Equation of Continuity and Deduction of Euler’s Equation | Mechanics | Mathematical Physics | University Notes | Physics Notes | B.Sc. Physics Notes by Study Buddy Notes

Equation of Continuity and Deduction of Euler’s Equation

Equation of Continuity and Deduction of Euler’s Equation | Mechanics | Mathematical Physics | University Notes | Physics Notes | B.Sc. Physics Notes by Study Buddy Notes

The Equation of Continuity and Euler’s Equation are fundamental concepts in fluid mechanics. The continuity equation expresses the principle of conservation of mass in fluid flow, while Euler’s equation describes the balance of forces acting on a fluid element, leading to the derivation of the equations of motion for an ideal (non-viscous) fluid.

1. Equation of Continuity

The Equation of Continuity states that the mass of fluid entering a given volume per unit time must equal the mass exiting that volume per unit time, assuming no accumulation or depletion of mass within the volume. For a steady, incompressible fluid, this principle is mathematically expressed as:

$$\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (\rho \vec{v})=0$$

where:

• $\rho$ is the fluid density,
• $\vec{v}$ is the velocity vector of the fluid,
• $\nabla \cdot (\rho \vec{v})$ is the divergence of the mass flux.

In incompressible flow ($\rho$ constant), the equation reduces to:

$$\mathrm{\nabla }\cdot \vec{v}=0$$

This form implies that the volume flow rate remains constant across any cross-section of the flow.

Derivation for In compressible Flow

Consider a fluid element in a three-dimensional coordinate system with velocity components $u, v, w$ in the $x, y,$and $z$ directions, respectively. For an incompressible fluid, the rate of change of volume of this element must be zero, which leads to:

$$\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}+\frac{\mathrm{\partial }w}{\mathrm{\partial }z}=0$$

This equation expresses the conservation of mass in differential form, indicating that the divergence of the velocity field is zero.

2. Euler’s Equation of Motion

Euler’s Equation of Motion describes the balance of forces on an infinitesimal fluid element and is derived from Newton’s second law. For an ideal (inviscid) fluid, the forces considered are:

• Pressure forces,
• Body forces (such as gravity),
• No viscous forces (ideal fluid assumption).

Let’s derive Euler’s equation for an incompressible fluid in the $x$-direction. The same process can be applied to the $y$ and $z$ directions.

Derivation

1. Consider a Fluid Element: Consider a small fluid element with dimensions $dx, dy, dz$ and density $\rho$.

2. Forces in the x-direction:

   • Pressure Force: Let $p$ be the pressure at the center of the element. The pressure force in the $x$-direction on the left face is $p \, dy \, dz$, and on the right face, it’s $\left(p + \frac{\partial p}{\partial x} dx \right) dy \, dz$. The net pressure force in the $x$-direction is:

     $$F_p=-\frac{\mathrm{\partial }p}{\mathrm{\partial }x}dxdydz$$

   • Body Force (e.g., Gravity): If $f_x$ represents the body force per unit mass in the $x$-direction, the total body force on the element is:

     $$F_b=\rho f_{x}dxdydz$$

3. Applying Newton’s Second Law:

   According to Newton’s second law, the net force on the element equals the mass of the element times its acceleration. The acceleration in the $x$-direction is $\frac{Du}{Dt}$, where $\frac{D}{Dt}$ is the material derivative:

   $$\frac{Du}{Dt}=\frac{\mathrm{\partial }u}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+w\frac{\mathrm{\partial }u}{\mathrm{\partial }\mathrm{z}}$$

   Thus, Newton’s second law gives:

   $$\rho dxdydz\cdot \frac{Du}{Dt}=-\frac{\mathrm{\partial }p}{\mathrm{\partial }x}dxdydz+\rho f_{x}dxdydz$$

4. Simplifying: Dividing through by $\rho \, dx \, dy \, dz$, we obtain:

   $$\frac{Du}{Dt}=-\frac{1}{\rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }x}+f_{\mathrm{x}}$$

Repeating similar steps for the $y$ and $z$ directions, we get the vector form of Euler’s Equation:

$$\frac{D\vec{v}}{Dt}=-\frac{1}{\rho }\mathrm{\nabla }p+\vec{f}$$

where $\vec{f}$ represents the body force per unit mass, such as gravitational acceleration $\vec{g}$.

3. Interpretation of Euler’s Equation

Euler’s equation shows that the acceleration of a fluid element is due to the pressure gradient and the body forces acting on it. For incompressible flow, it’s common to analyze the equation by breaking down the terms:

• Acceleration Term $\frac{D\vec{v}}{Dt}$: Represents the change in velocity of a fluid element as it moves through the flow field.
• Pressure Gradient Term $-\frac{1}{\rho} \nabla p$: Indicates that fluid moves from regions of high pressure to low pressure.
• Body Force Term $\vec{f}$: Often represents gravitational forces (e.g., $\vec{f} = \vec{g}$).

Special Case: Steady Flow with No Body Forces

For steady flow ($\frac{\partial \vec{v}}{\partial t} = 0$) and no body forces, Euler’s equation simplifies to:

$$\vec{v}\cdot \mathrm{\nabla }\vec{v}=-\frac{1}{\rho }\mathrm{\nabla }p$$

This form of Euler’s equation is particularly useful in applications such as potential flow theory, where it is further simplified using assumptions about the flow field.

Summary

• Equation of Continuity: $\nabla \cdot \vec{v} = 0$ for incompressible flow, expressing conservation of mass.
• Euler’s Equation: $\frac{D\vec{v}}{Dt} = -\frac{1}{\rho} \nabla p + \vec{f}$, governing the motion of an ideal fluid by relating acceleration to pressure gradient and body forces.