Give me a example.

When is the differential equation in this project guaranteed a unique solution?

UHU $7009 FLUIDS? when studying fluid mechanics, physicists often use three dimensional equations of vector fields to describe fluid f law. These equations often employ the use partial derivatives, and can be solved using differential equation methods. we were motivated to research fluid dynamics after learning the physics definition of Bernoulli's equation, which concerns itself with the velocity and Pressure of a .newtonian fluid. while the aforementioned equation has no direct relation to the set of Bernoulli's equations as defined in our diffErential equations class, we uBl'B nevertheless interested the ways that differential equations can be used to solve problems in fluid mechanics. we have used differential equations in physics classes in the past, so we felt confident in researching this sub- discipline within the physics field. * O , ' SOME BACKGROUND: the velocity of a fluid can be described by u, v, and H, which correspond to its moement in the x, y, and 2' directions respectively. GivEn a newtonian, incompressible {constant density} flaw, ihe equation for the conservation of mass of the fluid is: OH. + 01' + 0w 0.1' ill; r'J: :lJ OUR PROBLEM: n fluid is incompressible, such that is flows with constant density and prEssure. it has the folIOwins H' = 2: Jr J' m 5] velocity components as shown to the right. ASSume that 9 il u 9'2 + 4.:' whEn the fluid is at point (2,3,5) it is flowing with a velocity of J mfs in the y direction. Determine ir, the 3 component of velocity, which satisfies the conservation of mass. , \\' Hint: use the Fundamental. theorem of calculus and @- check if the differential equation is separable. . a: 5:2 OUR SOLUTION: 6v 6:: Given that our fluid is incompressible, us can use il-e (9y 0y conservation of mass equation as described above. we will plug in our known values and isolate the remaining partial derivative, such that the equation is separable. f 6:: = / ay After utilizing the Fundamental Theorem of tiaICulus and plugging in initial values, we can find the value of our constant and write our solution: v=6y+C (3) = 6(3)+C v = 6y +21 0:21