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The Continuity Principle by Eka Oktariyanto Nugroho Basic Equation 2 Eka O. N. 2.1. CONTINUITY EQUATION The principle of continuity expresses the conservation of mass in a given space occupied by a fluid. The simplest, well-known form of the continuity relationship in elementary fluid mechanics expresses that the discharge for steady flow in a pipe is constant; that is, VA constant, where A is the cross-sectional area of the pipe and V is the mean velocity'. In the case of an incompressible fluid (ρ = constant) in a uniform pipe (A = constant), the continuity relationship becomes simply: V = constant. If the X axis is taken as the axis of the pipe, then V u , and the continuity principle expressed in differential form becomes: dV u 0 dx x Since no fluid is being added or subtracted during the motion, the quantity of fluid involved is constant. This may be expressed mathematically in the case of two dimensional incompressible motions. The development of the mathematics follows. Consider a rectangular element in two-dimensional fluid motion as shown in Fig. 2.1. The rectangular boundaries have sides of length a and b and are considered to be fixed with respect to the axes. It is not a moving fluid element. Figure 2. 1 Rectangular element of an incompressible fluid. The volume of fluid entering the left-hand boundary by unit of time is au1 , and at the same instant, the amount leaving the right-hand boundary is is thus: a u2 u1 au . au2 . The difference in amount in the OX direction Similarly, the difference in amount in the OY direction is: b v2 v1 bv Since the total mass of fluid within the boundaries is constant, the total loss must be zero: au bv 0 ; u b v a 0 . In the limit, when b and a approach zero, one obtains u x v y 0 . This differential form is permitted because of the assumption of a continuous fluid. It should be noted that u x and v y are the rates of linear deformation of a fluid particle; hence, in an incompressible fluid, the total sum of linear deformation is nil, as has been previously noted. The Continuity Principle Page -9 Basic Equation 2 Eka O. N. 2.2. THE CONTINUITY RELATIONSHIP IN THE GENERAL CASE w v u C z G H D z u F B E A x x- 2 u dx x y x y u x x+ x 2 Figure 2. 2 Coordinate system for continuity equation. Conservation of Mass at a Point in for a Molecularly Averaged Continuum Mass Can neither Appear nor Disappear Mass stored in a CV = Mass of fluid into the CV - Mass of fluid out of the CV By applying a fixed CV (Eulerian analysis) as shown in figure 2.2 with geometries properties of the CV, result: Volume of CV dV dx dy dz Area Face ACBD = dy dz Area Face EFGH = dy dz Area Face AEHD = dx dz Area Face BFGC = dx dz Area Face AEFB = dx dy Area Face DHGC = dx dy Assume a hypothetical continuum with properties that average out the molecular scales, as follows , describes mass per volume down to a point in space Density, Velocities: u, v, w, describe motion of small parcels of fluid with molecular level motions averaged out Consider a fixed volume of fluid of which the edges dx, dy , dz are parallel to the axes OX , OY , OZ respectively (Fig. 2.2). The continuity relationship is obtained by considering that the change of fluid mass inside the volume dx dy dz during the time dt is equal to the difference between the rates of influx into and efflux out of the considered volume during the same interval of time (conservation of mass). Mass of the CV can be described by : dM dV The Continuity Principle Page - 10 Basic Equation 2 Eka O. N. The change of mass stored within the CV. dM t dV t Expanding this equation dM t dV dV t t Since the boundary of the control volume is fixed, dM t dV dV t 0 , and t Density with respect to time The fluid mass at the time t is: dx dy dz After a time dt , because of the change of density with respect to time, the quantity of fluid mass becomes dt dxdydz t Hence the change of fluid mass in a time dxdydz dt is dt dxdydz dtdxdydz t t On the other hand, if one takes into account the change in velocity and in density with respect to space coordinates, the quantity of fluid mass entering through the section ABCD during a time dt , parallel to the OX axis, is the product u times the area perpendicular to OX (ABCD) and the time dt . Since, ABCD = dy dz , the quantity of fluid mass entering is u dy dz dt . The derivative of u along AB and AD with respect to dz and dy is of an infinitely small order and can be neglected. Now the quantity of fluid mass coming out during the same interval of time through the section EFGH is: u dx dydzdt u x Velocities with respect to time In the general case, both the density p and velocity u are assumed to be changed along Hence the difference is dx . u u dx dydzdt udydzdt dxdydzdt u x x Similarly, the differences due to the components of motion parallel to the OY and OZ axes are, respectively, v dxdydzdt due to the difference of discharge across the sections BFGC and AEHD y dx dz The Continuity Principle Page - 11 Basic Equation 2 Eka O. N. w z dx dy dxdydzdt due to the difference of discharge across the sections AEFB and DHGC The total change of mass contained within the elementary region during the time dt is u v w dxdydzdt y z x Equating Density and Velocities Equating density and velocity equationas yields u v w dxdydzdt dxdydzdt t x y z Dividing both sides by dx dy dz dt yields u v w 0 t x y z Derivation continuity according to Taylor’s Series We assume that the variables may be expanded using Taylor series, as shown in Figure 2.2. Mass flux into face ACBD at x 0 : u dy dz u x Mass flux out of face EFGH at x dx : u Mass flux into face at AEFB z 0 : 2 u x 2 dx ... dy dz 2 w dx dy w z Mass flux out of face DHGC at z dz : w Mass flux into face at AEHD y 0 : dx dz 2 w z 2 dz 2 ... dx dy v dx dz v y Mass flux out of face BFGC at y dy : v dy 2 v y 2 dy 2 ... dx dz Apply the Conservation of Mass Law in a time rate form Time rate of change of mass stored in a CV = Net rate of mass flux into the CV in the x, y & z directions dV u 2 u 2 u dy dz u dx dx ... dy dz 2 t x x w 2 w 2 dz dz ... w dx dy w dx dy z z 2 v 2 v 2 dy dy ... v dx dz v dx dz y y 2 The Continuity Principle Page - 12 Basic Equation 2 Eka O. N. Collecting terms: 2 u u v w 2 v 2 w dV dV dV dV dx dy dz ... dV 0 2 2 2 x t x y z y z Factoring out dV 2 2 v 2 w u v w u dx dy dz ... 0 x 2 t x y z y 2 z 2 Letting the CV shrink to a point such that dx0, dy0, dz0 and thus dV0, lead to the Conservation of Mass Equation at a point space and time u v w 0 t x y z Since u x u x u x , and similarly for the terms v y and w z , the continuity relationship becomes u v w u v w 0 t x y z x y z u v w D 0 Dt x y z du dv dw d d d v w 0 u t dx dy dz dx dy dz divV+V grad 0 t or D V=0 Dt Where V uiˆ vjˆ wkˆ and ˆ ˆ ˆ i j k x y z The first term, Dρ/Dt, is the derivative of the density with time at a given point. This term is nil in the case of (1) incompressible fluid, since p is a constant and (2) a steady motion of a compressible fluid. This term has to be considered when sound, water hammer, shock waves, etc. are studied. 2.3. APPLICATION OF MASS CONSERVATION TO AN INCOMPRESSIBLE FLUID It can be shown that the rate of volumetric dilatation for a fluid is 1 D dV u v w 0 dV Dt x y z Thus for an incompressible fluid, the continuity equation is valid: u v w 0 x y z The Continuity Principle Page - 13 Basic Equation 2 Eka O. N. This equation describe conservation of volume for the fluid which the CV must remain immersed within the fluid In vector notation, the continuity equation at a point is written as: V=0 An incompressible fluid does not imply that density is contant; the conservation of mass equation reduces to D =0 Dt The Continuity Principle Page - 14