Bernoulli's Equation | Fluid Mechanics for Mechanical Engineering PDF Download

Equations of Motion

According to Newton’s second law of motion, the net force F_x acting on a fluid element in the direction of x is equal to mass m of the fluid element multiplied by the acceleration ax in the x − direction. Thus, mathematically F_x  = m a_x In the fluid flow, the following forces are present: 

1. F_g, gravity force 
2. F_p, the pressure force  
3. F _ν , the force due to viscosity 
4. F_c, force due to compressibility 
5. F_t, fore due to turbulence of flow

Thus , net force  F_x, = (F_g )_x +  (F_p)_x + (F_ν)_x + (F_t)_x + (F_c)_x 

• If the force due to compressibility, FC is negligible, the resulting net force
  F_x =(F_g)_x + (F_p)_x + (F_ν)_x + (F_t)_x and equations of motion are called “Reynold’s equations of motion” 
• For flow where (F_t) is negligible, the resulting equations of motion are known as “Navier −  Stokes equation”. 
• F_x = (F_g )_x + (F_p )_x+ (F_v )_x  (Navier Stoke’s equation.)  This equation is useful in analysis of viscous flow. 
• If the flow is assumed to be ideal, viscous force (F_ν) is zero and equations of motion are known as “Euler’s equations of motion”.
  F_x = (F_g )_x + (F_p )_x

Euler’s Equation of Motion

This is Equation of motion in which the force due to gravity and pressure are taken into consideration  Euler’s Equation is Bernouli’s  equation is obtained by integrating the Euler’s equation of motion as  This equation is applicable for steady, irrotational flow of incompressible fluids.
If flow is incompressible, ρ is constant i.e. independent of pressure (hence p is treated as constant).

Above equation is Bernoulli’s equation in which: 

 = pressure energy per unit weight of fluid or pressure head
v²  / 2g = Kinetic energy per unit weight or kinetic head
z = potential energy per unit weight or potential head  

Assumptions:

The following are the assumptions made in the derivation of Bernoulli’s equation:  
(i) The fluid is ideal, i.e. viscosity is zero
(ii) The flow is steady
(iii) The flow is incompressible
(iv) The flow is irrotational

Bernoulli’s Equation for Incompressible Real Fluid

Bernoulli's equation is originally derived under the assumption that the fluid is inviscid (non-viscous) and, therefore, frictionless. However, real fluids are viscous and resist flow, resulting in energy losses. These losses must be accounted for when applying Bernoulli's equation to real fluids. The modified Bernoulli's equation for real fluids, accounting for these losses between points 1 and 2, is given as:
Where h_L is loss of energy per unit weight of fluid between points 1 and 2.

Practical Applications of Bernoulli’s Equation

Bernoulli's equation is applied in all problems of incompressible fluid flow where energy considerations are involved e.g.
(i) Venturimeter
(ii) Orifice m eter
(iii) Pitot −  tube  

The Momentum Equation

The momentum equation is derived from the law of conservation of momentum, which states that the net force acting on a fluid mass is equal to the change in momentum of the fluid flow per unit time in that direction. The momentum equation can be expressed as follows:
Where F is the force on a fluid mass m.  
Above equation is known as the “momentum principle” Equation (1) may also be written as  
F. dt = d (mv)
Which is known as the “impulse − momentum equation”.

Force exerted by a flowing fluid on Pipe − Bend

The impulse momentum equation is used to determine the resultant force exerted by a flowing fluid on a pipe bend. Consider two sections (1) and (2) as shown in fig.
Let,  V₁  = Velocity of flow at section (1)  
p₁ = pressure intensity at section (1)  
A₁  = area of cross section of pipe at section (1) and  
V₂, p₂.,A₂ = corresponding values of velocity, pressure and area at section (2)
Let F_x and F_y be the components of the forces exerted by the flowing fluid on the bend in x and y direction respectively. Then  
F_x = ρ Q (V₁ − V₂ cosθ ) + p₁ A₁ − p₁ A₂ cosθ
F_y  = ρ Q (−V₂ sin θ) − p₂ A₂ sin θ
The resultant force (FR) acting on the bend =  
And the angle made by the resultant force with horizontal direction is given by   tanθ  = F_y / F_x 

Solved Numericals

Q1. Water flows in a horizontal tube (see figure). The pressure of water changes by 700 Nm⁻² between A and B where the area of cross section are 40 cm² and 20 cm², respectively. Find the rate of flow of water through the tube. (density of water = 1000 kgm⁻³)
Ans: In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy.
From Bernoulli's equation
PA  + 1/2ρV²A = PB + 1/2ρV²B
Calculations:
Given:
Δ P = 700 Nm^-2
a_A = 40 cm²
a_B = 20 cm²
From the continuity equation
a_AV_A = a_BV_B
Putting values of area, we will get,
2V_A = V_B
From Bernoulli's equation
P_A  + 1/2ρV²_A = P_B + 1/2ρV²_B
Δ P = 1/2ρ(V²_B - V²_A)
= 1/210³(4V²_A - V²_A)
700 × 2 = 10³ (3 V²_A)  
V²_A = $\frac{700}{\times }$
V_A = 7/15m/s
= 7/15$\times 100$ cm/s
The volume of the flow = a_A × v_A = 40 × 7/15$\times 100$
= 2732.5 cm³/s

Q2. If the flow speeds of the upper and lower surfaces of the wings of an aeroplane are 260m/s and 250m/s, the wings cover an area of 500 m² then what would be the lift generated (in kN)? (take density of air as 1kg/m³) 
Ans: 

• Bernoulli's principle states that the sum of pressure energy, kinetic energy, and potential energy per unit volume of an incompressible, non- viscous fluid in a streamlined irrotational flow remains constant along a streamline.
• This means that in steady flow the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline.
  

Assume P_top = pressure at the top, P_bottom = Pressure at the bottom, V_top = Velocity at the top of the wing, V_bottom = Velocity at the bottom

• Given - V_top = 260 m/s, V_bottom = 250 m/s, A = 500 m², and ρ = 1 kg/m³
• Assuming the height difference between the upper and lower surface of the wings to be zero.
• Hence the gravitational potential energy can be neglected then the equation of Bernoulli's theorem becomes 

• In order for the lift to occur P_bottom > P_top
• Hence equation 1 can be rewritten as

The lift force can be found out using the equation
⇒ F = ΔP A        -----(3)

Q3. The pressure of water in a pipe when water is not flowing is 3 × 10⁵ Pa and when the water flows the pressure falls to 2.5 × 10⁵ Pa. Find the speed of flow of water (in m/s)?
Ans:

• Bernoulli's principle states that the sum of pressure energy, kinetic energy, and potential energy per unit volume of an incompressible, non- viscous fluid in a streamlined irrotational flow remains constant along a streamline.

This means that in steady flow the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline.
From Bernoulli's principle

Given:
Initial velocity (v₁) = 0 m/s, Initial pressure (P₁) =  3 × 10⁵ Pa, and Final pressure (P2) =  2.5 × 10⁵ Pa  
According to Bernoulli's principle

Above equation can be written as