Viscous Force in Fluid Mechanics
Explore the concept of viscous force in fluid mechanics with Dr. Ali Alhafiz. Learn about the impact of friction between fluid molecules, calculation methods for viscous force, and the role of shearing stress in maintaining uniform motion. Dive into the depths of Cartesian and spherical coordinates to grasp the dynamics of viscous forces in fluid systems.
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Presentation Transcript
LECTURE 4 THE VISCOUS FORCE Dr.Ali alhafiz Third stage
SECOND LOW OF NEWTON ?? ??? + ?? ??? + ?? ??? (CARTESIAN COORDINATE) ? = ?? ??= 2 ? 1 ??? + ? + ?? (Spherical coordinates)
THE VISCOUS FORCE THE VISCOUS FORCE IS DUE TO THE FRICTION CAUSED BY INTERACTIONS OF MOLECULES OF THE FLUID. uo and lower plates will moved as the following: At z=L the fluid moves at speed u(L)=uo in x-direction At z=0 the fluid is motionless u(0)=0. The force exerted on the upper plate is: F (Auo)/L The layer in contact with the upper z=L u(L)=uo u(z) u(0)=0 z=0 where A is area of the plate, L is the distance (depth) between the plates. F= (Auo)/L where is the dynamic viscosity coefficient.
For a state of uniform motion, every horizontal layer of fluid of depth z must exert the same force F on the fluid below. This may be expressed in the form: F= A u/ z _zx=lim ( z 0) u/ z= u/ z ?? ?? ? ? where subscripts indicate that zx is the component of the shearing stress in the x-direction due to vertical shear of the x velocity component.
Calculating the Viscous Force CONSIDER A DIFFERENTIAL VOLUME ELEMENT (PARCEL) OF FLUID CENTERED AT (X, Y, Z) WITH SIDES X Y Z AS SHOWN IN FIGURE (4.2). IF THE SHEARING STRESS IN THE X DIRECTION ACTING THROUGH THE CENTER OF THE ELEMENT IS DESIGNATED ZX, THEN THE STRESS ACTING ACROSS THE UPPER BOUNDARY ON THE FLUID BELOW MAY BE WRITTEN APPROXIMATELY AS ???+???? ?? 2 ?? while the stress acting across the lower boundary on the fluid above is: ??? ???? ?? 2 ??
THE NETVISCOUS FORCE ON THE VOLUME ELEMENT ACTING IN THE X DIRECTION IS THEN GIVEN BY THE SUM OF THE STRESSES ACTING ACROSS THE UPPER BOUNDARY ON THE FLUID BELOW AND ACROSS THE LOWER BOUNDARY ON THE FLUID ABOVE: ???= ? ??? ???+???? ?? 2 ???=???? ???? ??? ???? ?? 2 ???? ?? ?? ?????? ?? Dividing by mass ???????, ???=1 ???? ?? (4.1) ? ??2? ???=1 ? ?? ??? ?? ???? ??2 (4.2) ?
where ? =? ? is the kinematic viscosity coefficient For standard atmospheric conditions ? = 1.46 10 5?2? 1 ???? ?? and for z-direction ???= 1 ? 1 ? ???? ?? Now for y-direction ???= The total force per unit mass: ? = ???+ ???+ ??? ???? ?? ? =? ???? ?? +? +? ???? ?? ? ? ? ? =? ?? ?? ?
HOMEWORK: 1. Why the fluid is considered incompressible in most atmospheric dynamics? When can we take into account the hypothesis that the fluid is compressible? 1. Derive the viscous force.
