Mie334 numerical methods project assignment derive the

Question

Numerical Methods Project Assignment

Introduction:

Early in the morning on a hazy day, you have probably noticed the deposition of dew on the surface of flowers and their leaves in the form of tiny drops. Depending on the surface characteristics of a leaf, the shape of the drops may be different. On a lotus leaf, specifically, a water drop looks almost spherical with minimum adhesion to the surface. This is due the hydrophobic nature of the lotus leaf which repels water drops. The term "hydrophobic" is made of two words, hydro + phobic, which means having little or no affinity for water. In fact a lotus leaf is considered a "super-hydrophobic" surface which is why if you slightly move the lotus leaf, the drops will fall. Usually surfaces or substances are hydrophobic due to the non-polar nature of their bonding which is opposite from the polar nature of bonding in water molecules. Super-hydrophobic materials usually are both non-polar in nature and also have micro-or nano-scale roughness, as shown in Figure.

Project Requirements: Paper Report Submissions:

1. Derive the relations for the coefficients of an m^th - polynomial.

2. Next, develop a function in MATLAB capable of fitting the m^th- polynomial to the data-points. In this procedure, your function should first calculate the coefficients a₀, a₁, . . . ,a_m for the polynomial as in Equation 2. Then the function should use the equation r = f(x) obtained for the polynomial to interpolate 1000 new-data points. This way, instead of only 22 data-points, you will have 1000 points to increase the accuracy of calculations. In your function, you are allowed to use the inv command to solve the system of equations you obtain for the coefficients.

3. Use each of your 2^nd, 4^th, 6^th, and 8^th order fits to estimate the contact angle. Usually the contact angle is measured within some distance from the surface and not exactly at the contact point of drop with the surface. As a result, use the data points between x = 0 mm and x = 0.1 mm, fit a straight line (1^st order polynomial) to these points, and estimate the contact angle based on the slope of the straight line. You may use your curve-fitting function developed in the previous part to find the slope of the line.

4. Finally, develop a program in MATLAB wherein the Trapezoidal rule is used to calculate the integral of Equation 3 for the new data-points. As a result, you will estimate the volume of the drop.

Attachment:- Project Assignment.rar