Viscosity and Surface Tension (SY-2024-2025 First Term) PDF

VISCOSITY FLUID FRICTION  ALL REAL FLUIDS HAVE SOME VISCOSITY. TO KEEP A VISCOUS FLUID FLOWING, A NET FORCE DUE TO FLUID PRESSURE MUST PUSH THE LIQUID FORWARD TO COMPENSATE FOR THE VISCOUS FORCES THAT OPPOSE THE FLOW. A PRESSURE DIFFERENCE BETWEEN THE ENDS OF THE PIPE MUST BE MAINTAINED TO KEEP...

VISCOSITY FLUID FRICTION  ALL REAL FLUIDS HAVE SOME VISCOSITY. TO KEEP A VISCOUS FLUID FLOWING, A NET FORCE DUE TO FLUID PRESSURE MUST PUSH THE LIQUID FORWARD TO COMPENSATE FOR THE VISCOUS FORCES THAT OPPOSE THE FLOW. A PRESSURE DIFFERENCE BETWEEN THE ENDS OF THE PIPE MUST BE MAINTAINED TO KEEP THE FLUID MOVING. THE PRESSURE DIFFERENCE IS IMPORTANT – IN EVERYTHING FROM BLOOD FLOWING THROUGH THE ARTERIES TO OIL PUMPED THROUGH A PIPELINE. IN VISCOUS FLOW, THE FLUID SPEED DEPENDS ON THE DISTANCE FROM THE TUBE WALLS. THE FASTEST FLOW IS AT THE CENTER OF THE TUBE. LAYERS CLOSER TO THE WALL OF THE TUBE MOVE MORE SLOWLY. THE OUTERMOST LAYER OF A FLUID, WHICH IS IN CONTACT WITH THE TUBE, DOES NOT MOVE.  A LIQUID IS MORE VISCOUS IF THE COHESIVE FORCES BETWEEN MOLECULES ARE STRONGER. THE VISCOSITY OF A LIQUID DECREASES WITH INCREASING TEMPERATURE BECAUSE THE MOLECULES ARE LESS TIGHTLY BOUND. A DECREASE IN TEMPERATURE OF THE HUMAN BODY IS DANGEROUS BECAUSE THE VISCOSITY OF THE BLOOD INCREASES AND THE FLOW OF BLOOD THROUGH THE BODY IS HINDERED. GASES HAVE AN INCREASE IN VISCOSITY FOR AN INCREASE IN TEMPERATURE. AT HIGHER TEMPERATURES THE GAS MOLECULES MOVE FASTER AND COLLIDE MORE OFTEN WITH EACH OTHER.  VISCOSITIES OF SOME FLUIDS Substance Temperature (0C) Viscosity (Pa.s) Gases: Water vapor 100 1.3 X 10-5 Air 0 1.7 X 10-5 20 1.8 X 10-5 30 1.9 X 10-5 100 2.2 X 10-5 Liquids: Acetone 30 0.30 X 10-3 Water 0 1.8 X 10-3 100 0.28 X 10-3 Blood plasma 37 1.3 X 10-3 Blood, whole 20 3.0 x 10-3 37 2.1 X 10-3 Glycerin 30 0.63 UNITS OF VISCOSITY Viscosity(η):units Symbol pascal-seconds Pa․s poise P (1 P = 0.1 Pa.s) centipoise cP (1cP = 0.001 Pa.s)  POISEUILLE’S LAW Δ𝑉 THE VOLUME FLOW RATE ( ) FOR LAMINAR FLOW OF A VISCOUS FLUID Δ𝑡 THROUGH A HORIZONTAL CYLINDRICAL PIPE DEPENDS ON THE FOLLOWING FACTORS: 1. IT IS PROPORTIONAL TO PRESSURE DROP PER UNIT LENGTH (ΔP/L), ALSO CALLED THE PRESURE GRADIENT. 2. IT IS INVERSELY PROPORTIONAL TO THE VISCOSITY ( η ) OF THE FLUID. 