B737 Undifferentiated or Other Model
Flight Crew Air Transport Pilot (ATP)
Flight Crew Last 90 Days 46; Flight Crew Total 12000; Flight Crew Type 46
Confusion; Time Pressure; Human-Machine Interface
Aircraft Equipment Problem Less Severe; Inflight Event / Encounter Fuel Issue
Aircraft Equipment Problem Dissipated; General Declared Emergency
Aircraft
Approach set us up for a mild slam-dunk; with a short turn to final (inside ZZZZZ); asking us to keep our speed up as much as possible (210 or greater until on final; then 180 or greater to ZZZZZ1). During configuration I noticed that the flaps still indicated 5 after I had called for 15. I called again for Flaps 15; and I thought I saw the First Officer (FO) move or check the flap handle; but they still indicated 5 on the gauge. With our airspeed under 180; I asked for Flaps 30. I saw him move the flap handle but confirmed the gauge still at 5. We agreed that there was a disagreement between the flap indicator gauge and the flap handle. We initiated a go-around inside ZZZZZ1 at about 2000 feet MSL (1300 feet AGL). We climbed out configured gear up; flaps 5. The fuel gauge showed 6.0.I took the radios and FO began to troubleshoot with QRH. ATC vectored us out on a 220 heading and an initial climb to 4000 feet then 5000 feet.  They kept asking us to climb (to 6000; then 7000) until I said 5000 feet was high enough and advised them we were at a low fuel state (did not use the words 'minimum fuel' at this point). I asked for a downwind turn and got a vector to the east. FO and I agreed that this warranted [advising ATC.] I advised approach; using the words 'Mayday; Mayday; Mayday'. I asked for the longest runway and was told to expect XXR. Here I must add that the 737 checklist is confusing on the entire Flaps Disagree section. It took us a couple of starts before we arrived at the right one (Flaps - Trailing Edge Flaps Disagree). This ate up time; fuel; and downwind distance. Placing the flaps up to '2' brought the flap indicator gauge up to 2. The flaps moved as commanded to 5; and then to 15. We turned base over the shoreline and extended the landing gear. It was a little far out for the gear; but the horn was sounding (additional confirmation that the flaps were at 15) plus we didn't want to put the flaps up and take the chance on them getting stuck again. We intercepted the glide slope for XXR at 4000 feet MSL. Shortly afterward extended flaps to 30 with a good '30' indication. We informed ZZZ Approach and then ZZZ Tower that the remainder of the approach would be normal.Landing was uneventful. We cleared the runway with fuel indicating 4.0; and we taxied to the gate. After parking the printer spit out a report 'ADVISORY FOR CREW USE ONLY; APPROACH REPORT BELOW 500 feet'. Parameters were normal; except for FLAP @ 1000 feet which read '4'. FLAP @ TD read '30&'. I don't know if that referenced the first or second approach; but on the second approach the flaps were at 30 all the way from 3000 feet on the glide slope.
When their flaps refused to extend beyond five degrees on approach the flight crew of a B737 opted to go around to troubleshoot the problem.  When fuel remaining became an issue they [advised ATC] and; during the subsequent approach; the flaps this time extended normally.