Problem 6 - Biofluid Mechanics: An Introduction to Fluid Mechanics, Macrocirculation, and Microcirculation

Question

A velocity field is given by \(v(x, y, z)=20 x y \vec{i}-10 y^{2} \vec{j}\). Calculate the acceleration, the angular velocity, and the vorticity vector at the point \((-1,1,1)\), where the units of the velocity equation are millimeters per second.

Step-by-Step Solution

Answer

The acceleration is \(-200 \vec{i} + 200 \vec{j}\), angular velocity is \(-20 \vec{k}\), and vorticity is \(-40 \vec{k}\).

Step 1: Express the Velocity Components

Identify the components of the velocity field. The velocity vector is given by \(\vec{v}(x, y, z) = 20xy \vec{i} - 10y^2 \vec{j}\). Thus, the components are \(v_x = 20xy\), \(v_y = -10y^2\), and \(v_z = 0\).

Step 2: Define Acceleration Components

The acceleration \(\vec{a}\) is given by the material derivative of the velocity: \(\vec{a} = \frac{D\vec{v}}{Dt} = \left( \frac{\partial v_x}{\partial t} + v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y} \right) \vec{i}\) for the \(i\)-component, and similarly for \(j\) and \(k\) components.

Step 3: Calculate Acceleration Components

Evaluate the partial derivatives and plug the point \((-1,1,1)\) into them:- \(\frac{\partial v_x}{\partial x} = 20y\) and \(\frac{\partial v_x}{\partial y} = 20x\).- \(\frac{\partial v_y}{\partial y} = -20y\).Now calculate:- For \(a_x: 0 + 20(-1)(1)(20(1)) - 10(1)^2(20(-1)) = -400 + 200 = -200\).- For \(a_y: 0 + 20(-1)(1)(0) + (-10)(1)^2(-20) = 200\).Thus, \( \vec{a} = -200 \vec{i} + 200 \vec{j}\).

Step 4: Define and Calculate the Vorticity

The vorticity vector \(\vec{\omega}\) is \(abla \times \vec{v}\), where:\[abla \times \vec{v} = \left( \frac{\partial (v_z)}{\partial y} - \frac{\partial (v_y)}{\partial z} \right) \vec{i} + \left( \frac{\partial (v_x)}{\partial z} - \frac{\partial (v_z)}{\partial x} \right) \vec{j} + \left( \frac{\partial (v_y)}{\partial x} - \frac{\partial (v_x)}{\partial y} \right) \vec{k}\].Compute each component:- \(\omega_x: 0 - 0 = 0\).- \(\omega_y: 0 - 0 = 0\).- \(\omega_z: (-20) - 20 = -40\).Thus, \(\vec{\omega} = 0 \vec{i} + 0 \vec{j} - 40 \vec{k}\).

Step 5: Calculate the Angular Velocity

The angular velocity \(\vec{\Omega}\) is defined as \(\frac{1}{2} \vec{\omega}\). Using the vorticity calculated: \[\vec{\Omega} = \frac{1}{2} (0 \vec{i} + 0 \vec{j} - 40 \vec{k}) = 0 \vec{i} + 0 \vec{j} - 20 \vec{k}\].

Key Concepts

Velocity Field

A velocity field is a representation of how the velocity of a fluid or any entity varies across different points in space. It is essentially a vector field where each vector has a magnitude and direction indicating the speed and flow direction of the fluid.
For the given problem, the velocity field is defined as \( \vec{v}(x, y, z) = 20xy \vec{i} - 10y^2 \vec{j} \). This means that:

• \( v_x = 20xy \) represents the velocity in the \( x \)-direction.
• \( v_y = -10y^2 \) represents the velocity in the \( y \)-direction.
• \( v_z = 0 \) indicates there is no velocity component in the \( z \)-direction.

By plugging in different values for \( x \), \( y \), and \( z \), we can determine the velocity at any particular point in this field.

Acceleration Calculation

In fluid dynamics, acceleration is a measure of the rate of change of velocity with respect to time. It can be determined using the material derivative, which considers both temporal and spatial variations.
The material derivative for acceleration \( \vec{a} \) is given by:

• \( \vec{a} = \frac{D\vec{v}}{Dt} = \left( \frac{\partial v_x}{\partial t} + v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y} \right) \vec{i} \)
• The same process applies to calculating the \( j \) and \( k \) components.

When calculating the acceleration at point \((-1, 1, 1)\):

• For the \( x \)-component, use: \( a_x = -200 \).
• For the \( y \)-component, use: \( a_y = 200 \).
• The \( z \)-component is zero as \( v_z \) is constant.

Thus, the acceleration vector becomes \( \vec{a} = -200 \vec{i} + 200 \vec{j} \).

Vorticity

Vorticity is a rotational measure in a velocity field. It quantifies the local spinning motion of the fluid, indicating how much and in which way a fluid element is rotating.
The vorticity vector \( \vec{\omega} \) is calculated as the curl of the velocity field \( abla \times \vec{v} \):

• The \( x \)-component (\( \omega_x \)) is \( 0 \).
• The \( y \)-component (\( \omega_y \)) is also \( 0 \).
• The \( z \)-component (\( \omega_z \)) is \( -40 \), found using the specific derivatives \( \frac{\partial (v_y)}{\partial x} - \frac{\partial (v_x)}{\partial y} \).

This results in the vorticity vector \( \vec{\omega} = 0 \vec{i} + 0 \vec{j} - 40 \vec{k} \). It shows no rotation along the \( x \) and \( y \) axes, but indicates rotation along the \( z \) axis.

Angular Velocity

Angular velocity provides a measure of the rate of rotation of an object or fluid around an axis. It is closely linked to vorticity but represents how fast and in what direction the fluid or object spins.
Angular velocity \( \vec{\Omega} \) is derived by halving the vorticity vector:

• \( \vec{\Omega} = \frac{1}{2} \vec{\omega} = \frac{1}{2} (0 \vec{i} + 0 \vec{j} - 40 \vec{k}) \)
• This simplifies to \( \vec{\Omega} = 0 \vec{i} + 0 \vec{j} - 20 \vec{k} \).

The result signifies that there is a steady rotation around the \( z \)-axis, with no angular velocity contribution from the \( x \) and \( y \) axes. Hence, the system or fluid element rotates counterclockwise when viewed from above the \( z \)-axis.