## Abstract

A necessary optimality condition for the minimum-drag shape for a non-magnetic solid body immersed in the uniform flow of an electrically conducting viscous incompressible fluid under the presence of a magnetic field is obtained. It is assumed that the flow and magnetic field are uniform and parallel at infinity, and that the body and fluid have the same magnetic permeability. The condition is derived based on the linearized magnetohydrodynamic (MHD) equations subject to a constraint on the body’s volume, and generalizes the existing optimality conditions for the minimum-drag shapes for the body in the Stokes and Oseen flows of a non-conducting fluid. It is shown that for any Hartmann number *M*, Reynolds number *Re* and magnetic Reynolds number *Re*_{m}, the minimum-drag shapes are fore-and-aft symmetric and have conic vertices with an angle of 2*π*/3. The minimum-drag shapes are represented in a function-series form, and the series coefficients are found iteratively with the derived optimality condition. At each iteration, the MHD problem is solved via the boundary integral equations obtained based on the Cauchy integral formula for generalized analytic functions. With respect to the equal-volume sphere, drag reduction as a function of the Cowling number S=*M*^{2}/(*Re*_{m} *Re*) is smallest at S=1. Also, in the considered examples, the drag values for the minimum-drag shapes and equal-volume minimum-drag spheroids are sufficiently close.

## 1. Introduction

There has been significant theoretical interest in minimum-drag shapes for a solid body immersed in the uniform flow of a non-conducting viscous incompressible fluid. Pironneau (1973, 1974) derived the necessary optimality condition for the minimum-drag shape subject to a volume constraint under the Stokes approximation of the Navier–Stokes equations, whereas Mironov (1975) extended the condition for the Oseen approximation and the full Navier–Stokes equations. It is well known by now that the ratio of the drag of the minimum-drag body to the drag of an equal-volume sphere decreases with the Reynolds number *Re* (drag reduction is considerably more significant for high *Re*), that the minimum-drag shape becomes more prolate with an increase in *Re*, and that for any *Re*, the minimum-drag shape has conic vertices, each with an angle of 2*π*/3. How would these results change if the fluid is electrically conducting and is under the presence of a magnetic field? These questions are also of direct interest to several industrial magnetohydrodynamic (MHD) applications including MHD flow control schemes for hypersonic vehicles (Petit *et al.* 2009), torpedo drag reduction with MHD boundary-layer control (Anderson & Wu 1971), MHD propulsion systems (Swallom *et al.* 1991), etc.

This work studies the minimum-drag shapes for a non-magnetic constant-volume solid body immersed in the uniform flow of an electrically conducting viscous incompressible fluid under the presence of a uniform magnetic field aligned with the undisturbed flow. It is particularly motivated by the fact that in this MHD problem, the drag for a non-magnetic sphere as a function of the Cowling number^{1} S has a minimum at S=1 and, remarkably, is non-smooth at S=1 (see (Zabarankin 2011, fig. 1). If the sphere is replaced by an arbitrary non-magnetic body of constant volume, how does the minimum-drag shape depend on S and would it be qualitatively different for S<1, S=1 and S>1? To address these questions, we derive a necessary optimality condition for the minimum-drag shape subject to a volume constraint analytically, represent the shape in the form of a function series and find the series coefficients iteratively. This study is the followup work of Zabarankin (2011), in which a solution to the MHD problem for an arbitrary body of revolution is represented in terms of four generalized analytic functions, and the MHD problem is reduced to boundary integral equations based on the generalized Cauchy integral formula. Here, we will also use the minimum-drag spheroids obtained in Zabarankin (2011) as initial approximations in an iterative procedure for finding the minimum-drag shapes.

For convenience, we restate the MHD problem formulated in Zabarankin (2011). Let (*r*,*φ*,*z*) be a cylindrical coordinate system with the basis (**e**_{r},**e**_{φ},**k**), and let the undisturbed flow of the fluid and the applied magnetic field be both uniform and aligned with the *z*-axis, i.e. and , where and are constants. Suppose a non-magnetic solid body of revolution is immersed in the flow with the body’s axis of revolution being parallel to the *z*-axis. Let and be the regions occupied by the body and fluid, respectively, and let *S* be their common piece-wise smooth boundary (surface of the body). Then, the fluid velocity **U** and magnetic filed **H** can be represented by in and in , where **u** and **h**^{±} are the disturbances of the fluid velocity and magnetic field, respectively. On the boundary *S* and at infinity, **u** and the fluid pressure ℘ satisfy the conditions
1.1
whereas under the assumption that the fluid and body have the same magnetic permeability, the disturbance of the magnetic field is continuous across S and should vanish at infinity,
1.2

This problem is axially symmetric: the velocity, pressure and electromagnetic field are independent of the angular coordinate *φ*, i.e. **u**=*u*_{r}(*r*,*z*)**e**_{r}+*u*_{z}(*r*,*z*)**k**, ℘=℘(*r*,*z*), and the electric field is zero everywhere (see Zabarankin 2011 for details). In this case, **u**, ℘ and **h**^{±}, being rescaled by , and , respectively, satisfy the linearized dimensionless MHD equations^{2}
1.3
in ,
1.4
in , where *a* is the half of the diameter of the body,^{3} is the Reynolds number, is the Hartmann number, is the magnetic Reynolds number, *ν* is the kinematic viscosity, *ρ* is the fluid density, *μ* is the magnetic permeability and *σ* is the electric conductivity (see also Cabannes 1970). All constants are measured in metre, kilogram and second (MKS) units.

