Streamline. Fluid tube. Flow filament

Fig. 2.2. Drawing of streamlineThe Euler's method feature is the concept of streamlines.

Let's consider an arbitrary point in space filled with fluid in arbitrary moment of time (fig. 2.2). Let is the vector of velocity at point . Let us put aside a small line segment from the point in the direction of vector and mark point . is the vector of velocity in the point . Having put aside a small line segment from point along the vector we'll receive point at the end of the line segment. Continuing such drawing further we shall receive in the aggregate some polyline . Increasing number of line segments and considering length of each segment as infinitesimal we shall receive a smooth curve instead of a polyline, which will be one of streamlines.The streamline is a totality of fluid particles, which velocity vectors are tangent to it at given point of time.The fluid particle draws a trajectory during its movement. Streamlines do not coincide with trajectories during the unsteady movement and they are identical, when the movement is steady. It is necessary to note, that it is possible to draw only one streamline through each point of the space filled with moving fluid at the fixed moment of time. The trajectory of a particle fixes changing position of the same particle eventually, and the streamline specifies a velocity direction of various particles at the same moment of time. Trajectories can intersect. Streamlines neither intersect themselves, nor any another, as the velocity vector would have two various directions in the cross point at given time, that is physically impossible. Exception is only so-called flow special points, where the value of velocity equals zero or is theoretically infinite.The set of streamlines gives a picture of flow at present time that is an instant photograph of flow velocity directions. A number of such pictures for various moments of time represent the geometrical flow image corresponding to the Eulers method.

Fig. 2.3. Streamline

Hence, it has been established, that velocity vectors are tangent to the streamline at each moment of time. Therefore if we should take any point on the curve, and an elementary line segment located close to it with projections , , (fig. 2.3) on corresponding axes, then the velocity vector and the direction of this line segment in the given point would coincide, or in other words they would be parallel.Let's find the differential equation of a streamline. It follows from parallelism condition of streamline and vector of velocity in the given point

.

Let's represent the vector product as determinant

,deploying the first line of determinant we shall obtain

,and from this equation

,

,

.The obtained expressions can be written as follows

.(2.14)The system (2.14) is called as differential equations of streamlines.Thus, the problem of streamline determination by the given field of velocities is reduced to integrating the system of differential equations.Let's introduce the concepts of the fluid tube. For this purpose we shall draw some closed contour within fluid (fig.2.4), which is not a streamline, and draw streamlines through each point of the contour. A set of these streamlines forms a surface, which is named fluid tube. Fluid flowing inside the fluid tube is called as flow filament.

Fig. 2.4. Fluid tubeIt is necessary to note, that fluid tube formed by streamlines in stationary motion, does not change with respect to time and is similar to impermeable tube in which the fluid flows as in a tunnel with solid walls, which limit its contents. The fluid does not flow out from the fluid tube through lateral surface and is not added to it as in all points of flow filament actual velocity is directed along the streamline.Within elementary flow filament velocities in all points of the same cross section can be assumed as equal to each other and to local velocities. Elementary flow filament is a visual kinematic representation, which significantly facilitates studying fluid motion, and it is put in the basis of so-called jet model of fluid flow.According to this model, entered in aerohydrodynamics even in a period of its scientific formation, the space filled with moving fluid is considered as a set of many elementary flow filaments. The set of elementary flow filaments, which flow through the area large enough, forms a fluid flow. Today the flow filament model is one of the basic fluid models.