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Inertia Forces by Eka Oktariyanto Nugroho Basic Equation 3 Eka O. N. 3.1. MASS, INERTIA, AND ACCELERATION 3.1.1. The Newton Equation To cause the motion of a constant mass M , or, more generally, to change the state of an existing motion, it is necessary to apply to this mass a force F , which causes an acceleration dV dt such that F M dV dt . This is a vector relationship, i.e., true for both magnitude and direction. The M dV dt is the inertia force, which characterizes the natural resistance of matter to any product change in its state of motion. The considered mass M is the mass of a unit of volume of fluid M unit of volume is the density. Hence the fundamental equation of momentum has the form F dV dt . Its three components along the three coordinate axes OX , OY , OZ where are 3.1.2. du dt , dv dt , and dw dt , respectively. Relationships between the Elementary Motions of a Fluid Particle and the Inertia Terms To each kind of motion of the fluid particles (chapter 1) there corresponds an inertia force. The relationship between the kind of motion described and the corresponding inertia force is straightforward. The elementary components of velocity of a fluid particle as given in chapter 1 are, in the case of a two-dimensional motion, Translation u, v Dilatational deformation u dx x Shear deformation 1 u v dy 2 y x 1 u v dx 2 y x Rotation 1 v u dy 2 x y 1 v u dx 2 x y v dy y To each of these velocity components corresponds a component of acceleration, which multiplied by , yields a component of inertia force. Two types of inertia forces may be distinguished, depending on the type of acceleration or elementary motion considered. These are: 1. Local acceleration-corresponding to a variation of the velocity of translation or the derivative of velocity with respect to time. 2. Convective acceleration-corresponding to a variation of velocity of deformation and rotation or derivative of velocity with respect to space. The physical meaning of these accelerations and the corresponding inertia forces is first examined: then their mathematical expression is demonstrated. Chapter 4 deals with the applied forces F which have to be equated to these inertia forces to obtain the momentum equation. Inertia Forces Page - 15 Basic Equation 3 Eka O. N. 3.2. LOCAL ACCELERATION Local acceleration characterizes any unsteady motion, i.e., motion where the velocity at a given point changes with respect to time. Local acceleration results from a change in the translatory motion of a fluid particle imposed by external forces F. Mathematical Expression of Local Inertia The mathematical expression of the inertia forces caused by a local acceleration is given by the change in the velocity of the translatory motion with respect to time only. The corresponding inertia V t force is equal to respectively: of which the components along the three axes are, u t , v t , and w t . The derivatives with respect to space are not taken into account. 3.3. CONVECTIVE ACCELERATION Convective acceleration characterizes any nonuniform flow, i.e., when the velocity at a given time changes with respect to distance. It is sometimes called field acceleration. Convective acceleration results from any linear or angular deformation, or from a change in the rotation of fluid particles, imposed by external forces F. 3.3.1. The Case of Linear Deformation In a convergent pipe, it has been seen that the velocity of a fluid particle, although constant with time at a fixed location, tends to increase along the converging streamlines. The velocity of the fluid particle increases with respect to space. This is a positive convective acceleration. The fluid tends to resist this acceleration by convective inertia. In a divergent conduit, the velocity decreases and the fluid tends to continue its motion with the same velocity because of its inertia. The applied forces cause a negative convective acceleration. Expansion or contraction of a compressible fluid is the sum of linear deformations and also results in corresponding inertia forces. It has been seen that the linear deformation velocity components are those given in Equation 3-1. u dx x v dy y w dz z (3-1) Two-dimensional motion Three-dimensional motion The expressions u x , v corresponding acceleration is y and w z are given time, as seen in section 1.3.1 The d u u dx dx dt x x dt Two similar expressions result for w and u . If u dx dy dz , v , and w are substituted in dt dt dt these expressions and the result is multiplied by the density, the inertia forces are obtained. They are: u 2 u 1 u x 2 x 2 v 1 v v y 2 y Inertia Forces Page - 16 Basic Equation 3 Eka O. N. w w 1 w z 2 z 2 It should be notice that the last group of expressions may be written as x u 2 . This 2 shows that the inertia force is equal to the derivative of the kinetic energy with respect to space along the three direction axes OX , OY , and OZ , respectively. 