The rest of this work is organized in five sections. Section 2 derives the necessary optimality condition for the minimum-drag shape subject to a volume constraint, which involves a solution of the problem adjoint to the MHD problem (1.1)–(1.4). Section 3 reduces the adjoint problem to boundary integral equations, with the approach of generalized analytic functions presented in Zabarankin (2011), and shows that solutions of the boundary integral equations and those derived for the MHD problem (1.1)–(1.4) in Zabarankin (2011) are closely related. Section 4 establishes the asymptotic behaviour of a solution to the MHD problem in the vicinity of a conic vertex. Section 5 shows that the minimum-drag shapes have conic vertices with an angle of 2*π*/3 for any *Re*, *M* and *Re*_{m}. It also obtains the minimum-drag shapes in a function series form based on the derived optimality condition and the iterative procedure suggested in Zabarankin & Nir (2011). Section 6 concludes the work.

## 2. Necessary optimality condition for minimum-drag shape

In the axially symmetric MHD problems (1.1)–(1.4), the drag is parallel to the *z*-axis and its absolute value is given by
2.1
where ** n** is the outward normal (see Zabarankin 2011, §3

*d*).

The shape-optimization problem is to minimize (2.1) with respect to *S* subject to (1.1)–(1.4) and the constraint on the volume of to be 4*π*/3 (the volume of a unit sphere).

To derive necessary optimality conditions for this problem, we use Mironov’s shape variation approach (Mironov 1975). Let ** r** be the radius vector representing the optimal shape

*S*, and let the shape variation

*S*

_{ϵ}be determined by where

*ϵ*is a positive small number, and

*f*(

**) is an arbitrary continuous bounded scalar function such that . The condition on**

*r**f*follows from the volume variation: if is the region bounded by the surface

*S*

_{ϵ}, then and since ,

*f*should satisfy .

Let **u**, ℘ and **h**^{±} be the velocity disturbance, pressure and magnetic field disturbances in the fluid and the body with the *optimal* shape, and let their variations be given by
where **u**_{1}, ℘_{1} and satisfy equations (1.3) and (1.4).

The boundary condition (1.1) should hold for **u**_{ϵ} on *S*_{ϵ}: **u**_{ϵ}=−**k**, which implies **u**_{ϵ}=**u**(** r**+

*ϵf*(

**)**

*r***)+**

*n**ϵ*

**u**

_{1}(

**+**

*r**ϵf*(

**)**

*r***)+**

*n**o*(

*ϵ*)=−

**k**, whence and, consequently,

*f*∂

**u**/∂

*n*+

**u**

_{1}=0 on

*S*, or, equivalently, 2.2

Similarly, the boundary condition (1.2) should hold for on *S*_{ϵ}: **h**^{+}_{ϵ}=**h**^{−}_{ϵ}, which implies **h**^{+}(** r**+

*ϵf*(

**)**

*r***)+**

*n**ϵ*

**h**

^{+}

_{1}(

**+**

*r**ϵf*(

**)**

*r***)+**

*n**o*(

*ϵ*)=

**h**

^{−}(

**+**

*r**ϵf*(

**)**

*r***)+**

*n**ϵ*

**h**

^{−}

_{1}(

**+**

*r**ϵf*(

**)**

*r***)+**

*n**o*(

*ϵ*), so that and, consequently, 2.3

The variation of the functional (2.1) can be determined as follows:
2.4
with
2.5
and
2.6
where is the region bounded by the surfaces *S* and *S*_{ϵ}. In obtaining equation (2.6), we used the divergence theorem, the first equation in (1.3), and the fact that **k**⋅[**k**×[**k**×(**u**−**h**^{−})]]=0.

A further transformation of *I*_{ϵ} relies on the equations adjoint to equations (1.3) and (1.4).

### Proposition 2.1 (adjoint equations)

*Let* *and* *satisfy the adjoint equations*
2.7
*in*
2.8
*with the boundary conditions on* *S* *and conditions at infinity given by*
2.9
*Then,*
2.10
*provided that*
*where* *S*_{R} *is a sphere centred at the origin with large radius* *R*.

### Proof.

Since **u**_{1}, ℘_{1} and **h**^{−}_{1} satisfy equations (1.3), we can write
which in view of the adjoint equations (2.7) implies
2.11
On the other hand, since **h**^{+} satisfies curl **h**^{+}=0 and div**h**^{+}=0, we have
which in view of equation (2.8) reduces to
2.12

Let be the region bounded by the surface *S* and sphere *S*_{R}. Integrating equations (2.11) and (2.12) over and , respectively, and then using the divergence theorem and adding the resulting equations, we obtain
which with the boundary conditions (2.9) and the assumption that as can be rearranged in the form (2.10).

With the relationship (2.10) and the boundary conditions (2.2) and (2.3), the functional (2.5) takes the form 2.13

Now since ∂**u**/∂*s*=0 on *S*, we observe that
2.14

Let on *S*. Then, the boundary condition (1.2) and equations div**h**^{±}=0 in imply that and on *S*. In this case, on *S* (see the proof of proposition 11 in Zabarankin (2008)). On the other hand, it follows from equations (1.4), the second equation in (1.3), and (1.1) that on *S*. Consequently,
2.15

Also, in the axially symmetric case, curl **u**=*ω*(*r*,*z*) **e**_{φ}, curl **w**=*ω**(*r*,*z*) **e**_{φ} and **g**^{−}=*g*^{−}(*r*,*z*) **e**_{φ}.

