Question

In: Mechanical Engineering

2) Find the Bernoulli Equation for Irrotational Flow.


2) Find the Bernoulli Equation for Irrotational Flow.

Solutions

Expert Solution

Irrotationality of flow field

Under some special condition, the constant C becomes invariant from streamline to streamline and the Bernoulli’s equation is applicable with same value of C to the entire flow field. The typical condition is the irrotationality of flow field.

Let us consider a steady two dimensional flow of an ideal fluid in a rectangular Cartesian coordinate system. The velocity field is given by

hence the condition of irrotationality is:

(1)

The steady state Euler's equation can be written as


(2a)
(2b)

We consider the y-axis to be vertical and directed positive upward. From the condition of irrotationality given by the Eq. (1), we substitute in place of in the Eq.2a and in place of in the Eq.2b. This results in

(3a)
(3b)

Now multiplying Eq.(3a) by 'dx' and Eq.(3b) by 'dy' and then adding these two equations we have  

(4)

The Eq. (4) can be physically interpreted as the equation of conservation of energy for an arbitrary displacement

. Since, u, v and p are functions of x and y, we can write


(5a)

(5b)
(5c)

With the help of Eqs (5a), (5b), and (5c), the Eq. (4) can be written as

(6)

The integration of Eq. 6 results in      

(7a)

For an incompressible flow,

(7b)

The constant C in Eqs (7a) and (7b) has the same value in the entire flow field, since no restriction was made in the choice of dr which was considered as an arbitrary displacement in evaluating the work.

Therefore, the total mechanical energy remains constant everywhere in an inviscid and irrotational flow, while it is constant only along a streamline for an inviscid but rotational flow.

The equation of motion for the flow of an inviscid fluid can be written in a vector form as

where is the body force vector per unit mass


Related Solutions

Solve the given differential equation using an appropriate substitution. The DE is a Bernoulli equation, A....
Solve the given differential equation using an appropriate substitution. The DE is a Bernoulli equation, A. dy/dx = y(xy^6 - 1) B. x dy/dx + y = 1/y^2 C. t^2 dy/dt + y^2 = ty
(Fluid Mechanics; Euler's equation and Bernoulli equation) As I know, in order to derive Euler's equation...
(Fluid Mechanics; Euler's equation and Bernoulli equation) As I know, in order to derive Euler's equation from Naveri-Stokes equation, the additional conditions are 1)Incompressible and 2)Inviscid Then, in order to derive Bernoulli's equation from Euler's equation, what additional conditions are needed? As I thought, Euler's equation already meets the incompressible and invsicid condtion, so only "steady state" is the additional condition to derive Bernoulli equation from Euler's equation. Am I right? In addition, are both Navier-stokes equation and Euler's equation...
What is the significance of Bernoulli equation and Reynolds number in Fluid Mechanics?
What is the significance of Bernoulli equation and Reynolds number in Fluid Mechanics?
1.Write the Bernoulli equation and state all the assumptions used in its derivation. 2.Write the vector...
1.Write the Bernoulli equation and state all the assumptions used in its derivation. 2.Write the vector form of the conservation of momentum equation for a control volume that is moving with a constant velocity. 3.) Water is flowing in a fire hose with a velocity of 1.0 m/s and a pressure of 200000 Pa. Use the Bernoulli equation to calculate the velocity of the water exiting the nozzle. Note: Derive the expression before plugging in numbers and clearly state all...
Introduce hydraulic loss   into the Bernoulli equation to account for the energy lost by the water flowing...
Introduce hydraulic loss   into the Bernoulli equation to account for the energy lost by the water flowing through the small horizontal pipe. Derive the expression for the hydraulic loss by considering two points at the inlet and outlet of the small horizontal pipe.
The probability of success in Bernoulli is 0.7. Find the expected value and variance of the...
The probability of success in Bernoulli is 0.7. Find the expected value and variance of the number of failures until the ninth success. (The problem is to find the mean and variance of the number of failures in the negative binomial distribution given the Bernoulli probability of success.)
For the equation x^2+xy+y^2=0, find the equation of the normal to the tangent line at the...
For the equation x^2+xy+y^2=0, find the equation of the normal to the tangent line at the point (−3,−1)(-3,-1).
12a Find an equation of the tangent plane to the surface ? = 2? 2 +...
12a Find an equation of the tangent plane to the surface ? = 2? 2 + ? 2 − 5?, ?? (1, 2, −4). 12b If ? = ? 2 − ?? + 3? 2 and (?, ?) changes from (3, −1) to (2.96, −0.95), compare ∆? and ??. Calculus 3 question. Please help.
1) concepts of water flow in piping system? 2) concepts of energy equation Bernonlli equation? 3)...
1) concepts of water flow in piping system? 2) concepts of energy equation Bernonlli equation? 3) concepts of energy head losses? 4) concepts of energy equation momentum? I hope not written in handwriting?
How is the macroscopic mechanical energy balance related to the Bernoulli equation for inviscid fluids? How...
How is the macroscopic mechanical energy balance related to the Bernoulli equation for inviscid fluids? How is it derived?
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT