The expression you provided represents a double integral in polar coordinates, $V_z$, which looks like it's related to fluid mechanics or a similar field, possibly calculating a velocity or volume through an area. Here's the breakdown of the given integral:

$V_z = \frac{\int_0^{2\pi} \int_{kR}^{R} \frac{(P_0 - P_L)R^2}{4 \mu L} \left[ 1 - \left(\frac{r}{R}\right)^2 - \frac{1 - k^2}{2 \ln (1/k)} \left(\frac{R}{r} - \frac{r}{R}\right)\right] r \, dr \, d\theta}{\int_0^{2\pi} \int_{kR}^{R} r \, dr \, d\theta}$

This is a form of the integral in cylindrical coordinates, with $r$ and $\theta$ as the radial and angular coordinates, respectively.

General Approach:

To solve this, you will want to:

1. Evaluate the angular part of the integrals (which should be straightforward since it ranges over $0$ to $2\pi$).
2. Evaluate the radial part by integrating the provided function over $r$ from $kR$ to $R$.
3. Once both integrals (numerator and denominator) are computed, the ratio gives the final expression for $V_z$.

Step 1: Angular Integral

The $\theta$-dependent parts are simple integrals from $0$ to $2\pi$. Since there are no terms dependent on $\theta$, the integration just multiplies the remaining expression by $2\pi$.

Step 2: Radial Integral

The radial part involves integrating a function of $r$. The terms inside the integral depend on $r$, $R$, and $k$.

1. First Integral (Numerator): You will integrate: $\frac{(P_0 - P_L)R^2}{4 \mu L} \left[ 1 - \left(\frac{r}{R}\right)^2 - \frac{1 - k^2}{2 \ln (1/k)} \left(\frac{R}{r} - \frac{r}{R}\right)\right] r$ over $r$ from $kR$ to $R$.

2. Second Integral (Denominator): The denominator is simpler, as you're integrating $r$ over the same limits, without the more complex term. Specifically: $\int_{kR}^{R} r \, dr$

Tip:

• The logarithmic term $\ln(1/k)$ and the factor $(P_0 - P_L)$ suggest that the final result will likely involve simplifying expressions with logs and fractions. These terms arise frequently in problems involving fluid flow through pipes (Hagen-Poiseuille type problems) or heat transfer.

Would you like me to proceed with the full step-by-step evaluation of the radial integrals? I can also focus on any specific part you may need more help with.