3.3.2. The Case of Shear Deformation In a bend, where the fluid particles are angularly deformed, the fluid paths are curved and because of its inertia, the fluid tends to continue along a straight line. This causes a centrifugal force proportional to the change of direction which is imposed by the applied forces. It is possible for the velocity of a fluid particle to keep the same magnitude along its path, but with a change in direction. This is the case of free vortex motion. It has been seen that the velocity components of angular deformation for a two-dimensional motion are 1 u v dy 2 y x 1 u v dx 2 y x Hence, as in the previous case, using the substitutions u dx dy , v , the corresponding inertia dt dt forces become: 1 u v v 2 y x 1 u v u 2 y x 3.3.3. The Case of a Change of Rotation In the entrance to a pipe (Fig. 3.1), because of the change in friction forces, there is a variation of rotation of the fluid particles. Hence there are inertia forces corresponding to the natural resistance of the fluid to change its rotational motion. In a uniform pipe, the rotation of particles exists but there is no change in rotational magnitude and the corresponding acceleration is zero. Figure 3. 1 Zone of acceleration of rotation. Inertia Forces Page - 17 Basic Equation 3 Eka O. N. As in the two previous cases, since 1 v u dy 2 x y 1 v u dx 2 x y are the velocities of the components of rotation in a two dimensional motion, the corresponding inertia forces obtained are 1 v u v 2 x y 1 v u u 2 x y It has been shown that it is possible to assume that the motion is irrotational when friction effects are negligible. It is evident that the same conditions lead to neglect of rotational inertia forces. 3.4. GENERAL MATHEMATICAL EXPRESSIONS OF INERTIA FORCES 3.4.1. Local and Convective Acceleration In the general case both local acceleration and convective acceleration occur at the same time. A simple example is when a fluid oscillates in a nonuniform curved pipe. Hence, in the general case, V and its components u, v, and w are functions of both time and space coordinates. For example, u x, y, z, t . The total differential of u is du u u u u dt dx dy dz t x y z The acceleration in the x direction is thus given by the total differential of u , with respect to time: du u u dx u dy u dz dt t x dt y dt z dt Similar expressions occur for dv dt and dw dt . Substituting u : dxldt, u : dyldt, and w : dzldt, and multiplying by the density p, the inertia forces given by Equation 3-2 are obtained. u u u u u v w x y z t v v v v u v w x y z t w t u Local acceleration terms 3.4.2. (3-2) w w w v w x y z Convective accleration terms Elementary Acceleration Components Following a procedure similar to that used in the study of the elementary motions of fluid particles (Section 1.5.2), that is, adding and subtracting 1 1 v v x and w w x to the first line 2 2 above, gives Equation 3-3, which emphasize the previous physical considerations. Similar forms can be obtained for the y and z components of the forces. Inertia Forces Page - 18 Basic Equation 3 Eka O. N. u t u 1 v u 1 u w 1 u w 1 v u (3-3) v w w v 2 x y 2 z x 2 z x 2 x y Acceleration in linear deformation Local Acceleration resulting in a change in translatory motion 3.4.3. u x Acceleration in angular deformation Acceleration in rotation Separation of Rotational Terms It is often useful to transform the acceleration terms to a form which emphasizes both the kinetic energy terms and the rotational terms. Adding and subtracting v v x w w x to the first line, gives the following expression, valid along the OX axis: u u v w u v u w u v w v w x x y x z x t x But u u v w 1 2 2 V2 v w u v w2 x x x 2 x x 2 When the coefficients of the rotational vector u w 2 z x v u 2 x y Are introduced, the following expression for the inertia forces along the OX axis results u V 2 2 w v t x 2 Similarly, it may be found that the inertia forces along the OY and OZ axes are v t w t V 2 2 u w y 2 V 2 2 v u z 2 These three expressions may be written more concisely in vector notation as shown in Eq. 3-4. V t Local acceleration grad V2 2 Kinetic energy term curl V x V (3.4) Rotational term Convective acceleration grad V 2 V2 V 2 V 2 i j k 2 z 2 x 2 y 2 It has to be noticed that the convective inertia term is, in fact, the derivative with respect to space of the kinetic energy, Inertia Forces V 2 2 , of the particle. Page - 19