Finally, in view of equations (2.6) and (2.13) with equations (2.14) and (2.15), the variation (2.4) reduces to

The necessary optimality condition requires , which yields . However, since arbitrary bounded *f* should satisfy , we conclude that *ωω**+*Re*_{m}*h*^{−}_{r}*g*^{−} is constant on *S*.

Thus, the following result has been proved.

### Theorem 2.2

*Under the assumptions of proposition 2.1, the necessary optimality condition for the minimum-drag shape subject to the volume constraint is given by
*
2.16
*where ω=***e**_{φ}⋅curl **u***, ω***=***e**_{φ}⋅curl **w***, h*^{−}_{r}*=***e**_{r}*⋅***h**^{−} *and g*^{−}*=***e**_{φ}*⋅***g**^{−}*, and* **u** *and* **h**^{−} *solve the MHD problem (1.1)–(1.4), whereas* **w** *and* **g**^{−} *solve the adjoint problem (2.7)–(2.9).*

In the case of a non-conducting viscous incompressible flow governed by either Stokes equations, Oseen equations or Navier–Stokes equations, the necessary optimality condition for the minimum-drag shape subject to a volume constraint is given by *ωω**=const. on *S*, where *ω* and *ω** are those as defined in theorem 2.2 (Pironneau 1973; Mironov 1975). Although for *Re*_{m}=0, (2.16) simplifies to *ωω**=const. on *S*, for the electrically conducting flow, the applied magnetic field still enters the condition (2.16) through *ω* and *ω** as seen from the first equations in (1.3) and (2.7).

Now the task is to obtain the minimum-drag shapes based on the condition (2.16) that requires solving the MHD problem (1.1)–(1.4) along with its adjoint problem (2.7)–(2.9). In three complementary cases: (a) *Re*_{m}≠0, *Re*_{m}*Re*≠*M*^{2}(*S*≠1); (b) *Re*_{m}=0; and (c) *Re*_{m}*Re*=*M*^{2}(*S*=1), all assuming *M*≠0,^{4} the MHD problem (1.1)–(1.4) is reduced to boundary integral equations based on the Cauchy integral formula for generalized analytic functions (see theorems 3.3, 3.10, 3.13 in Zabarankin (2011)). The next section obtains similar boundary integral equations for the adjoint problem (2.7)–(2.9) for each case separately and shows that solutions to the boundary integral equations and those in the theorems 3.3, 3.10, 3.13 in Zabarankin (2011) are closely related.

## 3. Solving the adjoint magnetohydrodynamic problem

The adjoint problem (2.7)–(2.9) is axially symmetric. In the cylindrical coordinate system (*r*,*φ*,*z*), in which the *z*-axis is the axis of revolution, the functions **w**, *q*, **g**^{±} and *v*^{±} are independent of the angular coordinate *φ*,

As in the study of Zabarankin (2011), a solution to the problem (2.7)–(2.9) can be represented in terms of four generalized analytic functions from two classes: *r*-analytic and *H*-analytic. A complex-valued function is *H*-analytic if its real and imaginary parts *U*=*U*(*r*,*z*) and *V* =*V* (*r*,*z*), respectively, satisfy the system
3.1
with a real-valued non-zero constant *λ*. For *λ*=0, the system (3.1) defines so-called *r*-analytic functions. The generalized Cauchy integral formula and series representations in the spherical coordinates for these two classes of functions are presented in Zabarankin (2008, 2010, 2011). In case (a), a solution to the problem (2.7)–(2.9) involves two *H*-analytic functions and two *r*-analytic functions; in case (b), it has only two *H*-analytic functions; whereas in case (c), it is represented by one *H*-analytic function and three *r*-analytic functions.

Let *ζ*=*r*+*i* *z* be a complex variable, where . For brevity, a function *f*(*r*,*z*) will be denoted by *f*(*ζ*) without assuming its analyticity. Let be the open regions corresponding to the interiors of the cross sections of in the right-half *rz*-plane (*r*≥0), and let ℓ be the positively oriented common boundary of and (ℓ is the cross section of *S* in the right-half *rz*-plane and is either a closed curve or an open curve with the endpoints lying on the *z*-axis). In this case, ℓ′ will denote the reflection of ℓ over the *z*-axis.

### (a) Case (a): *M*≠0, *Re*_{m}≠0 and *Re*_{m}*Re*≠*M*^{2}

### Theorem 3.1 (solution representation for the adjoint equations, case (a))

*Let* *as in theorem 2.1 in* Zabarankin (2011)*. In the axially symmetric case with M≠0, Re*_{m}*≠0 and Re*_{m}*Re≠M*^{2}*, a solution to the adjoint equations (2.7) and (2.8) is given by
*
3.2
*where G*_{1} *and G*_{2} *are H-analytic functions in* *that satisfy system (3.1) with λ=−λ*_{1,2}*, respectively, and vanish at infinity;* *are r-analytic functions in* *respectively, with G*^{−}_{3} *vanishing at infinity; and* *.*

### Proof.

Proved similarly to theorem 2.1 in Zabarankin (2011).

