Describing law of motion of flexible inextensional shell in gravity force field

This work makes use of Navier-Stokes equations to describe an analytical method of finding the motion speed of a flexible inextensional shell falling down to the ground from a preset height and determines the duration of this fall. The soft shell in question is a fabric body of aerodynamic shape or an item of clothes, an airborne vehicle element, etc. Analytical relations are presented for the speed at which the shell moves in the air, taking account of the air resistance and the shell fall duration. The boundary problem of the soft shell vertically falling in the air is solved.


RESUMEN
Aquí se usan las ecuaciones de Navier-Stokes para describir un método analítico de calcular la velocidad del movimiento de una cascarilla plegable no expansible que cae en tierra de una altura predeterminada y calcular la duración de la caída. La cascarilla plegable en cuestión es un objeto de tejido de configuración aerodinámica o una prenda de vestir, un elemento de un vehículo aéreo, etc. Se presentan las relaciones analíticas para calcular la velocidad con que la cascarilla cae en el aire. Se tienen en cuenta en las fórmulas la resistencia del aire y la duración de la caída. Se resuelve el problema de límites para la cascarilla que cae verticalmente en el aire.

INTRODUCTION
In modern mechanics of continua an important place belongs to the actively evolving section of soft aircraft dynamics; this section is dedicated to solving aircraft motion control problems. Soft airborne vehicles include aerodynamic shapes, ram-air parachutes, hand gliders, parachutes, and other devices that can be modeled in the form of flexible inextensional shell. To solve of the soft shell freely moving in the air is necessary for studying the motion of various airborne vehicles with a density exceeding the air density; as a result, a soft shell freely moving in the air falls down to the ground by gliding from a preset height.
A major application of the falling shell problem is the intrusion of an airborne vehicle onto the air strip (Perl, 2006). The summary showing how severe this issue is can be found in the hearings held by the US Subcommittee on Aviation in 2001. While landing an airborne vehicle has to enter into a gliding fall. Another topical problem is the safe landing control after a long supposed fall of humanoid robots (Bingham et al., 2014). Humanoid robots face a high risk of fall, when walking or working in an undefined medium, and for which it is necessary to ensure their safe controllable contact with technical object surface (Zhang, et al., 2017). The autonomous control of microsize airborne vehicles imitating flying insects in unknown media is a complex problem because these artificial subjects have small dimensions and are, therefore, exposed to wind influences. Similarly to insects, microsize airborne vehicles shall have highly efficient visual motor control systems (Kerhuel, Viollet & Franceschini, 2007). The issues related to controlling microsize airborne vehicles and humanoid robots led to the creation of hybrid routing models with semi-autonomous control during robot-assisted operations in unknown remote, dangerous, or otherwise unavailable areas (Lin et al., 2014).
The motion of technical objects in the air is described by Navier-Stokes equations, taking account of air viscosity and turbulent airflow stresses (Indenbirken et al, 2000;Rocha et al., 2000). That said, the forces with which the airflow contacts with the moving body surface are also taken into account (Paivanas & Hassan, 1981). However, the applied methods of calculating moving body speeds and pressures are defined either numerically or by experiment.
This work proposes an analytical method of finding the motion speed of a soft inextensional shell falling down to the ground from a preset height and determining the duration of this fall. The shell's fall down to the ground is modeled, taking into account the airflow motion resistance forces but without taking into account the forces with which the wind affects the shell.
The task of describing the soft shell motion law involves such major aspects as designing the shell and its forming in the field of gravity and elasticity forces (Byrdina, Bekmurzaev & Mitsik, 2017a;2017b). Another topical issue is the vertical stability of the shell with an airflow discharged inside it (Bekmurzaev, Mitsik & Byrdina, 2018). It should be noted that, similarly to the flow of fluid, the airflow can be described not only by Navier-Stokes equations but also by other equations, e.g., by Euler equations when the air viscosity is not taken into account (Kocharenko, Mitsik & Aleynikova, 2016). This work is an important part of describing dynamical behavior of flexible inextensional shells used in various environments.

SOFT SHELL MOTION EQUATIONS
Let us consider the common patterns typical of the considered physical model that are described in (Loytsyanskiy, 1978). The flow of viscous gas will be analyzed proceeding from the following assumptions: 406 1.Gas is perfect, i.e., its pressure р, density  , and absolute temperature Т meet the Clayperon-Mendeleev equation known as the ideal gas law where 0 R  is the universal gas constant and m  is the gas molecular weight.
2. Dynamic viscosity coefficient  is the function of Т only. Various laws of this relation can be used hereafter. In the Sutherland formula recorded as Ts is the Sutherland constant the value of which for air is around 122 К; То and 0  are the absolute temperature and the viscosity coefficient correspondent to some initial gas state.
3. Specific heat capacitance coefficient c does not depend on the absolute gas temperature and is a physical gas constant.  ; 3 .

Gaseous heat conduction coefficient
The assumption that can be considered for this model is 0; 0; , where g  is the gravity acceleration.
The physical meaning of this assumption is that it presupposes the absence of any external force action on the shell freely falling down through the air and moving only because of its higher density. Boundary conditions of the problem: 1. At the initial time instant the speed in each point of the shell is zero. 2. At the initial time instant of free fall, the speeds of the airflow outside the shell are zero in each point in space.

SIMPLIFYING THE FREE FALLING SHELL PROBLEM
In the first phase of simplifying the freely falling shell problem let us additionally assume that the process is isothermal, i.e., the airflow temperature is constant and equal to 15 0 С in each point of the airflow. Actually, this condition means that heat losses on friction are excluded. Let us also suppose that in this problem the air flow is incompressible. Proceeding from these assumptions, the airflow temperature will be constant and the air density in each point of the airflow will be constant as well.
These circumstances lead to constant pressures in each point of the airflow according to the Clayperon-Mendeleev equation. i.e., Since the airflow is supposed incompressible, then By virtue of condition (9), the latter two summands in each equation of system (5) are zero; taking account of formulas (1) and (6) and assumption (7), system (5) will be changed to a simpler form recorded as ; After the admitted assumptions, system (10) is a system of three nonlinear equations in partial derivatives relative to three unknown functions, i.e., (10) is a closed system. Let us make system (10) even simpler by assuming that the shell freely falls without moving horizontally, i.e., the fabric points motion paths have constant x and y coordinates and the shell configuration changes only along the OZ axis (Fig. 1).