Question

Bestimmen Sie eine divergenzfreie Lösung \(\mathbf{u}(x, y)=(u(x, y), v(x, y))\), d.h. \(\operatorname{div} \mathbf{u}=0\), des Differentialgleichungssystems $$ \begin{aligned} u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y} &=-\frac{\partial p}{\partial x}+\frac{1}{R e} \Delta u \\ u \frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y} &=-\frac{\partial p}{\partial y}+\frac{1}{R e} \Delta v \end{aligned} $$ in einem Rechteckgebiet \(\Omega=\\{(x, y) \mid 0

Short answer

A possible solution is a parabolic velocity profile \( u(y) = \alpha(y)(H-y) \).

Step-by-step solution

Analyze the Problem

We are given a system of differential equations describing a 2D flow in a rectangular domain \( \Omega \) with boundary conditions on \( u \) and \( v \). The flow must be divergence-free, indicating incompressible flow.

Recognize Boundary and Divergence Conditions

The boundary condition specifies \( u = v = 0 \) on the upper and lower sides of the rectangle. Additionally, since the flow is divergence-free, we recognize the condition \( \operatorname{div} \mathbf{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \), meaning the net flow in any infinitesimally small volume of the fluid remains constant.

Pressure Function Simplification

The pressure \( p(x, y) \) is given as \( x(p_1 - p_0) \). Since it varies linearly in the \( x \)-direction and is constant in the \( y \)-direction, the pressure gradient \( -\frac{\partial p}{\partial x} \) will be \( -(p_1 - p_0) \) and \( -\frac{\partial p}{\partial y} = 0 \).

Solution Ansatz Considering a Parallel Flow

Consider a parallel flow \( u(x, y) = u(y) \) and \( v(x, y) = 0 \). This satisfies \( \operatorname{div} \mathbf{u} = 0 \) since \( \frac{\partial u}{\partial x} = 0 \) and \( \frac{\partial v}{\partial y} = 0 \).

Setup the Simplified Equations

Substitute \( u(y) \) into the equations: Given \( v = 0 \), the equations simplify to \( u \frac{du}{dy} = - \frac{\partial p}{\partial x} + \frac{1}{Re} \Delta u \).

Solve for u(y)

With the given form of \( \frac{\partial p}{\partial x} = (p_1 - p_0) \) and \( \Delta u = \frac{d^2u}{dy^2} \), we have the ordinary differential equation \( u \frac{du}{dy} = -(p_1 - p_0) + \frac{1}{Re} \frac{d^2u}{dy^2} \). Given the boundary conditions: \( u(0) = u(H) = 0 \), this equation can be solved for \( u(y) \) using techniques such as separation of variables or numerical solutions.

Boundary Solution and Pressure

Ensure \( u(y) \) satisfies \( u(0) = 0 \) and \( u(H) = 0 \). The pressure is consistent with given conditions as it does not depend on \( y \). One possible solution for \( u(y) \) is a parabolic profile, \( u(y) = \alpha (y)(H-y) \) with an appropriate constant \( \alpha \).

Verification and Conclusion

Verify that the derived solution \( \mathbf{u}(x, y) \) satisfies all the original equations and boundary conditions. If \( \operatorname{div} \mathbf{u} = 0 \) and the boundary conditions are met, the solution is valid.

Key concepts

Divergence-free

When dealing with fluid dynamics, a key concept is being divergence-free. This term is often used to describe incompressible flow situations. In mathematical terms, a divergence-free condition for a vector field \( \mathbf{u} = (u, v) \) can be expressed as \( \operatorname{div} \mathbf{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \). This implies that the amount of fluid entering any region is equal to the amount leaving it. It ensures the conservation of mass, a vital principle for understanding flows like water or air, where density changes are negligible.
This condition was essential in the given exercise. A flow that meets this condition keeps the volume flow constant in any part of the fluid medium. For students, it's important to remember that divergence-free implies a balance, much like ensuring all people entering a room also find their way out, maintaining a constant number of people in the room over time.

• Shows incompressible flow.
• Mass conservation remains constant.
• Critical for understanding stable, uniform flows.

Boundary Conditions

Boundary conditions are constraints necessary for solving differential equations that describe physical phenomena, like fluid flow. They provide the necessary information about how the fluid behaves at the boundary of the domain considered, like the edges of a pipe or a channel.
In this case, the boundary conditions are given as \( u = 0 \) and \( v = 0 \) on the horizontal edges \( y = 0 \) and \( y = H \). These are referred to as no-slip conditions, commonly applied in fluid dynamics. They imply that the fluid at these boundaries does not move relative to the surface.
Boundary conditions help define how a fluid interacts with its environment. For students, remembering that these conditions typically aim to match real-world constraints is crucial. Knowing these can help confidently approach solving complex flows as they reduce the potential solutions to only those meeting realistic constraints.

• Describes fluid behavior at the edges.
• No-slip conditions mean fluid does not move along boundaries.
• Crucial for obtaining realistic solutions.

Pressure Gradient

The pressure gradient in fluid dynamics is an important concept that describes how pressure changes across a distance within the fluid. It affects how the fluid flows from areas of high pressure to low pressure and is a fundamental driving force behind fluid movement.
In the exercise, the pressure \( p(x, y) \) is linear in the \( x \)-direction, which means \( \frac{\partial p}{\partial x} = (p_1 - p_0) \) and \( \frac{\partial p}{\partial y} = 0 \). This indicates that pressure pushes the fluid horizontally, with no vertical pressure variation.
Understanding the pressure gradient's role can help students grasp why fluid moves a certain way. The concept is akin to a hill: fluids naturally "fall" or flow in the direction of the slope, from high to low pressure.

• Indicates fluid flow direction.
• Fluid moves high to low pressure.
• Key factor driving flow in channels.

Navier-Stokes Equations

The Navier-Stokes Equations are a set of partial differential equations describing the motion of fluid substances. They form the basis for many calculations in fluid dynamics.
In our problem, they are used to model the steady flow of a viscous, incompressible fluid. The main equations are:
\[ u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} = -\frac{\partial p}{\partial x} + \frac{1}{Re} \Delta u \]
\[ u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} = -\frac{\partial p}{\partial y} + \frac{1}{Re} \Delta v \]
These equations capture the complex balance between various forces acting on a fluid element: inertia forces, pressure forces, and viscous forces. Solving these equations typically requires simplifying assumptions or numerical techniques.
For students, understanding Navier-Stokes is crucial, as it helps decode how fluid behaves under different forces. It's like having a detailed guide on how fluid elements push and pull on each other, ensuring the dynamics are consistent with observed flows.

• Foundation of fluid dynamic analysis.
• Handles complex force interactions.
• Requires simplification for practical use.