In order to illustrate this concept, we consider a typical fluid element of certain volume at any arbitrary time as shown in Fig. 3.2.1. After certain time interval, it has moved and changed its shape as well as orientation drastically. However, when we limit our attention to an infinitesimal particle of volume . at time t and . within the fluid element, it may be observed that the change of its shape is limited to only stretching/shrinking and rotation with its sides remaining straight even though there is a drastic change in the finite fluid element. Thus, the particle motion in a fluid flow can be decomposed into four fundamental components i.e. translation, rotation, linear strain and shear strain as shown in Fig. 3.2.2. When the fluid particle moves in space from one point to another, it is referred as translation. Rotation of the fluid particle can occur in any of the orthogonal axis. In the case of linear strain , the particle's side can stretch or shrink. When the angle between the sides of the particle changes, it is called as shear strain .
Fig. 3.2.1: Schematic representation of motion of finite fluid element and infinitesimal particle mass at two different time steps.
Fig. 3.2.2: Basic deformations of fluid mass: (a) Linear deformation; (b) Angular deformation.
Linear Motion and Deformation
Translation is the simplest type of fluid motion in which all the points in the fluid element have same velocity. As shown in Fig. 3.2.3-a, the particle located as point O will move to O' during a small time interval . . When there is a presence of velocity gradient, the element will tend to deform as it moves. Now, consider the effect of single velocity gradient . on a small cube having sides . and volume . . As shown in Fig. 3.2.3-b, the x -component of velocity of O and B is u. Then, x- component of velocity of points A and C would be, . , which causes stretching of AA' by an amount . as shown in Fig.3.2.3-c. So, there is a change in the volume element .
Fig. 3.2.3: Linear deformation of a fluid element.
Rate at which the volume . changes per unit volume due to the velocity gradient . is
In the presence of other velocity gradients . , Eq. (3.2.1) becomes,
If we look closely to the unit of velocity gradients . , then they resemble to unit of strain rate and the deformation is associated in the respective directions of orthogonal coordinates in which the components of the velocity lie. Thus, the linear strain (Fig. 3.2.2-a) is defined as the rate of increase in length to original length and the linear strain rates are expressed as,
The volumetric strain rate/volumetric dilatation rate is defined as the rate of increase of volume of a fluid element per unit volume.
In an incompressible fluid, the volumetric dilatation rate is zero because the fluid element volume cannot change without change in fluid density.
Angular Motion and Deformation
The variations of velocity in the direction of velocity is represented by the partial derivatives . , which causes linear deformation in the sense that shape of the fluid element does not change. However, cross variations of derivatives such as . will cause the fluid element to rotate. These motions lead to angulardeformation which generally changes the shape of the element.
Fig. 3.2.4: Angular deformation of a fluid element.
Let us consider the angular motion in x-y plane in which the initial shape is given by OACB , as shown in Fig. 3.2.4-a. The velocity variations cause the rotation and angular deformation so that the new positions become OA' and OB ' after a time interval . . Then the angles AOA' and BOB' are given by . , respectively as shown in Fig. 3.2.4-b. Thus, the angular velocities of line OA and OB are,
When, both . are positive, then both . will be in counterclockwise direction. Now, the rotation of the fluid element about z-direction (i.e x-y plane) . can be defined as the average of . . If counterclockwise rotation is considered as positive, then
In a similar manner, the rotation of the fluid element about x and y axes are denoted as . , respectively.
These three components can be combined to define the rotation vector . in the form as,
It is observed from Eq.(3.2.6) that the fluid element will rotate about z- axis, as an undeformedblock, only when, . . Otherwise it will be associated with angular deformation which is characterized by shear strain rate. When the fluid element undergoes shear deformation (Fig. 3.2.2-b), the average shear strain rates expressed in different cartesian planes as,
Strain rate as a whole constitute a symmetric second order tensor i.e.
In a flow field, vorticity is related to fluid particle velocity which is defined as twice of rotation vector i.e.
Thus, the curl of the velocity vector is equal to the vorticity. It leads to two important definitions:
- • If . at every point in the flow, the flow is called as rotational. It implies that the fluid elements have a finite angular velocity.
• If . at every point in the flow, the flow is called as irrotational. It implies that the fluid elements have no angular velocity rather the motion is purely translational.
In Eq.(3.2.11) , if . is zero, then the rotation and vorticity are zero. The flow fields for which the above condition is applicable is known as irrotational flow. The condition of irrotationalityimposes specific relationship among the velocity gradients which is applicable for inviscid flow. If the rotations about the respective orthogonal axes are to be zero, then, one can write Eq. (3.2.11) as,
A general flow field would never satisfy all the above conditions. However, a uniform flow field defined in a fashion, for which . , is certainly an example of an irrotational flow because there are no velocity gradients. A fluid flow which is initially irrotational may become rotational if viscous effects caused by solid boundaries, entropy gradients and density gradients become significant.
It is defined as the line integral of the tangential velocity component about any closed curve fixed in the flow i.e.
where, . is an elemental vector tangent to the curve and with length ds with counterclockwise path of integration considered as positive. For the closed curve path OACB as shown in Fig. 3.2.4-a, we can develop the relationship between circulation and vorticity as follows;
Hence, circulation around a closed contour is equal to total vorticity enclosed within it. It is known as Stokes theorem in two dimensions.
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