\n Principle of continuity is a consequence of conservation of mass.\r\n Principle of continuity requires that during the flow of a fluid, the fluid must flow in such a way that its mass remains conserved.\r\n This imposes certain restrictions that the fluid velocity vector should obey when the fluid is flowing through a duct or a channel. Consider two points 'A' and 'B' along the flow of the fluid.\r\n The principle of continuity then implies:\r\n $$ \\rho_A A_A v_A = \\rho_B A_B v_B$$\r\n where \\(\\rho_A\\) and \\(\\rho_B\\) are the fluid densities, \\(A_A\\) and \\(A_B\\) are the cross-section areas and \\(v_A\\) and \\(v_B\\) are the fluid velocities at the two points, respectively.\r\n
\r\n\r\n To understand how conservation of mass places restrictions on the velocity field, we consider the steady flow of a fluid through a duct. The cross-section of the duct varies along its length i.e. the cross-section at the inlet is different from the cross-section at the outlet.\r\n Further, since the flow of the fluid is one-dimensional, the velocity and the density are assumed not to change over a cross-section of the duct.\r\n
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\r\n Let \\(v_1\\) denote the speed of the fluid at the inlet and \\(v_2\\) denote the speed of the fluid at the outlet.\r\n Let \\(\\rho_1\\) denote the density of the fluid at the inlet and \\(\\rho_2\\) denote the density of the fluid at the outlet.\r\n Let \\(A_1\\) denote the cross-section area of the fluid at the inlet and \\(A_2\\) denote the cross-section area of the fluid at the outlet.\r\n
\r\n The mass of the fluid entering the duct in a time interval \\(\\Delta t\\) is given as:\r\n $$\\Delta m_i = \\rho_1 A_1 v_1 \\Delta t$$\r\n The mass of the fluid leaving the duct in the same interval is given as:\r\n $$\\Delta m_o = \\rho_2 A_2 v_2 \\Delta t$$\r\n The conservation of mass requires that mass entering the duct be same as the mass leaving the duct. Thus, we have:\r\n $$\\Delta m_i = \\Delta m_o$$\r\n which leads to\r\n $$ \\rho_1 A_1 v_1 \\Delta t = \\rho_2 A_2 v_2 \\Delta t$$\r\n or\r\n $$ \\rho_1 A_1 v_1 = \\rho_2 A_2 v_2$$\r\n
\r\n For incompressible flows, the density of the fluid remains constant during the flow, i.e. \\(\\rho_1 = \\rho_2\\). Thus, the continuity equation for such flows can be written as:\r\n $$ v_1 A_1 = v_2 A_2$$\r\n
\r\n\n This MagicGraph lets you visualize how the velocity of a fluid element changes as it flows through a duct with variable cross-section.\r\n Drag points A, B, and C around to change the cross-section geometry of the duct. Then, tap on the blower to start the MagicGraph.\r\n
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