Answer :
Explanation
To find the strain tensor at the point (2, 1, -1) for the given velocity field \( \mathbf{V} = (0.2x^2 + 2y + 2.5)\mathbf{i} + (0.5x + 2y^2 - 6)\mathbf{j} + (0.15x^2 + 3y^2 + z)\mathbf{k} \), we need to calculate the components of the strain tensor \( \epsilon_{ij} \).
The strain tensor \( \epsilon_{ij} \) is related to the velocity components \( V_i = (V_x, V_y, V_z) \) as follows:
\[ \epsilon_{ij} = \frac{1}{2} \left( \frac{\partial V_i}{\partial x_j} + \frac{\partial V_j}{\partial x_i} \right) \]
Where \( \frac{\partial V_i}{\partial x_j} \) is the partial derivative of \( V_i \) with respect to \( x_j \).
### Steps to Find the Strain Tensor:
1. **Identify Velocity Components:**
Given:
\[ V_x = 0.2x^2 + 2y + 2.5 \]
\[ V_y = 0.5x + 2y^2 - 6 \]
\[ V_z = 0.15x^2 + 3y^2 + z \]
2. **Calculate Partial Derivatives:**
Calculate \( \frac{\partial V_x}{\partial x}, \frac{\partial V_y}{\partial y}, \frac{\partial V_z}{\partial z} \), and other relevant partial derivatives.
3. **Compute the Components of Strain Tensor:**
The strain tensor components \( \epsilon_{ij} \) are calculated using the formula:
\[ \epsilon_{ij} = \frac{1}{2} \left( \frac{\partial V_i}{\partial x_j} + \frac{\partial V_j}{\partial x_i} \right) \]
Substitute the derivatives into this formula to find each component of the strain tensor at the point (2, 1, -1).
### Conclusion:
To determine the exact values for the strain tensor components at (2, 1, -1), you would need to compute the partial derivatives of the velocity components with respect to \( x, y, \) and \( z \), and then evaluate them at the given point (2, 1, -1). This process involves detailed calculus calculations which are essential for accurately determining the strain tensor components \( \epsilon_{ij} \).