Start-up laminar flow in a narrow slit. A Newtonian fluid of constant viscosity and constant density...

 

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Start-up laminar flow in a narrow slit. A Newtonian fluid of constant viscosity and constant density p is initially at rest between two large horizontal plates a distance 2B apart. x Z Ap p(z=z+AL)-p(z=zo); AL AL At time t = 0 a constant pressure gradient is imposed on the system, which sets the fluid in motion. Here, we study how the velocity profile develops with time. 1- Using the relevant equation of motion, show that the velocity profile is governed by the following PDE. ,where Vmax 2 2 L in which = 2- Using no-slip boundary conditions, show that the steady-state velocity profile in above system is: Vzoo = Vmax [-6] 1- Vz Vmax = T= 3- Show that the PDE derived in part 1 can be put into dimensionless form: +2 2 ut PB2 P and = at = ax L B 4- Show that the steady-state velocity profile in dimensionless form is: 2 = 1 -? 5- Define pt by = po - t and solve the differential equation for pt by the method of separation of variables and determine all the constants using the relevant boundary conditions and initial condition. 6- Plot as a function of n for t = 0.01, 0.05, 0.1, 0.5, 1.0, 2.0, and 3.0. Start-up laminar flow in a narrow slit. A Newtonian fluid of constant viscosity and constant density p is initially at rest between two large horizontal plates a distance 2B apart. x Z Ap p(z=z+AL)-p(z=zo); AL AL At time t = 0 a constant pressure gradient is imposed on the system, which sets the fluid in motion. Here, we study how the velocity profile develops with time. 1- Using the relevant equation of motion, show that the velocity profile is governed by the following PDE. ,where Vmax 2 2 L in which = 2- Using no-slip boundary conditions, show that the steady-state velocity profile in above system is: Vzoo = Vmax [-6] 1- Vz Vmax = T= 3- Show that the PDE derived in part 1 can be put into dimensionless form: +2 2 ut PB2 P and = at = ax L B 4- Show that the steady-state velocity profile in dimensionless form is: 2 = 1 -? 5- Define pt by = po - t and solve the differential equation for pt by the method of separation of variables and determine all the constants using the relevant boundary conditions and initial condition. 6- Plot as a function of n for t = 0.01, 0.05, 0.1, 0.5, 1.0, 2.0, and 3.0.