ℓ is known. Then on ℓ, and with the Cauchy integral formulae for analytic and h-analytic functions, (3.11) yields representations for u, p and ω in D−: and , ζ∈D−, where the operator is given by with σ=|λ| |τ−ζ| and Km(⋅) being the mth-order modified Bessel function of the second kind. Observe that the operator has no Cauchy-type singularity on ℓ. Consequently, is continuous as ζ approaches ℓ from within D−, and on ℓ implies that g satisfies the boundary-integral equation 3.12
Let λ≠0, then a homogeneous solution of (3.12) is any imaginary constant and , ζ∈ℓ, where is a solution of (3.12), and c is the imaginary constant determined by ζ∈ℓ.
The proof follows the approach of Muskhelishvili (1992). The necessary and sufficient condition for a complex-valued function g on ℓ to be the boundary value of an analytic function in that vanishes at infinity is given by the Sokhotski–Plemelj formula 3.13
Let , ζ∈ℓ, be a solution of (3.12), then the Cauchy-type integrals define ordinary analytic and h-analytic functions, respectively, in D+. Then the boundary-integral equation (3.12) is equivalent to g1+eλxh1=0 on ℓ (through Sokhotski–Plemelj formulae as ζ approaches ℓ from within D+). Observe that similar to (3.11), is a formal solution of the two-dimensional Oseen equations (1.5) in D+. However, (1.5) with u=0 on ℓ yields u≡0 in D+. Thus, g1≡−eλxh1 in D+, which holds only if g1≡c1 in D+, where c1 is an imaginary constant. As ζ approaches ℓ from within D+, the Cauchy-type integral that defines g1 implies for any ζ∈ℓ. Since g should satisfy (3.13), the solution , ζ∈ℓ, follows. Now if is another solution of (3.12) with a corresponding imaginary constant c*, then because g is uniquely determined (remark 3.1), , ζ∈ℓ or, equivalently, , ζ∈ℓ, and thus, a homogeneous solution of (3.12) is only an imaginary constant. ■
It follows from (3.11) and on ℓ that and , which yields the drag exerted on the cylinder per unit length of cylinder’s span: 3.14where is the outward normal for ℓ and ds is the curve length element. Since Fx is unchanged by adding a constant to g, it holds for any solution of (3.12). Observe that 2λμ=ρvx and (3.14) bears a close resemblance to the Kutta–Joukowski formula (see §3a).
As a verification, (3.12) is solved for a circular cylinder with radius 1. In this case, and , t∈[−π,π], with ak and bk found by the quadratic error minimization technique (see §3b(ii)). The obtained drag coefficient CD=Fx/(2λμvx) for Reynolds numbers R=4λ=0.2, 2, 20, 200 and 2000 is 34.6541, 8.0847, 3.4658, 2.5387, 2.3237, respectively, versus 34.655, 8.090, 3.469, 2.545 and 2.324, respectively, computed in Miyagi (1974).
(ii) Translation of solid bodies of revolution
Suppose a solid axisymmetric finite body with the z-axis of revolution translates in the quiescent fluid at constant velocity −vzk, so that u=−vzk on the body’s surface S, and u and p vanish at infinity. In this case, u and p are independent of the angular coordinate φ: u=ur(r,z)er+uz(r,z)k, uφ≡0, p=p(r,z) and ω=ω(r,z)eφ, and proposition 1 in Zabarankin (2010) states that 3.15where λ=ρvz/(2μ)≠0, and G and H are zero-order r-analytic and zero-order H-analytic functions, respectively, that both vanish at infinity.
Let open regions and be the interior and exterior of the body’s cross section in the rz-half plane (r≥0), respectively, with common positively oriented boundary ℓ. Then (3.15) implies that G+eλzH=−vz on ℓ. If the boundary value of G on ℓ is known, then H=−e−λz(vz+G) on ℓ and with the Cauchy integral formula (2.9), (3.15) furnishes representations for u, p and ω in : and , , where is determined by
The operator has no Cauchy-type singularity on ℓ, so that is continuous as ζ approaches ℓ from within , and the boundary condition on ℓ implies that G on ℓ satisfies the boundary-integral equation 3.16
The representation (3.15) and on ℓ yield and and proposition 10 in Zabarankin (2010) shows that the drag exerted on the body is given by Observe that a real constant added to G has no effect on the drag.
As an illustration, Zabarankin (2010) solved the boundary-integral equation (3.16) with the quadratic error minimization technique for sphere, spheroids and bi-spheroids, and showed that the use of the conjugate kernels in the Cauchy integral formula for zero-order H-analytic functions improves technique’s running time by several times. Zabarankin & Molyboha (2011) used (3.16) with the conjugate kernels in an iterative procedure for finding minimum-drag shapes for solid bodies under the low Reynolds number assumption subject to constraints on body’s volume and body’s shape.
The approach of generalized analytic functions to the MHD problem formulated in example 1.4 is demonstrated in Zabarankin (2011a). With the generalized Cauchy integral formulae, the MHD problem is reduced to boundary-integral equations, which are then employed in an iterative procedure for finding minimum-drag shapes for solid nonconducting bodies in the MHD flow under the assumption of small R, Rm and M; see Zabarankin (2011b).
We are grateful to the anonymous referees for their valuable comments and suggestions, which greatly helped to improve the quality of the paper.
For |β|<1 and , formula (10) in §2.4 in Bateman & Erdelyi (1953) yields whereas formula (24) in §2.1.5 in Bateman & Erdelyi (1953) implies Consequently, where , so that where . For any positive a, b and c, formulae (20) and (22) in §2.8 in Bateman & Erdelyi (1953) yield so that , where , and thus, I1 simplifies to (2.19). Similarly, Now, (34) in §2.8 in Bateman & Erdelyi (1953) implies that which, with formula (39) in §2.8 in Bateman & Erdelyi (1953), results in (2.20).
↵2 The fundamental solution Φ(x) solves , where δ(⋅) is the Dirac delta function.
↵4 A function f(x) is Hölder continuous on ∂D, if ∥f(x2)−f(x1)∥≤c ∥x2−x1∥α for any x1,x2∈ ∂D, some α∈(0,1] and non-negative constant c.
↵5 A zero-order r-analytic function G vanishing at infinity behaves as G=o(|ζ|−1) as (Zabarankin 2008a). Since the second equation in (3.2) implies that G1 vanishes at infinity, i.e. G1=o(|ζ|−1) as , the first one in (3.2) yields G2=o(1) as .
- Received June 1, 2012.
- Accepted July 13, 2012.
- This journal is © 2012 The Royal Society