With the representations (3.2), the adjoint problem (2.7)–(2.9) reduces to the boundary-value problem for the generalized analytic functions *G*_{1}, *G*_{2} and introduced in theorem 3.1,
3.3
The problem (3.3) can be reduced to integral equations for the boundary values of *G*_{1} and *G*_{2} based on the generalized Cauchy integral formula for *H*-analytic and *r*-analytic functions.

### Theorem 3.2 (adjoint integral equations, case (a))adjoint integral equations, case (a)

*Let M≠0, Re*_{m}*≠0 and Re*_{m}*Re≠M*^{2}*. The problem (3.3) yields two integral equations for the boundary values of F*_{k}*(ζ)=e*^{−λkz}*G*_{k}*(ζ), k=1,2,
*
3.4
*where τ=r*_{1}*+iz*_{1}*, λ*_{1} *and λ*_{2} *are defined in theorem 3.1, and* *and* *are the generalized Cauchy kernels corresponding to r-analytic and H-analytic functions, respectively, and defined in theorem 2 in* Zabarankin (2008) *and theorem 1.1 in* Zabarankin (2011)*, respectively. A solution to equations (3.4) is determined up to a real-valued constant c, i.e. F*_{k}*(ζ)=c, k=1,2, are a homogeneous solution to equations (3.4). Let* *k=1,2, solve equations (3.4), then G*_{k}*(ζ), k=1,2, are determined by
*
3.5
*where
*
3.6

### Proof.

Proved similarly to theorem 3.3 in Zabarankin (2011).

In fact, the only difference in the integral equations (3.4) here and the integral eqns (3.14) in Zabarankin (2011) is in the sign of *λ*_{k} in the kernels. The next proposition shows that solutions of equations (3.4) here and eqns (3.14) in Zabarankin (2011) are closely related.

### Proposition 3.3

*Let* *be the reflection of the curve ℓ over the* *r**-axis, and let* *F*_{k}(*ζ*), *k*=1,2, *be a solution to the integral eqns (3.14) in* Zabarankin (2011) *for* *then* *k*=1,2, *ζ*∈ℓ, *is a solution to the integral equations (3.4), where* *and* *are complex conjugates of* *ζ* *and* *F*_{k}, *respectively*.

### Proof.

With the change of variables , , and , , in equation (3.4) and with the properties and , the integral equation (3.4) here reduces to integral eqns (3.14) in Zabarankin (2011) for unknown functions , *k*=1,2, .

### (b) Case (b): *M*≠0 and *Re*_{m}=0

For *Re*_{m}=0, the magnetic field in the MHD problem (1.1)–(1.4) is uncoupled from the velocity field, and corollary 2.2 in Zabarankin (2011) shows that in this case, it is constant everywhere. In the adjoint equations (2.7)–(2.8), when *Re*_{m}=0, **w** and *q* are uncoupled from **g**^{±} and *v*^{±}, and their representation follows directly from representation (3.2).

### Corollary 3.4 (solution representation for the adjoint equations, case (b))

*Let* . *In the axially symmetric case with* *M*≠0 *and* *Re*_{m}=0, *a solution to the adjoint equations (2.7) and (2.8) is given by*
3.7
*where* *G*_{1}, *G*_{2} *and* *are defined in theorem 3.1 for the case of* *Re*_{m}=0.

In this case, both equations in (3.3) reduce to the same boundary-value problem
3.8
Once *G*_{1} and *G*_{2} are determined from the problem (3.8), the boundary values of can be found based on the last two formulae in representation (3.7) and the fact that *v*^{+}+i*g*^{+}=*v*^{−}+i*g*^{−} on ℓ. Namely, *G*^{+}_{3}−*G*^{−}_{3}=(2/(*λ*_{2}−*λ*_{1}))(*λ*_{2} *e*^{−λ1z}*G*_{1}−*λ*_{1}*e*^{−λ2z}*G*_{2}) on ℓ, and are given by the Cauchy integral formulae (24)–(27) for *r*-analytic functions in Zabarankin (2008).

### Theorem 3.5 (adjoint integral equations, case (b))

*Let M≠0 and Re*_{m}*=0, and let F*_{1}*(ζ)=e*^{−λ1z}*G*_{1}*(ζ), where λ*_{1} *is defined in corollary 3.4, then (3.8) reduces to the integral equation for the boundary value of F*_{1}*,
*
3.9
*for ζ∈ℓ. A solution to equation (3.9) is determined up to a real-valued constant c. Let* *solve equation (3.9), then G*_{1}*(ζ) in problem (3.8) is given by
*
3.10
*where
*
3.11

### Proof.

Proved similarly to theorem 3.10 in Zabarankin (2011).

### Proposition 3.6

*Let* *be the reflection of the curve ℓ over the* *r*-*axis, and let* *F*_{1}(*ζ*) *be a solution to the integral eqn (3.26) in* Zabarankin (2011) *for* *then*, *ζ*∈ℓ, *is a solution to the integral equation (3.9)*.

### Proof.

The proof is similar to that of proposition 3.3.