3. IT IS PROPORTIONAL TO THE FOURTH POWER OF THE PIPE RADIUS. THE EQUATION FOR POISEUILLE’S LAW (FOR VISCOUS FLOW) IS SHOWN BELOW: Δ𝑉 π Δ𝑃 𝑟4 = Δ𝑡 8 η𝐿  APPLICATION OF VISCOUS FLOW: HIGH BLOOD PRESSURE THE STRONG DEPENDENCE OF FLOW RATE ON RADIUS IS IMPORTANT IN BLOOD FLOW. A PERSON WITH CARDIOVASCULAR DISEASE HAS ARTERIES NARROWED BY PLAQUE DEPOSITS. TO MAINTAIN THE NECESSARY BLOOD FLOW TO KEEP THE BODY FUNCTIONING, 1 THE BLOOD PRESSURE INCREASES. IF THE DIAMETER OF AN ARTERY NARROWS TO OF ITS 2 ORIGINAL VALUE DUE TO PLAQUE DEPOSITS, THE BLOOD FLOW RATE WOULD DECREASE TO 1 OF ITS ORIGINAL VALUE IF THE PRESSURE DROP ACROSS IT WERE TO STAY THE SAME. TO 16 COMPENSATE FOR THIS DECREASE IN BLOOD FLOW, THE HEART PUMPS HARDER, INCREASING THE BLOOD PRESSURE. HIGH BLOOD PRESSURE IS NOT GOOD EITHER: IT INTRODUCES ITS OWN SET OF HEALTH PROBLEMS, NOT LEAST OF WHICH IS THE INCREASED DEMANDS ON THE HEART MUSCLES. (GENERAL PHYSICS I, MCGRAW HILL 3RD EDITION)  HIGH BLOOD PRESSURE CAN, OF COURSE, BE DANGEROUS. IT CAN RUPTURE BLOOD VESSEL WALLS, PERHAPS CAUSING A STROKE OR HEART ATTACK. FURTHERMORE, THE HEART MUST EXERT MORE FORCE TO CREATE HIGH BLOOD PRESSURE AND IS STRAINED. HIGH BLOOD PRESSURE OFTEN OCCURS WITH AGING AND MAY BE DESIGNED TO MAINTAIN FLOW IN VESSELS NARROWED BY PLAQUE (ARTERIOSCLEROSIS). HIGH BLOOD PRESSURE IS CONTROLLED BY MEDICATION, AND NARROWED BLOOD VESSELS ARE SOMETIME REPLACED SURGICALLY. THE BODY ADJUSTS BLOOD FLOW BY CHANGING VESSEL RADII, SINCE RESISTANCE TO FLOW IS MORE SENSITIVE TO THIS PARAMETER THAN TO ANY OTHER. MOST VESSEL DILATION (VASODILATION) AND CONSTRICTION (VASOCONSTRICTION) TAKE PLACE IN THE SMALL ARTERIES AND ARTERIOLES, WITH SOME OCCURRING IN THOSE CAPILLARIES THAT HAVE SPHINCTERS. FLOW IS ALSO ADJUSTED BY CHANGES IN ARTERIAL BLOOD PRESSURE, WHILE VENOUS BLOOD PRESSURE REMAINS CONSTANT. FOR EXAMPLE, DURING STRENUOUS EXERCISE, BLOOD FLOW MAY QUADRUPLE WITH 50% INCREASE IN ARTERIAL BLOOD PRESSURE AND SIGNIFICANT VASODILATION OF THE ARTERIAL BLOOD SYSTEMS THROUGHOUT THE BODY. ATHLETIC CONDITIONING MAKES VASODILATION MORE EFFECTIVE AND BLOOD PRESSURE INCREASES NEED NOT BE AS GREAT.  APPLICABILITY OF THE POISEUILLE’S LAW POISEUILLE’S LAW DOES NOT QUANTITATIVELY DESCRIBE BLOOD FLOW VERY ACCURATELY BECAUSE OF THE FOLLOWING REASONS: FIRST, BLOOD IS NOT AN IDEAL FLUID. IT CONTAINS BLOOD CELLS, WHICH ARE NOT FLUID IN CHARACTER AND WHOSE SIZE IS LARGE ENOUGH TO AFFECT FLOW IN ARTERIOLES, CAPILLARIES, AND VENULES. SECOND, VESSEL WALLS ARE NOT RIGID, SO FLOW IS AFFECTED AS THEY EXPAND AND CONTRACT WITH EACH HEARTBEAT. THIRD, POISEUILLE’S LAW IS ONLY VALID FOR NONTURBULENT FLOW. HIGH BLOOD VELOCITY, SHARP BENDS OR CONSTRICTIONS, AND BLOOD CELLS CAN ALL CAUSE TURBULENCE. NEVERTHELESS, POISEUILLE’S LAW IS WIDELY APPLIED TO BLOOD FLOW AND DOES GIVE A GOOD QUALITATIVE DESCRIPTION OF THE DEPENDENCE OF FLOW ON RADIUS AND VISCOSITY. IT WILL CONTINUE TO BE USED QUANTITATIVELY TO DESCRIBE BLOOD FLOW BECAUSE IT GIVES ACCEPTABLE APPROXIMATIONS IN VIEW OF THE LACK OF A SIMPLE BUT SUPERIOR THEORY. (PHYSICS WITH HEALTH AND SCIENCE APPLICATIONS, URONE)  