Continuous Assessment – 2
(Report Writing Skill)
Momentum equation in Cartesian co-
ordinate system and Bernoulli’s
Equation
Subject-Water Resource Engineering
CODE- OE-ME701I
ARUP KUMAR SHEE (12000721083)
Section:ME-2Y
Date of Submission:07/09/2023
Department of Mechanical Engineering
DR. B. C. ROY ENGINEERING COLLEGE
Jemua road, Fuljhore, Durgapur-713206 (West Bengal), India.
Contents
• Momentum equation in Cartesian co-ordinate system &
Bernoulli’s equation
• Reference
1. Momentum Equation (Newton's Second Law for
Fluids):
The momentum equation is a fundamental equation in fluid
mechanics and describes the conservation of linear
momentum for a fluid. In a Cartesian coordinate system (x,
y, z), the momentum equation can be derived from Newton's
second law, which states that the rate of change of
momentum of a fluid element is equal to the sum of the
forces acting on it.
Let's consider a small fluid element in a steady flow with
velocity components (u, v, w) in the x, y, and z directions,
respectively. The density of the fluid is ρ, and the pressure at
the location of the element is P. The forces acting on this
element are:
• Pressure force in the x-direction: -∂P/∂x (negative
because it acts in the opposite direction of x).
• Pressure force in the y-direction: -∂P/∂y (negative
because it acts in the opposite direction of y).
• Pressure force in the z-direction: -∂P/∂z (negative
because it acts in the opposite direction of z).
• Viscous forces in the x, y, and z directions: These are
proportional to the gradients of velocity components
∂u/∂x, ∂v/∂y, and ∂w/∂z, and are given by -μ(∂u/∂x), -
μ(∂v/∂y), and -μ(∂w/∂z), respectively, where μ is the
dynamic viscosity of the fluid.
Using Newton's second law, we can write the momentum
equation in the x, y, and z directions:
x-direction:
ρ(u∂u/∂x + v∂u/∂y + w∂u/∂z) = -∂P/∂x + μ(∂²u/∂x² + ∂²u/∂y² +
∂²u/∂z²)
y-direction:
ρ(u∂v/∂x + v∂v/∂y + w∂v/∂z) = -∂P/∂y + μ(∂²v/∂x² + ∂²v/∂y² +
∂²v/∂z²)
z-direction:
ρ(u∂w/∂x + v∂w/∂y + w∂w/∂z) = -∂P/∂z + μ(∂²w/∂x² +
∂²w/∂y² + ∂²w/∂z²)
These are the three components of the momentum equation in
a Cartesian coordinate system.
Bernoulli’s Equation:
Bernoulli's equation is a fundamental equation in fluid
mechanics that describes the conservation of energy along a
streamline in an inviscid, incompressible flow. It can be
derived from the principle of conservation of energy.
The basic form of Bernoulli's equation is as follows:
P + 0.5ρv² + ρgh = constant
Where:
P is the pressure of the fluid.
ρ is the density of the fluid.
v is the velocity of the fluid.
g is the acceleration due to gravity.
h is the height above a reference point.
To derive Bernoulli's equation, you can use the work-energy
principle and the fact that the sum of mechanical energy
(kinetic energy + potential energy) remains constant along a
streamline. Here's a simplified derivation:
• Start with the work-energy principle:
∆KE + ∆PE + ∆W = 0
• For an incompressible fluid (ρ is constant), consider a
streamline where ∆KE = 0 (constant velocity) and ∆W is
the work done by external forces (usually negligible).
• Simplify the work-energy equation:
∆PE + ∆W = 0
• Consider two points along the streamline at different
elevations (h₁ and h₂). The change in potential energy
(∆PE) is given by ρgh₁ - ρgh₂, where g is the acceleration
due to gravity.
• Plug this into the equation:
(ρgh₁ - ρgh₂) + ∆W = 0
• Assuming ∆W is negligible, you get Bernoulli's equation:
P₁ + 0.5ρv₁² + ρgh₁ = P₂ + 0.5ρv₂² + ρgh₂
This is Bernoulli's equation in its basic form. It relates pressure,
velocity, and elevation along a streamline in an idealized,
inviscid, and incompressible fluid flow.
Conclusion:
In conclusion, the momentum equation in a Cartesian coordinate
system and Bernoulli's equation are fundamental equations in
fluid mechanics that describe different aspects of fluid
behaviour. the momentum equation focuses on the conservation
of momentum and the forces acting on fluid elements, while
Bernoulli's equation focuses on the conservation of energy
along a streamline. These equations are fundamental tools in
fluid mechanics and are used to solve a wide range of
engineering and scientific problems related to fluid flow.
Reference:
https://eng.libretexts.org/Bookshelves/Civil_Engineering/Book%
3A_Fluid_Mechanics_(Bar-
Meir)/08%3A_Differential_Analysis/8.5%3A_Derivations_of_t
he_Momentum_Equation.
http://www1.maths.leeds.ac.uk/~kersale/2620/Notes/m2620.pdf.