### (c) Case (c): *M*≠0 and *Re*_{m}*Re*=*M*^{2}

### Theorem 3.7 (solution representation for the adjoint equations, case (c))

*Let λ=(Re+Re*_{m}*)/2. In the axially symmetric case with M≠0 and Re*_{m}*Re=M*^{2}*, a solution to the adjoint equations (2.7) and (2.8) is given by
*
3.12
3.13
*where G*_{1} *is an H-analytic function in* *satisfying system (3.1) with λ=(Re+Re*_{m}*)/2 and vanishing at infinity; G*_{2} *and G*^{−}_{3} *are r-analytic functions in* *and vanishing at infinity; and G*^{+}_{3} *is an r-analytic function in* *.*

### Proof.

Proved similarly to theorem 2.5 in Zabarankin (2011).

In this case, the adjoint problem (2.7)–(2.9) reduces to the boundary-value problem for the generalized analytic functions introduced in theorem 3.7, 3.14

### Theorem 3.8 (adjoint integral equations, case (c))

*Let M≠0 and Re*_{m}*Re=M*^{2}*, and let λ=(Re+Re*_{m}*)/2, then the boundary-value problem (3.14) reduces to two integral equations for F*_{1}*(ζ)=e*^{−λz}*G*_{1}*(ζ) and F*_{2}*(ζ)=G*_{2}*(ζ),
*
3.15
*where ζ∈ℓ. Equations (3.15) determine F*_{1} *and F*_{2} *up to c and 2c, where c is a real-valued constant. Let* *and* *solve (3.15), then the solution to the boundary-value problem (3.14) is given by
*
3.16
*where
*
3.17

### Proof.

Proved similarly to theorem 3.13 in Zabarankin (2011).

### Proposition 3.9

*Let* *be the reflection of the curve ℓ over the* *r*-*axis, and let* *F*_{k}(*ζ*), *k*=1,2, *be a solution to the integral eqns (3.32) in* Zabarankin (2011) *for* *then* *k*=1,2, *ζ*∈ℓ, *are a solution to the integral equations (3.15)*.

### Proof.

The proof is similar to that of proposition 3.3.

## 4. Asymptotic behaviour at conic vertices

It is known that for a non-conducting viscous incompressible fluid governed by either the Stokes equations, Oseen equations or Navier–Stokes equations, the minimum-drag shapes subject to a volume constraint have conic vertices with an angle of 2*π*/3 (see Pironneau (1973) and Mironov (1975)). This section analyses the asymptotic behaviour of a solution to the MHD problem (1.1)–(1.4) on the body’s surface near the vicinity of a possible conic vertex.

In the *rz*-plane, let one of the endpoints of ℓ lie on the *z*-axis and have coordinates *r*=0, *z*=*c*, and let (*ρ*,*θ*) be local polar coordinates with the pole at *r*=0, *z*=*c* and with the angle *θ* counted from the *z*-axis, so that and , *θ*∈[0,*π*]. Let *θ*_{0} be the angle between the *z*-axis and the tangent to ℓ at the endpoint. It is assumed that the body is in the cone *θ*≥*θ*_{0}.

### (a) Cases (a) and (b): *M*≠0 and *Re*_{m}*Re*≠*M*^{2}

### Proposition 4.1 (asymptotic behaviour of *G*_{1} and *G*_{2}, cases (a) and (b))

*Let* *M*≠0 *and* *Re*_{m}*Re*≠*M*^{2}, *and let* *G*_{1} and *G*_{2} *be* *H*-*analytic functions defined in theorem 2.1 in* Zabarankin (2011). *The asymptotic behaviour of* *G*_{1} *and* *G*_{2} *on ℓ in the vicinity of the endpoint is given by*
4.1
*as* *where* *a*_{k} *and* *b*_{k}, *k*=1,2, *are some real-valued constants*, *with* *t*∈[−1,1], *being the associated Legendre polynomial of the first kind of order* *n* *and rank* *m* *(for* *m*=0, *the superscript is omitted), and* *α* *is the only zero of the equation*
4.2
*in the interval* (0,1) *for given* .^{5}

### Proof.

The proof is conducted similar to the proof of proposition 1 in Zabarankin & Molyboha (2011). First, let *Re*_{m}≠0.

In the local polar coordinates (*ρ*,*θ*), the *H*-analytic functions *G*_{1} and *G*_{2} and *r*-analytic functions can be represented in the form
4.3
for 0<*α*<1 as , where *k*=1,2, and *a*_{1}, *b*_{1}, *c*_{1}, *a*_{2}, *b*_{2}, *c*_{2}, , , are real-valued constants. Substituting representation (4.3) into the boundary-value problem of eqns (3.1) in Zabarankin (2011) with and equating corresponding coefficients at *ρ*^{α−1}, *ρ*^{0} and *ρ*^{α}, we obtain six equations,
4.4
and
4.5
where . The complex-valued equations (4.4) and (4.5) are equivalent to a system of four real-valued equations that can have non-zero *a*_{1} and *a*_{2} only if the system’s determinant is zero, i.e.
4.6
With the relationships and , the condition (4.6) reduces to equation (4.2), which has a zero *α* in (0,1) only for *θ*_{0}>2*π*/3. The zero is single on (0,1), and as .