STOKE’S LAW WHEN AN OBJECT MOVES THROUGH A FLUID, THE FLUID EXERTS A DRAG FORCE ON IT. WHEN THE RELATIVE VELOCITY BETWEEN THE OBJECT AND THE FLUID IS LOW ENOUGH FOR THE FLOW AROUND THE OBJECT TO BE LAMINAR, THE DRAG FORCE DERIVES FROM VISCOSITY AND IS CALLED VISCOUS DRAG. THE VISCOUS DRAG FORCE IS PROPORTIONAL TO THE SPEED OF THE OBJECT ( FD α v ). FOR LARGER RELATIVE SPEEDS, THE FLOW BECOMES TURBULENT AND THE DRAG FORCE IS PROPORTIONAL TO THE SQUARE OF THE OBJECT’S SPEED ( FD α v2 ).  STOKE’S LAW EQUATION THE VISCOUS DRAG FORCE DEPENDS ALSO ON SHAPE AND SIZE OF THE OBJECT. FOR A SPHERICAL OBJECT, THE VISCOUS DRAG FORCE IS GIVEN BY STOKE’S LAW: FD = 6πηrv WHERE r IS THE RADIUS OF THE SPHERE, η IS THE VISCOSITY OF THE FLUID, AND V IS THE SPEED OF THE OBJECT WITH RESPECT TO THE FLUID.  SURFACE TENSION THE SURFACE OF A LIQUID HAS SPECIAL PROPERTIES NOT ASSOCIATED WITH THE INTERIOR OF THE LIQUID. THE SURFACE ACTS LIKE A STRETCHED MEMBRANE UNDER TENSION. THE SURFACE TENSION (SYMBOL γ , THE GREEK LETTER GAMMA) OF A LIQUID IS THE FORCE PER UNIT LENGTH WITH WHICH THE SURFACE PULLS ON ITS EDGE. THE DIRECTION OF THE FORCE IS TANGENT TO THE SURFACE AT ITS EDGE. SURFACE TENSION IS CAUSED BY THE COHESIVE FORCES THAT PULL THE MOLECULES TOWARD EACH OTHER.  EVERYDAY PHYSICS DEMO: SURFACE TENSION PLACE A NEEDLE (OR A FLAT PLASTIC-COATED PAPER CLIP) GENTLY ON THE SURFACE OF A GLASS OF WATER. IT MAY TAKE SOME PRACTICE, BUT YOU SHOULD BE ABLE TO GET IT TO “FLOAT” ON TOP OF THE WATER. NOW ADD SOME DETERGENT TO THE WATER ANS TRY AGAIN. THE DETERGENT REDUCES THE SURFACE TENSION OF THE WATER SO IT IS UNABLE TO SUPPORT THE NEEDLE. SOAPS AND DETERGENTS ARE SURFACTANTS – SUBSTANCES THAT REDUCE THE SURFACE TENSION OF A FLUID. THE REDUCED SURFACE TENSION ALLOWS THE WATER TO SPREAD OUT MORE, WETTING MORE OF A SURFACE TO BE CLEANED.  SURFACE TENSION: APPLICATION 1 HOW INSECTS CAN WALK ON THE SURFACE OF A POND THE HIGH SURFACE TENSION OF WATER ENABLES WATER STRIDERS AND OTHER SMALL INSECTS TO WALK ON THE SURFACE OF A POND. THE FOOT OF THE INSECT MAKES A SMALL INDENTATION IN THE WATER SURFACE; THE DEFORMATION OF THE SURFACE ENABLES THE WATER TO PUSH UPWARD ON THE FOOT AS IF THE WATER SURFACE WERE A THIN SHEET OF RUBBER. VISUALLY IT LOOKS SIMILAR TO A PERSON WALKING ACROSS THE MAT OF A TRAMPOLINE. OTHER SMALL CREATURES, SUCH AS MOSQUITO LARVAE OR PLANARIA, HANG FROM THE SURFACE OF WATER, USING SURFACE TENSION TO HOLD THEMSELVES UP. IN PLANTS, SURFACE TENSION AIDS IN THE TRANSPORT OF WATER FROM THE ROOTS OF THE LEAVES.  