# hw_5 - ME152A Homework #5 ...

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Unformatted text preview: ME152A Homework #5  Due FRIDAY, NOVEMBER 14th, 2008    Question #1:  In this problem, assume gravity acts downwards in the y‐direction and x is the  horizontal direction. List all other assumptions and SHOW ALL WORK.   You are on the planet Zoezeb, similar to Zumeratron from the midterm exam, but  oceans are filled with zoe‐water and the atmosphere is zoe‐air. The surface of the  planet (h = 0) is known as zoe‐level (as opposed to sea‐level) Properties of these  fluids are below:  zoe‐water:        zoe‐air:  ρ = 100 kg/m3   ν = 0.01 m2/s  p0 = 100003 N/m2    ρ = p/N0   where: p = zoe‐ atmospheric pressure (N/m2)      p0 = zoe‐atmospheric pressure at zoe‐level  N0 = 12.5 (N.m/kg) (constant)    The acceleration of gravity on this planet is the same as earth (and acts in the same  direction, g0 = ‐ 9.8 m/s2. However additionally there is a linear electric field on the  planet that acts IN THE SAME DIRECTION AS GRAVITY:    E = ‐Ky+Eo      where: K = 0.23 V/m2              y = height above zoe‐level              Eo = electric field at zoe‐level = ‐1 V/m  Note that the body force per unit volume that attributed to an electric field is ρeE,  where ρe is the volumetric charge density, C/m3. In zoe‐water, ρe, is 10 C/m3, and in  zoe‐air, ρe is 0.05 C/m3.   a) Develop an expression for the pressure variation as a function of y, p(y), in zoe‐ air. Express the answer in terms of N0, g0, K, ν, ρe, E0 and p0. Do not substitute  numerical values for constants.  b) Develop an expression for the hydrostatic pressure variation in zoe‐water as a  function of y, p(y) in terms of N0, g0, K, ρ, ρe, E0, ν and p0.  What is the pressure at h =  5 km below zoe‐level?  c) Solve for the pressure distribution, p(y) in terms of N0, g0, K, ρe, E0, ρ, ν and p0, in  zoe‐water if the velocity field is                       d) A tank filled with zoe‐water at zoe‐level is subject to constant linear acceleration  ax = 5 m/s in zoe‐air. Calculate the height of zoe‐water in the tank as a function of x,  when the 15 m radius tank filled with zoe‐water of H = 2 m, is subject to the  constant acceleration ax. Assume that the electric field does not affect the velocity  distribution in the tank.      Question #2:  Derive the Navier‐Stokes equations using the differential control volume approach  that we used in class to derive linear motion, deformation, angular deformation, and  conservation of mass. Assume the flow is incompressible and that the forces that are  acting on the element are pressure, shear stress, and gravity.    Question #3:  A viscous, incompressible fluid flows in the horizontal direction in a circular tube  radius R due to a pressure gradient, dp/dx. Determine, by the use of the Navier‐ Stokes equations, an expression from the velocity distribution in the x‐direction, u,  as a function of r, u(r). (HINT: Use cylindrical coordinates).  Next, determine the  average velocity and the corresponding average flow rate. Express you answer in  terms of dp/dx, the coefficient of dynamic viscosity µ, and the channel radius, R.    Question 4:  Parallel flow in a two‐dimensional microfluidic channel shown below is induced by  pulling two thin plates through a liquid shown. The plates are each pulled with a  constant force F and eventually attain a velocity V0. The liquid has a dynamic  viscosity µ and the area of plates is A. Apply the Navier‐Stokes equations to find a  relation between the plate velocity V and the channel flow rate Q attained for given  values of a, b, µ, F, and A.        Question 5:  A long, thin, cylindrical tube connects two large reservoirs as shown below. The  walls of the tube are negatively charged and the liquid in the tube has a uniform net  positive charge density , ρe, (C/m3). A potential difference V is applied along the  length L of the tube using two electrodes.      a.  Express the Force per unit volume (body force) in terms of ρe, L and V  b.  Assuming zero pressure gradient in the channel, find the velocity profile in  the tube as a function of the parameters shown (assume the no‐slip  conditions holds)  c. After some time, the electric‐field‐induced flow generates a pressure  difference along the channel as the liquid levels in the two tanks change,  thereby causing backflow due to pressure. Find the maximum pressure that  can exist. (Hint: How much backflow can there possibly be? This would be  the point of maximum pressure).    ...
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