Steady flow of an incompressible perfectly conducting fluid past a thin airfoil
© Carabineanu; licensee Springer. 2013
Received: 23 August 2013
Accepted: 21 October 2013
Published: 8 November 2013
We consider the linearized Euler and Maxwell equations and Ohm’s law. We calculate the fundamental matrix and give integral representations for the velocity, magnetic induction and pressure. We use the boundary (slip) condition to obtain an integral equation for the jump of the pressure. We give some graphic representations of the velocity and magnetic induction for the case of the flat plate.
MSC:76X05, 45E05, 35E05.
Keywordslinearized system fundamental matrix integral representation
In papers dedicated to the motion of a wing in an electro-conductive fluid, the lift, drag and moment coefficients were calculated. Recent technological advances claim also for the study of the velocity and electromagnetic fields. We mention two examples: the plasma actuators for aircraft flow control (see ) and concealing aircrafts from radar using interaction between ionized gas and electromagnetic radiation. In the present paper, we study the steady two-dimensional flow of an ideal perfectly conducting incompressible fluid around a thin insulating airfoil. We consider the linearized partial differential equations of magnetohydrodynamics (consisting of Euler’s and Maxwell’s equations and Ohm’s law) and calculate the corresponding fundamental matrix. In order to obtain the integral representations for the velocity, the magnetic induction and the pressure fields (which represent the original result of this work), we perform the convolution of the components of the fundamental matrix with the simple layer distributions determined by the jump of the functions we are looking for. We notice that every integral representation has an elliptic as well as a hyperbolic part, this last one being determined by the presence of simple waves bounded by straight characteristics (weak shocks). From the integral representation of the velocity and the boundary conditions (linearized slipping condition and the continuity of the magnetic induction), we rediscover the singular integral equation for the jump of the pressure across the airfoil. We consider the particular case of the flat plate for which the solution of the integral equation is known. Then we perform some numerical integrations to calculate the velocity and the magnetic induction at the points of a two-dimensional grid in order to provide graphic representations for the velocity and magnetic induction fields.
Fundamental matrix of the linearized equations for the two-dimensional incompressible flow of perfectly conducting fluids
where is Dirac’s distribution and
Some components of the fundamental matrix
The integral equation for the jump of the pressure
We may take if we impose Kutta-Joukovsky’s condition. Other choices of the constant C were considered by Stewartson .
The flat plate
, are the Jacobi polynomials and , , are the roots of . We considered .
We have calculated the velocity, the pressure and the magnetic induction for the steady flow of an incompressible perfectly conducting fluid past a thin airfoil. The integral representations of the velocity, pressure and magnetic induction contain respectively an elliptic part and a hyperbolic one. Some calculations were performed for the flat plate. In the graphic representation of the velocity and magnetic induction from Figures 1 and 2, we may observe the simple waves which are determined by the influence of the magnetic field. In Figure 3 we represent the lift coefficient against Alfvén’s number A. We notice that the lift coefficient increases when A decreases, i.e., when the value of the magnetic induction at infinity increases.
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