Now when , reduces to *ρ*^{0} in representation (4.3). In this case, *G*_{1}, *G*_{2} and are represented by
4.7
with complex-valued functions *A*_{1}, *B*_{1}, *C*_{1}, *A*_{2}, *B*_{2}, *C*_{2}, , , given by
where *k*=1,2,
and *a*_{1}, *b*_{1}, *c*_{1}, *a*_{2}, *b*_{2}, *c*_{2}, , , are real constants. Observe that *A*_{1}, *B*_{1}, *C*_{1}, *A*_{2}, *B*_{2}, *C*_{2}, *A*^{−}_{3}, *B*^{−}_{3}, *C*^{−}_{3} are finite for *θ*∈[0,*π*), whereas *A*^{+}_{3}, *B*^{+}_{3}, *C*^{+}_{3} are finite for *θ*∈(0,*π*].

For brevity, let **a**_{k}=(*a*_{k},*b*_{k},*c*_{k}), *k*=1,2, and . Substituting representation (4.7) into the boundary-value problem (3.1) in Zabarankin (2011) with and equating corresponding coefficients at , , *ρ*^{0} and *ρ*, we obtain, after some transformations, *a*^{+}_{3}=0, *b*^{+}_{3}=0,
4.8
4.9
As in the previous case, equations (4.8) and (4.9) are a system of four real-valued equations that can have non-zero *a*_{1} and *a*_{2} only if the system’s determinant is zero,
4.10
which reduces to . This condition implies *θ*_{0}=0 and *θ*_{0}=2*π*/3, which proves the asymptotic behaviour (4.1) for *θ*_{0}=2*π*/3. For *Re*_{m}=0, the proof is completely analogous.

### (b) Case (c): *M*≠0 and *Re*_{m} *Re*=*M*^{2}

### Proposition 4.2 (asymptotic behaviour of *G*_{1} and *G*_{2}, case (c))

*Let* *M*≠0 *and* *Re*_{m}*Re*=*M*^{2}, *and let* *G*_{1} *and* *G*_{2} *be* *H*-*analytic functions defined in theorem 2.5 in* Zabarankin (2011). *The asymptotic behaviour of* *G*_{1} *and* *G*_{2} *on ℓ in the vicinity of the endpoint is given by formulae (4.1) and (4.2)*.

### Proof.

The proof is similar to that of proposition 4.1, though there are some differences. In the vicinity of the conic endpoint, let *G*_{1} and have the asymptotic behaviour (4.3) with *λ*_{1}=(*Re*+*Re*_{m})/2. However, *G*_{2} here is an *r*-analytic function and is represented by
Substituting these representations into the boundary-value problem (3.29) in Zabarankin (2011) and equating corresponding coefficients at *ρ*^{α}, *ρ*^{0} and *ρ*^{α}, we have
4.11
where . The complex-valued equations (4.11) have non-zero *a*_{1} and *a*_{2} if the determinant of each equation is zero. For the first equation in (4.11), this condition coincides with equation (4.6), whereas for the second, it is given by
Remarkably, as equation (4.6), this condition also reduces to equation (4.2), which proves the asymptotic behaviour (4.1) for 0<*α*<1.

When , the functions *G*_{1}, *G*_{2} and are represented in the form (4.7) with *λ*_{1}=(*Re*+*Re*_{m})/2 and with complex-valued functions *A*_{1}, *B*_{1}, *C*_{1}, , , as defined in the proof of proposition 4.1, and with
Substituting these representations into the boundary-value problem (3.29) in Zabarankin (2011) and equating corresponding coefficients at , , *ρ*^{0} and *ρ*, we obtain, after some transformations, *a*^{+}_{3}=0, *b*^{+}_{3}=0, 2*e*^{λ1c}*a*_{1}−*a*_{2}=0, 2*e*^{λ1c}*b*_{1}−*b*_{2}=0, 2*e*^{λ1c}*c*_{1}−*c*_{2}+(*Re*+*Re*_{m})(*c*^{+}_{3}+1)=0, *ca*_{2}+*a*^{−}_{3}=0, *a*_{2}/2+*cb*_{2}+*b*^{−}_{3}=0, *Re*_{m}(*cc*_{2}+*c*^{−}_{3}+1)−*Re* *c*^{+}_{3}=0,
4.12
The complex-valued equations (4.12) have non-zero *a*_{1} and *a*_{2} if the condition (4.10) holds, and the rest of the proof is similar to that of proposition 4.1.

### (c) The adjoint problem

The asymptotic behaviour of solutions to the adjoint problem (2.7)–(2.9) near a conic vertex on the boundary is analysed similarly.

### Proposition 4.3 (asymptotic behaviour of *G*_{1} and *G*_{2} for the adjoint problem)

*Let* *M*≠0, *and let* *G*_{1} *and* *G*_{2} *be defined as in either theorem 3.1 or theorem 3.7. In both cases, the asymptotic behaviour of* *G*_{1} *and* *G*_{2} *on ℓ in the vicinity of the endpoint of ℓ is given by formulae (4.1) and (4.2)*.

### Proof.

The proof is completely analogous to the proofs of propositions 4.1 and 4.2.

## 5. Analysis of the minimum-drag shape

The results obtained in §§2–4 have the following implications.

Using the representations (3.2), (3.7), (3.12) and (3.13), we can prove that in proposition 2.1, as . The proof is similar to showing that as in the proof of proposition 3.1 in Zabarankin (2011), and consequently, the assumption of proposition 2.1 holds true. With the representations (2.10), (2.21) and (2.26) in Zabarankin (2011) for the MHD problem (1.1)–(1.4) and with the corresponding representations (3.2), (3.7), (3.12) and (3.13) for the adjoint problem (2.7)–(2.9), the optimality condition (2.16) is specialized for each case (a), (b) and (c).

