A certain flow field is described by the velocity potential ϕ=A ln r+B r cosθ where A and

step 1 So we're told in, told, or given the following information. Certain flow field is described by t

A certain flow field is described by the velocity potential...

Key Concepts

Polar Coordinates in Potential Flow

Polar coordinates are frequently used in problems involving potential flow, particularly when the geometry of the problem suggests radial symmetry or the use of logarithmic and trigonometric functions. Expressing the velocity potential and stream function in terms of polar coordinates (r and ?) can simplify the mathematical formulation and solution of the governing equations, making it easier to analyze the behavior of the flow field.

Stagnation Points

Stagnation points are locations in the flow where the fluid velocity becomes zero. These points are found by setting the velocity components, which are derived from the derivatives of the potential or stream function, to zero. The identification of stagnation points is important for understanding behavior near boundaries or obstacles, as they often indicate regions of significant pressure changes and flow separation.

Stream Function

The stream function is a scalar function whose contours represent the streamlines of the flow, indicating the paths followed by fluid particles. In two-dimensional flow, the stream function is related to the velocity potential by the Cauchy-Riemann equations, ensuring that the flow is both incompressible and irrotational. It provides an alternative means to describe the flow field and is particularly useful for visualizing flow patterns.

Velocity Potential

The velocity potential is a scalar function whose gradient gives the velocity field of an irrotational flow. It is a useful tool in fluid dynamics because if the potential is harmonic (satisfies Laplace's equation), the resulting flow is automatically incompressible. This concept simplifies the analysis of fluid flows by reducing the vector field problem to a study of a single scalar function.

Transcript

00:01 So we're told in, told, or given the following information.

00:03 Certain flow field is described by the velocity potential that i've listed here.

00:08 A and b are given as positive constants.

00:10 We're asked to determine the corresponding stream function and locate any stagnation points in the flow field.

00:16 Let's begin.

00:18 So the relationship between steam function velocity best vectors will be vr and this will be negative.

00:48 Okay.

00:49 Okay, and the relationship between vector potential and velocity, so let's label this, this is the relation between stream function and velocity vectors in radial coordinates.

01:26 Okay, and then the relationship in between potential and velocity potential, sorry, and velocity vectors are given as, let's run, and how can you? so now let's go ahead and start substituting.

02:36 My first substitution will be, and this will be over, and vr will then equal a over r plus b.

03:04 And then we'll find the velocity vector in the tangential direction.

03:24 Okay, so this will be okay over this, and this will equal 1 over r times negative vr, times the sign of theta, and this will equal negative b r or b times the sign of theta.

04:20 Okay.

04:23 Now the stream function, let me go to the next thing.

04:35 We'll take our br, and we're going to integrate this as follows, a over r plus b times the cosine of theta equals 1 over r, times this over this.

05:04 Okay.

05:06 Now let's keep going.

05:21 Okay, then i'll get, i got somebody going along and i'm sitting outside, so i hope it's not too distracting.

05:49 There we go.

05:52 And then we'll integrate the tangential velocity vector, which colors for this.

06:11 And this will equal negative b...

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