Velocity potential and stream function are fundamental concepts in fluid dynamics that are commonly utilized to depict fluid motion. These concepts are interdependent and are frequently applied in tandem to describe fluid dynamics.
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What is Velocity Potential?
The velocity potential is a scalar function that is used to describe the motion of a fluid. It is defined as the scalar function ϕ such that the velocity vector v is equal to the gradient of ϕ:
v = ∇ϕ
where v is the velocity vector and ∇ is the gradient operator. In other words, the velocity potential is the scalar field that describes the rate at which the fluid is moving in a particular direction.
The velocity potential is often used to describe irrotational flows, where the fluid is moving without any rotation. In such cases, the curl of the velocity vector is zero, and the velocity potential can be used to describe the motion of the fluid. The velocity potential can be expressed mathematically as:
ϕ = -∫ v · dr
where the integral is taken along any path from a fixed point to a point in the fluid, and v · dr is the dot product of the velocity vector and the differential element of the path.
What is Stream Function?
The stream function is a scalar function that is used to describe the motion of a fluid. It is defined as the scalar function ψ such that the velocity vector v is equal to the curl of ψ:
v = ∇ × ψ
where ∇ is the gradient operator and × is the curl operator. In other words, the stream function is the scalar field that describes the circulation of the fluid around a point.
The stream function is often used to describe rotational flows, where the fluid is moving with rotation. In such cases, the curl of the velocity vector is not zero, and the stream function can be used to describe the motion of the fluid. The stream function can be expressed mathematically as:
ψ = ∫ v · ds
where the integral is taken along a closed path in the fluid, and v · ds is the dot product of the velocity vector and the differential element of the path.
Relationship between Velocity Potential and Stream Function:
The velocity potential and stream function are related to each other through the following equation:
v = ∇ϕ = ∇ × ψ
where v is the velocity vector, ϕ is the velocity potential, and ψ is the stream function. This equation is known as the Helmholtz equation.
In two-dimensional flows, the velocity potential and stream function are related to each other through the following equations:
u = ∂ψ/∂y and v = -∂ψ/∂x
where u and v are the x and y components of the velocity vector. These equations are known as the continuity equations.
The velocity potential and stream function are both useful in fluid dynamics and are often used to describe the motion of fluids in different scenarios. In addition, they can be used to solve the Navier-Stokes equations, which describe the motion of fluids under different conditions.