Question

If an incompressible fluid flows in a corner bounded by walls meeting at the origin at an angle of \(60^{\circ}\), the streamlines of the flow satisfy the equation \(2 x y d x+\left(x^{2}-y^{2}\right) d y=0\). Find the streamlines.

Short answer

The streamlines are given by \(|x^2 - y^2| = Ce^{-y}\).

Step-by-step solution

- Identify the Given Differential Equation

The given differential equation for the streamlines is \[2 x y \, dx + (x^2 - y^2) \, dy = 0\]

- Separate Variables

To solve this differential equation, we first rearrange the terms to separate the variables: Move all terms involving \(dx\) to one side and \(dy\) to the other side:\[\frac{2xy \, dx}{x^2 - y^2} = -dy\]Now, we separate the variables:\[\frac{2xy}{x^2 - y^2} \, dx = -dy\]

- Integrate Both Sides

Integrate both sides to find the implicit solution:The left-hand side becomes:\[\int \frac{2xy}{x^2 - y^2} \, dx \quad\text{and the right-hand side becomes:}\quad \int -dy\]Using substitution method, let \(u = x^2 - y^2\), then \(du = 2x \, dx\). Thus, the integral simplifies to:\[\int \frac{y \, du}{u} = -\int dy\]

- Evaluate the Integrals

Now we can evaluate the integrals:\[\ln |u| = -y + C \quad\text{or}\quad \ln |x^2 - y^2| = -y + C\]

- Solve for y in terms of x

To solve for \(y\) explicitly, we have:\[|x^2 - y^2| = e^{-y + C'} = Ce^{-y} \quad \text{where} \quad C' \text{is a new constant}\]\(\ln(|x^2 - y^2|) = C - y\)

- Obtain the General Solution

Rewriting the expression we have:\[|x^2 - y^2| = Ce^{-y}\]Which represents the streamlines of the flow.

Key concepts

Incompressible Fluid Flow

In fluid dynamics, an incompressible fluid is a fluid whose density remains constant regardless of changes in pressure. This assumption simplifies the analysis of the flow since the continuity equation, which ensures mass conservation, boils down to the divergence of the velocity field being zero: \[ abla \bullet \bar{v} = 0 \] This means that the volume of any fluid parcel remains constant as it moves through the flow field. Thus, incompressible fluid flow is often used in fluid dynamics problems involving liquids like water, where density changes are negligible.
Understanding the nature of incompressible fluids is crucial because it allows for the application of Bernoulli's equation and simplifies the equations of motion, making the problem more tractable.

Streamline Equations

Streamlines are paths followed by tiny fluid particles as they move through a fluid flow. These lines help visualize the flow pattern of the fluid. The differential equation governing the streamlines can usually be written in the form: \[ P(x, y) dx + Q(x, y) dy = 0 \] In our problem, the streamline equation is given as: \[ 2xy \, dx + (x^2 - y^2) \, dy = 0 \] Streamline equations are derived from the components of the velocity field. Solving these equations allows us to determine the shape of the paths that fluid particles follow, which are critical in understanding the flow behavior, especially around corners or obstacles.

Separation of Variables

This technique is a powerful method for solving differential equations. It involves rearranging the equation so each variable appears on a different side of the equation, making it easier to integrate. For our streamline equation, this method looks like: \[ \frac{2xy}{x^2 - y^2} \, dx = -dy \] By separating the variables in this manner, we've taken the first crucial step in solving the streamline equation. Breaking it down in simpler, separate, integrable parts paves the way for finding solutions that describe fluid flow patterns.

Integration Techniques

Integrating each side of a separated equation is necessary to find the solution. Different integration techniques apply depending on the complexity of the integral. In our problem, we integrate as follows: \[ \frac{2xy}{x^2 - y^2} \, dx \rightarrow \text{substitution method} \rightarrow \frac{y}{u} \, du \] After separating variables, we integrate: \[ \text{Left-hand side:} \ \text{Integrate} \frac{2xy}{x^2 - y^2} \, dx \ \text{by substitution} \rightarrow \frac{y \, du}{u} \] \[ \text{Right-hand side:} \text{Integrate} -dy \] These integrations provide us with the necessary functions to form the complete solution describing the flow pattern.

Substitution Method

The substitution method simplifies complex integrals by changing variables. This is particularly useful in our streamline equation. Let's substitute as follows: \[ u = x^2 - y^2, \, du = 2x \, dx \] This transforms our integral into a more straightforward form: \[ \frac{y \, du}{u} \] By making the substitution, the original differential equation becomes manageable. This method is a key part of many solutions in differential equations, allowing us to transform the integral into a recognizable and solvable form.
Subsequently, integrating typically gives the natural logarithm function: \[ \text{ln} |u| = -y + C \rightarrow \text{ln} |x^2 - y^2| = -y + C \] Finally, the solution relates to the streamline equation, describing the flow visually and mathematically.