Can you help me with question 1 and 4 only
Problem 1 (3 Points) The problem of 2-D steady viscous incompressible flow past a circular cylinder of radius a involves finding the velocity field u = hu1(x, y), u2(x, y), 0i which satisfies u · ?u = - 1 ? ?p + ??2u together with the boundary conditions u = 0 on x 2 + y 2 = a 2 , u ? hU, 0, 0i as x 2 + y 2 ? 8 where ? = µ/? is called the kinematic viscosity. Rewrite this problem in dimensionless form by using the dimensionless variables r 0 = r/a, u 0 = u/U, p 0 = p/(?U2 ), where r 0 = hx 0 , y0 i and r = hx, yi. Without attempting to solve the problem, show that the streamline pattern can depend on ?, a, and U only through the combination Re = U a/?, where Re is called the Reynolds number. In other words, flows at equal Reynolds numbers are geometrically similar.
Problem 4 (5 Points) Consider the steady flow of an incompressible viscous fluid down the inclined plane shown below, along with coordinate axes aligned with the plane. 1. Assume that u = hu1(y), u2(y), 0i. Show that the incompressibility condition implies u2 = 0. 2. Compute u1(y) given that the fluid is subject to the gravity force (with gravity g pointing down in the vertical direction) and pressure and viscous forces, with p = p0 at the air-fluid interface and µ the dynamic viscosity. For the boundary condition on u1 at y = h, observe that the shear stress is zero at the air-fluid interface. Sketch the velocity profile for a fixed value of x. Remember to justify your answers!
Document Preview: Introduction to Fluid Dynamics { Problem Set 5
March 9, 2017
Due March 23, 2017 after class
Problem 1 (3 Points)
The problem of 2-D steady viscous incompressible
ow past a circular cylinder of radius a involves nding
the velocity eld u =hu (x;y);u (x;y); 0i which satis es
1 2
1
2
uru = rp +r u
together with the boundary conditions
XXXXXXXXXX
u =0 on x +y =a ; u!hU; 0; 0i as x +y !1
where == is called the kinematic viscosity.
0 0
Rewrite this problem in dimensionless form by using the dimensionless variables r = r=a, u = u=U,
XXXXXXXXXX
p = p=(U ), where r =hx;yi and r =hx;yi. Without attempting to solve the problem, show that the
streamline pattern can depend on ,a, andU only through the combination Re =Ua=, where Re is called
the Reynolds number. In other words,
ows at equal Reynolds numbers are geometrically similar.
Problem 2 (7 Points)
Consider the steady
owu(x;y) =hu (y); 0i of a viscous, incompressible
uid with dynamic viscosity and
1
with pressure p(x) between two stationary rigid boundaries at y =h, where x is the coordinate for the
horizontal direction, and y the coordinate for the vertical direction. The
ow is in nite in the z-direction,
and gravity is negligible.
1. Show that dp=dx is a constant. Let this constant quantity be dp=dx = K.
2. Given that the boundary conditions on u are u (h) = 0, compute the velocity pro le u(y) in terms
1 1
of K, h and .
3. Repeat the problem for a
ow down a pipe with circular cross-section r = a. Using cylindrical
coordinates (r;;z), the
ow is u =hu ;u ;ui =h0; 0;u (r)i and p = p(z). Compute the velocity
r z z
pro le u (r).
z
4. For each case (
at boundaries and circular pipe), compute the volumetric
ow rate per unit area
RR RR
Q = undA= dA, where A is the cross section perpendicular to the
ow. Set h = a and
A A
compare the answers you obtained in each case. Why, intuitively, is one greater than the other?
Note: The
ow pro les you have computed are called Poisseuille
ows, and have played...