Differential Momentum Balances V

Supplemental InstructionIowa State University

Leader: Shawn Van BruggenCourse: ChE 356 – Fluid Mechanics

Instructor: Dr. Kurt HebertDate: 11-5-2017

Introduction

For the pictures below, the flow is steady state, incompressible, laminar, and Newtonian. For the picture on the left, the liquid is flowing in the theta direction between two concentric cylinders where the outer cylinder is stationary, and the inner cylinder is turning at ω radians per second. For the picture on the right, the liquid is flowing down a stationary plate that is oriented at the angle β. You may assume that the continuity equation shows fully developed flow in both cases.

For each situation determine the simplified Navier Stokes equation and the boundary conditions.

Problems

Review and Resources

One assumption that we made in the first problem is that the die must be larger than the gap between the die and the wire. Why did we make that assumption?

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A wire is pulled horizontally through a bath of the coating liquid and through a die that “wipes” the liquid and leaves coating of the desired thickness. The flow is incompressible, laminar, steady state, and Newtonian. Assume that the pressure drop across the length of the die is negligible. The wire is being pulled at a velocity uw. The radius of the wire, coating, and die are Rw, Rc, and Rd, respectively. You may assume that the flow is only in the z-direction (cylindrical coordinates) and that the die is very long relative to the gap (Rd - Rw). Your goal is to (1) find an expression for the thickness of the coating on the wire and (2) find an expression for the force required to pull the wire through the die.