Start-up laminar flow in a narrow slit. A Newtonian fluid of constant viscosity and constant density is initially at rest between two large horizontal plates a distance 2B apart. At time t=0 a constant pressure gradient Lp=Lp(z=z0+L)p(z=z0) 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. tvz=x22vzLp 2- Using no-slip boundary conditions, show that the steady-state velocity profile in above system is: vz=vmax[1(Bx)2] , where vmax=2LB2p 3- Show that the PDE derived in part 1 can be put into dimensionless form: =22+2 in which vmaxvz,=B2t, and Bx 4- Show that the steady-state velocity profile in dimensionless form is: =12 5- Define t by =t and solve the differential equation for t 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 for =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 is initially at rest between two large horizontal plates a distance 2B apart. At time t=0 a constant pressure gradient Lp=Lp(z=z0+L)p(z=z0) 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. tvz=x22vzLp 2- Using no-slip boundary conditions, show that the steady-state velocity profile in above system is: vz=vmax[1(Bx)2] , where vmax=2LB2p 3- Show that the PDE derived in part 1 can be put into dimensionless form: =22+2 in which vmaxvz,=B2t, and Bx 4- Show that the steady-state velocity profile in dimensionless form is: =12 5- Define t by =t and solve the differential equation for t 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 for =0.01,0.05,0.1,0.5,1.0,2.0, and 3.0