Consider a flow field represented by the stream function ψ=-A / 2(x^2+y^2), where A=constant. Is

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Consider a flow field represented by the stream function...

Consider a flow field represented by the stream function $\psi=-\mathrm{A} / 2\left(x^{2}+y^{2}\right),$ where $A=\mathrm{constant.}$ Is this a possible twodimensional incompressible flow? Is the flow irrotational?

Key Concepts

Irrotational Flow

An irrotational flow is one in which the vorticity is zero throughout the flow domain. In the context of potential flows, this also implies that the velocity field can be expressed as the gradient of a scalar potential function. If the computed vorticity from a given stream function or velocity field is nonzero, the flow is not irrotational.

Vorticity

Vorticity is a measure of the local rotation of the fluid and is defined as the curl of the velocity field. In a two-dimensional flow, the vorticity is given by ? = ?v/?x - ?u/?y. This scalar quantity indicates the tendency of fluid elements to rotate, and a nonzero vorticity implies rotational motion in the flow.

Incompressible Flow

An incompressible flow is one in which the fluid density remains constant, leading to a divergence-free velocity field. In two dimensions, this condition is automatically met when the velocity components are expressed in terms of a stream function, ensuring that the sum of the partial derivatives of the velocity components with respect to their corresponding coordinates is zero.

Stream Function

The stream function is a scalar function used in two-dimensional flow fields to automatically satisfy the incompressibility condition. For a flow described by a stream function ?, the velocity components (u, v) are defined as u = ??/?y and v = -??/?x. This guarantees that the continuity equation for incompressible flow (zero divergence) is fulfilled.

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