Question

Zeigen Sie, dass für ein 3-mal stetig differenzierbares divergenzfreies Geschwindigkeitsfeld \(\mathbf{u}=(u, v)\) aus den instationären STOKES- Gleichungen $$ \frac{\partial \mathbf{u}}{\partial t}=-\operatorname{grad} p+\frac{1}{R e} \Delta \mathbf{u} $$ für den Druck die Gleichung \(\Delta p=0\) folgt.

Short answer

The pressure \(p\) satisfies the Laplace equation \(\Delta p = 0\).

Step-by-step solution

Understand the Problem

The problem provides a velocity field \(\mathbf{u} = (u, v)\) that is 3-times continuously differentiable and divergence-free (which means \(abla \cdot \mathbf{u} = 0\)). We need to show that \(\Delta p = 0\) follows from the given time-dependent Stokes equations.

Use the Divergence-Free Condition

Since the velocity field is divergence-free, \(abla \cdot \mathbf{u} = 0\). This implies that the time derivative of the divergence is also zero: \(abla \cdot \frac{\partial \mathbf{u}}{\partial t} = 0\).

Differentiate the Stokes Equation

The Stokes equation is \(\frac{\partial \mathbf{u}}{\partial t} = -\operatorname{grad} p + \frac{1}{Re} \Delta \mathbf{u}\). By taking the divergence of both sides, we have \(abla \cdot \frac{\partial \mathbf{u}}{\partial t} = abla \cdot ( -\operatorname{grad} p + \frac{1}{Re} \Delta \mathbf{u})\).

Simplify Using Known Identities

Use the identity \(abla \cdot (\operatorname{grad} p) = \Delta p\) and the fact that the divergence and Laplacian are linear operators. The expression becomes \(0 = -\Delta p + \frac{1}{Re} abla \cdot \Delta \mathbf{u}\), knowing that \(abla \cdot \Delta \mathbf{u} = \Delta (abla \cdot \mathbf{u}) = 0\) because \(abla \cdot \mathbf{u} = 0\).

Solve for \(\Delta p\)

From the result of the previous step, we have \(0 = -\Delta p\). Thus, \(\Delta p = 0\), showing that the pressure \(p\) satisfies the Laplace equation.

Key concepts

Divergence-Free Condition

The divergence-free condition is an important concept in fluid dynamics, particularly in the study of incompressible flows. When we say a velocity field is divergence-free, it means the field satisfies the equation \( abla \cdot \mathbf{u} = 0 \). This states that there is no net flow of the fluid out of any infinitesimally small volume within the fluid. Essentially, the fluid neither accumulates nor depletes at any point, which is characteristic of incompressible fluids.
In practical terms:

• A fluid parcel's volume remains constant as it moves through the flow.
• There are no sources or sinks within the flow field.

In our problem, the divergence-free condition simplifies the equations, allowing us to focus on the role of pressure and how it relates to the velocity field.

Laplace Equation

The Laplace equation, \( \Delta p = 0 \), is a fundamental equation in mathematical physics. It describes the behavior of scalar fields such as temperature, electric potential, or in our case, pressure \( p \). When applied to fluid dynamics, this equation signifies that the pressure field is harmonic, meaning it does not experience any local maxima or minima within the domain.
In the context of the exercise, deriving \( \Delta p = 0 \) from the Stokes equations suggests the pressure distribution is in a state of equilibrium under the given conditions. This relationship is crucial as it couples the velocity and pressure fields, ensuring the system's consistency when the fluid density remains constant.

Velocity Field

A velocity field, represented often as \( \mathbf{u} = (u, v) \), details the speed and direction of fluid particles at each point in space. For the Stokes equations, this field is not only three times continuously differentiable but also divergence-free.
Key attributes of such a velocity field:

• Continuity ensures smooth changes in velocity with no sudden jumps.
• Divergence-free nature ensures the fluid is incompressible, maintaining constant density.

Understanding the behavior of the velocity field is vital for modeling fluid movement precisely. Through the interplay of the velocity and pressure fields in the Stokes equations, we can predict how the fluid will behave under various conditions.

Pressure Field

The pressure field \( p \) in a fluid is a scalar field that influences how the fluid moves and interacts with its environment. Pressure helps drive the flow, countering forces such as viscosity as described by the Stokes equations.
In incompressible flow conditions:

• Pressure gradients (\( abla p \)) cause fluid acceleration or deceleration.
• Harmonic nature of pressure (\( \Delta p = 0 \)) ensures balanced fluid motion, avoiding any spontaneous volatility points.

The pressure field's harmony aligns with the smooth and regulated changes in the velocity field, indicating a balance between external forces and inherent fluid properties. Understanding \( \Delta p = 0 \) in this context highlights the importance of pressure in maintaining the consistency of the fluid system under the conditions set by the Stokes equations.