In addition, derive the "wave equation" for an incompressible fluid. Use the continuity equation and the linearized euler equation.
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In addition, derive the "wave equation" for an incompressible
fluid. Use the continuity equation and the...
Advanced Fluid Mechanics For an inviscid fluid we have Euler's equation (vectors are denoted by bold characters)...
Advanced Fluid Mechanics
For an inviscid fluid we have Euler's equation (vectors are denoted by bold characters) )= Vp - Vgx + V xu at and whether or not the fluid is incompressible, we also have the conservation of mass Dp +pv u 0. Dt Show that x Vp Dt Deduce that, if p is a function of palone, the vorticity equation is exactly as in the incompressible, constant density case, except that ois replaced by ap.
For an inviscid...
#1 The Equation of Continuity: Consider Figure 1 which illustrates a small mathernat ical box in...
#1 The Equation of Continuity: Consider Figure 1 which illustrates a small mathernat ical box in a fluid. The basic idea behind the equation of continuity is that the rate of mass flow into the box must equal the time rate of change of the mass in the box (pur)l Figure 1: A small mathematical box in a fluid A) Consider just the 2-faces of the box for now. Figure 1 shows ρυ.Ε entering at and leaving at + ρ...
For an ideal, incompressible fluid of density p, subject to a gravitational field g= -Vº' (here...
For an ideal, incompressible fluid of density p, subject to a gravitational field g= -Vº' (here Ø' is the gravitational potential) the Euler equation is: Div=-- Vp+g, Div = .v + (v.7)v Use vector identity v * (I xv) = -(v.7)v + rv2) to derive Bernoulli's equation: --v2--p - d' = const, along any streamline (dye path) in steady flow. (a) Use y (b) This question concerns free surface flow and hydraulic jumps. Incompressible inviscid fluid (water) is in steady,...
2) If we now set H(x,y.t)-H0(x,y)+n(x,y,t) and assume that we only have small- amplitude motions with...
2) If we now set H(x,y.t)-H0(x,y)+n(x,y,t) and assume that we only have small- amplitude motions with we obtain the linearized shallow-water equations Ot on O a) For the special non-rotating case (f -0 ) with constant depth (Ho - const.) show that the speed of gravity waves is c-VgHo Hint: set v-0 and derive a wave equation for the sea level η b) Given a harmonic wave η(x,t)=Asin(k-or) with amplitude A (again for f-0 and Ho= const.), derive the equation...
4. An incompressible fluid with viscosity u and density p was contained in pipe of length...
4. An incompressible fluid with viscosity u and density p was contained in pipe of length L and radius R. Initially the fluid is in rest. At t=0, a pressure difference of AP is applied across the pipe length which induces the fluid flow in axial direction (V2) Only varies with time (t) and pipe radius (r). There is no effect of gravity. To describe the fluid flow characteristics, after the pressure gradient is applied, answer the following questions: a)...
An incompressible fluid flows between two porous, parallel flat plates as shown in the Figure below....
An incompressible fluid flows between two porous, parallel flat plates as shown in the Figure below. An identical fluid is injected at a constant speed V through the bottom plate and simultaneously extracted from the upper plate at the same velocity. There is no gravity force in x and y directions (g-g,-0). Assume the flow to be steady, fully-developed, 2D, and the pressure gradient in the x direction to be a constant P = constant). (a) Write the continuity equation...
Question 6 A3 A gas moves according to the (non-dimensional) continuity and mnomentum 12 marks equations...
Question 6 A3 A gas moves according to the (non-dimensional) continuity and mnomentum 12 marks equations - ρυ) 0, at + v- at where is the density(v)is the velocity and thé stress is σ-cos (πρ ) . Verify that v, = 0 and p, = 1/2 is an equilibrium of the gas flow. Introduce small variations of the velocity about v, and density about p., linearise the PDES and derive the wave equation governing the evolution of the density variations....
Problem 3. Consider a pipe containing a steadily flowing inviscid fluid. It has one inlet and branches into two arms so...
Problem 3. Consider a pipe containing a steadily flowing inviscid fluid. It has one inlet and branches into two arms so that there are two outlets (see Fig. 1). Flow can be considered uniform and parallel to the walls when entering and exiting the pipe Inlet Pi Outlet ρ2 A2 p, Outlet Figure 1: Flow of fluid through a "T" -junction in a pipe, shown from above (not to scale) Part A (a) The Continuity equation, as given on the...
