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Consider a slow and steady laminar flow of liquid through a capillary or a narrow cylindrical tube of uniform cross-section; let a be the radius and l be the length of the tube. The flow is maintained by applying a constant pressure difference ΔP between the ends of the tube. The layer of the liquid at the wall of the tube is at rest. Velocity of flow increases from the wall towards interior, being maximum at the axis. During laminar flow co-axial cylindrical layers of liquid flow past each other. Consider the volume of liquid bounded by the cylindrical surface of radius r within the tube. The viscous force exerted on this volume of liquid by the surrounding liquid is given by where v (r) velocity of flow at radial distance r from the axis; dv/dr represents corresponding velocity gradient. During steady flow viscous force is balance by force applied by external pressure difference: Fext = (π r2) ΔP Hence, we get Integrating, we get Constant c is fixed by boundary condition : at r = a, v = 0. Hence, The velocity varies as square of the distance from the wall, to a maximum value at the axis (r = 0). Hence, it is said that liquid flowing in a tube possesses a parabolic velocity profile. The volume of liquid flowing per second through the cylindrical shell of radius r and r + dr is given as dV = (2 π r dr) v (r) where 2 π r dr is area of cross-section of the shell. Therefore the total volume of liquid flowing through the tube, per second is The above relation is known as Poiseuille’s formula for laminar flow of liquid through a narrow tube. Services: - Liquid Flow Through Capillary Homework | Liquid Flow Through Capillary Homework Help | Liquid Flow Through Capillary Homework Help Services | Live Liquid Flow Through Capillary Homework Help | Liquid Flow Through Capillary Homework Tutors | Online Liquid Flow Through Capillary Homework Help | Liquid Flow Through Capillary Tutors | Online Liquid Flow Through Capillary Tutors | Liquid Flow Through Capillary Homework Services | Liquid Flow Through Capillary
Consider a slow and steady laminar flow of liquid through a capillary or a narrow cylindrical tube of uniform cross-section; let a be the radius and l be the length of the tube. The flow is maintained by applying a constant pressure difference ΔP between the ends of the tube. The layer of the liquid at the wall of the tube is at rest. Velocity of flow increases from the wall towards interior, being maximum at the axis. During laminar flow co-axial cylindrical layers of liquid flow past each other. Consider the volume of liquid bounded by the cylindrical surface of radius r within the tube. The viscous force exerted on this volume of liquid by the surrounding liquid is given by where v (r) velocity of flow at radial distance r from the axis; dv/dr represents corresponding velocity gradient. During steady flow viscous force is balance by force applied by external pressure difference: Fext = (π r2) ΔP Hence, we get Integrating, we get Constant c is fixed by boundary condition : at r = a, v = 0. Hence, The velocity varies as square of the distance from the wall, to a maximum value at the axis (r = 0). Hence, it is said that liquid flowing in a tube possesses a parabolic velocity profile. The volume of liquid flowing per second through the cylindrical shell of radius r and r + dr is given as dV = (2 π r dr) v (r) where 2 π r dr is area of cross-section of the shell. Therefore the total volume of liquid flowing through the tube, per second is The above relation is known as Poiseuille’s formula for laminar flow of liquid through a narrow tube.
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