### Corollary 5.1 (optimality condition in cases (a)–(c))

*Let* *G*_{1} *and* *G*_{2} *be* *H*-*analytic functions defined in theorem 2.1, corollary 2.2 and theorem 2.5 in* Zabarankin (2011), *in cases (a), (b) and (c), respectively, and let* *and* *with* *G*_{1} *and* *G*_{2} *being* *H*-*analytic functions introduced in theorem 3.1, corollary 3.4 and theorem 3.7 here in the corresponding cases, respectively. Then*, *ωω**+*Re*_{m}*h*^{−}_{r}*g*^{−} *in the optimality condition (2.16) takes the form*
5.1

*Detail*. Case (a) of formula (5.1) follows from the representation (2.10) and the boundary-value problem (3.1) in Zabarankin (2011) and from the representation (3.2), the boundary-value problem (3.3) and the identity (*Re*−2*λ*_{1})(*Re*−2*λ*_{2})+*M*^{2}=0. Case (b) follows from the representation (2.21) and the boundary-value problem (3.22) in Zabarankin (2011) and from the representation (3.7) and the boundary-value problem (3.8). Finally, case (c) follows from the representation (2.26) and the boundary-value problem (3.29) in Zabarankin (2011) and from the representations (3.12) and (3.13) and the boundary-value problem (3.14). Observe that in case (b), Im[*G*_{1}]=Im[*G*_{2}] and on ℓ, and under these conditions, case (a) reduces to case (b).

### Corollary 5.2

*If* *Re*_{m}=0 *and* *Re*=0, *the adjoint equations (2.7) for* **w** *and* *q* *coincide with equations (1.3) and (1.4) for* **u** *and* ℘. *In this case, the boundary conditions (1.1), (1.2) and (2.9) imply* **w**=−**u** *and* *ω**=−*ω*, *and thus, the optimality condition (2.16) simplifies to* *ω*=const. *on* *S*, *or* e^{Mz/2}Im[*G*_{1}]=const. *on* ℓ, *where* *G*_{1} *is defined in corollary 2.3 in* Zabarankin (2011).

### Corollary 5.3 (fore-and-aft symmetry)

*Propositions 3.3, 3.6 and 3.9 imply that if ℓ is fore-and-aft symmetric, i.e. symmetric with respect to the* *r*-*axis, and ℓ admits a parametrization* *ζ*=*ζ*(*t*), *t*∈[−1,1], *such that* *then if* *F*_{k}(*t*), *t*∈[−1,1], *k*=1,2, *are a solution to eqns (3.14) or (3.26) or (3.32) in* Zabarankin (2011), *then* *t*∈[−1,1], *k*=1,2, *are a solution to equations (3.4), or (3.9) or (3.15), respectively*.

*Now if the minimum-drag shape is to be found iteratively, and the iteration process starts from a fore-and-aft symmetric shape, e.g. sphere or spheroid, then at each iteration, the expression (5.1) is symmetric with respect to the* *r*-*axis, which means that the minimum-drag shape will be fore-and-aft symmetric. In this case, the condition (2.16) with (5.1) takes the form* *υ*(*t*)=const. *for* *t*∈[−1,1], *where*
5.2
*and* *V*_{k}(*t*)=Im[*F*_{k}(*t*)], *k*=1,2.

Propositions 4.1 and 4.2 do not specify whether *a*_{1} and *a*_{2} for *G*_{1} and *G*_{2} in the asymptotic behaviour (4.1) are related. Since the kernels in the integral eqns (3.14) and (3.32) in Zabarankin (2011) have logarithmic singularities, the integrals in eqns (3.14) and (3.32) in Zabarankin (2011) with *G*_{1} and *G*_{2} having the asymptotic behaviour (4.1) in the vicinity of a conic endpoint are integrable and finite. This implies that *F*_{1}(*ζ*)−*F*_{2}(*ζ*) and 2*F*_{1}(*ζ*)−*G*_{2}(*ζ*) in eqns (3.14) and (3.32), respectively, in Zabarankin (2011) are finite, and consequently, in the asymptotic behaviour (4.1), *a*_{1}=*a*_{2} for cases (a) and (b), and 2*a*_{1}=*a*_{2} for case (c). This fact and corollary 5.3 yield the following result.

### Corollary 5.4 (conic vertices)

*In the setting of corollary 5.3, suppose* *t*=±1 *correspond to conic vertices. Then, in the vicinity of* *t*=±1, *υ*(*t*) *behaves as* −4*V*_{1}(*t*)*V*_{1}(−*t*) *in all three cases (a)–(c) with* *V*_{1} *defined in corollary 5.3. Then, the asymptotic behaviour (4.1) implies that in the vicinity of ±1, the condition* −4*V*_{1}(*t*)*V*_{1}(−*t*)=const.≠0 *holds only if* *θ*_{0}=2*π*/3. *This means that for any* *Re*, *Re*_{m} *and* *M*, *the minimum-drag shape has conic vertices, each with an angle of* 2*π*/3.