Consider the initial value problem for the one-dimensional wave equation Write as clear as Ou Ou...
Consider the initial value problem for the one-dimensional wave equation Write as clear as Ou Ou ot (4) possible , some work has been hard to follow Thanks! a(z,0) = e-r2 (a) Determine the solution u(r,t) of (4) (b) Sketch the solution in the xu-plane at t = 0, t = 1 , and (c) Which direction does the wave travel? 2
Bernoulli equation. The Bernoulli equation is a special case of conservation of linear momentum law of...
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...
Advanced Fluid Mechanics
For an inviscid fluid we have Euler's equation (vectors are denoted by bold characters) )= Vp - Vgx + V xu at and whether or not the fluid is incompressible, we also have the conservation of mass Dp +pv u 0. Dt Show that x Vp Dt Deduce that, if p is a function of palone, the vorticity equation is exactly as in the incompressible, constant density case, except that ois replaced by ap.
For an inviscid...
#1 The Equation of Continuity: Consider Figure 1 which illustrates a small mathernat ical box in a fluid. The basic idea behind the equation of continuity is that the rate of mass flow into the box must equal the time rate of change of the mass in the box (pur)l Figure 1: A small mathematical box in a fluid A) Consider just the 2-faces of the box for now. Figure 1 shows ρυ.Ε entering at and leaving at + ρ...
For an ideal, incompressible fluid of density p, subject to a gravitational field g= -Vº' (here Ø' is the gravitational potential) the Euler equation is: Div=-- Vp+g, Div = .v + (v.7)v Use vector identity v * (I xv) = -(v.7)v + rv2) to derive Bernoulli's equation: --v2--p - d' = const, along any streamline (dye path) in steady flow. (a) Use y (b) This question concerns free surface flow and hydraulic jumps. Incompressible inviscid fluid (water) is in steady,...
2) If we now set H(x,y.t)-H0(x,y)+n(x,y,t) and assume that we only have small- amplitude motions with we obtain the linearized shallow-water equations Ot on O a) For the special non-rotating case (f -0 ) with constant depth (Ho - const.) show that the speed of gravity waves is c-VgHo Hint: set v-0 and derive a wave equation for the sea level η b) Given a harmonic wave η(x,t)=Asin(k-or) with amplitude A (again for f-0 and Ho= const.), derive the equation...
4. An incompressible fluid with viscosity u and density p was contained in pipe of length L and radius R. Initially the fluid is in rest. At t=0, a pressure difference of AP is applied across the pipe length which induces the fluid flow in axial direction (V2) Only varies with time (t) and pipe radius (r). There is no effect of gravity. To describe the fluid flow characteristics, after the pressure gradient is applied, answer the following questions: a)...
An incompressible fluid flows between two porous, parallel flat plates as shown in the Figure below. An identical fluid is injected at a constant speed V through the bottom plate and simultaneously extracted from the upper plate at the same velocity. There is no gravity force in x and y directions (g-g,-0). Assume the flow to be steady, fully-developed, 2D, and the pressure gradient in the x direction to be a constant P = constant). (a) Write the continuity equation...
Question 6 A3 A gas moves according to the (non-dimensional) continuity and mnomentum 12 marks equations - ρυ) 0, at + v- at where is the density(v)is the velocity and thé stress is σ-cos (πρ ) . Verify that v, = 0 and p, = 1/2 is an equilibrium of the gas flow. Introduce small variations of the velocity about v, and density about p., linearise the PDES and derive the wave equation governing the evolution of the density variations....
Problem 3. Consider a pipe containing a steadily flowing inviscid fluid. It has one inlet and branches into two arms so that there are two outlets (see Fig. 1). Flow can be considered uniform and parallel to the walls when entering and exiting the pipe Inlet Pi Outlet ρ2 A2 p, Outlet Figure 1: Flow of fluid through a "T" -junction in a pipe, shown from above (not to scale) Part A (a) The Continuity equation, as given on the...
Consider the initial value problem for the one-dimensional wave equation Write as clear as Ou Ou ot (4) possible , some work has been hard to follow Thanks! a(z,0) = e-r2 (a) Determine the solution u(r,t) of (4) (b) Sketch the solution in the xu-plane at t = 0, t = 1 , and (c) Which direction does the wave travel? 2
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...
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