SURFACE TENSION: APPLICATION 2 SURFACTANT IN THE LUNGS THE HIGH SURFACE TENSION OF WATER IS A HINDRANCE IN THE LUNGS. THE EXCHANGE OF OXYGEN AND CARBON DIOXIDE BETWEEN INSPIRED AIR AND THE BLOOD TAKES PLACE IN TINY SACS CALLED ALVEOLI, 0.05 TO 0.15 MM IN RADIUS, AT THE END OF THE BRONCHIAL TUBES. THE MUCUS COATING OF THE ALVEOLI HAD THE SAME SURFACE TENSION AS OTHER BODY FLUIDS, THE PRESSURE DIFFERENCE BETWEEN THE INSIDE AND OUTSIDE OF THE ALVEOLI WOULD NOT BE GREAT ENOUGH FOR THEM TO EXPAND AND FILL THE AIR. THE ALVEOLI SECRETE A SURFACTANT THAT DECREASES THE SURFACE TENSION IN THEIR MUCUS COATING SO THEY CAN INFLATE DURING INHALATION.  TURBULENCE IN A VISCOUS FLUID, THE FLOW AT LOW SPEED CAN BE DESCRIBED AS LAMINAR, WHICH SUGGESTS LAYERS SLIDING SMOOTHLY OVER ONE ANOTHER. WHEN THE FLOW SPEED IS SUFFICIENTLY LARGE, THE MOTION BECOMES DISORDERED AND IRREGULAR; THIS IS TURBULENT FLOW. THESE ARE EXAMPLES OF TURBULENT FLUID FLOW: AFTER RISING A SHORT DISTANCE, THE SMOOTH COLUMN OF SMOKE FROM A CIGARETTE BREAKS UP INTO AN IRREGULAR AND SEEMINGLY RANDOM PATTERN. SIMILARLY, A STREAM OF FLUID PAST AN OBSTACLE BREAKS UP INTO EDDIES AND VORTICES, WHICH GIVE THE FLOW IRREGULAR VELOCITY COMPONENTS TRANSVERSE TO THE FLOW DIRECTION. LIKE, FOR INSTANCE, THE FLAPPING OF A FLAG IN A BREEZE – IF THE FLOW OF AIR WERE LAMINAR, THE FLAG WOULD OCCUPY A FIXED POSITION ALONG STREAMLINES, BUT THE FLAGPOLE BREAKS THE FLOW INTO AN IRREGULAR PATTERN, WHICH CAUSES THE TRANSVERSE FLAPPING MOTION OF THE FLAG. OTHER EXAMPLES INCLUDE THE WAKES LEFT IN WATER BY MOVING SHIPS AN IN AIR BY MOVING CARS AND AIRPLANES. THE SOUNDS PRODUCED BY WHISTLING AND BY WIND INSTRUMENTS RESULT FROM THE TURBULENT FLOW OF AIR. (PHYSICS BY RESNICK AND HALLIDAY, 5TH ED.)  REYNOLDS NUMBER WHAT IS THE CRITICAL SPEED AT WHICH THE FLOW BECOMES TURBULENT? OSBORNE REYNOLDS (1842- 1912) FOUND THAT THE CRITICAL SPEED DEPENDS ON THE VISCOSITY ( η ) AND DENSITY OF THE FLUID ( ρ ) AND THE DIAMETER D OF THE PIPE. HENCE, THE CRITICAL SPEED CAN BE WRITTEN AS η vc = R ρ𝐷 WHERE R IS THE REYNOLDS NUMBER WHICH A DIMENSIONLESS QUANTITY US ED TO PREDICT FLOW BEHAVIOR IN FLUIDS. IN THIS INTERPRETATION, THE REYNOLDS NUMBER CAN BE USED TO CHARACTERIZE ANY FLOW, AND WE CAN DETERMINE BY EXPERIMENT THE VALUE OF THE REYNOLDS NUMBER AT WHICH THE FLOW BECOMES TURBULENT. (ADAPTED FROM PHYSICS, RESNICK AND HALLIDAY 5TH EDITION)  REYNOLDS NUMBER: SAMPLE PROBLEM THE REYNOLDS NUMBER CORRESPONDING TO THE CRITICAL SPEED FOR CYLINDRICAL PIPES IS ABOUT 2000. HENCE FOR WATER FLOWING THROUGH A PIPE OF DIAMETER 2 CM (A TYPICAL GARDEN HOSE), THE CRITICAL SPEED IS − 3 1 𝑋 10 𝑃𝑎.𝑠 vc = 2000 𝑘𝑔 = 0.1 m/s = 10 cm/s (1000 3)(0.02𝑚) 𝑚 THIS IS QUITE A LOW SPEED, WHICH SUGGESTS THAT THE FLOW OF WATER IS TURBULENT IN ORDINARY HOUSEHOLD PLUMBING. ( THE FLOW SPEED FROM A TYPICAL HOUSEHOLD TAP IS ABOUT 1 M/S.) (FROM PHYSICS BY RESNICK AND HALLIDAY, 5TH EDITION, P. 363)