From Archimedes principle it follows directly that the buoyancy force must beF

From archimedes principle it follows directly that

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From Archimedes’ principle it follows directly that the buoyancy force must be F b = ρ water 3 4 h 3 g , and it must be pointing upward as indicated in the figure. The force due to the mass of the cube, i.e. , the weight, is given by F W = ρ cube h 3 g , with the minus sign indicating the downward direction of the force. Now in static equilibrium (the cube floating as indicated), the forces acting on the cube must sum to zero. Hence, F b + F W = 0 = ρ water 3 4 h 3 g ρ cube h 3 g , or ρ cube = 3 4 ρ water , the required result. 4.2 Bernoulli’s Equation Bernoulli’s equation is one of the best-known and widely-used equations of elementary fluid mechan- ics. In the present section we will derive this equation from the N.–S. equations again emphasizing the ease with which numerous seemingly scattered results can be obtained once these equations are available. Following this derivation we will consider two main examples to highlight applications. The first of these will employ only Bernoulli’s equation to analyze the working of a pitot tube for measuring air speed while the second will require a combination of Bernoulli’s equation and the continuity equation derived in Chap. 3.
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110 CHAPTER 4. APPLICATIONS OF THE NAVIER–STOKES EQUATIONS 4.2.1 Derivation of Bernoulli’s equation As noted above, we will base our derivation of Bernoulli’s equation on the N.–S. equations allowing us to very clearly identify the assumptions and approximations that must be made. Thus, we begin by again writing these equations. We will employ the 2-D form of the equations, but it will be clear as we proceed that the full 3-D set of equations would yield precisely the same result. The 2-D, incompressible N.–S. equations can be expressed as u t + uu x + vu y = 1 ρ p x + ν ( u xx + u yy ) , (4.8a) v t + uv x + vv y = 1 ρ p y + ν ( v xx + v yy ) g , (4.8b) where we have employed our usual notation for derivatives, and we are assuming that the only body force results from gravitational acceleration acting in the negative y direction. In addition to the incompressibility assumption embodied in this form of the N.–S. equations we need to assume that the flow being treated is inviscid. Recall from Chap. 2 that this implies that the effects of viscosity are negligible, and more specifically from Chap. 3 we would conclude that viscous forces can be considered to be small in comparison with the other forces represented by these equations, namely inertial, pressure and body forces. In turn, this implies that we will not be able to account for, or calculate, shear stresses in any flow to which we apply Bernoulli’s equation, and furthermore, the no-slip condition will no longer be used. Dropping the viscous terms in Eqs. (4.8) leads to u t + uu x + vu y = 1 ρ p x , (4.9a) v t + uv x + vv y = 1 ρ p y g , (4.9b) a system of equations known as the Euler equations . The compressible form of these equations is widely used in studies of high-speed aerodynamics.
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