Question

Bernoulli equation. The Bernoulli equation is a special case of conservation of linear momentum law of conservation of energy) for steady frictionless flow. This equation can be arrived at in three different ways. The usual form of the Bernoulli equation is: 1. pv2 + P + ?9z-constant a) For frictionless flow at steady state, Euler's equation of conservation of linear momentum reduces to: Starting from this equation, derive the Bernoulli equation. Assume irrotational flow. Derive the Bernoulli equation using the law of conservation of energy for the flow shown below: b) Z, 1aconstant Z-0 c) A common engineering application of Bernoulli equation is to investigate conservation of energy along a streamline. This form of the equation arises in steady flow in which the particles of fluid move along fixed streamlines, as on rails, and are accelerated and decelerated by the forces acting tangent to the streamlines. Starting from the equation of motion, and the definition of a streamline, derive the Bernoulli equation

Answer 1

a) Euler equation of motion is given by P V nabla V = -nabla P + p g nabla n rightarrow p v nabla v + nabla p - p g_y nabla h = 0 rightarrow (1) * we are taking 'h' in +ve direction but gravity is acting in '_ve' direction so, g_y = -g equation (1) will become, p v nabla v + nabla p + pg nabla h = 0 for a small change nabla = alpha, Now, Euler equation will be, pv dv + dp + pg dn = 0 rightarrow (2) integrate the equation (2) integral pv dv + integral dp + integral pg dn = integral 0 p v^/2 + p + pgh = c rightarrow Bernoulli theorem * where pv^3/2 = kinetic energy per unit volume, p = pressure energy per unit volume, pgh = potential energy \r\n b) Bernoulli equation using law of conversation of energy * Let us consider a fluid flowing through a smooth pipe at time, t > 0, the fluid at point (1) is moved by a distance a_1 & fluid at point (2) is moved by a distance 'a_2'