Now we proceed with finding the minimum-drag shape in a function-series form. In the right-half *rz*-plane (*r*≥0), the cross section ℓ of the minimum-drag shape can be parametrized as in Zabarankin & Nir (2011),
5.3
for *t*∈[−1,1], where *T*_{j}(*t*) is the Chebyshev polynomial of the first kind; *a*_{0},…,*a*_{n1}, *b*_{0},…,*b*_{n2} are real coefficients; and *γ*=*γ*(*a*_{0},…,*a*_{n1},*b*_{0},…,*b*_{n2}) is the multiplier introduced to satisfy the volume constraint identically, i.e. if , then
5.4

Let , where is the average of *υ*(*t*) on *t*∈[−1,1]. Then, optimal ℓ can be found iteratively as in Pironneau (1973),
5.5
where *ζ*_{k}=*r*_{k}+*i* *z*_{k}, and *ϵ*_{k} represent ℓ, and step size, respectively, at step *k*. In this case, *a*_{0},…,*a*_{n1}, *b*_{0},…,*b*_{n2} and *γ* in the series (5.3) are updated as in algorithm 1 in Zabarankin & Nir (2011), in which should be replaced by , and initial *a*_{0} and *b*_{0} should correspond to the axes of the minimum-drag spheroid found in §4*b* in Zabarankin (2011) (for given *Re*, *Re*_{m} and S), whereas initial *a*_{1},…,*a*_{n1}, *b*_{1},…,*b*_{n2} should be set to zero.

Figure 1 shows the cross sections of three minimum-drag shapes of volume 4*π*/3 for *Re*=*Re*_{m}=3 and *S*=0, 1, 2 in the *zr*-plane: the shape for *S*=1 is shortest. In the case of *S*=0, which corresponds to the Oseen flow of a non-conducting fluid, the minimum-drag shape was obtained in Zabarankin & Molyboha (2011). For the three shapes, tables 1 and 2 present the corresponding drag coefficient , determined by eqns (3.36)–(3.38) in Zabarankin (2011); the constant in the optimality condition (2.16); -error in satisfying the condition (2.16); and the parameters in the representations (5.3) and (5.4). For comparison, table 1 also shows the drag coefficient *C*^{*}_{D} for a unit sphere and the drag coefficient *C*^{†}_{D} for a minimum-drag spheroid of volume 4*π*/3 for *Re*=*Re*_{m}=3 and *S*=0, 1, 2. It is seen that *C*^{†}_{D} and *C*_{D} for same *S* are sufficiently close, whereas the drag ratio (0.87177, 0.91126 and 0.86096 for *S*=0, 1 and 2, respectively) indicates that drag reduction is smallest for *S*=1 and becomes more significant for *S*≫1.

## 6. Conclusions

The necessary optimality condition (2.16) for the minimum-drag shape subject to a volume constraint has been obtained by using Mironov’s shape variation approach (Mironov 1975). It generalizes the necessary optimality conditions corresponding to non-conducting viscous incompressible flows governed by the Stokes and Oseen equations. For any *Re*, *Re*_{m} and S, the minimum-drag shapes are fore-and-aft symmetric and have conic vertices each with an angle of 2*π*/3.

The presented shape-optimization approach relies on the boundary integral equation method, which compared with the finite-element method (FEM) reduces the dimensionality of the MHD problem and involves no discretization and truncation of the interior–exterior regions (see Zabarankin & Molyboha (2010, 2011) for a discussion of *boundary integral equation-constrained optimization* versus FEM-based partial differential equation (PDE)-constrained optimization). In addition, it has two advantages: (i) the iterative procedure uses the exact (analytically derived) necessary optimality condition (2.16) and (ii) the minimum-drag shapes are found in a function-series form (not point-wise). This approach can be used for testing the accuracy of FEM-based optimization schemes. It is applicable, however, to linear/linearized PDEs and requires corresponding boundary integral equations.

In the numerical examples, the minimum-drag shape as a function of S has the smallest drag coefficient and the smallest aspect ratio both at S=1. This agrees with the behaviour of the sphere drag coefficient, with respect to which, drag reduction provided by the minimum-drag shape is smallest for S=1. Also, for the same *Re*, *Re*_{m} and S, the minimum-drag shapes and equal-volume minimum-drag spheroids used as in the initial approximations have sufficiently close drag coefficients.

Although the applicability of the linearized MHD equations is limited to small values of *Re*, *Re*_{m} and *M*, the drag for non-small *Re*, *Re*_{m} and *M* can be approximated based on the linearized MHD equations with so-called effective values of *Re*, *Re*_{m} and *M*. In this case, the approach of generalized analytic functions developed here and in Zabarankin (2011) can be useful in various MHD flow control schemes for drag reduction.

## Acknowledgements

I am grateful to the anonymous referees for their valuable comments and suggestions that helped to improve the quality of the paper. Research was supported by AFOSR grant FA9550-09-1-0485.

## Footnotes

↵1 In the conducting fluid,

*S*characterizes the ratio of the magnetic forces to the inertial forces and is also denoted by*β*.↵2 The first equation in (1.3) here and the first equation in (2.5) in Zabarankin (2011) are equivalent provided that div

**u**=0.↵3 In equations (1.3) and (1.4), linear dimensions are also rescaled by

*a*.↵4 For

*M*=0, the velocity field in the MHD problem (1.1)–(1.4) is uncoupled from the magnetic field and is governed by the Oseen equations.

- Received April 8, 2011.
- Accepted June 9, 2011.

- This journal is © 2011 The Royal Society